PART I TRUTH FUNCTIONS

§ 1. NEGATION AND CONJUNCTION

The peculiarity of statements which sets them apart from
other linguistic forms in that they admit of truth and falsehood,
and may hence be significantly affirmed and denied.

Instead of denying a given statement we may, equivantly,
affirm another statement, known as the negation or contradictory
of the given statement; and instead of affirming each of several
statements we may, equivalently, affirm a single statement which
is known to logicians as the conjunction of the given statements.
(Note that the word 'conjunction' means something else to gram-
marians.)

The commonest method of forming the negation of statements in
ordinary language is by attaching 'not' (or 'does not', etc.) to
the main verb; thus the negation of 'Jones is away' is 'Jones is
not away'. But if the verb is governed by 'sometimes' or 'always',
the negation is formed rather by substituting 'never', or 'not
always'. If the statement is compound and thus has no main verb,
its negation has to be phrased more elaborately; e.g., 'Is is not
the case both that ... and that ...'. But, despite such irreg-
ularities of ordinary language, a little care suffices for con-
structing a clear negation of any given statement, the guiding
consideration being simply this: the negation is to count as
false if the given statement is true, and the negation is to count
as true under any and all circumstances under which the given
statement is false.

In logical studies it is convenient to adopt a single sign of
negation, consisting of the prefix 'not', applied to statements as
wholes. Thus '(Jones is away)' means 'Jones is not away'; the
parentheses here serve to group, as a single whole, the statement
to which 'not' is applied. The sign 'not' might be translated into
words as 'it is not the case that'; briefly it may be pronounced
'not'.

The conjunction of two or more statements is commonly expressed
in English by linking the statements by 'and', or commas, or a
combination of the two: 'Some are born great, some achieve great-
ness, and some have greatness thrust upon them.' In logical studies
it is convenient to express the conjunction simply by writing the
component statements in juxtaposition; e.g., '(some are born great)
(some achieve greatness) (some have greatness thrust upon them) --
where, again, the parentheses serve merely to mark off the component
statements as wholes. Thus, if we think of 'p', 'q', and 'r' as
statements, their conjunction is represented as 'pqr'.

The sign of negation, when the statement to which it is
applied is represented as a single letter, will be placed
above instead of in front; thus we shall write 'p' instead of
'not-p' for the negation of 'p'. The negation of the conjunction,
'pq', on the other hand, is '-(pq); and the negation of 'pqr'

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is '-(pqr)'.

The need of parentheses may be appreciated by comparing
'-(pq)r' and '-(pqr)'. In '-(pq)r' we have a conjunction of two
components, '-(pq)' and 'r'; '-(pqr)', on the other hand, is the
negation of the whole three-way conjunction 'pqr'. In general the
negation sign is to be understood as applying to as little as
possible of what follows it; thus in '-(pq)r' it applies only to
'pq', not to '(pq)r', whereas in '-(pqr)' the parentheses compel
it to apply to 'pqr'.

The meanings of negation and conjunction are summed up in
these laws: The negation of a true statement is false; the nega-
tion of a false statement is true; a conjunction of statements
all of which are true is true; and a conjunction of statements not
all of which are true is false.

We see immediately that 'P'; the negation of 'P', will be
true if and only if 'P' is false, hence if and only if 'p' is
true; so there is no point in writing a double negation 'P',
amounting as it does simply to 'p'. It is equally evident that
the conjunction 'pp' amounts simply to 'p'.

A conjunction of more than two statements may be viewed as a
conjunction merely of two statements one of which happens to be a
conjunction in turn; thus 'pqr' may be viewed as the conjunction
of 'p' with the conjunction 'qr', or, if we prefer, as the con-
junction of 'pq' with 'r'. Both of these ways of analyzing 'pqr'
come to the same thing in the end. For, the conjunction 'p' and
'qr' will be true just in case 'p' and 'qr' are both true, and
'qr' will in turn be true just in case 'q' and 'r' are both true;
so the conjunction of 'p' and 'qr' is true just in case 'p', 'q',
and 'r' are all true. Similarly the conjunction of 'pq' and 'r'
will be true just in case 'pq' and 'r' are both true--hence just
in case 'p', 'q' and 'r' are all true.

Because the complex conjunctions '(pq)r' and 'p(qr)' thus
turn out equivalent, we write simply 'pqr', which may be viewed
at will as the conjunction of 'pq' and 'r', the conjunction of 'p'
and 'qr', and the conjunction of 'p', 'q', and 'r'. Conjunction
is, in mathematical jargon, _associative_: internal groupings are
inmaterial.

Conjunction is also _commutative_; i.e., order is immaterial,
there being no need to distinguish between 'pq' and 'qp'. In
being associative and commutative, conjunction resembles the
familiar arithmetical operations of addition and multiplication.
(Note, in contrast, that division lacks these properties;
(x☆y)☆z and x☆(y☆z) are not in general equal, nor are
x☆y and y☆x.) But conjunction was seen also to have a con-
venient third property, not enjoyed by addition and multiplication;
viz. (if we would persist in the jargon) _idempotence_, meaning that
'pp' reduces to 'p'. Taken together, these three properties come
simply to this: once we have an inventory of all the distinct com-
ponents of a continued conjunction, no further details of the con-
stitution of the conjunction need concern us.

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But grouping becomes important when conjunction and negation occur in combination. We have no right, e.g., to equate $-(pq)r'$ with $-(pqr)$. The conjunction $-(pq)r'$ is true just in case $-(pq)'$ and $r'$ are both true; hence just in case 'pq' is false and $r'$ is true; hence just in case 'p' and 'q' are not both true but $r'$ is true. On the other hand $-(pqr)'$ is true just in case 'pqr' is false; hence just in case not all of 'p', 'q' and $r'$ are true.

It is important to remember that $-(1q)'$ is the negation of the whole conjunction 'pq', and not of 'p' and 'q' separately; it is not to be confused with $\overline{pq}$, which is the conjunction of 'p' and $\overline{q}$. The conjunction 'pq' is true just the one case where $\overline{p}'$ and 'q' are both true; hence in just the one case where 'p' and $\overline{q}'$ are both false. On the other hand the negation $-(pq)'$ is true in all of the three cases where 'pq' is false; viz., in the case where 'p' is false and 'q' is true, the case where 'p' is true and 'q' false, and the case where 'p' and 'q' are both false. The conjunction 'pq' amounts to the words 'Neither p nor q'; e.g., 'Neither are some born great nor do some achieve greatness'. On the other hand the negation $-(pq)'$ amounts to the words 'Not both p and q'; e.g., 'It is not the case both that some are born great and that some achieve greatness'. This latter compound will still be true though some be born great, if none achieve greatness; and it will still be true though some achieve greatness, if none are born great.

Not only must we distinguish thus between $-(pq)'$ and $\overline{pq}'$, but we must also distinguish 'pq' and $\overline{pq}'$ from each of those and from each other. The conjunction 'pq' is true just in case 'p' and $\overline{q}'$ are both true, hence just in case 'p' is false and $\overline{q}'$ true; 'pq' is true just in case 'p' is true and $\overline{q}'$ false; $\overline{pq}'$ is false just in case 'p' is false and $\overline{q}'$ true; and $-(pq)'$ is true just in all three cases. To sum up:

<table><tr><td>p</td><td>true</td><td>false</td><td>true</td><td>false</td></tr><tr><td>q</td><td>true</td><td>true</td><td>false</td><td>false</td></tr><tr><td>pq</td><td>false</td><td>true</td><td>false</td><td>false</td></tr><tr><td>pq</td><td>false</td><td>false</td><td>true</td><td>false</td></tr><tr><td>\overline{pq}</td><td>false</td><td>false</td><td>false</td><td>true</td></tr><tr><td>-(pq)</td><td>false</td><td>false</td><td>false</td><td>true</td></tr></table>

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## § 2. ALTERNATION

The connective 'or', or 'either-or', is subject in ordinary
discourse to conflicting usage. One sense is the _inclusive_,
according to which the compound is true so long as at least one of
the components is true. Under this usage the statement:

(Either) Jones is ill or Smith is away

is true if Jones is ill and Smith is away; true again if Jones is
not ill but Smith is away; true again if Jones is ill but Smith is
not away; and false only in case Jones is neither ill nor Smith
away. The other sense in which 'or' is sometimes used, called the
exclusive, construes the compound as true just in case exactly one
of the components is true. In this sense of 'or', the compound
becomes false not only when the components are both false (Jones
neither ill nor Smith away) but also when the components are both
true (Jones ill and Smith away).

The ambiguity of 'or' is commonly resolved, in ordinary usage,
by adding the words 'or both' or 'but not both'. Thus the inclusive
sense is expressible in the unambiguous fashion:

Jones is ill or Smith is away or both,

and the exclusive sense thus:

Jones is ill or Smith is away but not both.

When we are confronted with 'p or q' by itself, we do not, in
general, know which interpretation to assign to it. Often the choice
is immaterial, in that either sense would serve equally. For example,
consider the expression 'x ≤ y', i.e., 'x < y or x = y'. It makes no
difference whether 'or' here is understood in the inclusive or the
exclusive sense. The only difference between the two senses occurs
in the case where both components are true; but when the components
concerned are 'x < y' and 'x = y', the case of joint truth does not
arise, either in fact or in the mind of the speaker.

It is a common error to believe that examples like 'x < y' or
'x = y' are clear cases of the use of 'or' in the exclusive sense,
and in consequence of this error there is a tendency to over-
estimate the role which the exclusive sense of 'or' plays in every-
day language. The clauses 'x < y' and 'x = y' are, of themselves,
mutually exclusive or incompatible clauses; but this incompatibility,
far from establishing that the context 'x < y' or 'x = y' uses 'or'
in the exclusive sense, deprives us if the one case in which we might
be able to distinguish between the exclusive and inclusive senses.
Since the clauses 'x < y' and 'x = y' are already of such nature as
to exclude each other, it is indifferent whether we understand 'or'
as repeating this exclusion or not.

If we want to establish indisputable instances of the exclusive
use of 'or', we must imagine circumstances in which the person who
uses 'or' has a positive purpose of denying, explicitly within the
given statement, the joint truth of the components. Such examples
are rare, but they exist. In an example given by Tarski it is sup-
posed that a child asks his father to take him to the beach and (2

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afterwards to the movie. The father replies, in a tone of refusal, "We will go either to the beach or to the movie." Here the exclusive use is clear; the father means simultaneously to promise and to refuse. But it is much easier to find cases in which the inclusive interpretation is obligatory. For example, when it is decreed that passports will be issued only to persons who were born in the country or who are married to natives of the country, this does not mean that passports will be refused to persons who were born in the country and are married to natives.

Most use of 'or' in everyday language is either of this type which admits only of the inclusive interpretation, or of the type of 'x < y or x = y', which admits both interpretations indifferently.

Latin has distinct words for the two senses of 'or': vol for the inclusive, and aut for the exclusive. In modern logic it is customary to write 'v', reminiscent of 'vel', for 'or' in the inclusive sense: 'p v q'. This mode of compounding statements is called alternation. When the ambiguous 'or' if ordinary language appears hereafter in the book, let us agree to construe it in this inclusive sense. If occasions arise where the exclusive sense of 'or' is really wanted, it is easy enough to express it explicitly:

p or q but not both,

or equivalently:

Either p and not q or q and not p,

i.e., in symbols:

$$p \ddot{q} v \ddot{p} \cdot 1$$

The meaning of alternation, then, is given by this rule: An alternation is true if at least one of the components is true, and otherwise false. Whereas a conjunction is true if and only if its components are all true, an alternation is false if and only if its components are all false. In a metaphor from genetics, conjunction and alternation may be contrasted thus: in conjunction, truth is recessive and falsity dominant; in alternation, truth is dominant and falsity recessive.

Because the explanation of alternation is just the same as that of conjunction except for interchanging the roles of truth and falsity, it is evident that the formal properties of conjunction must reappear as properties of alternation; thus alternation, like conjunction, in associative, commutative, and idempotent. We can render '(p v q) v r' and 'p v (q v r)' indifferently as 'p v q v r'; we can interchange 'p v q' with 'q v p'; and we can reduce 'p v p' to 'p'. All that matters in a continued

1 After § 14, this may also be written 'p ≡ q'.

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alternation, as in a continued conjunction, is an inventory of the
distinct components.

Though the grouping of components is irrelevant within a con-
tinued conjunction and within a continued alternation, it is impor-
tant where conjunction and alternation are mixed; we must distin-
guish, e.g., between '(pq) v r' and 'p(q v r)'. The latter, being
a conjunction with 'p' as a component, cannot be true unless 'p' is
true; on the other hand '(pq) v r', being an alternation with 'r'
as one component, will be true so long as 'r' is true, even if 'p'
be false.

Some conventions will now be adopted to spare excessive use
of parentheses. One such convention has already been noted (§1):
the negation sign is to be understood as governing as little as
possible of what follows it. A second convention is this: the sign
'v' will be treated as marking a greater break than conjunction;
thus 'pq v r' will be understood as having the grouping '(pq) v r',
and not 'p(q v r)' -- as is pretty well suggested, indeed, by the
typographical pattern itself. If on the other hand we want to con-
vey the meaning 'p(q v r)', and thus create a greater break at the
point of conjunction than at the point of alternation, we shall in-
sert a dot at the point of conjunction thus: 'p. q v r'. Next, if
at some point of alternation we want to create a still greater
break than is expressed by the dot of conjunction, we insert a dot
alongside the alternation sign. Just as the undotted 'v' marks a
greater break than the undotted conjunction, so the dotted 'v' marks
a greater break than the dot of conjunction. The dot which is
thus added to reinforce 'v' goes

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on the side where the reinforcement is needed; thus $!((p \cdot q \vee r) \vee s!$ becomes $!p \cdot q \vee r \vee s!$, but $!s \vee ((p \cdot q \vee r),)!$ becomes $!s \vee p \cdot q \vee r!$. Again $!((p \cdot q \vee r) \vee (p \vee q \cdot r)!)$, calling for reinforcement on both sides of the central $!v!$, becomes $!p \cdot q \vee r \vee v \cdot p \vee q \cdot r!$. When we want to create a still greater break at some point of conjunction than is expressed by the dotted $!v!$ of a neighboring alternation, we put a double dot `:**` for the conjunction; when we want to create a still greater break than this at an alternation sign, we put a double dot beside the alternation; and so on to larger groups of dots. What might be written in terms of parentheses as:

$sv((p(q \vee r) \vee (p \vee q)r)t,$

e.g., is written with help of dots as follows:

$sv: p \cdot q \vee r \vee v \cdot p \vee q \cdot r : t.$

In general, thus, a group of dots at the right of `!v!` means that the component of alternation which starts there extends to the first larger group of dots, if any, and otherwise to the end of the context; and a group of dots at the left of `!v!` means that the component of alternation which ends there begins at the last previous larger group of dots, if any, and otherwise at the beginning of the context. When a group of dots stands alone as a sign of conjunction, the components of the conjunction extend in either direction until they reach an equal or greater group of dots or the limits of the context.

Parentheses will continue to be used to enclose a compound governed by a negation sign; the notations `!-(pq)!` and `!-(p \vee q)!`, thus, persist unchanged. Dots have no power, of course, to transcend parentheses; in `!-(p \vee q \cdot r)s!`, e.g., the dot is powerless to group the `!s!` with the `!r!`. What is spoken of as "the context", in the preceding paragraph, is merely the contents of the nearest enclosing parentheses if such there be.

### § 3. TRUTH VALUES

Statements which are both true or both false are said to be alike in *truth value*; and statements one of which is true and the other false are said to be unlike in *truth value*. We are said to know the truth value of a statement when we know whether the statement is true or false. These convenient manners of speaking hint of two entities of some sort, so-called "truth values", one of which is said to be "had" by all true statements and the other by all falsehoods. We may identify these entities arbitrarily with, say, the numbers 1 and 0; thus the truth value of truths will be said to be 1, and that of falsehoods will be said to be 0. Affirming that the truth value of a statement is 1 amounts merely to affirming that the statement is true, and hence morely to affirming the statement. Affirming that the truth-value of a statement is 0 amounts merely to affirming that the statement is false, and hence merely to denying the statement, or affirming its negation.

The characterizations of conjunction, alternation, and negation

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in foregoing sections may now be restated in terms of truth values
as follows:

(I) The truth value of a conjunction is 0 unless the truth
values of all its components are 1, in which case it is 1.

(II) The truth value of an alternation is 1 unless the truth
values of all its components are 0, in which case it is 0.

(III) The truth value of a negation is opposite to that of
its component.

A compound constructed from given components, whether from
actual statements or from letters 'p', 'q', etc. as in foregoing
sections, will be called a truth function of those components,
provided that the construction proceeds exclusively by means of
negation, conjunction, and alternation. Thus 'pq', 'Pq',
'P v -(pq)', etc. are truth functions of 'p' and 'q'; and 'P',
'-(p v P)', etc. are truth functions of 'p'. We also count 'p'
itself as a truth function of 'p'.

The peculiarity of truth functions which gives rise to the
name is this: in order to know the truth value of a truth function
of given statements, it is sufficient to know the truth values of
those statements. This property is by no means shared by all our
everyday ways of compounding statements. The connective 'because',
though like 'and' and 'or' it joins statements to form a state-
ment, lacks the property; from the truth values of the component
statements we can not always divine the truth value of the
'because' compound. Through knowing, e.g., that Jones ate fish
with ice cream and died, we may still dispute over the truth value
of the compound 'Jones died because he ate fish with ice cream'.
There can be no dispute, on the other hand, over the truth value
of an alternation, conjunction, or negation, once the truth values
of the components are agreed upon.

By a series of computations we can determine the truth value
of any truth function of given statements if we know the truth
values of those statements. E.g., assuming a true statement for
'p' and a false one for 'q', we can evaluate 'pq v pq' thus:

p: 1; q: 0; ḉ: 1; pq: 1; Ḋ: 0; Fq: 0; pq v Fq: 1.

The truth value of 'q' here is inferred from that of 'q' by
(III). Then the truth value of 'pq' is inferred from those of 'p'
and 'q' by (I). The truth value of 'Fq' is obtained by a similar
pair of steps. Finally the truth value of 'pq v Fq' is inferred
from those of its components 'pq' and 'Fq' by (II).

In computing truth values it is convenient simply to put the
digits in place of the letters throughout the whole compound, and

1. The scope of this terminology undergoes a superficial
extension in §§ 13-14, below.

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The table tells us that 'pq v pq' is false where 'p' and 'q' are both true or both false, and that it is true where 'p' is false and 'q' true or vice versa.

When only one letter is concerned, instead of two as above, a truth table runs to only two cases instead of four as above; thus the truth table for '-(p v q)' is:

$$-(1 \lor I) \quad -(0 \lor \bar{O})$$

$$-(1 \lor 0) \quad -(0 \lor 1)$$

$$\begin{array}{c|c} I & I \\ \hline 0 & 0 \end{array}$$

The story told by the table is told by the tops and bottoms of the columns; the intermediate entries in a column are merely stops in computing the bottom entry. Since such stops are very obvious, and readily reconstructed at will, it will be customary hereafter to omit them from the printed page, and to record only the top and bottom entries. The above truth table for 'pq v pq', e.g., would be rendered thus:

$$\begin{array}{c|c|c|c} 1I & v & 1I & 0I \lor v \bar{0}I & 1\bar{0} \lor v \bar{I}0 & 0\bar{0} \lor v \bar{0}0 \\ \hline 0 & & 1 & & 1 & 0 \end{array}$$

The truth table for '-(p v q)' would be rendered thus:

$$\begin{array}{c|c} -(1 \lor I) & -(0 \lor \bar{O}) \\ \hline 0 & 0 \end{array}$$

Where three letters are concerned, it takes eight cases to exhaust the ways of assigning truth values to the letters; thus the truth table for 'pq v -(pr)' is as follows:

$$\begin{array}{c|c|c|c|c|c|c|c} llv-(11) & 0lv-(01) & 10v-(11) & 00v-(01) & 11v-(10) & 01v-(00) & 10v-(10) & 00v-(00) \\ \hline 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$$

Where four letters are concerned, 16 cases are needed; where five, 32; and so on. A good system for exhausting the combinations of truth values for the letters, however many there may be, is illustrated in the above example: use '1' and '0' for 'p' in alternate cases, use '1' and '0' for 'q' in alternate pairs of cases, use '1' and '0' for 'r' in alternate fours of cases, and so on. But the preferential order of letters may of course be changed to suit the problem in hand.

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Another example of inconsistency is 'p -(p v q)':

1 -(1 v 1)

0 -(0 v 1)

1 -(1 v 0)

0 -(0 v 0)

0

0

0

0

A test of validity may be stopped short, with negative outcome, as soon as we come to a case yielding '0'; and a test of consistency may be stopped, with affirmative outcome, as soon as we come to a case yielding '1'. Thus, consider '-(p v q)(q v r)'.

-(1 v 1)(1 v 1)

-(0 v 1)(1 v 1)

-(1 v 0)(0 v 1)

-(0 v 0)(0 v 1)

0

0

0

1

We may infer consistency at this point, without pursuing the remaining four cases; and we could have inferred non-validity after the first case.

A schema is inconsistent if and only if its negation is valid; and a schema is valid if and only if its negation is inconsistent. For clearly all instances of a schema will be false if and only if all instances of its negation are true, and all instances of a schema will be true if and only if all instances of its negation are false.

The terms 'valid' and 'inconsistent' may be applied not only to truth-functional schemata, as above, but also to any of the examples obtainable by substitution for the letters in valid or inconsistent truth-functional schemata. E.g., the instance:

(1) Tufa floats or does not float

of 'p v p' is called valid, and the instance:

(2) Tufa floats and does not float

of 'pp' is called inconsistent. When the terminology is thus extended, however, a qualifying adverb 'truth-functionally' is needed (with a view of later developments) as a reminder that the underlying schema was a truth-functional one rather than a schema of some other sort subsequently to be considered. So (1) is called truth-functionally valid and (2) is called truth-functionally inconsistent. A truth-functionally valid statement is a true statement whose truth is assured by the more truth-functional structure of the statement (structure in terms of conjunction, alternation, and negation), quite independently of the particular content or even the truth or falsity of its component clauses; and a truth-functionally inconsistent statement is a false statement whose falsity is assured by the more truth-functional structure of the statement.

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'Validity' is not to be thought of as a term of praise. Precisely because it is valid, 'Tufa floats or does not float' gives us no useful information. It is too idle a statement to warrant consideration at all, unless perhaps within the covers of a logic book. Truth-functionally valid statements and truth-functionally inconsistent ones could be banished from the language altogether, and we should be the worse off for their banishment in only two respects: the rule of banishment would increase the complexity of our grammar, and some useful logical techniques to come later would suffer in point of neatness and convenience of formulation.

An expression is called *truth-functionally consistent* if it is not truth-functionally inconsistent. Note that it is by no means true that every instance of a consistent truth-functional schema is truth-functionally consistent. The example (2) is an instance of the consistent schema 'pq' and of the consistent schema 'p'; but it is also an instance of the inconsistent schema 'pp', and this makes it truth-functionally inconsistent. If we are to be assured of the truth-functional consistency of a statement, the schema which we must test is not just any schema, whether the statement is an instance, but one which reveals the structure of the statement in as full detail as possible.

What has been said of consistency applies in parallel fashion, obviously, to non-validity: an instance of a non-valid truth-functional schema may or may not be truth-functionally valid.

## § 5. EQUIVALENCE

Equivalence between schemata is: relation which has been touched on in foregoing sections, nor for less implicitly, and has yet to be defined. It was remarked (§§ 1-2), e.g., that the schemata 'p', 'p v p', and 'P' are all equivalent to 'p'. On the other hand the schemata 'Pa', 'pq', 'Pq', and '-(pq)' were contrasted (§ 1) with a view to showing that they are not equivalent to one another. Their non-equivalence was established by pointing out that, where a false statement is put for 'p' and a true one for 'q', 'Pq' and '-(pq)' become true while 'pq' and 'Pq' become false; and that, where a true statement is put for 'p' and a false one for 'q', 'pq' and '-(pq)' become true while 'Pq' and 'Pq' become false. From these remarks the intended sense of 'equivalence' is evident: equivalent schemata are such that, no matter what statements we put for the letters (always putting the same statement, of course, for all occurrences of the same letter), the two resulting instances will agree with each other in truth value. More briefly: two schemata are equivalent just in case no assignment of truth values to their letters issues in unlike truth values for the two schemata.

Schemata can be tested for equivalence simply by setting up comparative truth tables for them, one beneath the other, and noting their agreement or disagreement case by case. The fact that

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none of '$\overline{pq}$', '$\overline{pq}$', '$\overline{pq}$', and $-(pq)$' are equivalent is checked by aligning their four truth tables thus:

<table><thead><tr><th>Im</th><th>01</th><th>10</th><th>00</th></tr></thead><tbody><tr><td>0</td><td>1</td><td>0</td><td>0</td></tr><tr><td>1</td><td>0</td><td>1</td><td>0</td></tr><tr><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>1</td><td>0</td><td>1</td><td>0</td></tr><tr><td>0</td><td>0</td><td>0</td><td>1</td></tr><tr><td>-(11)</td><td>-(01)</td><td>-(10)</td><td>-(00)</td></tr><tr><td>0</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

No two of these four tables agree throughout; no two of '$\overline{pq}$', '$\overline{pq}$', '$\overline{pq}$', and $-(pq)$', therefore, are equivalent to each other.

On the other hand $-(pq)$' and '$\overline{p v q}$' prove equivalent:

<table><thead><tr><th>-(11)</th><th>-(01)</th><th>-(10)</th><th>-(00)</th></tr></thead><tbody><tr><td>0</td><td>1</td><td>1</td><td>1</td></tr><tr><td>I v I</td><td>O v I</td><td>I v O</td><td>O v O</td></tr><tr><td>0</td><td>1</td><td>1</td><td>1</td></tr></tbody></table>

So also do '$p(q v r)$' and '$pq v pr$'$:

<table><tbody><tr><td>1(1 v 1)</td><td>0(1 v 1)</td><td>1(0 v 1)</td><td>0(0 v 1)</td><td>1(1 v 0)</td><td>0(1 v 0)</td><td>1(0 v 0)</td><td>0(0 v 0)</td></tr><tr><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>11 v 11</td><td>01 v 01</td><td>10 v 11</td><td>00 v 01</td><td>11 v 10</td><td>01 v 00</td><td>10 v 10</td><td>00 v 00</td></tr><tr><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

To insure a proper comparison it is essential, of course, that the cases vertically aligned with each other actually represent the same substitution of digits for letters. If, e.g., we are testing the equivalence of '$p(q v r)$' with '$\overline{rp v qp}$', and we start off the table for '$p(q v r)$' in the fashion:

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1(1 v 1)

0(1 v 1)

1

0

as previously, then we must start off the table for 'rp v qp' in this fashion:

11 v 11

10 v 10

...;

1

0

the second of its eight cases must, for proper comparison with the second case in the table for 'p(q v r)', be formed by putting '0' for 'p' and '1' for 'q' and 'r'. Confusion over this point can be kept at a minimum by working vertically: doing the first case of the first schema, then moving down to do the first case of the second under it, then up and over to do the second case of the first schema, then down to do the second case of the second schema, and so on. This vertical approach saves labor, moreover, when schemata prove non-equivalent; for we can discontinue our work at the first discrepancy. Thus, suppose we are testing 'p(q v r)' and 'p v qr' for equivalence. We begin thus, working vertically.

1(1 v 1)

0(1 v 1)

1

0

1 v 11

0 v 11

1

1

From this disagreement we conclude non-equivalence without having to touch the remaining six cases.

Occasionally it may happen that two schemata which are to be tested for equivalence do not contain exactly the same letters--e.g., 'pq' and 'p(q v qr)'. Here the most efficient strategy is to exhaust the combinations of truth values for only those letters which appear in the schema having the fewer letters. The odd letters of the other schema are simply carried along, unsupplanted by '1' and '0'; commonly, however, they will prove no obstacle to the evaluations. Thus, the test of the equivalence of 'pq' and 'p(q v qr)' appears as follows:

11

01

10

00

1

0

0

0

1(1 v 1r)

0(1 v 1r)

1(0 v 0r)

0(0 v 0r)

1

0

0

0

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The surviving 'r' was no obstacle to those evaluations, since 'l' v 'lr' and 'Or' reduce to 'l' and '0' by (iii) and (iv) of § 3 regardless of 'r'.

Sometimes, however, the odd letter or letters may persist through the reductions, instead of disappearing everywhere as 'r' did above. E.g., let us test 'p v q' and 'pr v pr v (q v r)(q v r)' for equivalence:

<table><thead><tr><th>1 v 1</th><th>0 v 1</th><th>1 v 0</th><th>0 v 0</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>0</td></tr><tr><td>1</td><td>1</td><td>r v r v r<sup>F</sup></td><td>r<sup>F</sup></td></tr></tbody></table>

Here we have an unresolved 'r' in the last two cases. If the equivalence is to be sustained, the 'r v F v r<sup>F</sup>' of the third case must come out in agreement with the 'l', above it, both when 'l' and when '0' is put for 'r'; in short, 'r v F v r<sup>F</sup>' must be valid to sustain the equivalence. The 'r<sup>F</sup>' of the fourth case, similarly, must come out in agreement with the '0' above it, both when 'l' and when '0' is put for 'r'; in short, 'r<sup>F</sup>' must be inconsistent to sustain the equivalence. But 'r v F v r<sup>F</sup>' and 'r<sup>F</sup>' are respectively valid and inconsistent, as can be checked by the method of § 4 (insofar as not already evident); so the equivalence holds.

In general, thus, when schemata do not have exactly the same letters, the method of testing them for equivalence is as follows. We set up a comparative table in which all letters that appear in the poorer schema are supplanted by 'l' and '0' in all possible ways, but the extra letters of the richer schema are left unsupplanted. The reductions of all expressions are then compared; if in every case where the poorer schema reduces to 'l' the richer one reduces to 'l' or a valid schema, and in every case where the poorer schema reduces to '0' the richer one reduces to '0' or an inconsistent schema, then the equivalence holds; otherwise not. Subsidiary tests of validity and consistency are thus sometimes required; usually, however, these subsidiary points are simple enough to be decided by inspection.

One more example: let us test 'p v q' and '(p v r)(q v r)' for equivalence.

<table><thead><tr><th>1 v 1</th><th>0 v 1</th></tr></thead><tbody><tr><td>1</td><td>1</td></tr><tr><td>(1 v r)(1 v r)</td><td>(0 v r)(1 v r)</td></tr><tr><td>1</td><td>r</td></tr></tbody></table>

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Here we may stop; since 'r' is not a valid schema, the equivalence fails. By working vertically we have spread ourselves the necessity of taking up the remaining two cases.

The relationship of equivalence becomes quite trivial in connection with schemata which are valid or inconsistent. If we set up an equivalence test for two schemata as above, and the two schemata happen to be valid, obviously the test will show them equivalent; for, both schemata will reduce to 'l' in all cases and will therefore agree. Thus all valid schemata are equivalent. A similar relection shows that all inconsistent schemata are equivalent. Conversely, also, a valid schema is equivalent to valid schemata only, and an inconsistent schema is equivalent to inconsistent schemata only.

The term 'equivalent', like 'valid' and 'inconsistent' (cf. § 4), is extended--subject to the qualifier 'truth-functionally'--to the special examples obtainable by substitution for the letters in equivalent truth-functional schemata. E.g., the statements 'The dean and the president will not both speak' and 'Either the dean will not speak or the president will not speak' are called truth-functionally equivalent, for they proceed from the equivalent truth-functional schemata'--(pq)' and 'P v q' by putting 'the dean will speak' for 'p' and 'the president will speak' for 'q'. (The substitutions must, of course, as in this example, affect each letter uniformly in all its occurrences.)

## § 6. IMPLICATION

One of the conspicuous purposes of logic, in its application to science and everyday discourse, is to transform one statement into another which "says the same thing" in another form -- a form which is simpler, perhaps, or more convenient for the particular purposes in hand. Thus it is that techniques for establishing or checking the equivalence of statements are an objective of logical theory. Now there are degrees of equivalence, depending on what aspects of logical structure are taken into consideration. When the truth-functional structure of two statements -- the structure in terms of alternation, conjunction, and negation -- happens to be enough by itself to make the statements agree in truth-values, the statements are truth-functionally equivalent. A technique for determining such equivalence has been set forth.

But another purpose of logic, yet more conspicuous than transformation, is deduction. Deduction, unlike transformation, need not be reversible. Whereas transformation leads from a statement to an equivalent which "says the same thing", deduction leads from a statement to a consequence, which merely "says no more": which is merely implied by the original statement.

There are degrees of implication, as of equivalence, depending on what aspects of logical structure are taken into consideration. When the truth-functional structure of the two statements happens to be enough by itself to make the truth of the one statement guarantee that of the other, the relation is truth-functional implication.

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'p v r', e.g., is checked thus:

(O v q)(q v O)
qq

1 v 1      0 v 1      1 v 0      0 v 0
1          1          1          0

Here we had to evaluate the upper and more complex schema only in the fourth case, since in the other three cases the lower schema reduced to '1'. Since in the fourth case, finally, we come out with the inconsistent 'qq' above, we conclude that the implication holds.

That $(p \lor q)(p \lor r)'$ does not imply $(q \lor r)(q \lor \bar{r})'$ is found thus:

$(p \lor O)(p \lor 1)$

p

$(1 \lor 1)(1 \lor \bar{1})$ $(0 \lor 1)(0 \lor \bar{1})$

1          0

Having found a case which yields '0' below and the consistent 'p' above, we have disproved the implication and may ignore the 'remaining' two cases. We can render '(p v q)(p v r)' true and '(q v r)(q v \bar{r})' false by taking 'p' as true, 'q' as false, and 'r' as true.

The strategy of working thus upward, instead of downward, obviously will tend to a saving of labor even when the two schemata have exactly the same letters, so long as the lower one is more complex than the upper one.

Clearly implication is "reflfoxivo"; in the sense that every schema implies itself; and "transitive", in the sense that if one schema implies a second and the second implies a third then the first also implies the third. Equivalence, of course, is likewise reflexive and transitive. Furthermore equivalence is "symmetrical": If one schema is equivalent to a second then the second is equivalent to the first; but implication lacks this property.

The behavior of implication in connection with conjunction is worth noting. If a schema implies each of several schemata, it also implies their conjunction. More generally, given several implications, the conjunction of the implying schemata will imply the conjunction of the implied schemata. These points are evident from a little reflection on the testing techniques.

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Such reflection shows also that equivalent schemata imply each other, and conversely, schemata that imply each other are equivalent. Equivalency is mutual implication. From the nature of the tests it is likewise apparent that a valid schema is implied by every schema and implies only valid schemata, and an inconsistent schema implies every schema and is implied only by inconsistent ones.

The term 'implication', like 'equivalence' and the rest, is extended -- subject to the qualifier 'truth-functional' -- to the special examples obtainable by uniform substitution upon the letters in truth-functional schemata one of which implies the other. Thus, the statement 'Sodium is a metal and lighter than water' truth-functionally implies the statement 'Sodium is either a metal or soluble, and sodium is either lighter than water or soluble'. For, we have seen that 'pq' implies '(p v r)(q v r)'; and the two statements in question are obtainable from those schemata by substituting 'sodium is a metal' for 'p', 'sodium is lighter than water' for 'q', and 'sodium is soluble' for 'r'.

## § 7. INTERCHANGE, SUBSTITUTION, RESOLUTION

The importance of equivalence lies in the interchangeability of equivalent schemata. If two schemata are equivalent, they behave alike under truth-table tests of all sorts; therefore they are both valid or both non-valid, they are both consistent or both inconsistent, they both imply the same schemata, and they are both implied by the same schemata.

Furthermore, if a schema is supplanted by an equivalent schema within a longer schema which is valid, the result will be valid. E.g., if in the valid schema 'pq v p v q' we supplant the part 'pq' by its equivalent '-(p v q)', the result '-(p v q) v p v q' must, like the original, be valid. The outcome of a truth-table computation of validity cannot be affected by our having changed the part 'pq' to '-(p v q)', since those, being equivalent, always behave alike under truth-table tests.

Similarly, if a schema is supplanted by an equivalent within a longer schema which is non-valid, or consistent, or inconsistent, the result will correspondingly be non-valid, or consistent, or inconsistent. More generally, if in any schema we supplant a part by an equivalent, the whole original schema and the whole result will be equivalent to each other. From the equivalence of '-(pq) v (p v q) v (-((pq) v r))s' with '(p v q v r)s' we may immediately infer, without a separate test, the equivalence of '-(pq) v r)s' with '(p v q v r)s'.

Such putting of equivalents for equivalents is not to be confused with that other kind of substitution, no less important, which consists in putting a schema for a letter in order to get a special case. The validity of '-(p v q) v p v q', noted above could have been arrived at by substituting the schema 'p v q' for the letter 'r' in the valid schema 'T v r'; but this is not because of any equivalence on the part of 'r' and 'p v q'. The reasoning here is

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rather as follows: since 'T v r' is bound to come out true no matter what statement be put for 'S', it follows, as a special case, that '-(p v q) v p v q' will come out true no matter what statement 'p v q' may represent; hence no matter what statements be put for 'p' and 'q'.

In putting equivalents for equivalents, we are free to supplant some occurrences and leave others untouched. Given 'pā v -(pā)' (a case of 'r v T') as valid; we may supplant just the first 'pā' by its equivalent '-(p v q)', and thus infer that '-(p v q) v -(pā)' is valid; or we may supplant just the second 'pā', and thus infer that 'pā v --(p v q)' is valid; or we may supplant both, and thus, infer that '-(p v q) v --(p v q)' is valid. In the other kind of substitution, however -- substituting a schema for a letter to get a special case -- we must take care to supplant all occurrences of the letter uniformly. From the validity of 'T v r' we may infer that '-(p v q) v p v q' is valid, but not that 'T v p v q' is valid, nor that '-(p v q) v r' is valid. The need of this precaution is evident: partial substitution ordinarily yields a new form which is no more special case of the original form.

We have seen why substitution of any one schema for all occurrences of a letter, in a valid schema, yields a valid schema. The reason was that if all instances of the original more general schema are true, all instances of the special case will be true. Similarly, substitution of any one schema for all occurrences of a letter in an inconsistent schema will yield an inconsistent schema; for, if all instances of the original were false, all instances of the special case will be false. Similar reasoning shows that substitution of any one schema for all occurrences of a letter throughout two equivalent schemata will yield schemata which are equivalent to each other. Similarly, again if a given schema implies another, and we substitute some one schema for all occurrences of a letter throughout the two given schemata, the first of the two resulting schemata will imply the second.

It is not to be supposed, however, that the substitution of a schema for all occurrences of a letter in a consistent schema will always yield a consistent schema. The more fact that the more general schema has some true instances (which is what consistency means) gives us no reason to suppose that the special case will share any of the true instances. The schema 'p y pq', o.g., is consistent (as may be verified by the method of § 4), but substitution of 'T' for 'p' therein yields an inconsistent schema 'T v rTq'. Similarly, substitution for a letter in a non-valid schema cannot be depended upon to yield a non-valid schema; it may yield a valid or a non-valid one.

The procedures of *interchanging equivalents* and *substituting* for letters have been explained and contrasted. Now a third type of activity which often proves rewarding is that of *resolving* valid or inconsistent parts of schemata. To see how this works, consider a schema S within which a valid clause occurs as a part; and suppose we are working out a truth table for S. No matter how we put '1' and '0' for the letters of S, once we proceed to make reductions by (i)-(v) of § 3 the valid part of S is bound to reduce to '1'; and then this '1' in its turn will disappear by (ii) if in

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a conjunction, or will engulf a whole alternation, by (iii), if in an alternation. So the net truth table of S would have come out differently if, prior to working out the table, we had treated the valid clause of S as a '1' and simply deleted it from conjunction, and deleted all but it from alternation, as in (ii) and (iii) of § 3. Within any schema S, thus, from a conjunction we may delete any valid component, and from an alternation with a valid compound we may delete all other components; and what is left of S will be equivalent to S. Simplest examples: 'p(q v q)' is equivalent to 'p', and 'p v q v q' to 'q v q'.

Parallel remarks apply to inconsistent parts; the reasoning is as above, but based on (i) and (iv) instead of (ii) and (iii). Within any schema S, thus, from an alternation we may delete any inconsistent component, and from a conjunction with an inconsistent component we may delete all other components, and what is left of S will be equivalent to S. Simplest examples: 'p v qq' is equivalent to 'p', and 'pqq' is equivalent to 'qq'.

Thus it is that any conjunction or alternation having a valid or inconsistent component can be simplified by deletion, one way or another. But how can a valid or inconsistent schema ever occur as a part of a longer schema, save as component of a conjunction or alternation? There is one other way: it may occur negated. But the negation of a valid schema is inconsistent, and the negation of an inconsistent one is valid (cf. § 4); hence negation is no obstacle to the above technique of deletion. Any schema having a valid or inconsistent schema as a part can be reduced, by progressive deletions, until either the part in question has disappeared or else we are left with a valid or inconsistent remainder. This process will be called resolution of the valid or inconsistent component. E.g., consider the schema 'pq v -(r v -(qq))', which contain as a part the inconsistent schema 'qq'. Since the negation '-(qq)' of that inconsistent schema is valid, we can delete 'r' from the alternation 'r v -(qq)'; the whole thus reduces to 'pq v --(qq)', or 'pq v qq'. From this alternation we delete the inconsistent component 'qq', and are left with 'pq'.

Again, consider 'pq v p(qq v r)'. Here we delete the inconsistent component 'qq' from the alternation 'qq v r', thereby reducing the whole to 'pq v pr'.

Next consider 'pq v -(p v q v q)', which contains as a part the valid schema 'q v q'. Deleting 'p' from the alternation 'p v q v q', we reduce the whole to 'pq v -(q v q)'. Since the negation '-(q v q)' of the valid 'q v q' is inconsistent, we can delete it from the alternation 'pq v -(q v q)', and are left thus with 'pq'.

In general, confronted with any schema in which we detect a valid or inconsistent part, we may be sure that the whole is reducible to a simpler equivalent; and we can reduce it forthwith, by progressive deletions according to the above rules of resolution.

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## § 8. DUALITY

Consider any schema built up of letters 'p', 'q', etc. by means of any or all of the operations of conjunction, alternation, and negation. Then consider a new schema, built up exactly like the old one except that alternation is used in the new schema wherever conjunction was used in the old schema, and conjunction is used in the new schema wherever alternation was used in the old schema. The two schemata are said to be duals; each is the dual of the other. E.g., 'p(q v r)' and 'p v qr' are duals; so are 'pq v pr' and '(p v q)(p v r)'; and so are 'pq v pq' and '(p v q)(p v q)'. A more letter or its negation, e.g., 'p' or 'F', is counted as the dual of itself, since there are no conjunctions or alternations here to be changed.

A conjunction, we know, is true when its components are all true, and otherwise false; and an alternation is false when its components are all false, and otherwise true. Conjunction is to truth as alternation is to falsity. Consequently the following important law holds, which I shall call the first law of duality: *Dual schemata take on opposite truth values under opposite assignments of truth values to their letters.* E.g., having evaluated 'pq v pq' as I where 1 is assigned to 'p' and 0 to 'q' (cf. § 3), we may rest assured that the dual '(p v q)(p v q)' will reduce to '0' where 0 is assigned to 'p' and 1 to 'q'. Similarly, having evaluated '(p-(qpr))' as 0 where '1' is assigned to 'p' and to 'q' and 0 is assigned to 'r', we may rest assured that the dual '(p v -(q v pvr))' will reduce to '1' where 0 is assigned to 'p' and to 'q' and 1 is assigned to 'r'.

Conspicuous among the consequences of the above law is this second law of duality: If one schema implies another, the dual of the latter implies the dual of the former. The reasoning is as follows: Suppose S1 is a schema implying S2, and that S3 and S4 are the respective duals. We want to show that S4 implies S3; i.e., that every assignment of truth values to the contained letters which makes S4 true makes S3 true. Consider, then, any assignment of truth values to the contained letters which makes S4 true. By the first law of duality, the opposite assignment to the letters will make S2 false. Hence this opposite assignment to the letters will also make S1 false, since S1 implies S2 (and therefore cannot be true where S2 is false). Then, by the first law of duality, the original assignment of truth values to the letters will make 3 true, q.o.d.

Let us repeat this reasoning in application to an example. Given that 'pq' implies '(p v r)(q v r)' (cf. § 6), to show that the dual of the latter, viz. 'pr v qr', implies the dual 'p v q', of the former. Given any assignment of truth values which makes 'pr v qr' true (e.g., 1 to 'p' and 'r', and 0 to 'q'), we know by the first law of duality that the opposite assignment (0 to 'p' and 'r', and 1 to 'q') must make '(p v r)(q v r)' false, and hence must make 'pq' false (since 'pq' implies '(p v r)(q v r)'). But then, by the first law of duality, the original assignment (1 to 'p' and 'r', and 0 to 'q') must make the dual 'p v q' of 'pq' true.

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That 'pr v qr' implies 'p v q' could of course be verified directly by the method of § 6. But the convenience of the second law of duality is that it assures us immediately of this implication, without need of special verification, once we have verified as in § 6 that 'pq' implies '(p v r)(q v r)'. Similarly, having determined in § 6 that '(p v q)(q v r)' implies 'p v r', we can conclude that 'pr' implies 'pq v qr'. For every implication, established by the method of § 6, a dual implication is provided gratis by the second law of duality.

A third law of duality is that if two schemata are equivalent to each other, their duals are equivalent to each other. Reflecting that equivalence is mutual implication (cf. § 6), we see that this law follows immediately from the second law of duality.

For every equivalence established by the method of § 5, another is forthcoming by the above law. From the equivalence of 'pp' with 'p' we might have concluded immediately, by the above law, that 'p v p' (which is the dual of 'pp') is equivalent to 'p' (which is the dual of 'p'). From the equivalence of 'pq' with 'qp', similarly, we can infer the equivalence of 'p v q' with 'q v p'. From the equivalence of '-(pq)' with 'p v q' we can infer the equivalence of '-(p v q)' with 'pq'. From the equivalence of 'p(q v r)' with 'pq v pr' we can infer the equivalence of 'p v qr' with '(p v q)(p v r)'.

A fourth law of duality is that a schema is valid if and only if its dual is inconsistent. This is apparent from the first law of duality, in view of which, if a schema comes out true under all assignments of truth values to its variables, the dual must come out false under all assignments, and vice versa.

For each validity or inconsistency which is established by the method of § 4, thus, a companion inconsistency or validity is provided by the above law. From the validity of 'p v p' we can infer the inconsistency of 'pq', and from the validity of 'pq v p v q' (of § 4) we can infer the inconsistency of '(pvq)pq'.

A further important law, essentially one of duality, is thus:

(i) '-(p v q v ... v s)' is equivalent to 'pq...s'.

For, by the first law of duality, if a given assignment of truth values to the letters yields a given truth value for 'p v q v ... v s' the opposite assignment must yield the opposite value for the dual 'pq...s'. But the effect of reversing the assignments to 'p', 'q', etc. can be gained by changing 'p' to 'p', 'q' to 'q', etc. So any assignment of truth values to 'p', 'q', etc. yields opposite values for 'p v q v ... v s' and 'pq...s'; therefore (1) follows: (The truth of (1) is seen yet more quickly by reflecting directly upon (I)-(III) of § 3; but the above argument brings out the relationship to duality.)

By the third law of duality we may conclude further, from (1), that

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(ii) $1'-(pq...s)'$ is equivalent to $\overline{1}p \vee \overline{q} \vee \dots \vee \overline{s}'$.

This and (i) are known as **DeMorgan's laws**, because they were stated by Augustus DeMorgan in the nineteenth century (as well as by William of Ockham in the fourteenth century and by Petrus Hispanus in the thirteenth).¹

Since $1-(pq)'$ is equivalent by (i) to $\overline{1pq}'$, we see, by negating each side, that $\overline{1p} \vee \overline{q}'$ is equivalent to $1-(\overline{pq})'$. (This could be verified also, of course, by the regular routine of § 5.) Now the important thing about this equivalence is that it shows the sign $1'$ of alternation to be strictly superfluous. Instead of writing an alternation, e.g., $\overline{1p} \vee \overline{q}'$, we could always limit ourselves to the notation of conjunction and negation and write $1-(\overline{pq})'$.

Conversely, also, given the notations of alternation and negation, we could get along without that of conjunction. For, by duality, since $\overline{1p} \vee \overline{q}'$ is equivalent to $1-(\overline{pq})'$, $\overline{1pq}'$ is equivalent to $1-(\overline{p} \vee \overline{q})'$.

The reason for using the three-fold notation of conjunction, alternation, and negation, when two would suffice, is that the three-fold notation enables us to avoid negating compounds. We never need apply negation to the whole of an alternation, since $1-(\overline{1p} \vee \overline{q} \vee \dots \vee \overline{s})'$ is equivalent to $\overline{1pq...s}'$; and we never need apply negation to the whole of a conjunction, since $1-(\overline{pq...s})'$ is equivalent to $\overline{1p} \vee \overline{q} \vee \dots \vee \overline{\overline{s}}'$. Also of course we never need apply negation to a negation, since $\overline{1p}'$ is equivalent to $\overline{1p}'$. Therefore any schema built up by means of alternation, conjunction, and negation can be put over into an equivalent schema in which negation never applies to anything but individual letters. Transformation of this kind is generally conducive to easy intelligibility. The schema $1-(\overline{1p}-(\overline{q} \vee \overline{p})')$, e.g., becomes less bewildering when transformed into $\overline{1p} \vee \overline{1q} \vee \overline{p}'$. Method: by (ii), $1-(\overline{1p}-(\overline{q} \vee \overline{p})')$ is equivalent to $\overline{1p} \vee 1-(\overline{q} \vee \overline{p})'$, i.e., to $\overline{1p} \vee \overline{q} \vee \overline{p}'$.

## § 9. ALTERNATIONAL NORMAL SCHEMATA

The following important equivalence, verified in § 5, is known as the **law of distributivity of conjunction through alternation**:

$\overline{1p} \vee \overline{q} \vee \overline{r}'$ is equivalent to $\overline{1pq} \vee \overline{pr}'$.

The law can be given this more general form:

$\overline{1p} \vee \overline{q} \vee \overline{r} \vee \dots \vee \overline{s}'$ is equivalent to $\overline{1pq} \vee \overline{pr} \vee \dots \vee \overline{ps}'$.

¹ Cf. Lukasiowicz, "Zur Geschichte der Aussagenlogik," *Erkenntnis*, vol. 5, pp. 122, 130.

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For, whenever '0' is put for 'p' or for all of 'q', 'r', ..., 's',
clearly both 'p' · 'q' · 'q' · ... 'v' · s' and 'pq' · 'pr' · ... 'v' · s' will
reduce to '0'; and whenever 'l' is put for 'p' and for at least one
of 'q', 'r', ..., 's', clearly both 'p' · 'q' · 'q' · ... 'v' · s' and
'pq' · 'pr' · ... 'v' · s' will reduce to 'l'.

The law of distributivity can be applied not only when an
alternation appears in the position 'p · qvr v ... vs', but also
when it appears in the position 'qvr v ... vs · t', or
'p · qvr v ... vs · t'; for, the order and grouping of the com-
ponents of a conjunction are known to be immaterial. In every
case we may simply multiply out, as we do in algebra when a sum
occurs as a factor of a product. Thus 'qvr v ... vs · t' becomes
'qt vrt v ... vst', and 'p · qvr v ... vs · t' becomes
'pqt v prt v ... vps't'.

If several alternations are joined in conjunction, we can
eliminate them one at a time by multiplying out. Thus, consider:

p v q v ... v s · p' v q' v ... v s' ·

Multiplying 'p v q v ... v s' bodily through 'p' v q' v ... v s'', we get:

p v q v ... v s · p' v q' v ... v s · q' v ... v s ·

p v q v ... v s · s'.

Then the part 'p v q v ... v s · p'', multiplied out in turn, becomes 'pp' v qp' v ... sp'', and correspondingly for the other parts; the final result is therefore:

pp' v qp' v ... v sp' v pq' v qq' v ... v sq' v ... v ps' v qs'

v ... v ss'.

What the distributivity of conjunction through alternation shows
is that we never need a conjunction having an alternation as one
of its components. We can expand any schema into an equivalent
in which no conjunction has an alternation as a component. Also,
by DeMorgans' laws, we can avoid applying negation otherwise than
to single letters. Combining these two kinds of reduction, we
can reduce any schema to an equivalent in which negation applies
only to single letters and conjunction applies only to letters and
negation of letters. A schema thus reduced is said to be an *alter-
national normal schema*.

Thus, let us reduce 'p · -(qr) · s v st' to an alternational
normal schema. The successive stages are as follows:

p · -(qr) · s v st,

p · q v r · s v st,

p · q v r · s · v · p · q v r · st,

pq s v prs v pqst v prst.

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It will be convenient, here and at later points, to have a general word covering letters and negations of letters. Let us call them **literals**. Thus 'p', 'q', 'p', etc. are literals. An alternational normal schema, then, may be a literal, a conjunction of literals, or an alternation whose components are literals and/or conjunctions of literals. The component schemata of which an alternational normal schema S is an alternation will be called the primary parts of S; or, if S is not an alternation, it counts as its own sole primary part. Thus the primary parts of 'pqr v qr v prs' are 'pqr', 'qr', and 'prs'; the primary parts of 'p v q' are 'p' and 'q'; and the primary part of 'pq' (or of 'F', or of 'p') is simply 'pq' (or 'F', or 'p') itself. Primary parts of alternational normal schemata are always literals or conjunctions of literals.

The advantage of alternational normal schemata is that we can tell at a glance just what truth values for the component letters are compatible with truth of the whole. Thus 'pqsv prsv pqst v prst', above, affirms that one of these four situations holds: either 'p' and 's' true and 'q' false, or else 'p' and 's' true and 'r' false, or else 'p' and 't' true and 'q' false, or else 'p' and 't' true and 'r' and 's' false. In the original form, 'p . -(qr) . sv et', the net import of the compound 'qr' is much less evident.

This advantage can be enhanced, and at the same time any contemplated computations as of §§ 3-6 can be rendered less laborious, by certain abridgments to which an alternational normal form can be subjected on sight. To begin with, we can go through it crossing out any primary part which, like 'pqpr', is visibly inconsistent through exhibiting a letter both plain and negated (cf. § 7), and also crossing out any repetitions of literals in a primary part, e.g., the second 'p' in 'pqpr'. The alternational normal schema:

$$ (1) \quad \text{prp}^s v \text{qr}^d v \text{rq}^r v \text{pqr}^qs v \text{psr}^q v \text{srp}^p v \text{qr}^r, $$

e.g., thus reduces, by deletion of the inconsistent primary parts 'qr' and 'psr', and deletion of the repeated 'p' from 'prps' and the repeated 'T' from 'rq', to:

$$ (2) \quad \text{pr}^s v \text{Fq} v \text{psr}^q v \text{sr}^p v \text{qr}^r. $$

In this way almost any alternational normal schema reduces to an equivalent one in which (as in (2)) no primary part exhibits a letter more than once. The sole exception arises when every primary part of an alternational normal schema exhibits a letter both plain and negated; in this case every primary part but one can disappear under the above procedure, but we have to keep one if we are not to end up with an absolute blank. In this case, accordingly, we come out with a primary part standing along and visibly inconsistent. We can reduce it arbitrarily to 'pp' if we like, since any two inconsistent schemata are equivalent (cf. § 5).

A literal, or a conjunction of literals wherein no letter appears twice, will be called a fundamental schema. What we have seen, then, is how, simply by progressive deletions, to reduce any alternational normal schema to an equivalent which is either (I) a conjunction of literals containing some letter both plain and

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negated or (II) an alternational normal schema whose primary parts are fundamental schemata.

A result of kind (I) is visibly inconsistent. A result of kind (II), on the other hand, is always consistent; for, we can make it come out true simply by assigning '1' to those letters which occur plain in the first primary part, and '0' to those which occur negated in the first primary part. This assignment makes that primary part come out true, and therewith the alternation.

So every consistent alternational normal schema reduces to one whose primary parts are fundamental schemata, and every inconsistent normal schema reduces to a visibly inconsistent conjunction of alternals. Here we have a mechanical test of inconsistency, alternative to that of § 4: convert the schema to alternational normal form and reduce as above.

Abridgment was what we were after; the new test of inconsistency came as a by-product. Now an alternational normal schema which has been reduced, as above, to one whose primary parts are fundamental schemata, will still commonly admit of further abridgment of one or another of the following types:

<table><tr><td>(a)</td><td>'p v p'</td><td>to</td><td>'p',</td></tr><tr><td>(b)</td><td>'pq v pq'</td><td>to</td><td>'p',</td></tr><tr><td>(c)</td><td>'p v pq'</td><td>to</td><td>'p',</td></tr><tr><td>(d)</td><td>'p v pq'</td><td>to</td><td>'p v q'.</td></tr></table>

(The equivalences upon which (b), (c), and (d) depend are readily checked by the method of § 5.) In making such abridgments, the order in which the components of a conjunction follow one another is conveniently ignored; and similarly for an alternation. Thus abridgment of type (a) gets rid of the 'qF' at the end of (2), since this is a repetition (disregarding order) of an 'Fq' earlier in (2). The result is:

(3) prs v Fq v psrq v srp.

By (c), next (taking the 'p' of (c) as 'prS'), we can reduce 'prS v prsq' to 'prS'; in other words, we can simply cross out 'prsq' (or 'psrq') in (3). Result:

(4) prs v Fq v srp.

But 'prS' here amounts to 'SrP'; and, by (b), 'SrP v SrP' reduces to 'Sr'. So (4) becomes:

(5) sr v Fq.

In practice the reduction of (1) to (5) would proceed by a mere crossing out of various letters and segments of (1), until only (5) remained.

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These rules of abridgment could be extended, but we should soon reach a point of diminishing returns because of rising complexity in the rules themselves. Let us content ourselves with one more example of the use of the rules already at hand. Given:

$$ (6) \qquad pq \vee pq \vee \bar{p}r \vee \bar{r} \vee qst, $$

we first reduce 'pq \vee pq' therein to 'p' by (b), and then reduce 'p \vee \bar{p}r' to 'p \vee r' by (d), obtaining:

$$ (7) \qquad p \vee r \vee \bar{r} \vee qst. $$

Now the interesting thing about this result is that it is visibly valid, because of its part 'r \vee \bar{r}' (resolution of which, by § 7, would reduce the whole to 'r \vee \bar{r}') . If we had been wondering about the validity of (6), we have the answer in (7) without computation. But it must be borne in mind that this, while a common enough outcome, cannot be depended upon to be forthcoming in all cases of validity; we have here no general alternative to the method of § 4 for testing validity.¹ The schema:

$$ pq \vee \bar{p}r \vee \bar{p}r \vee \bar{p}s \vee qr \vee \bar{r}s $$

is valid, as may be verified by the method of § 4, but it is not visibly valid, nor is it further reducible by the rules at hand.

## § 10, CONJUNCTIONAL NORMAL SCHEMATA

From the law of distributivity of conjunction through alternation (§ 9) there follows, by duality (§ 8, third law), a law of distributivity of alternation through conjunction:

'p \vee qr...s' is equivalent to 'p \vee q \vee pvr \vee ... \vee pvs'.

Thus conjunction and alternation behave much more symmetrically than multiplication and addition. In arithmetic we have 'x(y \oplus z)' = xy \oplus xz' but not 'x \oplus yz' = xy \oplus xz'; in the case of conjunction and alternation, on the other hand, distributivity works both ways.

It is consequently possible to reduce any schema not only to an alternational normal schema, but also to a conjunctional normal schema -- i.e., to a schema in which negation applies only to single letters and alternation applies only to literals. A conjunctional normal schema may be a literal, or an alternation of literals, or a conjunction whose components are literals and/or alternations of literals. Examples are:

p \vee q \vee r \vee q \vee p \vee r \vee s, p \vee q \vee r \vee s, p \vee q, \bar{p}q, p, \bar{r}.

Conjunctional normal schemata are the duals (cf. § 8) of alternational normal schemata.

1 We shall have in § 10, however, along other lines.

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The step-by-step reduction of a schema to conjunctional normal form is exactly parallel to the reduction to alternational normal form, the only difference being that the roles of alternation and conjunction are interchanged. Thus, let us go back to the example 'p • -(qr) • svst' of § 9, this time reducing it to conjunctional normal form. The successive stages are now as follows:

p • -(qr) • svst,

r • qvr • svst,

p • vr • svst • svt.

Every schema S is reducible equally, we see, to an alternational normal schema S' and a conjunctional normal schema S'', S' and S'', being equivalent to S, will of course be equivalent to each other. Duals, however, are in general not equivalent; and it must not be expected that S' and S'' will be duals of each other. S', being an alternational normal schema, will have as its dual some conjunctional normal schema, but ordinarily both other than S'', and S'', being a conjunctional normal schema, will have as its dual some alternational normal schema, but ordinarily one other than S''. Jc.g., where S is p • -(q1) • svst, we have seen that it reduces to the alternational normal schema:

pqs v prs v pqst v prst (S')

and to the conjunctional normal schema:

p • qvr • svst • svt. (S'')

S' and S'' here are therefore equivalent; but they are by no means duals of each other. The respective duals of S' and S'' are rather:

pvqs • pvrvs • pvqvsvt • pvrvsvt,

pvqrvssvst.

These two latter schemata are by no means equivalent to S' and S''. (They are, however, equivalent to each other, by the third law of duality, § 8.)

The components whereof an alternational normal schema is an alternation (or the whole, if it is not an alternation) were called the *primary parts* of the alternational normal schema. Similarly the components whereof a conjunctional normal schema is a conjunction (or the whole, if it is not a conjunction) may be called the *primary parts* of the conjunctional normal schema. Thus the primary parts of 'pvqvr • qvr • pvrvs' are 'pvqvr', 'qvr', and 'Fvrsvs'; the primary parts of 'p}' are 'p' and 'q'; and the primary part of 'pvq' (or of 'p', or of 'q') is simply 'pvq' itself (or 'p', or 'q').

Actually this terminology leads to an ambiguity, because of the rest that certain schemata, such as 'p}' and 'pvq', qualify simultaneously as alternational and conjunctional normal schemata. Thus 'p}' has 'p' and 'q' as primary parts from the point of view

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of a conjunctional normal schema, but only 'p$\overline{q}$' as primary part from the point of view of an alternational normal schema; and conversely for 'p v $\overline{q}$'. A double terminology, 'alternational primary part' and 'conjunctional primary part', would hence be more proper; but in practice the ambiguity is resolved well enough by the purposes in hand.

Any alternational normal schema which is inconsistent reduces, we saw (§ 9), to a visibly inconsistent conjunction of literals; a conjunction containing a letter both plain and negated. The method consisted simply in dropping other visibly inconsistent primary parts. Now the related behavior on the part of conjunctional normal schemata is, by considerations of duality, this: any conjunctional normal schema which is valid reduces to a visibly valid alternation of literals: an alternation containing a letter both plain and negated. The method consists simply in dropping other visibly valid primary parts.

Here, then, we have a mechanical test of validity, alternative to that of § 4: convert to conjunctional normal form and reduce as above.

We saw (§ 9) that any alternational normal schema which was consistent reduced to one in which no primary part contained a letter twice. The method consisted in dropping visibly inconsistent primary parts and dropping duplications of literals within a primary part. Now with conjunctional normal schemata the situation is, by considerations of duality, this: any conjunctional normal schema which is not valid reduces to one in which no primary part contains a letter twice. The method consists in dropping visibly valid primary parts and dropping duplications of literals within a primary part.

Further reductions of consistent alternational normal schemata were possible by (a)-(d) of § 9. Similarly, further reductions of non-valid conjunctional normal schemata are possible by the duals of (a)-(d):

<table><tr><td>'pp'</td><td>to 'p',</td></tr><tr><td>'p v q . p v $\overline{q}$'</td><td>to 'p',</td></tr><tr><td>'p . p v q'</td><td>to 'p',</td></tr><tr><td>'p . $\overline{p}$ v q'</td><td>to 'pq'.</td></tr></table>

## § 11. DEVELOPED NORMAL SCHEMATA

The equivalence of 'pq v $\overline{pq}$' with 'p' was used in § 9 (b) as a basis for abridging alternational normal forms. For some purposes, however, it is convenient to use this equivalence in the opposite way, as a means of turning a shorter alternational normal form into a longer one. If a primary part '--' of a given alternational normal schema lacks a certain letter, say '$q$', and if we want to

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supply that lack, we may, by the above equivalence, expand `|---|` into `|---q v ---q'`. Such expansion is called *development* of `|---|` with respect to `|q|`.

Every consistent schema S is reducible, we have seen (§ 9), to an alternational normal schema S' whose primary parts are fundamental schemata. Now if a primary part of S' lacks a letter appearing elsewhere in S, let us develop that part with respect to the missing letter; and let us continue thus until we have turned S' into a schema S" in which each primary part exhibits each letter of S". Finally we may rid S" by § 9 (a), of any duplication of primary parts. The result S"† is an alternational normal schema in which each primary part contains exactly one occurrence of each letter of the schema, and all the primary parts are different (different not merely in order, but in the incidence of negation signs). Such a schema will be called a *developed normal schema*.

Every developed normal schema is consistent; for, it is an alternational normal schema whose primary parts are fundamental schemata, and we saw (§ 9) that any such alternational normal schema is consistent. So, whereas every consistent schema S has been seen to be transformable into a developed normal schema S"†, no inconsistent schema can be so transformed.

As a specific example now of the transformation of S into S"†, let us put the schema:

$$ (1) \qquad p \cdot -(qr) \cdot r v \bar{q}s $$

through the pages. The S of the above exposition is now (1). By the method of § 9, (1) goes over into an alternational normal schema by way of the following stages:

p · q v r · r v q̅s,

p · q v r · r · v · p · q v r · q̅s,

pqr v prr v pqqs v prqqs.

This last reduces in turn, as explained in § 9, to an alternation of fundamental schemata:

$$ (2) \qquad pqr v pqs v prqqs, $$

which is our S'. Next we develop the primary part `pqr` of (2) with respect to the missing letter 's', thus turning (2) into:

$$ (3) \qquad pqrs v pqrs v pqs v prqqs $$

Developing `pqqs` here, in turn, with respect to the missing letter 'r', we turn (3) into:

$$ (4) \qquad pqrs v pqrs v pqsr v pqsr v prqqs $$

which is S". Now the primary part `pqsr` of (4) is a duplicate of an earlier `pqrs` (disregarding order), and the part `pqrs` at the

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For every equivalence established by the method of § 5, another is forthcoming by the above law. From the equivalence of 'pp' with 'p' we might have concluded immediately, by the above law, that 'p v p' (which is the dual of 'pp') is equivalent to 'p' (which is the dual of 'p'). From the equivalence of 'pq' with 'qp', similarly we can infer the equivalence of 'p v q' with 'q v p'. From the equivalence of '-(pq)' with 'p v q' we can infer the equivalence of '-(p v q)' with 'pq'. From the equivalence of 'p(q v r)' with 'pq v pr' we can infer the equivalence of 'p v qr' with '(p v q)(p v r)'.

For each validity or inconsistency which is established by the method of § 4, thus, a companion inconsistency or validity is provided by the above law. From the validity of 'p v p' we can infer the inconsistency of 'pp', and from the validity of 'pq v p v q' (of § 4) we can infer the inconsistency of '(pvq)pq'.

(i) '-(p v q v ... v s)' is equivalent to 'pqqq...s'.

For, by the first law of duality, if a given assignment of truth values to the letters yields a given truth value for 'p v q v ... v s' the opposite assignment must yield the opposite value for the dual 'pq...s'. But the effect of reversing the assignments to 'p', 'q', etc. can be gained by changing 'p' to 'P', 'q' to 'Q', etc. So any assignment of truth values to 'p', 'q', etc. yields opposite values for 'p v q v ... v s' and 'pq...s'; therefore (1) follows: (The truth of (1) is seen yet more quickly by reflecting directly upon (I)-(III) of § 3; but the above argument brings out the relationship to duality.)

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For, whenever '0' is put for 'p' or for all of 'q', 'r', ..., 's',
clearly both 'p . q v r v ... v s' and 'pq v pr v ... v ps' will
reduce to '0'; and whenever 'l' is put for 'p' and for at least one
of 'q', 'r', ..., 's', clearly both 'p . q v r v ... v s' and
'pq v pr v ... v ps' will reduce to 'l'.

The law of distributivity can be applied not only when an
alternation appears in the position 'p . q v r v ... v s', but also
when it appears in the position 'q v r v ... v s . t', or
'p . q v r v ... v s . t'; for, the order and grouping of the com-
ponents of a conjunction are known to be immaterial. In every
case we may simply multiply out, as we do in algebra when a sum
occurs as a factor of a product. Thus 'q v r v ... v s . t' becomes
'qt v rt v ... v st', and 'p . q v r v ... v s . t' becomes
'pqt v prt v ... v ps't'.

p v q v ... v s . p' v q' v ... v s' .

p v q v ... v s . p' v q . p v . q v ... v s . q' v ... v s'.

p v q v ... v s . s'.

Then the part 'p v q v ... v s . p

v ... v ss'.

What the distributivity of conjunction through alternation shows
is that we never need a conjunction having an alternation as one
of its components. We can expand any schema into an equivalent
in which no conjunction has an alternation as a component. Also,
by DeMorgans' laws, we can avoid applying negation otherwise than
to single letters. Combining those two kinds of reduction, we
can reduce any schema to an equivalent in which negation applies
only to single letters and conjunction applies only to letters and
negation of letters. A schema thus reduced is said to be an *alter-
national normal schema*.

Thus, let us reduce 'p . -(qr) . s v st' to an alternational
normal schema. The successive stages are as follows:

p . -(qr) . s v st,

p . q v r . s v st,

p . q v r . s . v . p . q v r . st,

pq's v pr's v pq'st v pr'st.

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It will be convenient, here and at later points, to have a general word covering letters and negations of letters. Let us call them **literals**. Thus 'p', 'q', 'p', etc. are literals. An alternational normal schema, then, may be a literal, a conjunction of literals, or an alternation whose components are literals and/or conjunctions of literals. The component schemata of which an alternational normal schema S is an alternation will be called the **primary parts** of S; or, if S is not an alternation, it counts as its own sole primary part. Thus the primary parts of 'pqr v qr v prs' are 'pqr', 'qr', and 'prs'; the primary parts of 'p v q' are 'p' and 'q'; and the primary part of 'pq' (or of 'F', or of 'p') is simply 'pq' (or 'F', or 'p') itself. Primary parts of alternational normal schemata are always literals or conjunctions of literals.

The advantage of alternational normal schemata is that we can tell at a glance just what truth values for the component letters are compatible with truth of the whole. Thus 'pqsv prsv pqst v prst', above, affirms that one of these four situations holds: either 'p' and 's' true and 'q' false, or else 'p' and 's' true and 'r' false, or else 'p' and 't' true and 'q' false, or else 'p' and 't' true and 'r' and 's' false. In the original form, 'p - (qr) . sv st', the net import of the whole is much less evident.

This advantage can be enhanced, and at the same time any contemplated computations as of §§ 3-6 can be rendered less laborious, by certain abridgments to which an alternational normal form can be subjected on sight. To begin with, we can go through it crossing out any primary part which, like 'pqr', is visibly inconsistent through exhibiting a letter both plain and negated (cf. § 7), and also crossing out any repetitions of literals in a primary part, e.g., the second 'p' in 'pqpr'. The alternational normal schema:

(1) prps v qrpq v rqp v pqrqs v psrq v srp v qr,

e.g., thus reduces, by deletion of the inconsistent primary parts 'qr' and 'pqrs', and deletion of the repeated 'p' from 'prps' and the repeated 'r' from 'rq', to:

(2) prsv rq v psrq v srp v qr.

In this way **almost** any alternational normal schema reduces to an equivalent one in which (as in (2)) no primary part exhibits a letter more than once. The sole exception arises when every primary part of an alternational normal schema exhibits a letter both plain and negated; in this case every primary part but one can disappear under the above procedure, but we have to keep one if we are not to end up with an absolute blank. In this case, accordingly, we come out with a primary part standing along and visibly inconsistent. We can reduce it arbitrarily to 'pp' if we like, since any two inconsistent schemata are equivalent (cf. § 5).

A literal, or a conjunction of literals wherein no letter appears twice, will be called a **fundamental schema**. What we have seen, then, is how, simply by progressive deletions, to reduce any alternational normal schema to an equivalent which is either (I) a conjunction of literals containing some letter both plain and

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So every consistent alternational normal schema reduces to one whose primary parts are fundamental schemata, and every inconsistent normal schema reduces to a visibly inconsistent conjunction of literals. Here we have a mechanical test of inconsistency, alternative to that of § 4: convert the schema to alternational normal form and reduce as above.

(a) 'p v p' to 'p'

(b) 'pq v pq' to 'p'

(c) 'p v pq' to 'p'

(d) 'p v pq' to 'p v q'

(The equivalences upon which (b), (c), and (d) depend are readily checked by the method of § 5.) In making such abridgments, the order in which the components of a conjunction follow one another is conveniently ignored; and similarly for an alternation. Thus abridgment of type (a) gets rid of the 'q' at the end of (2), since this is a repetition (disregarding order) of an 'q' earlier in (2). The result is:

(3) prs v Fq v psrq v srp.

By (c), next (taking the 'p' of (c) as 'pr'), we can reduce 'prsv prsq' to 'pr'; in other words, we can simply cross out 'prsq' (or 'psrq') in (3). Result:

(4) prs v Fq v srp.

But 'prsv' here amounts to 'srp'; and, by (b), 'srp v Fpr' reduces to 'sr'. So (4) becomes:

(5) sr v Fq.

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(c)

$$ (6) \qquad pq \vee \overline{pq} \vee \overline{pr} \vee \overline{r} \vee qst, $$

we first reduce 'pq \vee \overline{pq}' therein to 'p' by (b), and then reduce 'p \vee \overline{pr}' to 'p \vee r' by (d), obtaining:

$$ (7) \qquad p \vee r \vee \overline{r} \vee qsr. $$

Now the interesting thing about this result is that it is visibly valid, because of its part 'r \vee \overline{r}' (resolution of which, by § 7, would reduce the whole to 'r \vee \overline{r}'). If we had been wondering about the validity of (6), we have the answer in (7) without computation. But it must be borne in mind that this, while a common enough outcome, cannot be depended upon to be forthcoming in all cases of validity; we have here no general alternative to the method of § 4 for testing validity.¹ The schema:

$$ pq \vee \overline{p} \vee \overline{p} \vee \overline{r} \vee \overline{p} \vee \overline{s} \vee \overline{q} \vee \overline{r} \vee \overline{r} $$

'p \vee qr...s' is equivalent to 'p \vee q \cdot p \vee r \cdot ... \cdot p \vee s'.

Thus conjunction and alternation behave much more symmetrically than multiplication and addition. In arithmetic we have 'x(y \oplus z)' = xy + xz' but not 'x \oplus yz' = xy \oplus xz'; in the case of conjunction and alternation, on the other hand, distributivity works both ways.

p \vee \overline{q} \vee r \cdot q \vee \overline{r} \cdot p \vee r \vee s, p \vee \overline{q} \vee r \cdot s, p \vee \overline{q}, \overline{p}q, p, \overline{r}.

¹ We shall have in § 10, however, along other lines.

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The step-by-step reduction of a schema to conjunctional normal form is exactly parallel to the reduction to alternational normal form, the only difference being that the roles of alternation and conjunction are interchanged. Thus, let us go back to the example 'p . -(qr) . svst' of § 9, this time reducing it to conjunctional normal form. The successive stages are now as follows:

p . -(qr) . svst,

t . dvF . svst,

p . K vF . svF . svt.

Every schema S is reducible equally, we see, to an alternational normal schema S' and a conjunctional normal schema S". S' and S", being equivalent to S, will of course be equivalent to each other. Duals, however, are in general not equivalent; and it must not be expected that S' and S" will be duals of each other. S', being an alternational normal schema, will have an its dual some conjunctional normal schema, but infinitely farther than S", and S", being a conjunctional normal schema, will have as its dual some alternational normal schema, but ordinarily one other than S'. D.g., where: § is p . -(q1) . svst, we have seen that it reduces to the alternational normal schema:

pqs v prs v pqst v prst (S')

p . qvr . svS . svt. (S")

S' and S" here are therefore equivalent; but they are by no means duals of each other. The respective duals of S' and S" are rather:

p v qvs . pvFvs . pvqvs vt . pvFvs vt,

p v qF v ss v st,

These two latter schemata are by no means equivalent to S' and S". (They are, however, equivalent to each other, by the third law of duality, § 8.)

The components whereof an alternational normal schema is an alternation (or the whole, if it is not an alternation) were called the *primary parts* of the alternational normal schema. Similarly, the components whereof a conjunctional normal schema is a conjunction (or the whole, if it is not a conjunction) may be called the *primary parts* of the conjunctional normal schema. Thus the primary parts of 'p v qvr . qvr . pvrvs' are 'p v qvr', 'qvr', and 'pvrvs'; the primary parts of 'p' are 'p' and 'q'; and the primary part of 'p v q' (or of 'p', or of 'q') is simply 'p v q' itself (or 'p', or 'q').

Actually this terminology leads to an ambiguity; because of the root that certain schemata, such as 'p' and 'q', qualify simultaneously as alternational and conjunctional normal schemata. Thus 'p' has 'p' and 'q' as primary parts from the point of view

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of a conjunctional normal schema, but only 'pq' as primary part from the point of view of an alternational normal schema; and conversely for 'p v q'. A double terminology, 'alternational primary part' and 'conjunctional primary part', would hence be more proper; but in practice the ambiguity is resolved well enough by the purposes in hand.

<table><tr><td>'pp'</td><td>to 'p',</td></tr><tr><td>'p v q . p v q'</td><td>to 'p',</td></tr><tr><td>'p . p v q'</td><td>to 'p',</td></tr><tr><td>'p . p v q'</td><td>to 'pq'.</td></tr></table>

The equivalence of 'pq v pq' with 'p' was used in § 9 (b) as a basis for abridging alternational normal forms. For some purposes, however, it is convenient to use this equivalence in the opposite way, as a means of turning a shorter alternational normal form into a longer one. If a primary part '--' of a given alternational normal schema lacks a certain letter, say 'q', and if we want to

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supply that lack, we may, by the above equivalence, expand `|---|` into `|---q v ---q'|`. Such expansion is called *development* of `|---|` with respect to `|q|`.

Every consistent schema S is reducible, we have seen (§ 9), to an alternational normal schema S' whose primary parts are fundamental schemata. Now if a primary part of S' lacks a letter appearing elsewhere in S, let us *develop* that part with respect to the missing letter; and let us continue thus until we have turned S' into a schema S'' in which each primary part exhibits each letter of S''. Finally we may rid S'' by § 9 (a), of any duplication of primary parts. The result S''† is an alternational normal schema in which each primary part contains exactly one occurrence of each letter of the schema, and all the primary parts are different (different not merely in order, but in the incidence of negation signs). Such a schema will be called a *developed normal schema*.

Every developed normal schema is consistent; for, it is an alternational normal schema whose primary parts are fundamental schemata, and we saw (§ 9) that any such alternational normal schema is consistent. So, whereas every consistent schema S has been seen to be transformable into a developed normal schema S'', no inconsistent schema can be so transformed.

As a specific example now of the transformation of S into S'', let us put the schema:

(1) $p \cdot -(qr) \cdot r v \bar{q}s$

p \cdot \bar{q} v \bar{r} \cdot r v \bar{q}s,
p \cdot \bar{q} v \bar{r} \cdot r \cdot v \cdot p \cdot \bar{q} v \bar{r} \cdot \bar{q}s,
p\bar{q}r v p\bar{r}r v p\bar{q}q\bar{s} v p\bar{r}q\bar{s}.

(2) $p\bar{q}r v p\bar{q}s v p\bar{r}q\bar{s}$

which is our S'. Next we develop the primary part `p\bar{q}r` of (2) with respect to the missing letter 's', thus turning (2) into:

(3) $p\bar{q}rs v p\bar{q}rs v p\bar{q}s v p\bar{r}q\bar{s}$

Developing `pqsr` here, in turn, with respect to the missing letter 'r', we turn (3) into:

(4) $p\bar{q}rs v p\bar{q}rs v p\bar{q}sr v p\bar{q}sr v p\bar{r}q\bar{s}$

which is S''. Now the primary part `pqsr` of (4) is a duplicate of an earlier `pqrs` (disregarding order), and the part `pqrs` at the

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end is a duplicate of an earlier 'pāsṛ'. Dropping these duplications by § 9 (a), we reduce (4) to:

pārs v pārṣ v pāsṛ,

which is S''', a developed normal schema. Incidentally it is well to impose alphabetical order upon each of the primary parts, thus:

(5) pārs v pārṣ v pārṣ.

Alphabetical arrangement helps us to recognize any duplications of primary parts, and also it expedites the special uses of developed normal schemata to which we are soon to turn our attention.

Developed normal schemata as defined above might be denominated, more explicitly, developed alternational normal schemata, to distinguish them from their duals the developed conjunctional normal schemata. We could, by considerations of duality, reiterate the whole foregoing discussion putting conjunction for alternation, alternation for conjunction, and non-validity for consistency. This, however, would serve little purpose. Developed normal schemata in the sense presented above -- the alternational sense -- are what are needed for the ensuing matters; no use will be made of their duals.

The important thing about a developed normal schema is that it is virtually a truth table in algebraic form. Thus, consider (5). Among the sixteen possible ways of assigning truth values to 'p', 'q', 'r', and 's', there are exactly three that make (5) true; viz.:

p: 1; q: 0; r: 1; s: 1 (making 'pārs' true),

p: 1; q: 0; r: 1; s: 0 (making 'pārṣ' true),

p: 1; q: 0; r: 0; s: 1 (making 'pārṣ' true),

If we were to construct a full sixteen-case table (of § 3) for the above schema, we should obtain '1' in the three cases picked out above and '0' in each of the remaining thirteen. Each primary part of a developed normal schema depicts an assignment of truth values to letters which makes the whole come out true. Transformation of a schema into a developed normal schema is tantamount to constructing a truth table, or listing the assignments which make the schema true.

Validity, in particular, can be determined by inspection when a schema is in developed normal form. A developed normal schema is valid if, and only if, its primary parts exhibit the letters plain and negated in all combinations. The developed normal schema:

pq v pq v pā v pā,

e.g., is valid; and so is:

pqr v pqr v pār v pār v pq v pq v pq v pq.

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of special treatment, outside the domain of developed normal
schemata; for, a schema which comes out true in no case whatever
is readily devised in the fashion 'ppqrs...', containing whatever
letters we please.

The schema constructed to fit a given schedule of truth values,
by the above technique, will not in general be the simplest possible.
The schema (6) above e.g., reduces by § 9 (b) to:

$$\overline{\mathfrak{P}}\mathfrak{q}\mathfrak{r} \lor \overline{\mathfrak{q}}\mathfrak{r} \lor \overline{\mathfrak{p}}\overline{\mathfrak{q}}\overline{\mathfrak{r}} \lor \overline{\mathfrak{p}}\mathfrak{q}\overline{\mathfrak{r}}$$

and thence to:

$$\overline{\mathfrak{p}}\mathfrak{q}\mathfrak{r} \lor \overline{\mathfrak{q}}\mathfrak{r} \lor \mathfrak{p}\overline{\mathfrak{r}}.$$

What is interesting, however, is that every schedule of truth
values is fulfilled by some schema, built up by means exclusively
of the operations of conjunction, alternation, and negation. These
three operations suffice for expressing any compound which is
"truth-functional" in the sense of being describable by a list of
the truth values which it is to take on for each assignment of
truth values to the component letters. In studying schemata built
up by conjunction, alternation, and negation we have been studying
truth-functional schemata in full generality. Conjunction and
negation would have sufficed for truth function theory, indeed, to
the exclusion even of alternation, since alternation is trans-
latable into conjunction and negation (cf. § 8); and alternation
and negation, likewise, would have sufficed to the exclusion of
conjunction.¹

¹ The basis can actually be reduced to a single connective,
'égé', described by the table:

$$\begin{array}{c|ccc}
1/1 & 0/1 & 1/0 & 0/0 \\
\hline
0 & 1 & 1 & 1
\end{array}$$

For, both negation and conjunction can be translated into terms
of this connective, '$\overline{P}$' being rendered as '$\overline{p} / p$' and '$\overline{pq}'$ as
'$(\overline{p} / q) / (p' / q)'$. Another connective which would suffice by
itself is '$\overline{e}$', described by the table:

$$\begin{array}{c|ccc}
1 \downarrow 1 & 0 \downarrow 1 & 1 \downarrow 0 & 0 \downarrow 0 \\
\hline
1 & 0 & 0 & 0
\end{array}$$

For, '$\overline{p}$' can be rendered as '$p \downarrow p$' and '$\overline{pq}'$ as '$(\overline{p} \downarrow p) \downarrow (q \downarrow q)'$.
These matters were first pointed out by H.M. Sheffer in 1913.

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## § 12. THE CONDITIONAL: INTERPRETATION

Besides 'and' and 'or', another connective of statements which plays an important part in everyday language is 'if-then'. A statement of the form 'if p then q' is called a conditional. The component in the position of 'p' here is called the antecedent of the conditional, and the component in the position of 'q' is called the consequent.

A conjunction of two statements is true, we know, just in case both components are true; and an alternation is true just in case one or both components are true. Now under what circumstances is a conditional true? Even to raise this question is to depart from everyday attitudes. An affirmation of the form 'if p then q' is commonly felt less as an affirmation of a conditional than as a conditional affirmation of the consequent. If, after we have made such an affirmation, the antecedent turns out true, then we consider ourselves committed to the consequent, and are ready to acknowledge error if it proves false. If on the other hand the antecedent turns out to have been false, our conditional affirmation is as if it had never been made.

Departing from this usual attitude, however, let us think of conditionals simply as compound statements which, like conjunctions and alternations, admit as wholes of truth and falsity. Under what circumstances, then, should a conditional as a whole be regarded as true, and under what circumstances false? Where the antecedent is true, the above account of common attitudes suggests equating the truth value of the conditional with that of the consequent; thus a condition with true antecedent and true consequent will count as true, and a conditional with true antecedent and false consequent will count as false. Where the antecedent is false, on the other hand, the adoption of a truth value for the conditional becomes rather more arbitrary; but the decision which proves most convenient is to regard all conditionals with false antecedents as true. Thus, writing 'p→q' for 'if p then q' in conformity with current logical practice, we have the truth table:

1→1      0→1      1→0      0→0
1         1         0         1

The conditional 'p→q', so construed, is called a material conditional. In meaning it differs none from what has hitherto been written '-(pq)', or 'p v q'; for, we have:

I v 1      0 v 1      I v 0      0 v 0
1          1          0          1

Now consider the statement:

(1) If anything is a vertebrate, it has a heart.

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This, to begin with, is not a conditional in the sense with which we have been concerned above, for it is not really a compound of two statements 'anything is a vertebrate' and 'it has a heart'. The form of words 'it has a heart' is not a statement, true or false, which can be entertained in its own right, and be mooted to be true in case anything is a vertebrate. Rather, (1) must be viewed as affirming a bundle of individual conditionals: If a is a vertebrate, a has a heart; if b is a vertebrate, b has a heart; and so on. In short;

(2) No matter what x may be, if x is a vertebrate then x has a heart.

But it is important to note that, of the bundle of conditionals which (2) affirms, each individual conditional can quite suitably be interpreted as a material conditional. For, if we reflect that the material conditional 'p → q' amounts to '-(pq)', and then rewrite (2) accordingly, we have:

> No matter what x may be, it is not the case that x both is a vertebrate and does not have a heart.

or briefly:

> Nothing is a vertebrate and yet does not have a heart

-- which does full justice to the original (1).

So the assignment of 1 to 'OD1' and 'OOD', represented hitherto as rather arbitrary, actually has indirect support in common usage, at least to this extent: a *generalized conditional*, such as (1), can in full accordance with common usage be construed as affirming a bundle of material conditionals. The assignment of 1 to 'OD1' and 'OOD' was spoken of as convenient; this relationship with the generalized conditional is largely the reason why.

We have seen that a generalized conditional, such as (1), is not to be construed directly as of the form 'p → q'. Its more explicit phrasing (2) makes it a topic for Part III; also it can be put in the form 'All vertebrates have hearts', whereupon it becomes a topic for Part II; but in any case it lies beyond the present phase of analysis, which concerns only the compounding of statements explicitly from block-like components which are self-contained statements in turn.

Another use of 'if-then' which is certainly not to be construed in the fashion 'p → q' is the *contrafactual conditional*; e.g.:

(3) If Bizet and Verdi had been compatriots, Bizet would have been Italian.

Whoever affirms a conditional thus in the subjunctive mood is already prepared in advance to maintain also, unconditionally, the falsehood of the antecedent, but still he thinks the conditional adds

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some information; surely, then, he does not consider that such a conditional is automatically verified (like 'p → q') simply by the falsity of the antecedent. This kind of conditional, moreover, is not subject to the earlier remark to the effect that in ordinary usage a conditional is dropped from consideration, as empty and uninteresting, once its antecedent proves false.

The contrafactual conditional is best dissociated from the ordinary conditional in the indicative mood, as a separate problem and one upon which we shall not here embark. Whatever the proper analysis of the contrafactual conditional may be, we may be sure in advance that it cannot rest merely on truth tables; for, obviously ordinary usage demands that some contrafactual conditionals with false antecedents and false consequents be true and that other contrafactual conditionals with false antecedents and false consequents be false. Any adequate analysis of the contrafactual conditional must go beyond mere truth values and consider causal connections, or kindred relationships, between matters spoken of in the antecedent of the conditional and matters spoken of in the consequent. It may be wondered, indeed, whether any really coherent theory of the contrafactual conditional of ordinary usage is possible at all, particularly when we compare (3) with its equally plausible but seemingly incompatible counterpart:

> If Bizet and Verdi have been compatriots, Verdi would have been French.

The problem of contrafactual conditionals is in any case a perplexing one,¹ and it belongs not to pure logic but to the theory of meaning or possibly the philosophy of science. We shall not recur to it here.

So the foregoing truth table for 'p → q' is put forward not as an analysis of general conditionals such as (1); nor as an analysis of contrafactual conditionals such as (3), but, at most, as an analysis of the ordinary singular conditional in the indicative mood. Even as an analysis of such conditionals the truth table is sometimes felt to be unnatural, for it directs us to construe a conditional as true no matter how irrelevant its antecedent may be to its consequent, so long as it is not the case that the antecedent is true and the consequent false. The following conditionals, e.g., would qualify as true:

(4) If France is in Europe then the sea is salt,

(5) If France is in Australia then the sea is salt,

(6) If France is in Australia then the sea is sweet.

No doubt this result seems strange; but I do not think it would be any less strange to construe (4)-(6) as false. The strangeness is intrinsic rather to the statements (4)-(6) themselves, regardless of their truth or falsity; for it is not usual in practice to form conditionals out of component statements whose truth or false-

¹ See Nelson Goodman, "The problem of counterfactual conditionals," Journal of Philosophy, 1947, pp. 113-128.

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hood is already known unconditionally. The reason this is not usual
is readily seen: Why affirm a long statement like (4) or (5) when
we are in position to affirm the shorter and stronger statement
'The sea is salt'. And why affirm a long statement like (5) or (6)
when we are in position to affirm the shorter and stronger state-
ment 'France is not in Australia'?

In practice, one who affirms 'If p then q' is ordinarily un-
certain as to the truth or falsehood individually of 'p' and of
'q' but has some reason merely for disbelieving the combination
'p and not q' as a whole. We say:

(7) If Jones has malaria then he needs quinine,

because we know about malaria but are in doubt both of Jones' ail-
ment and of his need of quinine. Thus only those conditionals are
worth affirming which follow from some manner of relevance between
antecedent and consequent -- some law, perhaps, connecting the mat-
ters which these two component statements describe. But such con-
nection underlies the useful application of the conditional without
necessary to participate in its meaning. Such connection would, as
we have seen, underlie all useful application of the conditional
even though the meaning of the conditional be understood precisely
as given in the truth table.

The situation is quite similar, indeed, in the case of the
connective 'or'. The statement:
> If and only if one is a Frenchman,
> then one must be a French citizen.
> (8)
> or
> one is not a Frenchman.

France is in Europe or the sea is sweet

is as little worth affirming as (4), and for the same reason: we
can save breath and yet convey more information by affirming simply
'Trance is in Europe'. In practice, one who affirms 'p or q' is
ordinarily uncertain as to the truth or falsehood individually of
'p' and of 'q', but believes merely that at least one of the two is
true, because of some law or some other manner of relevance connect-
ing the matters which the two component statements describe. Yet
clearly no meaning need be imputed to 'or' itself beyond the mere
matter of truth and falsehood of component statements: "not both
false".

Despite the observed parallelism between 'if - then' and 'or',
people feel differently about the two connectives. Many are in-
clined to impute a strong connection, something like causality, to
the very meaning of 'if - then', whereas there is no corresponding
tendency with regard to the meaning of 'or'. This situation can be
accounted for by the influence of other uses of 'if-then', viz, the
use in (1) and the use in (3).

It may still be felt that the truth table does not do justice
to the ordinary indicative 'if-then'; but the issue is in any case
one of linguistic analysis, and of little consequence for our
purposes. What is important to note is that 'p ⊃ q', the so-called

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material conditional, is to have precisely the meaning given by the
above truth table; and it will become evident enough, as we proceed,
how well adapted this concept is to purposes for which the idiom
'if-then' naturally suggests itself. In particular, as already
noted, the material conditional is precisely what is wanted for the
individual instances covered by a general conditional of the type (1).

### § 13. THE CONDITIONAL: LAWS AND TECHNIQUES

Truth functions were represented, hitherto, as constructed by
means exclusively of conjunction, alternation, and negation. Now
the connective '∴' of the material conditional has been added. But
the addition is purely notational, since 'p ∴ q' amounts simply to
'p v q'. We know, indeed, that no essential addition is possible
at the truth-functional level; negation and conjunction, or equally
negation and alternation, suffice for expressing any function for which
a truth table could be devised (cf. § 11).

The techniques of duality and of transformation into normal
schema (§§ 8-11) need no supplementation to accommodate '∴', since
we can eliminate '∴' at will by translation into the form 'p v q'.
For the same reason, the truth-table techniques for determining
validity, consistency, equivalence, and implication (§§ 3-6) remain
adequate as they stand. However, in the case of truth-table tech-
niques it is convenient, instead of demanding translation of con-
ditionals, to accommodate them on the ground floor by supplementing
(i)-(v) of § 3 as follows:

(vi) If a conditional has '1' as antecedent or consequent,
drop the antecedent.

(vii) If a conditional has '0' as antecedent or consequent,
negate the antecedent and drop the consequent.

These rules reduce to '1 ∴ q' to 'q', 'p ∴ 1' to '1', 'p ∴ 0' to
'p', and '0 ∴ q' to '0' (and hence to '1'); then thus merely do
what (i), (iii), and (v) of § 3 would do if '1 ∴ q', 'p ∴ 1', 'p ∴ 0',
and '0 ∴ q' were first translated as '1 v q', 'p v 1', 'p v 0',
and '0 v q'.

Given any schema constructed by conjunction, alternation, and
negation, (i)-(v) enabled us to evaluate it for any assignment of
truth values to the component letters. Now (i)-(vii) enable us to
do the same when conditionals are thrown in. Accordingly the tests
of validity, consistency, equivalence, and implication in §§ 4-6
can now be performed directly upon schemata built up by means of not
only conjunction, alternation, and negation, but also the conditional.
E.g., the validity of 'p ∴ p' may be verified by (vi)-(vii) as
follows:

1 ∴ 1
0 ∴ 0

1
1

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Again, the equivalence of 'pq > r' and 'ps. q > r' may be verified as follows:

<table><tr><td>11</td><td>1</td><td>01</td><td>1</td><td>10</td><td>1</td><td>00</td><td>1</td><td>11</td><td>0</td><td>01</td><td>0</td><td>10</td><td>0</td><td>00</td><td>0</td></tr><tr><td>.</td><td>1</td><td>.</td><td>1</td><td>.</td><td>1</td><td>.</td><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0</td><td></td><td>1</td><td></td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>13</td><td>1</td><td>07</td><td>1</td><td>13</td><td>0</td><td>03</td><td>1</td><td>13</td><td>1</td><td>03</td><td>1</td><td>13</td><td>0</td><td>03</td><td>0</td></tr><tr><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>1</td></tr><tr><td>1</td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td><td>0</td><td></td><td>1</td><td></td><td></td><td>1</td><td></td><td></td></tr></table>

In practice the intermediate types of reduction which are shown above as aids to understanding, viz. '1 > 0', '0 > 0', etc., would normally be omitted in conformity with the policy set down in §§ 4-6.

Paralleling the above extension (vi)-(vii) of (i)-(v) of § 3, there is, of course, an extension of the rules of resolution of valid and inconsistent parts (§ 7). Those rules were as follows: Delete any valid component from a conjunction and any inconsistent component from an alternation; and from a conjunction with an inconsistent component, or an alternation with a valid component, delete all other components. Now the added rules for conditionals are as follows: If a conditional has a valid antecedent or consequent, drop the antecedent; and if a conditional has an inconsistent antecedent or consequent, negate the antecedent and drop the consequent.

Dots were used in connection with 'D', above, without comment. Let us agree that so far as the dot conventions of § 2 are concerned, 'D' is to be treated exactly on a par with 'v'. Similarly for '$\neg$', which comes in the next section.

One reason for adding the strictly superfluous notation 'D' to those with which we had contented ourselves through §§ 1-11 is that it corresponds, at least in important ways, to the common idiom 'if-then'. But the main reason is that the material conditional is peculiarly useful, more so even than alternation itself, in connection with some logical techniques which are to follow (especially in Part III). The reason for its special usefulness lies in the intimate relationships which obtain between the material conditional and implication. The basic relationship is this:

(I) One schema implies another if and only if the conditional having the one schema as antecedent and the other as consequent is valid.

For, S implies S' unless there is an assignment of truth values which makes S true and S' false; but such an assignment is the only kind, also, that can falsify the conditional formed of S and S' (cf. the truth table of 'p > q', § 12).

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like this (cf. § 6):

100.000

qq

101

001

100

000

1

1

0

1

The important connection between the conditional and implication has engendered a tendency, among many writers on logic, to confuse the two notions and read 'implies' as 'implies'. Then, since the truth table of § 12 represents 'p → q' as holding wherever a false statement is put for 'p' or a true one for 'q', it is charged that that table requires every false statement to imply all statements, and every true statement to be implied by all statements. It is not perceived that an attempted approximation to 'if-then' is one thing and an attempted approximation to 'implies' is quite another.

There is, to begin with, this grammatical difference between 'if-then' and 'implies': 'if-then', like 'or' and 'and', is a connective which unites statements to form a statement, whereas 'implies', being a transitive verb, is a connective which unites names to form a statement. Implying is indeed a relation between statements or schemata and not between names, just as loving is a relation between persons and not between names; but to say that one person loves another we write 'loves' not between the persons but between their names, and similarly to say that one statement or schema implies another we write 'implies' between names of the statements or schemata (those names being formed, usually, by adding single quotation marks).

The distinction is important not merely for grammatical priority, but for clarity of thought. What is concerned is the contrast between expressions talked about, on the one hand, e.g., 'p → q', and, on the other hand, the things we say about them: that this one is valid, that that one implies that other, that this one is equivalent to that. It is the distinction between the "object language" and what has come to be called, in a barbarous coinage, "motalanguage".

It should be noted, finally, that implication as construed in this and foregoing sections is quite a strict relation; it requires that the corresponding conditional not merely be true, but be valid. Besides this particular sort of implication, which depends on truth-functional structure, others may be defined which turn on other structural aspects; some such will be taken up later. But there is nothing in the concept of the material conditional 'p → q' to support the absurd notion that every falsehood "implies" everything and every truth is "implied" by everything, in any normal or interesting sense of the word 'implies'.

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§ 14. THE BICONDITIONAL

The idiom 'p if and only if q', called the *biconditional*, amounts obviously to the conjunction of two conditionals, 'if p then q' and 'if q then p'. All that was said in § 12, regarding the interpretation of the conditional, applies mutatis mutandis to the biconditional; whatever use 'if-then' may be put to, and whatever meaning it may be conceived to have, a corresponding use and a corresponding meaning must accrue to 'if and only if'. When in particular, the condition is construed as the material conditional 'p ∨ q', the corresponding biconditional is called the *material biconditional* and written 'p ≡ q'. Since 'p ≡ q' may be regarded simply as an abbreviation of 'p ∨ q · q ∨ p', its truth table is readily derived from that of 'p ∨ q'.

<table><tr><td>1 ≡ 1</td><td>0 ≡ 1</td><td>1 ≡ 0</td><td>0 ≡ 0</td></tr><tr><td>101 · 101</td><td>001 · 100</td><td>100 · 001</td><td>000 · 000</td></tr><tr><td>11</td><td>10</td><td>01</td><td>11</td></tr><tr><td>1</td><td>0</td><td>0</td><td>1</td></tr></table>

We see then that a material biconditional is true if the components are alike in truth value (both true or both false), and that it is false if the components differ in truth value.

The computation rules (i)-(v) of § 3, supplemented by (vi)-(vii) in § 13 to provide for the conditional, may now be further supplemented as follows to provide for the biconditional:

(viii) *Drop '1' as component of a biconditional.*

(ix) *Drop '0' as component of a biconditional and negate the other side.*

These rules reduce 'p ≡ 1' to 'p', and 'p ≡ 0' to 'p'; they thus merely do what (vi), (vii), (ii), and (iv) would do if 'p ≡ 1' were translated as 'p ∨ 1', and 'p ≡ 0' as 'p ∨ 0 · 0 ∨ p'. Correspondingly for '1 ≡ p' and '0 ≡ p'.

So now the tests of validity, consistency, equivalence, and implication as of §§ 4-6 can be performed directly upon schemata built up by any of these devices: negation, conjunction, alternation, conditional, biconditional. E.g., the validity of 'p ≡ p' may be verified by (viii)-(ix) as follows:

<table><tr><td>1 ≡ 1</td><td>0 ≡ 0</td></tr><tr><td>1</td><td>1</td></tr></table>

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The equivalence of '$p \equiv q$' and '$q \equiv p$' may be verified thus:

<table><tr><td>1</td><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>1</td><td></td><td>0</td><td></td><td>0</td><td></td><td>1</td></tr><tr><td>1</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td></tr><tr><td>1</td><td></td><td>0</td><td></td><td>0</td><td></td><td>1</td></tr></table>

The equivalence of '$p \equiv q$. $\equiv r$' and '$p \equiv q \equiv r$', not an obvious matter, may be verified similarly.

Those two equivalences show that for the material biconditional, just as for conjunction and for alternation, the order and grouping of components is immaterial. Note that this is by no means the case for the conditional; '$p \supset q$' is not equivalent to '$q \supset p$', nor is '$p \supset q$. $\supset r$' equivalent to '$p \supset q$. $q \supset r$', as the reader can easily see by the techniques at his disposal.

Paralleling (viii)-(ix) above we can now add the following to the rules of resolutions of valid and inconsistent parts (summed up in § 13): We may drop a valid component from a biconditional; and we may drop an inconsistent component from a biconditional if we negate the other component. So now any schema, built up by any or all of the devices of negation, conjunction, alternation, the conditional, and the biconditional, and having a valid or inconsistent part, can be reduced step by step, in accordance with these extended rules of resolution, until either the part in question has disappeared or else we are left with a valid or inconsistent remainder. (Of. § 7.)

The relationship which was observed in (I) of § 13 to hold between implication and the conditional holds also between equivalence and the biconditional: Two schemata are equivalent if and only if their biconditional is valid. For, S and S' are equivalent unless there is an assignment of truth values to letters which makes S true and S' false or vice versa; but such an assignment is the only kind, also, that can falsify the biconditional of S and S' (in view of the truth table for '$p \equiv q$').

Thus equivalence, like implication, reduces to validity. Conversely, also, the validity of a biconditional can be tested by testing the components for equivalence.

The widespread confusion between implication and the conditional, noted in § 13, carries with it a confusion between equivalence and the biconditional. Actually of course the biconditional, like the conditional, conjunction, and alternation, is a way of combining statements, while equivalence, like implication, is a relation of statements (or schemata). To say that two statements or schemata are equivalent, we put 'is equivalent to' between names of the statements or schemata, just as, to say that one statement or schema

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implies another, we put 'implies' between names of the statements or schemata; and just as, to say that one person loves another, we put 'loves' between names of the persons. On the other hand, read 'if and only if', is written not between names of statements or schemata but between the statements or schemata themselves, on a par with 'I' and 'v'; it is not a verb but a statement connective. Saying that two statements are equivalent is, moreover, much stronger than affirming their biconditional; for it is to say that their biconditional is not just true, but valid.

## § 15. WORDS INTO SYMBOLS

Logical inference leads from *promisses* -- statements assumed or believed for whatever reason -- to *conclusions* which are implied by the promises. The implication is *grounded in logic*, but ordinarily the promises and conclusions themselves are not; and herein, precisely, lies the *application* of logic to fields other than it-self.

The promises and conclusions may treat of any topics and are couched, to begin with, in ordinary language rather than in the technical ideography of modern logic. It is only as an aid to establishing implications that we then proceed to mutilate and distort the statements, introducing schematic letters in order to bring out relevant skeletal structures, and translating varied words into a few fixed symbols such as 'D' and 'v' in order to gain a manageable economy of structural elements. The task of thus suitably paraphrasing a statement, and isolating the relevant structure, is just as essential to the application of logic as is the test or proof of implication for which that preliminary task prepares the way.

Truth-functional logic, actually, to which our attention has thus far been limited, is too elementary a fragment of logic to admit of application to ordinary discourse on any impressive scale; the importance of this part of logic for ordinary discourse is indirect, rather, consisting in its use as a foundation for the more elaborate logical theory which is to follow. But even truth-functional logic admits of direct application, albeit in a somewhat trivial vein, to ordinary discourse; and such application will afford us a laboratory, advantageous for its very simplicity, in which we may gain a first acquaintance with the considerations which govern the paraphrasing of statements and the isolating of structures for logical treatment.

An example of how such paraphrasing reduces varied idioms to uniformity has already been noted in the notation of negation (cf. § 1). A further example may be seen in the notation of conjunction; for in ordinary language conjunction is expressed not only by 'and' but also by 'but', by 'although', by unspoken punctuation (as in 'Sometimes I'm happy, sometimes I'm blue'), and in various other ways. Consideration of 'but' and 'although' is instructive, for it brings out a distinction between what may be called the logical and the rhetorical aspects of language. We are likely to

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say:

Jones is here but Smith is away,

rather than:

Jones is here and Smith is away,

because of the contrast between being here and being away; or, if
the contrast between 'Jones is here' and 'Smith is away' attains
such proportions as to cause surprise, as it might e.g., if Jones
is not in the habit of coming except to see Smith, we are likely to
say:

Jones is here although Smith is away.

But the circumstances which render the compound true are always
the same, viz., joint truth of the two components, regardless of
whether 'and', 'but', or 'although' is used. The compounds convey
the same information; use of one rather than another may make a
difference in naturalness of idiom and may also provide some inci-
dental evidence as to what is going on in the speaker's mind, but
it is incapable of making the difference between truth and false-
hood of the compound. The difference in meaning between 'and',
'but', and 'although' is rhetorical, not logical. Logical notation,
unconcerned with rhetorical distinctions, expresses conjunction uni-
formly by simple juxtaposition.

For a further example of the reduction of manifold idioms of
ordinary language to uniformity in logical notations, consider the
idiomatic variants of 'if-then':

If p then q, p only if q, q if p, q provided that p,
q in case p.

The notation 'p ⊃ q', insofar as it may be admitted as a version of
'if p then q' at all, is a version at once of all those variant
idioms.

Note that the antecedent of a conditional, corresponding to
the 'p' of 'p ⊃ q', is not always the part which comes first in the
vernacular. It is the part, rather, that is governed by 'if' (or
by 'in case', 'provided that', etc.), regardless of whether it
comes early or late in the conditional. Thus it is that 'p if q'
goes over into 'q ⊃ p', not 'p ⊃ q'. But whereas 'if' is thus
ordinarily a sign of the antecedent, the attachment of 'only'
reverses it; 'only if' is a sign of the consequent. Thus 'p only
if q' means, not 'p if q', but 'if p then q'; not 'q ⊃ p', but
'p ⊃ q'. E.g., 'You will graduate only if your bills have been
paid' does not mean 'If your bills have been paid you will graduate';
it means 'If you will graduate, your bills (will) have been paid.'

If the deplorable tendency to read ⊃ as 'implies' (cf. § 15)
depends at all on a desire for a word which will stand in the
middle like ⊃ and unlike 'if-then', then the above observations
on 'only if' call for special notice; ⊃ may be read 'only if'.

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It is particularly to be noted that 'only if' does not have the sense of '≡', which is 'if and only if'. As the words suggest, 'p if and only if q' is a conjunction of 'p if q' and 'p only if q' -- hence of 'q ∨ p' and 'p ∨ q'.

Among the linguistic variants of 'if p then q' listed above, one more might have been included: 'not p unless q'. This variant leads to the following curious reflection: if 'not p unless q' means 'p ∨ q', and 'p ∨ q' means 'p v q', then 'not p unless q' must mean 'p v q', which makes 'unless' answer to 'v' and hence to 'or'. Whatever strangeness there may be in equating 'unless' to 'or' is precisely the strangeness of equating 'if-then' to '∴'. It is often felt that 'if-then' suggests a causal connection, or the like; and, insofar as it does, so also does 'unless'. But when we distill a truth function out of 'if-then', we have '∴'; and when we distill a truth function out of 'unless' we have 'v', 'or'.

The evident commutativity of 'or', i.e., the equivalence of 'p or q' with 'q or p', is less evident with 'unless'. The statements:

(1) Smith will sell unless he hears from you,

(2) Smith will hear from you unless he sells

seem divergent in meaning. However, this divergence may be attributed in part to a subtle tendency, in 'unless' compounds, to mention the earlier event last when time relationships are important. Because of this tendency, we are likely to construe the vague 'hears from you' in (1) as meaning 'hears from you that he should not sell', and in (2) as meaning 'hears from you that he should have sold'. But naturally, if we are to compare (1) and (2) as genuine compounds of statements, we must first render each component unambiguous and durable in its meaning -- if not absolutely, at least sufficiently to exclude shifts of meaning within the space of the comparison. Thus if we elaborate (1) and (2) to read:

(3) Smith will sell unless you restrain him,

(4) Smith will be reprimanded by you unless he sells,

then our compared compounds are related not as 'p unless q' and 'q unless p', but as 'p unless q' and 'r unless p'; and there is no thought of equivalence.

The necessity, just now illustrated, of so rephrasing component statements as to render their meanings durable for the space at least of the argument in hand, may be illustrated also in simpler and more striking ways. The two conjunctions:

(5) He went to Pawcatuck and I went along,

(6) He went to Saugatuck but I did not go along

may both be true; yet if we represent them as of the forms 'pq' and

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as
'rq', seems superficially to fit the case, we come out with an in-
consistent combination 'pqrq'. Actually, of course, the 'I went
along' in (5) must be distinguished from the 'I went along' whose
negation appears in (6); the one is 'I went along to Pawatuck',
and the other is 'I went along so Saugatuck'. When (5) and (6) are
completed in this fashion, they can no longer be represented as
related in the manner of 'pq' and 'rq', but only in the manner of
'pq' and 'rs'; and the apparent inconsistency disappears.

In practice, statements whose meanings are unambiguous and in-
dependent of context or circumstances of utterance are decidedly in
the minority. The reference of the pronoun 'I' varies with the
speaker; the reference of 'here' varies with the place of utterance;
the reference of 'now', 'yet', 'still', and commonly of tense it-
self, varies with the time of utterance; and the reference of 'you',
'ho', and even many proper names like 'Smith' and 'Springfield'
varies with circumstances. And even when the intended Smith is un-
ambiguously identified, 'Smith stayed' may still vary between truth
and falsehood according to what particular occasion we have in mind.
If we specify the occasion by introducing date and hour, the state-
ment still remains ambiguous as between truth and falsehood until
we supplement 'stayed' to show whether we mean 'stayed rooted to
the spot', or 'stayed in the same town', or 'stayed with the firm',
etc. It is only after such points are decided that the truth or
falsehood of a statement comes to be a question purely of fact ra-
thor than of fact plus intended interpretation.

Ambiguity being thus rather the rule than the exception, we
may wonder how it is that logical techniques predicated on the con-
trary fiction are generally applicable at will. The answer is that
when we undertake a logical argument we assume that for each of the
ambiguous statements concerned some one interpretation has been
fixed upon. For purposes of the argument it does nor ordinarily
matter what those interpretations are, so long as they are singly
adhered to. Moreover, in practical cases this working assumption
is commonly sound; the circumstances giving occasion for a logical
argument are likely in practice to leave little doubt in one's
mind as to which Smith is intended, or when and in what sense
thought of as having stayed. The essential point, therefore, is
simply this: The trustworthiness of a logical argument depends on
our not giving one and the same expression different interpretations
in the course of the argument. Violation of this principle was
known traditionally as the fallacy of equivocation.

Insofar as the interpretation of ambiguous elements depends on
circumstances of the argument as a whole (speaker, hearer, scene,
date, and underlying problem and purpose), the fallacy of equivoca-
tion is not to be feared; for, circumstances shared by the argument
as a whole will not of themselves affect different occurrences of
one and the same statement in different ways. What we have to look
out for is rather the case where the interpretation of a statement
is affected differently by its different immediate contexts within
the argument. This was the trouble in (5)-(6); the natural in-
terpretation of the ambiguous statement 'I went along' was one thing
in the context (5), and another in the context (6). In such cases

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we have to rephrase before proceeding with logical analysis and
argument; not rephrase to the extent of resolving all ambiguity,
but to the extent of resolving such part of the ambiguity as might
be affected differently by different immediate contexts within
the proposed logical argument. The logical connectives by which
components are joined in compounds must not be allowed to transmit
interpretative clues from one component to another; each component
is to be wholly on its own, except insofar as its meaning may depend
on those broader circumstances which condition the meanings of words
in the compound as a whole, or in the logical argument as a whole.

It often becomes evident, when this warning is born in mind,
that a compound which superficially seems suited to the truth-
functional level of logical analysis really calls for logical devices
of a more advanced nature. The statement:

(7) We saw Stromboli but it was not erupting

is not adequately analyzed as a simple conjunction, for the con-
struction 'was...ing' in the second clause involves an essential
temporal reference back to the first clause. An adequate analysis
would construe (7) rather as:

There was a time x such that we saw Stromboli at x and
Stromboli did not erupt at x,

which involves logical devices taken up in later sections. Note
that (5) and (6) also might be handled, alternatively, along those
more advanced lines of temporal analysis; and similarly, indeed,
for (1) and (2).

§ 16. GROUPING

A conspicuous type of ambiguity in ordinary language is
ambiguity of grouping. In this respect logical notation, and math-
ematical notations generally, enjoy a great advantage over ordinary
language. The statement:

Rutgers will get the pennant and Hobart will be runner-up if Rzynski
is disqualified,

e.g., is hopelessly ambiguous in point of grouping; there is no
telling whether Rutgers' getting the pennant is supposed to be con-
tinent upon Rzynski's being disqualified. If so the logical form
is 'p > q', and otherwise it is 'q > p'. Or, where 'p' represents
'Rzynski is disqualified', 'q' represents 'Rutgers gets the pennant,
and 'r' represents 'Hobart is runner-up'.

In complex statements of ordinary language the intended group-
ing sometimes has to be guessed, as above, and sometimes to be in-
ferred from unsystematic cues, as illustrated by the following

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example:

Jones will sell his car and mortgage his home unless the new mail-order campaign breaks the Dripsweet monopoly and restores freedom of competition.

What component statements is the first 'and' supposed to connect? What ones is the 'unless' supposed to connect? What ones is the last 'and' supposed to connect? There are some pretty clear cues here. Naturally the first of the two statements which are joined by the first 'and' must in any case be 'Jones will see his car', since this is all there is before that 'and'. Now the words 'mortgage his home' which follow the 'and' are pretty clearly coordinate with 'sell his car'; our cue for this is that the words 'Jones will', instead of being repeated, were allowed to govern 'sell his car' and mortgage his home simultaneously. Thus we decide that the statements joined by the first 'and' are 'Jones will sell his car' and '(Jones will) mortgage his home'. Turning now to the other 'and', we know that the second of the two statements which it joins must in any case be '(the new mail order campaign) restores freedom of competition', since this is all there is after that 'and'. Accordingly the first of the two statements joined by that 'and' must be the coordinate construction 'the new mail-order campaign breaks the Dripsweet monopoly'. So now we have identified both of the conjunctions, neither of which embraces the whole statement; it remains to 'unless' to be the main break, or main connective, joining the two conjunctions to form the statement as a whole. The logical structure of the whole is 'pq v rs' (writing 'v' for 'unless' in view of § 15).

Now, if e.g. we want to argue from the above statement to the
conclusion:

If the new mail order campaign does not restore freedom of competition then Jones will sell his car,

we are ready; we simply verify that 'pq v rs' implies 's → p', by
the method of § 6:

Oq v r0

0

1 → 1

o → 1

1 → 0

o → 0

1

1

1

0

The cue to grouping which has thus far been stressed is the
suppression of the second occurrence of a repeated subject or other
constituent. This device reveals grouping through coordinateness

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of clauses. There are also more artificial ways of showing grouping through coordinateness of clauses. E.g., the vacuous phrase 'it is the case that' may be introduced, balanced with another 'that' to show coordination:

> It is the case that either Jones came or Smith stayed, and that Robinson left.

This is interpretable only in the fashion:

(Jones came or Smith stayed) and Robinson left,

since the devices 'it is the case that' and 'that' show the clauses 'Jones came or Smith stayed' and 'Robinson left' to be coordinate. If these devices had been left out, there would have been no way of telling whether to adopt the above interpretation or the alternative one:

> Jones came or (Smith stayed and Robinson left).

In some cases involving 'or', the insertion of an 'either' at the proper place is enough to settle the grouping. Thus, whereas the statement:

> Jones came and Smith stayed or Robinson left

is flatly ambiguous as to grouping, the variants:

> Either Jones came and Smith stayed or Robinson left, Jones came and either Smith or Robinson left

are unambiguously interpretable respectively as:

(Jones came and Smith stayed) or Robinson left, Jones came and (Smith stayed or Robinson left);

for, the first component of the 'or' compound is neatly marked off in each case as extending from the 'either' to the 'or'. But this device is of no avail where there is a question rather as to the extent of the *last* component of an alternation.

The idiom 'if-then' shares with 'either-or' this virtue of marking off the limits of the first component. 'If' is a verbal makeshift for the opening parenthesis of conditional, and 'either' is a verbal makeshift for the opening parenthesis of an alternation. Moreover, 'both' does a similar service for conjunction. Thus, in the example where 'it is the case' was resorted to, we might have accomplished the same purpose by putting in a 'both' to support the 'and':

> Both Jones came or Smith stayed and Robinson left

-- which, though it looks like nonsense, is quite unambiguous.

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Another device of ordinary language which sometimes serves to suggest intended groupings is that of inserting seemingly idle particles, e.g., 'else' after 'or', or 'also' or 'furthermore' after 'and'. Such reinforcement of a connective has the effect of suggesting that it is a major one.

In trying to extract the logical structure of a highly complex statement, it is a good plan to look for the outermost structure first and then paraphrase inward, stop by step. This procedure has the double advantage of dividing the problem up into manageable parts, and of keeping the complexities of grouping under control. E.g., consider the statement:

(1) If Jones is ill or Smith is away then neither will the Argus deal be concluded nor will the directors meet and declare a dividend unless Robinson comes to his senses and takes matters into his own hands.¹

First we seek the main connective of (1). It is not 'or', since this is visibly imprisoned between the 'if' and the 'then'.

Nor is it 'neither-nor', since these are obviously governed by 'then'. Nor is it the first 'and'; for, 'the directors' stands as subject for the verbs on either side of 'and', suggesting that this 'and'-compound is supposed to begin only at 'the directors' and not at the beginning of (1). Nor is it the second 'and', for a similar reason: 'Robinson' stands as subject for the verbs on either side of this 'and'. So the main connective of (1) must be either the 'If-then' or the 'unless'. Choice between these is, in the present ambiguous example, open to debate. Suppose, to cut matters short, that we decide on 'If-then'. The outward structure of (1), then, is that of a conditional; so let us impose just this much structure explicitly upon (1), postponing minuter analysis. We have:

(2) Jones is ill or Smith is away — neither will the Argus deal be concluded nor will the directors meet and declare a dividend unless Robinson comes to his senses and takes matters into his own hands.

Next we may consider, as if it were a separate problem removed from (2), just the long component 'neither...hands'. This is the statement whose main connective we are now to seek. We know, by the earlier reasoning, that its main connective will not be either of the 'ands', so it must be either 'neither-nor' or 'unless'. Between those choices there is perhaps room again for debate; but the choice of 'neither-nor' seems rather the less likely interpretation, if only because the two 'will'-clauses seem to balance best when the second one is cut short at 'unless'. So we adopt 'unless' as the main

¹ This example and others in this section are taken from my *Elementary Logic*.

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connective. Treating 'unless' as 'v', then, we turn (2) as a whole into:

(3) Jones is ill or Smith is away. D. neither will the Argus deal be concluded nor will the directors meet and declare a dividend v Robinson will come to his senses and take matters into his own hands.

Now we take up, as if it were a separate problem removed from (3), the longest component yet unanalyzed; viz., 'neither...dividend'. The main connective here is clearly 'neither-nor', since 'and' is excluded by earlier considerations. Reflecting then that 'neither nor s' goes into symbols as 'rs', we rewrite 'neither...dividend' accordingly; (3) thus becomes:

(4) Jones is ill or Smith is away. D. -(the Argus deal will be concluded) -(the directors will meet and declare a dividend) v Robinson will come to his senses and take matters into his own hands.

Directing our attention finally to the various short compounds which remain unanalyzed in (4), we turn the whole into:

(5) Jones is ill v Smith is away. D: -(the Argus deal will be concluded) -(the directors will meet, the directors will declare a dividend) v. Robinson will come to his senses. Robinson will take matters into his own hands.

Put schematically (with accents to increase our stock of statement letters), the total structure is:

p v p' .D. q -(rs) v tt'.

## PART II. TERMS

### § 17. TERMS

There are many simple and logically sound inferences for which the foregoing techniques are inadequate. An example is this:

No philosophers are wicked,

Some Greeks are philosophers;

Therefore some Greeks are not wicked.

Schematically: No G are H,

Some F are G;

Some F are not H.

### 1 What are spoken of simply as terms in the present pages my, in view of developments in Parts III-IV, be designated more accurately as general absolute terms.

It will be noted that 'F' and 'G' here stand not for statements, after the manner of 'p' and 'q' of Part I, but for common nouns.

-- or, in logical parlance, terms.¹ Whether those nouns be thought of as in the singular or the plural is a logically insignificant question of phrasing; thus there is no need to distinguish between 'No philosopher is wicked' and 'No philosophers are wicked', nor between 'All philosophers are wise' and 'Every philosopher is wise'.

¹ What are spoken of simply as terms in the present pages my, in view of developments in Parts III-IV, be designated more accurately as general absolute terms.

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There is no need even to distinguish between 'Some Greek is a philosopher' and 'Some Greeks are philosophers', provided that, as will be our practice here, we understand 'some' always to mean simply 'at least one'. Quite commonly, as in those examples, the distinction between singular and plural is the mostest accident of the vernacular; and, even in cases where the distinction is normane to the meaning of a statement, in logical analysis it is customary to resort to other devices.

Whether the nouns be substantive or adjective is likewise an insignificant question of phrasing. 'G' appears as a substantive in the above example, viz. 'philosophers', and 'H' as an adjective, 'wicked'; but we could rewrite the adjective as a substantive, 'wicked individuals', if we liked. In the same spirit we can even treat intransitive verbs as general terms, in effect, thus reckoning 'Some fishes fly' as a case of 'Some F are G'; for the difference between 'Some fishes fly' and 'Some fishes are flying things' is purely notational. The nouns or verbs which figure as general terms may also, of course, be complex phrases such as 'employed for ten years by Sunnyrinse', 'wear brass rings in their noses', etc.

But, for all the latitude accorded to the concept of general term, it remains clear that general terms are never statements; and this is why the techniques of Part I, limited as they were to the analysis of statements into component statements, are inadequate to the above example. The example belongs to that part of logic in which statements are analyzed not into statements but into terms; and it is upon this part of logic that we are now embarking.

It is the peculiarity of a statement to be true or false. A term, on the other hand, has as its business the denoting of many objects, one, or none. The term 'Greek' denotes each Greek, and the term 'wicked' each wicked individual, and nothing else. The term 'satellite of the earth' denotes each satellite of the earth and nothing else; hence it denotes but one object, the moon. The term 'centaur' denotes each centaur and nothing else; hence it does not denote at all, there being no centaurs.

Negation, conjunction, and alternation, known thus far as operations upon statements, admit of exact analogues which are operations upon terms. These analogues will be called by the same names and expressed by the same notations, except that parentheses will be used instead of dots to indicate grouping; e.g., 'F (GvH)''. We may explain them as follows, writing 'Fx' to mean 'x is an F':

(1) (FG)x ≡. Fx. Gx,

(2) (FvG)x ≡. Fx v Gx,

(3) Fx ≡ -(Fx).

Thus the term-conjunction 'FG' denotes all and only those objects which the component terms 'F' and 'G' denote in common; it means, in words, 'Fs which are Gs'. If 'F' is taken to denote

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Frenchmen and 'F' generals, then 'F' denotes French generals. The term-alternation 'FvF', on the other hand, goes clumsily into words as 'things which are either Fs or Gs'; it denotes all those objects which are denoted by 'F' or by 'G' (or by both).

"Thereas the word 'and' as a statement connective represents conjunction of statements, on the other hand as a term connective the word 'and' in ordinary English usage usually represents not term-conjunction but term-alternation, thus answering to the term connective 'v'. The phrase 'Frenchmen and Germans' refers, in likeliest usage, not just to those few Frenchmen who (by mixed parentage or confused citizenship) are also Germans, but rather to all those persons who are either Frenchmen or Germans or both; it answers not to 'F' but to 'FvF'. Thus, whereas 'pq' and 'pvq' may safely be read 'p and q' and 'p or q', we must be on our guard not to equate 'F' uncritically with 'F and G', nor 'FvF' with 'F or G'. The notations 'FG' and 'FvG' are quite clear and precise in meaning, but they are to be understood through the foregoing explanations and not through any brief and infallible English equivalents. The formulation of one's ideas in this logical notation must proceed by a clear re-thinking of just what it is that one wants to say. The notation would lose much of its purpose if it were simply to reflect the vagaries of ordinary language.

The term-negation 'F', finally, according to (3), denotes just those objects which 'F' does not denote. If 'F' is taken to denote apples, then 'F' denotes non-apples. 'F' means, in words, 'non-F', 'things other than Fs', 'what 'F' will denote depends not only on what 'F' is taken to denote, but also on what we conceive to be the limits of reality; for 'F' is to denote everything in the universe which is not denoted by 'F'. Now it is convenient to allow with some explicitness for variations in the choice of the universe. This is convenient not only because philosophers disagree regarding the limits of reality, but also because -- as we shall see -- it is often possible to simplify logical arguments by deliberately limiting the universe, e.g. to persons or to the employees of a given firm, for the space of a particular problem.

The schemata with which we were occupied in Part I were statement schemata, and indeed statement schemata of a special kind called truth-functional schemata: truth functions of schematic statement letters 'p', 'q', etc. But the notation introduced in the foregoing section engenders schemata of another kind. The letters 'F', 'G', etc. are schematic term letters; they stand in place of terms just as 'p', 'q', etc. stand in the place of statements. From schematic term letters, by means of conjunction, alternation, and negation, we build up compounds which are term schemata, in that they depict the patterns not of complex statements but of complex terms.

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§18. NULLITIES

Let us write '$F$' to mean 'There are no $F$. Thus, when '$F$' is taken as denoting nothing whatever, and only then, '$F$' becomes true. "There '$F$' means 'centaurs', e.g., $F$. On the other hand '$F$', affirming as it does that there are no $F$, says that everything in the universe is an $F$.

"When the sign '$F$' precedes a conjunction it is to be understood, without help of parentheses, as applying to the conjunction as a whole. Thus '$FG$' means that there are no $FG$; in other words, that no $F$ are $G$. Similarly '$FG$' means that no $F$ are $G$; i.e., that all $F$ are $G$. This form of statement, which may also be phrased 'If anything is an $F$, it is a $G$, was already touched on in §12 under the name of generalized conditional; however, the line of analysis which was hinted in §12 corresponds more directly to the methods to which we shall turn in Part III than to those of the present Part.

$F$' denies that there are no $F$, and thus affirms that there are $F$. In particular then '$F$' affirms that there are $FG$; i.e., that some $F$ are $G$ (if we understand 'some' in the sense 'at least one'). Similarly '$F$' says that some $F$ are not $G$. The example at the beginning of the preceding section may now be rewritten in the new notation:

$$ (1) \qquad \begin{array}{c} \square F; \\ \sim \square F; \\ \vdots \\ \sim \square F \vdash . \end{array} $$

When '$X$' is applied to a term schema, the result is a statement schema. It is not, however, a truth-functional schema, but a statement schema of a new form, and will be called a **nullity**. From nullities we can generate a whole realm of statement schemata, viz. truth functions of nullities, just as we generated truth-functional schemata from statement letters. The truth functions of nullities comprise the nullities themselves and everything constructible from them by negation, conjunction, alternation, and the conditional and biconditional.

The concept of validity, explained hitherto for truth-functional schemata, can be extended now to truth functions of nullities. A truth function of nullities is **valid** (for a given universe) if, no matter how we interpret '$F$' (i.e., no matter what objects of the universe we regard '$F$' as denoting, if any), and no matter how we interpret '$G$', '$H$', etc., the schema comes out true. Any truth function of nullities which is truth-functionally valid (cf. §4), e.g. '$F \supset \square F$', will obviously be valid in the above sense, regardless of the choice of universes. Also some truth functions of nullities will be valid in all universes though not truth-functionally valid; an example is '$F \supset \square FG$'. Its validity depends not merely on the properties of the truth-functional connective '$\supset$', as was the case with '$F \supset \square F$', but also on the properties of '$X$' and term-conjunction. Another and simpler example is the nullity '$\neg F$' itself; clearly this is valid in all universes, since, no matter what objects we let '$F$' denote, '$\neg F$' will denote nothing.

We can find truth functions of nullities which are valid in a

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universe of given size and not valid in a universe of larger size. If, to take another extreme case, we adopt a universe of one object, then 'F v ∃F' becomes valid; for, 'F' must be interpreted either as denoting nothing (in which case ∃F) or as denoting the one object (in which case ∃F). But 'F v ∃F' is not valid in any universe of more than one object, since in such a universe we can interpret 'F' as denoting something and still have objects left over for 'F' to denote. Again, to take a very extreme case, if we adopt a wholly empty universe then 'F' itself becomes valid; but clearly it is not valid in any non-empty universe. Again, the schema:

$$∃FG v ∃FG v ∃F'$$

is valid in any universe of less than three objects, but in no larger one -- a fact of which it may amuse the reader to persuade himself.

The notions of consistency, equivalence, and implication may be extended to truth functions of nullities in similar fashion. A truth function of nullities is called *consistent* in a given universe when there is a way of so interpreting 'F', 'G', etc. in that universe as to make the schema come out true. Two truth functions of nullities are called *equivalent* in a given universe if, for every way of interpreting 'F', 'G', etc. in that universe, the two schemata come out alike in truth value. One truth function of nullities is said to *imply* another in a given universe if every way of interpreting 'F', 'G', etc. which makes the one schema come out true makes the other come out true also. Or, what comes to the same thing, a truth function of nullities is *consistent* in a given universe when its negation is not valid in that universe; two truth functions of nullities are *equivalent* in a given universe when their biconditional is valid in that universe; and one truth function of nullities *implies* another in a given universe when the conditional having the one as antecedent and the other as consequent is valid in that universe.

Just as validity in any universe follows from truth-functional validity, so inconsistency in any universe follows from truth-functional inconsistency, and equivalence and implication in any universe follow from truth-functional equivalence and implication.

The discovery and use of equivalence in Part I often depended on the three procedures which were justified in §7: interchange of equivalents, substitution for schematic letters, and resolution of valid and inconsistent parts. When equivalence, validity, and the rest are redefined for truth functions of nullities as above, those three procedures continue to hold good; the reader will find he can adapt the reasoning of §7 to the new definitions without difficulty.

To say that (1) above is a sound argument is to say, in the terminology just developed, that '∃GH ... ∃Fg' implies '¬∃FH' in every universe. An easy method of verifying implications of this and closely related forms will be explained in the next two sections. Afterward more general methods will be presented for implication, validity, etc.

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S19. CATEGORICAL STATEMENTS AND VENN'S DIAGRAMS

In the long tradition stemming from Aristotle, statements of the special forms 'All F are G', 'No F are G', 'Some F are G', and 'Some F are not G' were called categorical. The four forms were distinguished by special nomenclature and by code letters 'A', 'E', 'I', and 'O', as follows. The corresponding nullities have been added at the right.

<table><tr><td>A (Universal affirmative)</td><td>All F are G</td><td>✓F<span style="text-decoration: overline;">G</span></td></tr><tr><td>E (Universal negative)</td><td>No F are G</td><td>✓E<span style="text-decoration: overline;">G</span></td></tr><tr><td>I (Particular affirmative)</td><td>Some F are G</td><td>-✓I<span style="text-decoration: overline;">G</span></td></tr><tr><td>O (Particular negative)</td><td>Some F are not G</td><td>-✓O<span style="text-decoration: overline;">G</span></td></tr></table>

The first of the two terms of a categorical statement, hence 'F' above, was known in traditional terminology as the subject, and the other term, 'G' here, as the predicate.

In a diagrammatic method due to Venn (1880), overlapping circles are used to represent the subject and predicate of a categorical statement. The lens, as geometers call the area in which the two circles overlap, represents the objects which are at once F and G. Accordingly the statement 'No F is G', which says that that realm is empty, is represented by shading the lens. On the other hand the statement 'Some F are G' is represented by putting a cross in the lens to indicate that it is occupied. Again, the crescent-shaped area -- or lune, as geometers call it -- which lies within the F-circle and outside the G-circle represents the objects which are F and not G; so the statement 'All F are G', which says that that realm is empty, is represented by shading the lune. 'Some F are not G', finally, is represented by putting a cross in the lune to show that it is occupied. These representations of the four categorical statements are shown in Diagrams 1-4.

Diagram 1

Diagram 2

A and O are mutual contradictories, or negations; and so are E and I. This situation becomes explicit when nullities are used; '-✓F<span style="text-decoration: overline;">G</span>' denies '✓F<span style="text-decoration: overline;">G</span>', and '-✓E<span style="text-decoration: overline;">G</span>' denies '✓E<span style="text-decoration: overline;">G</span>'. In Diagrams 1-4 this opposition of A to O and E to I appears as an opposition between shading an area to mark it empty and putting a cross in it to mark it non-empty.

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Being mutual negations, A and O are of course incapable of
both true, or both false; and similarly for T and I. Note on the
other hand that A and T are quite commonly both false; e.g., 'All
Greeks are wise' and 'No Greeks are wise' are both false. Similarly
I and O, of course, are quite commonly both true.

I. Some F are G; -YG

Diagram 3

O. Some F are not G; -YFG

Diagram 4

It is less usual for I and O to come out both false, or for A
and T to come out both true; but these things will happen where
there are no F. Clearly, where there are no F, 'Some F are G' and
'Some F are not G' will both be false. Also, where there are no F,
'No F are G' will obviously be true; and yet 'All F are G' will
likewise be true, in that there will be no F which is not G.

A, 'All F are G', would seem at first glance to be stronger
than I, 'Some F are G', and to imply it; but it does not, because of
the possibility of there being no F. It may happen that all my
dimes are shiny (in that I have no dime to the contrary), and yet
be false that some of my dimes are shiny, simply because I have no
dimes at all. The most we can say, in general, is that if all F
are G and there are F then some F are G. The argument.

<table><tr><td>All F are G,</td><td>-YFG</td></tr><tr><td>There are F,</td><td>-YF</td></tr><tr><td>∴ Some F are G</td><td>-YFG</td></tr></table>

is sound. To restate the matter in the terminology of §18: the
conjunction 'YFG'. -YF' implies '-YFG' in every universe; '-YFG' by
itself, on the other hand, does not imply '-YFG' in every universe
(nor, in fact, in any universe at all, even the empty one).

If the reader thinks it odd to say that all of one's dimes are
shiny when one has no dimes, he is perhaps interpreting 'All F are
G' to mean, not simply 'There is no F that is not G', but 'There
are F and none of them is G'. This, however, even if it be one of
several defensible interpretations of an ambiguous idiom, is clearly
not the interpretation which would make A the simple contradictory,
or negation, of O: 'Some F are not G'. It is the general logical
practice, and a convenient one, to understand 'All F are G' simply
as the contradictory of O, or, in symbols, simply as '-YFG'; and it
is on this understanding that the foregoing paragraph is predicated.
An advantage of the notation '-YFG', as against 'All F are G', is
that it dodges any ambiguity inherent in the 'all' of the vernacular.

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Just as I is implied not by A alone but by the conjunction of A and 'There are F', so O is implied not by E alone but by the conjunction of E and 'There are F'. The argument.

<table><tr><td>No F are G,</td><td>YFG</td></tr><tr><td>There are F,</td><td>-YF</td></tr><tr><td>∴ Some F are not G</td><td>-YFG</td></tr></table>

is sound, but fails when the second premiss, 'There are F', is dropped. The situation here is more obvious, perhaps, than it was in the case of A and I, because here there is no ambiguity like that of 'all'.

The use of venn diagrams in checking simple implications may be illustrated in connection with the implications affirmed above. To show that I (or '-YF') is implied by A (or 'YFG') in conjunction with 'YF', we set down the diagram for A, viz. Diagram 1, and then enter 'YF' into the diagram by putting a cross in the 'F'-circle (where it remains unshaded). The result, showing a cross in the lens as it does, verifies I. Similarly, to show that O is implied by F in conjunction with 'YF', we put a cross in the 'F'-circle of Diagram 2 and observe that we have verified O.

Various relationships have now been noted among A, E, I and O. It is interesting to investigate, in corresponding fashion, the relationships among these four simpler analogues of A, E, I and O.

<table><tr><td>Everything is an F,</td><td>YF</td></tr><tr><td>There are no F,</td><td>YF</td></tr><tr><td>There are F,</td><td>-YF</td></tr><tr><td>There are non-F</td><td>-YF</td></tr></table>

The first and last of these negate each other, just as A and O did, and the second and third of these negate each other, just as E and I did. The first and second of these are capable of being both false, just as A and E were; and the third and fourth are capable of being both true, just as I and O were. However, whereas A and E were capable also of being both true (viz., through there being no F), on the other hand the top two of the above forms, 'Everything is an F' and 'There are no F', are quite incapable of being both true -- unless, indeed, the universe itself is taken to be empty. Similarly the bottom two, 'There are F' and 'There are non-F', cannot be both false unless the universe is empty. Again, whereas A implied I only in conjunction with 'There are F', on the other hand 'YF' implies '-YF' outright -- in all non-empty universes. Similarly for the other two: 'YF' implies '-YF' in all non-empty universes.

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820. SYLLOGISMS

"That are spoken of traditionally as syllogisms¹ are arguments wherein a categorical statement is derived as conclusion from two categorical statements as premisses; the three statements being so related as to contain just three terms altogether. The predicate of the conclusion, called the major term of the syllogism, turns up also in one of the premisses, called the major premiss; the subject of the conclusion, called the minor term, turns up also in the other premiss, called the minor premiss; and a third term, called the middle term, turns up in both premisses. Six examples follow; one among them was already noted in §§17-18. The annotations at the left will be explained.

<table><tr><td>AAA</td><td>All men are mortal,</td><td>All G are H,</td><td>✓GH</td></tr><tr><td>1</td><td>All Greeks are men;</td><td>All F are G;</td><td>✓FG</td></tr><tr><td></td><td>∴ All Greeks are mortal.</td><td>∴ All F are H.</td><td>✓FH</td></tr><tr><td>IAE</td><td>No men are perfect,</td><td>No G are H,</td><td>✓GH</td></tr><tr><td>1</td><td>All Greeks are men;</td><td>All F are G;</td><td>✓FG</td></tr><tr><td></td><td>∴ No Greeks are perfect.</td><td>∴ No F are H.</td><td>✓FH</td></tr><tr><td>ALL</td><td>All philosophers are wise,</td><td>All G are H,</td><td>✓GH</td></tr><tr><td>1</td><td>Some Greeks are philosophers;</td><td>Some F are G;</td><td>- ✓FG</td></tr><tr><td></td><td>∴ Some Greeks are wise.</td><td>∴ Some F are H.</td><td>- ✓FH</td></tr><tr><td>RIO</td><td>No philosophers are wicked,</td><td>No G are H,</td><td>✓GH</td></tr><tr><td>1</td><td>Some Greeks are philosophers;</td><td>Some F are G;</td><td>- ✓FG</td></tr><tr><td></td><td>∴ Some Greeks are not wicked.</td><td>∴ Some F are not H.</td><td>- ✓FH</td></tr><tr><td>ACO</td><td>All Greeks are men,</td><td>All H are G,</td><td>✓HG</td></tr><tr><td>2</td><td>Some mortals are not men;</td><td>Some F are not G;</td><td>- ✓FG</td></tr><tr><td></td><td>∴ Some mortals are not Greeks.</td><td>∴ Some F are not H.</td><td>- ✓FH</td></tr><tr><td>OAO</td><td>Some men are not Greeks,</td><td>Some G are not H,</td><td>- ✓GH</td></tr><tr><td>3</td><td>All men are mortal;</td><td>All G are F;</td><td>✓GF</td></tr><tr><td></td><td>∴ Some mortals are not Greeks.</td><td>∴ Some F are not H.</td><td>- ✓FH</td></tr></table>

In each example the minor term has been represented as 'F', the middle term as 'G', and the major term as 'H'; and in each example the major premiss has been set down first, then the minor premiss, and finally the conclusion.

The three letters at the left of each example indicate the so-called mood of the syllogisms. Thus the first example is in the mood AAA, which means that its premisses and conclusion are all of the form A (universal affirmative); the fourth example is in the

¹ Categorical syllogisms, more specifically, to distinguish them from hypothetical syllogisms, which were certain truth-functional arguments manageable by the methods of Part I.

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mood 'EIO', which means that the major premiss has the form E, the minor has the form I, and the conclusion has the form O. The mood of a syllogism specifies, thus, the respective forms of major premiss, minor premiss, and conclusion. But we need more than the mood in order to know the full form of a syllogism. We need also to know whether the major premiss has the major term as subject and the middle term as predicate, or vice versa; and correspondingly for the minor premiss. The four possibilities of arrangement which thus arise are called figures, and referred to by number as follows:

<table><thead><tr><th></th><th>1st</th><th>2nd</th><th>3rd</th><th>4th</th></tr></thead><tbody><tr><th>Major premiss</th><td>GH</td><td>HG</td><td>GH</td><td>HG</td></tr><tr><th>Minor premiss</th><td>FG</td><td>FG</td><td>GF</td><td>GF</td></tr><tr><th>Conclusion</th><td>FH</td><td>FH</td><td>FH</td><td>FH</td></tr></tbody></table>

The digits at the left of the examples previously listed indicate the figures. Thus the first example is in the mood AAA of the first figure; the last example is in the mood OAO of the third figure.

A "valid" syllogism, ordinarily so-called, is a syllogism of such form as to be incapable of leading from true premisses to a false conclusion. If we represent a syllogism in nullities and negations of nullities (as was done at the far right of the above examples), then its "validity" may be defined thus in terms of §18: the conjunction of the premisses implies the conclusion in every universe. In other words, the conditional having the conjunction of the premisses of a "valid" syllogism as antecedent and the conclusion as consequent is valid (in the sense of §18) in every universe.

An easy test of validity of syllogisms is afforded by Venn's diagrams. Three overlapping circles are used (as in Diagrams 5 et

Diagram 5

Diagram 6

Diagram 7

seq.) to represent the three terms 'F', 'G', and 'H', of the syllogism. We enter the content of the two premisses in the diagram by the method explained in connection with Diagrams 1-4, and then we inspect the diagram to see whether the content of the conclusion has automatically appeared in the diagram as a result of entering the two premisses. This is all there is to the method. Let us apply it to the fourth of the above six syllogisms.

We enter the first premiss by shading the lens common to the circles G and H, and we enter the second premiss by putting a cross in what remains of the lens common to F and G; the result is Dia-

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gram 5. Then we examine this result to see whether, in conformity with the desired conclusion, some F are not H. The conclusion is borne out: there is a cross in the circle F and outside H.

Let us try the method on the second of the foregoing examples, viz. EAE in the first figure. We enter the first premiss as before, and we enter the second premiss by shading the lune which lies in the circle F and outside G; the result is Diagram 6. The desired conclusion, 'No F are H', is established; the lens common to F and H is completely shaded.

Now to the third of the foregoing examples, viz. AII in the first figure. We enter the first premiss by shading the lune which lies in G and outside H, and we enter the second premiss by putting a cross in what remains of the lens common to F and G; the result is Diagram 7. The conclusion 'Some F are H' is established, there being a cross common to F and H.

It is left to the reader to construct diagrams verifying the remaining three of the above six syllogisms. (In the last of them, the second premiss should be handled first; note why.)

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## §21. SURVEY OF THE VALID MODES

**Simple conversion** is the operation of switching the two terms of a categorical statement, so that 'Some F are G' becomes 'Some G are F', and 'No F are G' becomes 'No G are F'. Obviously simple conversion has no effect on the meaning of an E or I. On the other hand it is not admissible for an A or O; 'all Greeks are men' is not equivalent to 'all men are Greeks', nor is 'Some men are not Greeks' equivalent to 'Some Greeks are not men'.

By simple conversion of the E and I statements, in the six valid modes of the syllogism considered above, we get further valid modes. Thus, consider EIO in the first figure, which was:

No G are H,
Some F are G;
∴ Some F are not H.

Simple conversion turns it into:

No H are G,
Some F are G;
∴ Some F are not H.

No G are H,
Some G are F;
∴ Some F are not H.

No H are G,
Some G are F;
∴ Some F are not H.

These are EIO in the second, third, and fourth figures. Note that the Venn diagrams for all these are identical with that for EIO in the first figure, viz. Diagram 5 of §20.

Again consider EAE and 'AII in the first figure, which were:

No G are H,
All F are G;
∴ No F are H.

All G are H,
Some F are G;
∴ Some F are H.

Simple conversion of the premises, in each case, turns these into:

No H are G,
All F are G;
∴ No F are H,

All G are H,
Some G are F;
∴ Some F are H,

which are respectively EAE in the second figure and AII in the

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third. The Venn diagrams for these are the same as before the conversion, viz. Diagrams 6 and 7 of §20.

Moreover, the conclusions of the above four syllogisms can all be converted into turn. If we do this, though, we must reletter the results in order that 'F' may continue to represent the minor term and 'H' the major term; also we must switch the order of the premises, so that the major premiss may continue to appear first. We got:

All H are G,
No G are F;
.. No F are H.

All H are G,
No F are C;
.. No F are H.

Some H are G,
All G are F;
.. Some F are H.

Some G are H,
All G are F;
.. Some F are H.

These are respectively AEE and IAI in the fourth figure, AEE in the second, and IAI in the third. The diagram of these two AEE moods is the mirror image of Diagram 6 (or, the result of switching the letters 'F' and 'H' in Diagram 6), and the diagram of the two IAI moods is the mirror image of Diagram 7.

Altogether, then, we have found fifteen valid moods:

First Figure Second Figure Third Figure Fourth Figure
AAA, EAB, AII, EIO EAB AEE, EIO, AOO IAI, AII, OAO, EIO AEE, IAI, EIO

Diagram 8

Diagram 9

The diagrammatic method can be used to determine not merely whether a given conclusion follows from given premises, but whether any conclusion at all (of a syllogistic kind) is capable of following from given premises. For, the conclusion -- in order to be the conclusion of a so-called syllogism at all -- would have to be a categorical statement with 'F' as subject and 'H' as predicate; hence, unless the two bottom circles of the finished diagram

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exhibit one of the four patterns shown in Diagrams 1-4 of §13, there is no conclusion. In Diagram 8, e.g., the bottom circles exhibit none of the four patterns of Diagrams 1-4; and this shows that the premisses AA in the second figure cannot be the premisses of any valid syllogism at all.

As a further example, consider the premisses AO in the first figure: 'All G are H', 'Some F are not G'. First we shade the lune which lies in G and outside H. It remains, in view of the minor premiss, to put a cross in the lune which lies in F and outside G. But this lune has two compartments, both surviving; which one should contain the cross? (See Diagram 9). One position would verify a conclusion I ('Some F are H'), and the other position would verify a conclusion O ('Some F are not H'). There is not one conclusion, I or O (much less A or E), of which we can be sure. So there is no valid first-figure syllogism with the premisses AO.

Viewed in terms merely of combinations and without regard to the existence of a valid conclusion, there are sixty-four possibilities for the premisses of a syllogism:

Of these sixty-four, the fifteen italicized cases are the only ones that actually yield valid syllogistic conclusions; they are the premisses of the fifteen valid moods arrived at earlier. If the reader constructs Venn diagrams for the remaining forty-nine (a shore which I do not urge), he will find that a valid syllogism is forthcoming in none of them. In four of them, however, marked with stars, a syllogism is forthcoming which is almost valid; and in each of the five italicized cases which are starred, an extra syllogism is forthcoming which is almost valid. These nine stepchildren will be dealt with in the next section.

## §22. Syllogisms needing an added existence premiss.

One of the additional moods which are almost valid is AAA in the third figure:

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There are two more moods which, similarly to the above one, fail of validity as they stand and become valid when a premiss 'There are G' is added. They are AEO in the third and fourth figures. Further, there are five moods which are not valid as they stand but become valid when a premiss 'There are F' (relating to the minor term) is added. They are AAI in the first figure, AEO in the first and second, and AEO in the second and fourth. Finally, there is one mood which, not valid as it stands, becomes valid when a premiss 'There are H' (relating to the major term) is added. This mood is AAI in the fourth figure. It is left to the reader to establish those by Venn diagrams.

Altogether, then, the previous list of fifteen valid moods can be supplemented by a list of nine moods which might be called semi-valid: not valid as they stand, but capable of being rendered valid by addition of one of the premisses 'There are F', 'There are G', 'There are H'. Those nine may be summed up as follows:

<table><thead><tr><th>1st Figure</th><th>2nd Figure</th><th>3rd Figure</th><th>4th Figure</th><th>Aided Premiss</th></tr></thead><tbody><tr><td>AAI, EAO</td><td>AEO, AEO</td><td>AAI, EAO</td><td>AEO<br>EnO<br>aaI</td><td>There are F<br>There are G<br>There are H</td></tr></tbody></table>

In traditional logic it was customary to prebound various rules whereby to test the validity of a syllogism. Examples: every valid syllogism has a universal premiss (A or E); every valid syllogism has an affirmative premiss (A or I); every valid syllogism with a particular premiss (I or O) has a particular conclusion; every valid syllogism with a negative premiss (E or O) has a negative conclusion. These rules can be verified in extensio by surveying the foregoing lists of valid moods. But as a practical method of appraising syllogisms, rules are less convenient than the method of diagrams.

Also, diagrams are more economical in point of conceptual apparatus. The very notion of a syllogism and its moods need never have been touched on in these pages, except out of consideration for its prominence in logic during the thousand years; for we can apply the diagram test to a given argument out of hand, without pausing to consider where the argument may fit in the taxonomy of syllogisms. The diagram test is equally available, indeed, for many arguments which do not fit any of the arbitrary delimited set of forms known as syllogisms; one example is the three-premiss argument tested in Diagram 11, and we shall see others.

1. There are further rules whose formulation depends on a concept of "distribution" which has been omitted from the present exposition.

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on the basis of the other premiss, we run a bar through as much of the unshaded F as lies within G or H. The meaning of the bar is that one or another part of the total region marked by the bar has something in it. But the bar lies wholly within F and H; so the conclusion is sustained.

The utility and versatility of Venn's diagrams are particularly evident from these last two examples. A shortcoming of the diagrams, however, is that they lend themselves less readily to arguments involving four or more terms. A diagram of overlapping ellipses can be constructed for four-term arguments, but it calls for careful drawing; and there is no way "whatever of constructing a diagram which will exhibit five or more ellipses or other simple regions in all combinations of overlapping. Where many terms are involved we may, however, try to break the argument down into parts involving manageably few terms. The following example is from Lewis Carroll:

Premises:
(1) The only animals in this house are cats;
(2) Every animal is suitable for a pet, that loves to gaze at the moon;
(3) When I detest an animal, I avoid it;
(4) No animals are carnivorous, unless they prowl at night;
(5) No cat fails to kill mice;
(6) No animals ever take to me, except what are in this house;
(7) Kangaroos are not suitable for pets;
(8) None but carnivora kill mice;
(9) I detest animals that do not take to me;
(10) Animals that prowl at night always love to gaze at the moon.

Conclusion: "I always avoid a kangaroo."

This argument can be broken down as follows. From (1) and (5) we get the **lemma** or intermediate conclusion:

(11) All animals in this house kill mice;

this is the sort of step to which a simple three-term diagram is adequate. From (8) and (11), similarly, we get:

(12) All animals in this house are carnivora.

Step by step in this fashion we can proceed to our desired conclusion, never using more than a three-term diagram for any one step. (If the reader cares to carry this through in detail, he should think of the universe as limited for purposes of the argument to animals -- thus never bothering with 'animals' itself as a term.)

So we see that the purely mechanical method of diagrams en-
counters an obstacle when an argument turns on a large number of
terms; a supplementary technique has to be invoked, such as that of
resolving the argument into parts. Now another place where the un-
aided method of diagrams bogs down is where there is an admixture

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of truth functions, as in the following example.

Premises: If all applicants are of the class of '00, then some applicants did not receive the second announcement. Either everyone in the class of '00 received the second announcement or else nobody received it.

Conclusion: If all applicants are of the class of '00, then nobody received the second announcement.

If we assign 'F', 'G', and 'H' in obvious fashion, the inference takes on the form:

All F are G. some F are not H, $\therefore F\bar{G} \supset \neg \mathcal{F}\bar{H}$

All G are H. v. there are no H, $\therefore \mathcal{G}\bar{H} \lor \mathcal{H}$

∴ All F are G. there are no H. $\therefore F\bar{G} \supset \mathcal{H}$

Diagrams are suited to handling the components 'All F are G', 'There are no H', etc., and the methods of Part I are suited to '$D$' and '$v$'; but just how may we splice the two techniques in order to handle a combined inference of the above kind?

The above premisses $\therefore F\bar{G} \supset \neg \mathcal{F}\bar{H}$ and $\therefore \mathcal{G}\bar{H} \lor \mathcal{H}$ and the conclusion $\therefore F\bar{G} \supset \mathcal{H}$ are truth functions of nullities (cf. §18). Turning our backs now on syllogistic theory and Venn diagrams alike, let us direct our steps toward a logical theory powerful enough for appraising any argument composed thus of truth functions of nullities, no matter how complex. We shall soon see how, given any truth functions of nullities as premisses and conclusion, to decide mechanically whether the conjunction of the premisses implies the conclusion in every universe.

## §24. VALIDITY OF CANONICAL SCHEMATA

Consider the eight conjunctions:

(1) $FGH, \bar{FGH}, \bar{FGH}, \bar{FGH}, \bar{FGH}, \bar{FGH}, \bar{FGH}$

They correspond to the eight compartments of Venn's diagram:

Diagram 14

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Now no matter what universe we adopt, and no matter how we interpret 'F', 'G', and 'H' within that universe (i.e., no matter which objects of the universe we let 'F' denote, if any, and which 'G', and which 'H'), each object in the universe is bound to be denoted by one and only one of the conjunctions in (1). This is apparent from Diagram 14 but can also be seen without help of the diagram, as follows. Since 'F' will denote those and only those things in the universe which 'F' does not denote, every object in the universe will be denoted by one and only one of 'F' and 'F'. Hence every object denoted by 'G' will be denoted by one and only one of 'FG' and 'FG'; and every object denoted by 'G' will be denoted by one and only one of 'FG' and 'FG'. Then, since every object is denoted by one and only one of 'G' and 'G', it follows that every object is denoted by one and only one of 'FG', 'FG', 'FG', and 'FG'. Bringing in 'H' now and repeating the reasoning, we see that every object in the universe is denoted by one and only one of the conjunctions in (1).

What has been established may be restated in two parts: the conjunctions in (1) are exhaustive in denotation, in that everything in the universe is bound to be denoted by one or another of them; and they are mutually exclusive in denotation, in that no two of them can denote anything in common. While these things are apparent from the diagram, it was desirable to argue them out independently of the diagram because the same arguments can then obviously be extended to show the mutual exclusiveness and exhaustiveness also of:

$$ (2) \quad \begin{array}{ccccccc} \text{FGHK,} & \overline{\text{FGHK,}} & \overline{\text{FGHK,}} & \overline{\text{FGHK,}} & \text{FG\overline{HK,}} & \overline{\text{FG\overline{HK,}}} & \overline{\text{FG\overline{HK,}}} \\ \text{FG\overline{HK,}} & \overline{\text{FG\overline{HK,}}} & \overline{\text{FG\overline{HK,}}} & \overline{\text{FG\overline{HK,}}} & \text{FG\overline{HK,}} & \overline{\text{FG\overline{HK,}}} & \overline{\text{FG\overline{HK,}}} \end{array} $$

and correspondingly for any higher number of term letters; the diagram, on the other hand, is not so readily extensible.

Let us return now for further reflections on the three-letter case. Since the conjunctions in (1) are exhaustive in denotation, no interpretation of 'F', 'G' and 'H' can deprive *all* these conjunctions of denotation unless the universe itself be empty. But, even in a non-empty universe, we can so interpret 'F', 'G', and 'H' as to withhold denotation from any of the eight conjunctions short of all. And we can also so interpret 'F', 'G', and 'H' as to endow any or all of the conjunctions with denotations, so long as our universe has enough objects to go around -- hence at least as many as there are conjunctions to be endowed with denotation.

Suppose, e.g., we want so to interpret 'F', 'G', and 'H' as to deprive 'FGH', 'FG\overline{H}', and 'FG\overline{H}' of denotation but endow the other five conjunctions with denotations. Suppose our universe has just a dozen objects, $a_1, a_2, ..., a_{12}$. Then these objects may be arbitrarily apportioned among the five conjunctions which are to have denotation -- say thus:

$$ \begin{array}{ccccccccc} \text{FGH} & \overline{\text{FGH}} & \overline{\text{FG\overline{H}}} & \overline{\text{FG\overline{H}}} & \overline{\text{FG}} & a_1, a_2 & a_3, a_4 & a_5, a_6 & a_7, a_8 & a_9 \text{ to } a_{12} \end{array} $$

What 'F' itself is to denote can now be read off: it must denote just those objects denoted by conjunctions having 'F', rather than 'F', as component. Correspondingly 'G' must denote just the objects denoted by conjunctions having 'G', rather than 'G', as component; and correspondingly for 'H'. The interpretations of 'F', 'G', and 'H' called for are therefore as follows:

$$ \begin{array}{ccc} \text{F} & \text{G} & \text{H} \\ a_1, a_2, a_5, a_6, a_9 & a_1, a_2, a_5 & a_9 \text{ to } a_8 \end{array} \quad \begin{array}{ccc} a_1 & \text{to } a_4, a_9 & \text{to } a_{12} \end{array} $$

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74.

The same end of depriving 'F', 'G', and 'H' of denotation, and endowing the other five with denotation, could obviously have been achieved with a universe of any size from five up. There had to be at least five objects to distribute among the five conjunctions, since the conjunctions must be mutually exclusive in denotation.

This much, then, is clear: so long as our universe has at least eight objects, we can so interpret 'F', 'G', and 'H' as to endow as many or as few of the eight conjunctions in (1) with denotations as we like, except that we cannot deprive them all of denotation. This exception can itself be accommodated in another universe, viz. the empty one; here, naturally, all the conjunctions are bound to be deprived of denotation.

These observations may be stated in another way, by reference to the eight nullities:

$$ (3) \quad \mathfrak{Y}F\overline{G}\overline{H}, \mathfrak{Y}\overline{F}\overline{G}\overline{H}. $$

What we have found is that we can so interpret 'F', 'G', and 'H', in one universe or another (perhaps empty, perhaps size eight or larger), as to make as many or as few of these nullities true as we like. Any assignment of truth values to the above eight nullities, in other words, can be realized by some interpretation or other 'F', 'G', and 'H', in some universe or other. If 1 is to be assigned to all the nullities, the universe must be empty; otherwise any universe of eight or more objects will serve.

Truth functions of any or all the nullities in (3) will be called *canonical schemata* in 'F', 'G', and 'H'. The test of truth-functional validity of such a schema consists, of course, simply in treating the component nullities as if they were single letters 'p', 'q', etc., and testing as in Part I. Thus a canonical schema in 'F', 'G', and 'H' fails of truth-functional validity just in case some assignment of truth values to its nullities makes the whole come out false. But we have seen that any assignment of truth values to the nullities listed in (3) is realizable by some interpretation of 'F', 'G', and 'H' in some universe. Hence whenever a canonical schema is not truth-functionally valid there will be some interpretation of 'F', 'G', and 'H' in some universe which makes it come out false; i.e., it will not be valid in the sense of § 18. Conversely, we know that any truth-functionally valid schema is valid in any universe (cf. §18). So, a canonical schema in 'F', 'G', and 'H' is valid in all universes if and only if it is truth-functionally valid. To decide whether it is valid in all universes we simply treat its nullities as if they were 'p', 'q', etc., and test it for validity as in Part I.

If under this test we find in particular that the *only* assignment which makes the whole come out false involves assigning 1 to every one of the nullities listed in (3), then we know that though our schema is not valid in all universes, it is valid in all except the empty universe. A canonical schema in 'F', 'G', and 'H' is valid in all non-empty universes if and only if it comes out true under every assignment of truth values to its nullities which does not involve assigning 1 to all the nullities of (3).

For concreteness we have been adhering to the case of three letters. Clearly, however, all that has been said would hold equally for four letters, with obvious adjustments -- (3) for (2), 'sixteen' for 'eight', etc.; and similarly for any other number of letters. So let us now state our results more generally. But we shall need some terminology.

By a *canonical nullity* will be meant a nullity consisting of 'Y' followed by a term literal ('F', 'F', 'G', etc.) or a conjunction of term literals, provided that the letters appear in alphabetical order and none appears twice. The nullities in

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(3), e.g., are canonical nullities; and so also are $F'$ and $\bar{F}'$. An assignment of truth values to nullities will be called annihilative if it assigns 1 to every one of the canonical nullities exhibiting a given set of letters (thus requiring the universe to be empty); the assignment of 1 to all the nullities in (3), e.g., is annihilative. By a canonical schema will be meant any truth function of canonical nullities all of which contain just the same letters. Now we may state our findings: a canonical schema is valid in all universes if and only if it is truth-functionally valid; and a canonical schema is valid in all non-empty universes if and only if it comes out true under every non-annihilative assignment of truth values to its nullities.

These tests of validity will next be extended to truth functions of nullities generally, by showing how to reduce all such schemata to canonical schemata.

## §25. VALIDITY OF TRUTH FUNCTIONS OF NULLITIES

Let us speak of a term schema as corresponding to a given truth-functional schema if it is like it except for containing 'F' in place of 'p', 'G' in place of 'q', etc.; thus $-(F(GvH))$ corresponds to $-(\bar{p} \cdot q \vee r)$. Classifications of truth-functional schemata in Part I may now be carried over to the corresponding term schemata. Thus we shall speak e.g. of fundamental term schemata, meaning term schemata corresponding to fundamental truth-functional schemata (§9).

When two truth-functional schemata are equivalent, the corresponding term schemata are bound to be alike in denotation under any interpretations of 'F', 'G', etc., in any universe. E.g., $\bar{p} \vee q'$ and $-(pq)$ being equivalent, $\overline{FvG'}$ and $-(FG)'$ are alike in denotation under any interpretations of 'F' and 'G', in any universe. For, consider any object x of the universe. To say that x is denoted by $\overline{FvG'}$ is to say that ($FvG$)x, and to say that x is denoted by $-(FG)$ is to say that $-(FG)x$. But, by (1)-(3) of §17, ($FvG)x$ is explained as $\overline{Fx v Gx'}$ and hence as $-(\overline{Fx} \vee Gx')$, and $-(FG)x$ is explained as $-(\overline{-(FG)x})$ and hence as $-(\overline{-(Fx} \vee Gx'})$; and these two, $\overline{-(Fx} \vee Gx')$ and $-(\overline{-(Fx} \vee Gx'})$, are bound to be both true or both false, being truth-functionally equivalent. The same reasoning applies, obviously, to all examples.

Next let S be any truth function of nullities, and let S' be like S except for containing $-(FG)$ where S contains $\overline{FvG}$. In view of the way in which the definition of equivalence in §18 turned on questions of denotation, clearly S and S' will be equivalent in any universe; for, we have seen that $\overline{FvG'}$ and $-(FG)'$ are always indistinguishable in denotation. Similarly in general; term schemata which (like $\overline{FvG'}$ and $-(FG)'$) correspond to equivalent truth-functional schemata are interchangeable, in this sense: when one is put for the other in any truth function of nullities S, the result S' is equivalent to S in every universe.

We can now freely exploit the techniques of equivalence in Part I, as a means of establishing the interchangeability of term schemata. In particular, we saw in §9 how to reduce any truth-functional schema to either $\overline{pp'}$ or a fundamental truth-functional schema or an alternation of such; in parallel fashion, then, we can reduce any term schema to $\overline{FF'}$ or a fundamental term schema or an alternation of such. The term schema:

¹The reason for requiring of canonical nullities that the letters appear in alphabetical order is simply that in testing for truth-functional validity we must be able to recognize the same nullity whenever it reappears.

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$FHFK v GH v HG v FGHCK v FKHG v KHF v GH$

e.g., is transformed into the alternation of fundamental term schemata:

$FHK v HG v FKHG v KHF v GH$

by paralleling the steps which led from (1) of §9 to (2). Also we can go on to simplify the result, paralleling the steps which led from (2) of §9 to (5); we come out thus with:

$KH v HG$.

Such transformation of term schemata will be an important step in transforming truth functions of nullities into canonical schemata. Other steps in that process will depend on the following laws:

(i) '$F\overline{F}$' is equivalent to '$F \supset \overline{F}$' in all universes.

(ii) '$F\overline{G}$' is equivalent to '$F \cdot \overline{G}$' in all universes.

(iii) '$F$' is equivalent to '$FG \cdot \overline{FG}$' in all universes.

In (i) we have a trivial consequence of the fact that '$F\overline{F}$' and '$F \supset \overline{F}$' are both valid in all universes. (ii) is obvious from the fact that '$F\overline{G}$' is bound to denote just what either of '$F$' and '$\overline{G}$' denotes, and hence will lack denotation if and only if '$F$' and '$\overline{G}$' both lack denotation. (iii) is established by similar reasoning, since '$F$' must denote just what either of '$FG$' and '$\overline{FG}$' denotes.

The following procedure, now, suffices to reduce any truth function of nullities to a canonical schema which is equivalent to it in all universes. Given any truth function of nullities, we first transform its term schemata into '$FF$' or fundamental term schemata or alternations of such, as noted above, and simplify them further as occasion may offer. Next, by (i), we change '$F\overline{F}$' everywhere to '$F \supset \overline{F}$'. Also, wherever we have a nullity containing an alternation, we transform it into a conjunction of nullities in conformity with (ii). Also, wherever we have a nullity lacking a letter which appears anywhere else in the context, we incorporate the missing letter by turning the defective nullity into a conjunction of nullities according to (iii). Finally we rearrange the components of each term-conjunction in alphabetical order, and come out thus with a canonical schema.

Though the above procedure suffices for the stated purpose, it is apt to eventuate in far more complex canonical schemata than need be, and apt also to entail a needlessly long series of transformations. A supplementary directive which rights matters is this: In the course of the transformations, interrupt proceedings and perform, immediately, any simplifications of truth-functional structure that suggest themselves. The main simplification to look out for is resolution, as in §7 (and §813-14), of the truth-functionally valid component '$F \supset \overline{F}$' whenever it turns up. Indeed, we may as well resolve out the valid clause '$F\overline{F}$' directly whenever it emerges; nothing is gained by first transforming it into '$F \supset \overline{F}$' according to (i). The only real occasion for resorting to (i) is where we have nothing left but '$F\overline{F}$' or '$\overline{F}F$' in isolation; in this trivial situation the transformation is wanted in order merely that our result conform to the definition of a canonical schema.

E.g., let us apply the technique to:

$$ (1) \qquad F \equiv FGH \cdot G - (FG)H \cdot G - G - (GxH). $$

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Transforming `-(FG)H` into `((FvG)H` and thence into `FHVGH`, and transforming `G-(GvH)` into `GGH` and thence into `FF`, we turn (1) into:

$$ \intertext{Interrupting proceedings now in order to resolve out the valid clause `FF`, we obtain:} \intertext{Resuming operations now with (ii), we get:} \intertext{Applying (iii) now to `FG`, `FH`, and `GH`, we get:} \intertext{Instead of imposing alphabetical rearrangement thus as a separate step, it is as well to maintain alphabetical order all along; thus the letters added by (iii) might have been inserted directly in their alphabetical positions.} \intertext{By truth-functional considerations, (2) can still be simplified to the extent of dropping the duplicate `FFGH`, and also reducing the negated conjunction to an alternation by DeMorgan's law. The result is:} \intertext{We can now test any truth function of nullities to see if it is valid in all universes, or in all non-empty universes; we have merely to transform it into a canonical schema, as above, and then test it as in §24. E.g., to determine whether (1) is valid in all universes, we see whether (3) is truth-functionally valid. To decide this we put 'p', 'q', etc. for the distinct nullities in it (an unnecessary but convenient step), and examine the result 'pq'. For the biconditional `FFGH` for validity by Part I standards. The quickest way to test a biconditional for validity is to test its two components for equivalence (cf. §14). We proceed to do so (as in §5):}$$

11

1

l v r v s v t

r v s v t

Here the process terminates; `r v s v t` not being valid, the equivalence fails. Therefore (1) is not valid in all universes.

We may still wonder whether it is valid in all non-empty universes. This will be the case if the only assignment of truth values which falsifies (3) is annihilative -- i.e., assigns 1 to every one of the eight nullities `FGH`, `FFGH`, etc.

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But this is clearly not so, since some of those eight do not turn up in (3) at all

Our technique for testing validity carries with it a technique for testing consistency, equivalence, and implication, since inconsistency is validity of the negation, equivalence is validity of the biconditional, and implication is validity of the conditional. E.g., let us verify by these methods that 'FCH - /FG' implies '-/FH' in all universes. This will be recognized as the principle of the syllogism in EIO of the first figure, and the same that was mentioned in §17-18. Now our problem, in verifying that this implication obtains in all universes, is to verify that the corresponding conditional:

$$ /GH . -/FG . \neg -/FH $$

is valid in all universes. To check this we first put the conditional over into canonical form, obtaining:

$$ /FGH . /FGH . -(/FGH . /FGH) . \neg -(/FGH . /FGH). $$

(Note that only (iii) is involved in this transformation.) Then we test this for truth-functional validity; i.e., we test the corresponding truth-functional schema 'pq -(pr) \rightarrow -(rs)' for validity. The quickest way (cf. §13) is to test its components 'pq -(pr)' and '(rs)' for implication, thus (cf. §6):

$$ pq -(pl) $$

$$ pq\bar{p} $$

-(11)

-(01)

-(10)

-(00)

0

1

1

1

## §26. EXTENSIONS

Often the notation 'F = G' is used by logicians to mean 'All F are G and all G are F'. Various valid equations are then obtained, such as:

$$ FG = GF, $$

$$ -(FG) = \overline{FvG}, $$

etc. Along these lines a simple logical discipline is developed which bears a resemblance to algebra, and is known as Boolean algebra (for George Boole, 1815-64). The term 'A' is introduced, which may be read 'non-entities' and is understood as denoting nothing; it figures in Boolean algebra as an analogue of zero. What we have been writing as '/F' is rendered then as an equation 'F = A'. If, for comparison with Boolean algebra, we think of the notation '/F' of foregoing sections simply as an abbreviation of 'F = A', then the work which we have been doing with nullities may be viewed as a study of a restricted form of Boolean algebra, the restriction being that all the equations dealt with have 'A' as their right side.

Though the techniques developed in foregoing pages have gained in simplicity by our not having burdened them with the full equational notation 'F = G', the restriction has been one only of form and not of content; for, the fact is that whatever could be expressed by adding full use of '=' can also be expressed in our notation of nullities as it stands. This is obvious from the interpretation given of 'F = G', viz. All F are G and all G are F'; for, we already know well how to express

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(5C)

G's, air. VII B are G and S are E; Ix, 'we strategy well now to express
G' of nullify as it stands. This is an opinion from the interpretation plan of I.
I am going to express nullify as nullify now as I: can also be expressed in our
statement has been one out of four and not of content; Ix, the fact is just au-
other nullify as it stands. This is an opinion from the interpretation plan of I.
This is clearly not so, since some of those eight to not turn to in (3) at all

EG = OK, - (AG) = EG,

pointing that all the equations gear with one V, so their right side are
just applied as a sum of a restricted form of Boolean algebra, the restriction
is an application of A = V, then the work which we have been doing with nullities
with Boolean algebra, we think of the notation 'A', of foregoing sections simply as
been writing as 'A', is remembered then as an equation 'A = V'. It's for comparison
generating working; ittgme in Boolean algebra as an algebra of zero. What we have
to do is introduced, which must be read now entities, and is understood as
compliance to algebra, and is known as Boolean algebra (for George Boole, 1812-1844).
You know these lines a simple logical fact that is developed which para a re-
lation is OK.

SEC. EXTENSIONS

0
I
I
I

-(IT) -(Of) -(10) -(00)

hd

hd -(br)

bd -(br), ang -(re), for, implication, pure (of 'sc):
bd -(br) C -(re), for, affinity. The dropout asl (of '812) is to test its components
trufp-functional affinity; i.e., we test the corresponding truth-functional scheme
(above first only (iii)) is involved in the transformation.) Then we test pure for

xch . xch . -(xch . xch) C -(xch . xch).

cautionary form, obtaining:
is affinity in all numbers. To check this we first but the conditional over into

xch . -xch . C -xch

that the corresponding condition:

problem, in verifying that this implication obtains in all variables, is to aerily
in EIO of the that time, and the same that was mentioned in 821-18. Now om
- xch, in all variables. This will be received as the brincline of the diagram
of the conditional. E.g., jet we verify p three times more that xch . xch, implies
affinity, adiavance, and implication, since incoherent is affinity of the
om, recognise for testing affinity carries with it a technique for testing

But this is clearly not so, since some of those eight to not turn to in (3) at all

18

79

that idea: $\overline{\notin} \notin$. This being the case, Boolean algebra is already covered by our techniques. To determine the validity of a Boolean equation or of a truth function of Boolean equations we have merely to translate it into a truth function of nullities and then go at it as in §25. Similarly for implication and the rest.

Boolean algebra, then, is only notationally and not logically an extension of the foregoing theory. But there is a more genuine extension of the foregoing theory which now bears consideration; viz., the introduction of statement letters into the company of nullities. In Part I we considered truth functions purely of statement letters; in Part II we have considered truth functions purely of nullities; but we have not yet taken up truth functions of statement letters and nullities -- e.g.:

$$ (1) \qquad p \lor \overline{\notin} \notin. \quad q \equiv \overline{\notin} \notin. $$

This, happily, is a topic to which justice can now be done in a couple of pages.

We defined a canonical schema in §24 as a truth function of canonical nullities all having the same letters. We may now define an *extended canonical schema* as a truth function of statement letters and canonical nullities, wherein the nullities all have the same letters. Thus (1) is, as it happens, an extended canonical schema.

A truth function of nullities was said to be *valid* in a given universe if it came out true for every interpretation of 'F', 'G', etc. within that universe (§18). We may now explain a truth function of statement letters and nullities as *valid* in a given universe if it comes out true for every assignment of truth values to 'p', 'q', etc. and every interpretation, within the given universe, of 'F', 'G', etc. Consistency, equivalence, and implication may be extended similarly.

We saw in §24 that every assignment of truth values to the canonical equalities made up of given letters 'F', 'G', ..., 'K' could be realized by some interpretation of 'F', 'G', ..., 'K' within some universe (the empty universe in the case of an annihilative assignment, and otherwise a non-empty universe). From these considerations it followed immediately, as seen in §24, that to test a canonical schema for validity in all universes we had only to test it for truth-functional validity. Now by the same considerations the same result follows for extended canonical schemata: they are valid in all universes if and only if truth-functionally valid. To test an extended canonical schema for validity in all non-empty universes, the procedure is again just as in §24: test it for truth-functional validity but except annihilative assignments. An annihilative assignment, now as before, is one which assigns 1 to each of the canonical equalities exhibiting a given set of letters.

The entire routine of §25, for transforming a truth function of nullities into a canonical schema, serves also to transform any truth function of statement letters and nullities into an extended canonical schema, there being nothing in that routine which the presence of 'p's and 'q's in the context might in any way obstruct.

In short, the techniques of foregoing sections carry over in full to truth functions of statement letters and nullities. In conclusion, let us try, by way of example, a case of implication.

**Premises:** If the Bissagos report is to be trusted then the *chargé d'affaires* is a more tool of the sisal interests and none of the natives really favored the coupon plan.

If the *chargé d'affaires* is a mere tool of the sisal interests then some of the natives either really favored the coupon plan or were actuated by a personal animosity against the deputy resident.

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Conclusion: If the Bissagos report is to be trusted then some who were actuated by a personal animosity against the deputy resident did not really favor the coupon plan.

Putting 'p' for 'the Bissagos report is to be trusted', 'q' for 'the charge d'affaires is a mere tool of the sidal interests', 'F' for 'natives', 'G' for 'those really favoring the coupon plan', and 'H' for 'those actuated to a personal animosity against the deputy resident', we have as premises:

$$ (2) \quad p \dashv. q. \ \overline{p} \overline{F}, \qquad q \dashv. \overline{p} \overline{F}(G \overline{v} H) $$

and as conclusion:

$$ (3) \quad p \dashv. \overline{p} \overline{H} $$

The question, then, is whether the conjunction of the premises (2) implies (3); i.e., whether the conditional:

$$ (4) \quad p \dashv. q. \ \overline{p} \overline{F} : q \dashv. \overline{p} \overline{F}(G \overline{v} H) : \begin{array}{l} \therefore p \dashv. \overline{p} \overline{H} \\ \text{and as conclusion:} \end{array} $$

is valid. (4) goes over into an extended canonical schema by the following stages:

$$ p \dashv. q. \ \overline{p} \overline{F} : q \dashv. \overline{p} \overline{(F \overline{v} F H)} : \begin{array}{l} \therefore p \dashv. \overline{p} \overline{H} \\ \text{and as conclusion:} \end{array} $$

$$ p \dashv. q. \ \overline{p} \overline{F} : q \dashv. \overline{p} \overline{(F \overline{v} G \ \overline{v} F H)} : \begin{array}{l} \therefore p \dashv. \overline{p} \overline{H} \\ \text{and as conclusion:} \end{array} $$

$$ p \dashv. q. \ \overline{p} \overline{F} \overline{G} \ \overline{p} \overline{F} : q \dashv. \overline{p} \overline{F} \overline{G} \ \overline{p} \overline{F} : \begin{array}{l} \therefore p \dashv. \overline{p} \overline{F} \overline{G} \\ \text{and as conclusion:} \end{array} $$

Dropping the repetition of $\overline{p} \overline{F} \overline{G} \overline{H}$ in the first parentheses, and then representing the various nullities concerned as 'r', 's', 't', and 'u', we have the truth-function-al schema:

$$ p \dashv. qrs. \ q \dashv. -(rst) \ . \ p \dashv. -(tu) $$

to examine for validity. To do so we inquire whether its antecedent $p \dashv. qrs. q \dashv. -(rst)$ implies its consequent $p \dashv. -(tu)$. The test, by the method of §6, is as follows:

$\overline{D}qrs.q \dashv. -(rs1)$

$qrs.q \dashv. -(rs)$

$\overline{D}-(11) \quad \overline{D}-(11) \quad \overline{D}-(01) \quad \overline{D}-(01) \quad \overline{D}-(10) \quad \overline{D}-(10) \quad \overline{D}-(00) \quad \overline{D}-(00)$

$0 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1$

The outcome here is favorable, in view of the inconsistency of 'qrs. q \dashv. -(rs)' -- which may be checked by a subsidiary test:

$\overline{111.D}-(11) \quad \overline{011.D}-(11) \quad \overline{101.D}-(01)$

$0 \quad 0 \quad 0$ (etc. to eight cases)

Therefore (4) is valid in all universes. So the conjunction of the premises (2) implies the conclusion (3) in all universes.

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In abstracting the schemata (2) - (3) out of the premisses and conclusion as originally given in words, it will be noted that certain component statements were left unanalyzed and represented by 'p' and 'q', while others were broken down into terms 'F', 'G', and 'H' and analyzed as nullities or negations of nullities. In deciding when to use one method and when the other, the best plan is to look to the superficial interrelationships of the stated premisses and conclusion as wholes. Since 'natives', 'actuated by ... resident ', and 'really favored the coupon plan' turned up in various combinations and with varying use of 'some' and 'none' in the course of the premisses and conclusion, it was reasonable to expect that these parts will need separate representation if the structure is to exhibit implication. Since on the other hand the clauses 'the Bissagos report is to be trusted' and 'the chargé d'affaires is a mere tool of the sisial interests' stayed intact throughout the premisses and conclusion, it was clear that no question of the internal structure of these clauses must be irrelevant to the question of implication.

# PART III. QUANTIFICATION

## §27. RELATIVE TERMS

Terms must be distinguished into two kinds, known traditionally as *relative* and *absolute* terms. The characteristic of a relative term is that it describes things only relatively to further things which have afterward to be specified in turn. Thus 'father' as in 'father of Isaac', or 'capital' as in 'capital of Peru', or 'author' as in 'author of Waverly', are relative terms. Other terms are absolute. Words capable of behaving as relative terms can regularly be used *also* as absolute terms, through what amounts to a tacit existential quantification in the context; thus we may say absolutely that Abraham is a father, or that Lima is a capital, meaning that there is something of which Abraham is father, or of which Lima is capital.

In English a convenient aearmark of the relative use of a term is the adjoining of an 'of'-phrase or possessive modifier whose sense is not that of ownership. Thus 'father of Isaac', or 'Isaac's father', has nothing to do with proprietorship on Isaac's part, but means merely 'that which bears the father-relation to Isaac'. The point is well illustrated by Aristotle:

> This dog is mine,
> This dog is a father;
> ∴ This dog is my father.

The 'mine' in the first premiss (or the 'my' of 'my dog') expresses ownership, and the 'father' in the second premiss is an absolute term; on the other hand the 'my' of the conclusion does not carry the sense of ownership but is an adjunct rather of 'father' as a relative term. The fallacy is one of equivocation (see §15).

A relative term, like an absolute one, may occur as a substantive, adjective, or verb, indifferently. In 'x is a helper of y' we use the substantive, in 'x is helpful toward y' the adjective, and in 'x helps y' the verb; but logically there is no need to distinguish the three. Logically the important thing about relative terms is that they denote objects pairwise. Whereas 'man', 'walks', etc. denote Caesar, Socrates, etc. one by one, on the other hand the relative term 'helps' denotes Jesus and Lazarus as a pair (or, Jesus with respect to Lazarus), and Farley and Roosevelt as a pair (or, Farley with respect to Roosevelt), and so on. If as in §17 we write 'Fx' for 'x is an F', then the analogous notation in connection with relative terms should be 'Fxy', 'x is F to y'.

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Besides relative terms in the sense just now touched upon, which are *dyadic*, we may recognize also *triadic* ones, *tetradic* ones, and so on; e.g., 'Gxyz' may mean 'x gives y to z', and 'Hxyzw' may mean 'x pays y to z for w'.

The so-called logic of terms studied in Part II may be more accurately described now as a logic of absolute terms. There are forms of inference, logically no less sound than those dealt with in Part II, which are insusceptible to the methods of Part II simply because their analysis calls for recognition of relative terms. A simple example is the following (which was propounded in the Middle Ages to illustrate the limitations of syllogistic method):

Premiss: All circles are figures;

Conclusion: All who draw circles draw figures.

The premiss can be represented in terms of Part II as '∠FG', where 'F' represents 'circle' and 'G' represents 'figure'; and the conclusion can be represented as '∠HK', where 'H' represents 'drawer of a circle' and 'K' represents 'drawer of a figure'. But obviously '∠FG' does not imply '∠HK' in all universes, nor, indeed, in any non-empty universe at all. The difficulty is that in representing the premiss and conclusion as '∠FG' and '∠HK' we have not exposed enough of their relevant structure; we have not preserved any connection between 'circle' ('F') and 'drawer of a circle' ('H'), nor between 'figure' ('G') and 'drawer of a figure' ('K').

To reveal these needed connections we must go beyond the scope of Part II and let 'H' represent, not the absolute term 'drawer of a circle', but the relative term 'draw' (or 'drawer of'). Then the combination of 'drawer of' ('H') and 'circle' ('F'), in 'drawer of circle', may be represented by a new notation: H"F, read 'H of an F'. This new logical notation, supplementary to those introduced in §17, may be explained in the spirit of (1)-(3) of §17, as follows:

(1) (H"F)x ≡. there is something y such that (Hxy . Fy).

E.g., where 'H' means 'draw' and 'F' means 'circle',

(H"F)x ≡. there is something y such that x draws y and y is a circle.

'All who draw circles draw figures' now becomes 'All H"F are H"G', and hence this:

(2) ∠ (H"F) -(H"G).

So we need to supplement the techniques of Part II in such a way as to be able to show that '∠FG' implies (2).

However, not to pause longer over the problems raised by the one example, let us turn to another:

Premiss: There is a painting that all critics admire;

Conclusion: Every critic admires some painting or other.

Here we may represent the absolute terms 'painting' and 'critic' as 'F' and 'G', and the relative term 'admirer of' as 'H'. The notation just previously introduced now enables us to express the conclusion as 'Every G is an H"F':

(3) ∠ G -(H"F).

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Expression of the premiss, however, still presents a problem: the problem of forming an absolute term from 'G' and 'H' to correspond to the words 'admired by all critics', or, in general, 'Hd by every G'. Let us imagine, then, a new notation adopted, say 'H;G', for 'Hd by every G'. Its sense may be conveyed in a formula thus:

(4) (H;G)x =. no matter what y may be, Gy ⊢ Hyx.

Now the premiss of the above example can be rendered 'There is an F that is an H;G':

(5) -∫F(H;G).

Our proposed extension of logic should be capable, then, of showing not only that '∫FG' implies (2), but also that (5) implies (3).

One more example:

Premiss: There is a philosopher whom all philosophers contradict;

Conclusion: There is a philosopher who contradicts himself.

Representing the absolute term 'philosopher' as 'F' and the relative term 'contradict (or 'contradictor of') as 'H', we can express the premiss thus:

(6) -∫F(H;F),

using the new notation just previously introduced. But the conclusion calls for yet a further new notation, whereby, from a relative term such as 'contradictor of', to form an absolute term 'contradictor of self'. Let us suppose the notation 'sH' adopted for this purpose. Its sense may be conveyed in a formula thus:

(7) sHx = Hxx.

Now the conclusion of our last example can be expressed:

(8) -∫F sH.

Our proposed extension of logic should be capable then, of showing also that (6) implies (8).

But this proposed extension of logic begins to look rather forbidding. Each of the three examples has demanded some additional structural device with which the extended logic must deal. We could go on and on adducing examples which would call for yet further such devices. In extending logic to relative terms we confront the problem of reducing this array of new notions -- 'H''F'', 'H;F'', 'sH'', and their intermix suite -- to a manageable few from which the rest may be generated; and the further problem of devising general techniques whereby implication and validity, in these new realms, can be tested or proved. The department of logical theory which is thus demanded seems to have received no serious study prior to DeMorgan (1860) and C. S. Peirce (1870). The theory which has since evolved, variously known as the "algebra of relatives" and the "calculus of relations", is far harder than the logic of absolute terms with which we have so easily become familiar. The translating of verbal examples into the notation of the algebra of relatives is often difficult and unnatural, and the laws of the algebra, even those that are brief and simple from a notational point of view, are seldom intuitive or obvious.

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A much more intuitive technique for dealing with inferences depending on relative terms is arrived at by abandoning the special notations 'H"F', 'H;F', 'sH', etc. and dealing, instead, directly with the explanatory notations such as appeared on the right-hand sides of (1), (4), and (7). The form of notation 'Fx' ('x is an F'), used in Part II only at an expository level (cf. §17), is thus elevated to the status of standard notation subject to rules of manipulation; and, along with it, the analogue 'Hxy' for relative terms ('x is H to y'). In addition the idioms 'there is something y such that' and 'no matter why y may be', appearing in (1) and (4), are accorded abbreviations '(Ey)' and '(y)' and subjected to rules of manipulation. Logic developed in these terms is known as quantification theory. In its essentials it dates from Frege (1879). It is to this that we now turn, taking leave of 'H"F', 'H;F', 'sH', and the algebra of relatives.

## §28. QUANTIFICATION

The so-called **universal quantifiers** '(x)', '(y)', etc. correspond to the words 'no matter what x may be', 'no matter what y may be', etc.; and the so-called **existential quantifiers** '(Ex)', '(Ey)', etc. correspond to the words 'there is something x such that', 'there is something y such that', etc. The purpose of allowing for different letters 'x', 'y', etc. is merely to key the cross-references. The need of such an expedient becomes evident as soon as we try translating a statement of the complexity of 'There is a painting which all critics admire'. This becomes:

(1) (Ex)(x is a painting . (y)(y is a critic .) y admires x)).

The purpose served by such recourse to distinct letters is often accomplished in ordinary language by use of the words 'former' and 'latter', or 'first', 'second', 'third', etc.; thus (1) might be put back into words rather literally as follows:

> There is something such that it is a painting and such that, no matter what you select, if the latter is a critic then the latter admires the former.

The application of '(x)' to an expression, say:

(2) x is a man → x is mortal,

in the fashion:

(3) (x)(x is a man → x is mortal),

is called **universal quantification** of (2) with respect to 'x'. Also the result, (3), is spoken of as the **universal quantification** of (2) with respect to 'x'. Correspondingly for existential quantification; thus, existential quantification of:

(4) x is a Greek.. x is wise

with respect to 'x' yields:

(5) (Ex)(x is a Greek . x is wise);

and (5) is called the existential quantification of (4) with respect to 'x'. Correspondingly, of course, for other letters and other examples.

The letter 'x' is called a variable, in quantifications as in algebraic formulae, but it is not to be thought of as an unknown, or as imparting any variability to statements in which it occurs. The quantification (3) is just as unambiguous as its

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equivalent 'All men are mortal'; and the purpose of the 'x' in it is merely to refer back to the quantifier, much as pronouns refer back to their grammatical antecedents in an ordinary sentence. Indeed, (1) may be thought of as corresponding to the words:

> No matter what may be selected, if it is a man then it is mortal

in which the pronoun 'it' plays the role of the last two occurrences of 'x' in (3). Similarly (5), read briefly 'Some Greeks are wise; corresponds more literally to the words:

> There is something such that it is a Greek and it is wise.

Expressions like (2) and (4), and also 'x is a man' and 'x is wise' individually are called *matrices*. A matrix is of the form of a statement but is not actually a statement, because it contains a *free variable* -- a variable without a corresponding quantifier. We cannot meaningfully speak of a matrix as true or false. It happens indeed in the case of (2) (unlike (4)) that what the matrix says of x *is true* of every particular object; still we do not speak of the matrix (2) itself true, but reserve this epithet rather for its universal quantification (3). Matrices are not statements, but fragments of statements; they can be built out into statements by applying quantifiers, as in (1), (3), and (5).

In Parts I and II the letters 'p', 'q', etc. and 'F', 'G', etc. were often used to take the place of statements and terms. These schematic letters are not to be confused with variables in the above sense. Schematic letters do not appear in quantifiers, nor, properly speaking, in statements or matrices at all. They are merely dummy statements and dummy terms, used in schemata which depict outward structures of statements in abstraction from content.

The schematic letters 'p', 'q', etc. can now represent matrices as well as statements. But when in schematically representing a matrix we want to show explicitly that it contains a particular variable, say 'x', we abandon 'p', 'q', etc. in favor of 'Fx' (or 'Gx', 'Hxy', etc.)

By starting with 'p', 'q', 'Fx', 'Gx', 'Fy', 'Gxy', etc., and applying quantifiers and the truth-functional notations, we obtain a category of schemata which will be called *quantificational schemata*. The schema:

$$ (6) \qquad (\mathbf{Ex})(\mathbf{Fx} \cdot (y)(\mathbf{Gy} \supset \mathbf{Hyx}))), $$

which depicts the form of (1), is a quantificational schema. So also are '(x)(p \supset Fx)', 'p \supset (Ex)Fx', etc., which contain statement letters. It is convenient to impose just this one restriction: no term letter is to appear twice in the same quantificational schema followed by different numbers of occurrences of variables. Thus 'Fx' and 'Fy' may appear in the same schema; also 'Fxx' and 'Fxy'; also 'Fx' and 'Gxy'; but not 'Fx' and 'Fxy', nor 'Fx' and 'Fxx'.

A quantificational schema will be called *open* if it contains one or more free variables, and *closed* if it contains none. Thus (6) is closed; on the other hand 'Fx · (y)(Gy \supset Hyx)' is open, and so is '(y)(Gy \supset Hyx)'. Closed schemata depict forms of statements, while open schemata depict forms of matrices. A closed schema which is formed from an open schema by universally quantifying it, once for each of its free variables, will be called a *closure* of that open schema; thus the open schema:

$$ (7) \qquad \mathbf{Fx} \supset (\mathbf{Ey})(\mathbf{Gyz} \cdot \mathbf{Hzx}), $$

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e.g., has:

(8) (z)(x)(Fx → (Ey)(Gyz . Hzx))

as a closure, and:

(9) (x)(z)(Fx → (Ey)(Gyz . Hzx))

as another closure.

The use of multiple quantifiers as in (8) and (9) is no new notation. The single quantifier '(x)' in (8) applies, as usual, to the parenthesized expression which follows it; and then the quantifier '(z)', in (8), applies to the whole schema '(x)(...)' which follows it. We may read (8) literally in the fashion: 'No matter what z may be, it will be the case, no matter what x may be, that if Fx then etc.'; and we may read (9) in the fashion: 'No matter what x may be, it will be the case, no matter what z may be, that if Fx then etc.' But it is obvious that the order of such consecutive universal quantifiers is bound to be indifferent to truth and falsehood, and that (8) and (9) could both be read alternatively in the simple fashion: 'No matter what z and x may be, if Fx then etc.' An open schema with several free variables has many closures -- as many as there are permutations of the added quantifiers; but they differ only notationally from one another.

## §29. VALIDITY, EQUIVALENCE, ETC.

The concept of validity, as defined for truth functions of multitudes and statement letters (§26), carries over verbatim to closed quantificational schemata: a closed quantification schema is valid in a given universe just in case it comes out true under every assignment of truth values to its statement letters and every interpretation, within that universe, of its term letters.

There is, however, the following difference in background. In Part II the term letters were used monadically only; i.e., they represented absolute terms exclusively, and so, from the point of view of present notations, would admit of being followed only by single variables in the fashion 'Fx', 'Ey', etc. But quantificational schemata admit term letters also in polyadic use: 'Fxy' (dyadic), 'Gxyz' (triadic), etc. The notion of an "interpretation" of a term letter has then to be extended to cover term letters in polyadic use. In the monadic case, an interpretation of 'F' in a given universe consisted in specifying just what things in the universe (if any) 'F' was to denote. In the dyadic case, an interpretation of 'F' in a given universe consists in specifying just what things in the universe (if any) 'F' is to denote relatively to just what other things in the universe; i.e., just what things are to be regarded as 'F' of just what things (cf. §27). Less cumbersomely stated: in the dyadic case an interpretation of 'F' in a given universe consists in specifying what ordered pairs of things in the universe (if any) 'F' is to denote. Similarly, in the triadic case an interpretation of 'F' in a given universe consists in specifying just what ordered triples of things in the universe 'if any' 'F' is to denote; and so on. Given this extension of the notion of interpretation, the preceding paragraph then defines validity of closed quantification schemata.

This definition of validity has to be limited to closed schemata simply because we cannot speak of an open schema as coming out "true" or "false" under interpretations of its term letters. The putting of terms for term letters (and statements for statement letters) would turn an open schema into a matrix rather than a statement (cf. § 28), and matrices are neither true nor false. However, the notion of validity will be extended to open schemata in an indirect way: An open schema will be called valid, in a given universe, just in case its closures are valid in that universe.

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This comes to the same thing: an open schema is valid (in a given universe) if and only if it comes out true under every assignment of truth values to its statement letters, every interpretation (within that universe) of its term letters, and every assignment of objects (from that universe) to its variables.

Validity has now been defined for quantificational schemata generally, closed and open. Consistency, equivalence, and implication are then definable in terms of validity in the usual way: a quantificational schema is consistent in a given universe just in case its negation is not valid in that universe; quantificational schemata are equivalent in a given universe just in case their biconditional is valid in that universe; and one quantificational schema implies another in a given universe just in case the conditional having the one schema as antecedent and the other as consequent is valid in that universe.

One example of equivalence was noted in (8) and (9) of §28; it may be put more generally thus:

(i) '$(x)(y)Fxy$' and '$(y)(x)Fxy$' are equivalent in all universes.

Similarly

(ii) '$(Ex)(Ey)Fxy$' and '$(Ey)(Ex)Fxy$' are equivalent in all universes.

Thus the order of consecutive quantifiers in immaterial as long as both are universal or both existential. Note however that the order becomes essential when one is universal and the other existential; '$(Ex)(y)(Fxy$' and '$(y)(Ex)Fxy$' are not equivalent in all universes. E.g., take the universe as comprising just the numbers, and cons true 'F' as $. (I.e., interpret 'F' as denoting just those pairs x, y such that x y.) Then '$(y)(Fx)Fxy$' says that for every nummber y there is a greater; and this is true, however, '$(Ex)(y)Fxy$' says there is some number which is, at once greater than all numbers; and this is false.

Since '$(x)Fx$' says that everything in the universe is an F, and '$(Ex)Fx$' says that something is an F, it is evident that

(iii) $-(x)Fx$' is equivalent to '$(Ex) -Fx$' in all universes.

(iv) $-(Ex)Fx$' is equivalent to '$(x) -Fx$' in all universes.

These principles show how negation of quantifications can be avoided, at will, in
favor of quantification of negations. They show also that the existential quantifier
is dispensable in favor of the universal, and vice versa; for, '$(Ex)Fx$' is translat-
able as $-(x) -Fx$' in view of (iv), and conversely '$(x)Fx$' is translatable as $-(Ex)
-Fx$' in view of (iii). The admission of the two kinds of quantifiers is a matter
merely of convenience.

The following is a useful law of distribution:

(v) '$(x)(Fx . Gx)$' is equivalent to '$(x)Fx . (x)Gx$' in all universes.

For, obviously, to say that everything is both an F and a G is the same as saying
that everything is an F and that everything is a G. A companion-piece is this
further law of distribution:

(vi) '$(Ex)(Fx v Gx)$' is equivalent to '$(Ex)Fx v (Ex)Gx$' in all universes.

This can be seen by direct reflection; also it can be substantiated on the basis
of (iii) — (v). For, '$Fx v Gx$' is truth-functionally equivalent to $'-(\neg Fx . \neg Gx)'$,

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so that $'(\mathfrak{E})(\mathfrak{F}v\mathfrak{G})'$ becomes $'(\mathfrak{E}) -(-\mathfrak{F} \cdot -\mathfrak{G})'$ and hence, by (iii), $'-(\mathfrak{X})$ ($-\mathfrak{F} \cdot -\mathfrak{G}$)'. But this in turn becomes $'-(\mathfrak{X})^2 -\mathfrak{F} \cdot (\mathfrak{X}) -\mathfrak{G}'$ by (v). Transforming its parts $'\mathfrak{X}$-$\mathfrak{F}'$ and $'\mathfrak{X}$-$\mathfrak{G}'$ then by (iv), we get $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} \cdot -(\mathfrak{E})\mathfrak{G}\mathfrak{x}'$, which is truth-functionally equivalent to $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} v (\mathfrak{E})\mathfrak{G}\mathfrak{x}'$.

Whereas a universal quantifier may thus be distributed through a conjunction, and an existential quantifier through an alternation, note on the other hand that a universal quantifier may not be distributed through an alternation, nor an existential quantifier through a conjunction. $'\mathfrak{X} (\mathfrak{F} v \mathfrak{G}\mathfrak{x})'$ and $'\mathfrak{X} \mathfrak{F} v (\mathfrak{X})\mathfrak{G}\mathfrak{x}'$ are not equivalent, for $'\mathfrak{X}(\mathfrak{X}$ is red $v \mathfrak{X}$ is not red')$'$ is true while $'\mathfrak{X}(\mathfrak{X}$ is red $v \mathfrak{X}$) ($\mathfrak{X}$ is not red')$'$ is false. Again $'-(\mathfrak{E})(\mathfrak{F}\mathfrak{x} \cdot \mathfrak{G}\mathfrak{x})'$ and $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} \cdot (\mathfrak{E})\mathfrak{G}\mathfrak{x}'$ are not equivalent, for $'-(\mathfrak{E})(\mathfrak{X}$ is red $\cdot \mathfrak{X}$ is not red')$'$ is false while $'-(\mathfrak{E})(\mathfrak{X}$ is red $\cdot (\mathfrak{E})$ ($\mathfrak{X}$ is not red')$'$ is true.

The following are two of four principles enabling us to confine a quantifier to the relevant portion of a compound:

(vii) $'\mathfrak{X}(\mathfrak{P} v \mathfrak{F}\mathfrak{x})'$ is equivalent to $'\mathfrak{P} v (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$ in all universes.

(viii) $'-(\mathfrak{E})(\mathfrak{P} \cdot \mathfrak{F}\mathfrak{x})'$ is equivalent to $'\mathfrak{P} \cdot (\mathfrak{E})\mathfrak{F}\mathfrak{x}'$ in all universes.

Let us consider (vii). What we want to show is that $'\mathfrak{X}(\mathfrak{P} v \mathfrak{F}\mathfrak{x})^2$ $\equiv$ $\mathfrak{P} v (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$ is valid in all universes; or, in other words, that $'\mathfrak{X}(\mathfrak{P} v \mathfrak{F}\mathfrak{x})'$ and $'\mathfrak{P} v (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$ come out alike in truth value under either assignment of truth value to $'\mathfrak{P}'$ and any interpretation of '$\mathfrak{F}'$ in any universe. Case 1: $\mathfrak{P}$ is assigned to $'\mathfrak{P}'$. Then $'\mathfrak{P} v (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$ becomes true; and also $'\mathfrak{P} v \mathfrak{F}\mathfrak{x}'$ becomes true for each object $\mathfrak{X}$, so that $'\mathfrak{X}(\mathfrak{P} v \mathfrak{F}\mathfrak{x})$ becomes true. Case 2: $\mathfrak{P}$ is assigned to $'\mathfrak{P}'$. Then $'\mathfrak{P} v \mathfrak{F}\mathfrak{x}'$ becomes true or false, for each object $\mathfrak{X}$, according as '$\mathfrak{F}'$ is interpreted as denoting that object or not; so that $'\mathfrak{X}(\mathfrak{P} v \mathfrak{F}\mathfrak{x})$ becomes true if '$\mathfrak{F}'$ is interpreted as denoting everything, and otherwise false. But so does $'\mathfrak{P} v (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$.

(viii) can be substantiated by a similar division into cases; or, equally, it can be deduced from (iii), (iv), and (vii) just as (vi) was deduced from (iii)-(v).

Two principles concerning the conditional are forthcoming as corollaries of (vii):

(ix) $'\mathfrak{X}(\mathfrak{P} \supset \mathfrak{F}\mathfrak{x})'$ is equivalent to $'\mathfrak{P} \supset (\mathfrak{X})\mathfrak{F}\mathfrak{x}'$ in all universes.

(x) $'\mathfrak{X}(\mathfrak{F}\mathfrak{x} \supset \mathfrak{P})'$ is equivalent to $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} \supset \mathfrak{P}'$ in all universes.

Since $'\mathfrak{P} \supset \mathfrak{q}'$ is equivalent to $'\mathfrak{P} v \mathfrak{q}'$, we see in (ix) merely a repetition of (vii) with '$\mathfrak{P}'$ for '$\mathfrak{P}'$. To justify $(\mathfrak{X})$ we put $'-(\mathfrak{F}\mathfrak{x})'$ for '$\mathfrak{F}\mathfrak{x}'$ in (vii), and then change $'\mathfrak{X} \cdot -\mathfrak{F}\mathfrak{x}'$ to $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x}'$ by (iv), thus concluding that $'\mathfrak{X}(\mathfrak{P} v -\mathfrak{F}\mathfrak{x})'$ is equivalent to $'\mathfrak{P} v -(\mathfrak{E})\mathfrak{F}\mathfrak{x}'$. But $'\mathfrak{P} v -\mathfrak{F}\mathfrak{x}'$ here is truth-functionally equivalent to '$\mathfrak{F}\mathfrak{x} \supset \mathfrak{P}'$, and $'\mathfrak{P} v -(\mathfrak{E})\mathfrak{F}\mathfrak{x}'$ to $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} \supset \mathfrak{P}'$.

With regard to grouping, quantifiers are subject to the same convention as the negation sign: a quantifier is to govern as little as possible of what follows it. Thus it is that $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x} \supset \mathfrak{P}'$, in $(\mathfrak{X})$, is understood as a conditional having $'-(\mathfrak{E})\mathfrak{F}\mathfrak{x}'$ as antecedent. To make $'-(\mathfrak{E})\mathfrak{F}'$ cover the whole conditional we should have to write $'-(\mathfrak{E})(\mathfrak{F}\mathfrak{x} \supset \mathfrak{P})'$.

The symmetry of (vii)-(viii) is rounded out by these two further laws, which, however, hold only for non-empty universes:

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["(50)", "however, would only for non-antibound universes:", "The symmetry of (aif)-(viii) is supposed out of these two upper laws, which,", "$$(Ex)(x \\to D \\ b).$$", "is equivalent. To write $((Ex) \\to D \\ b)$, cover the whole conditional we should have to write", "that it is true: $((Ex)(x \\to D \\ b), \\text{in } (x'),$ is equivalent to a conditional formula $((Ex)(x', \\text{condition sign})).$", "$b \\to ((Ex)(x', \\text{formula})).$", "But $b \\to (-Ex)(x', \\text{formula}((x') \\to D \\ b))$, here is definitely equivalent to $:Ex \\ D \\ b$, and", "$((x') \\to (-Ex)(x', \\text{formula}((x') \\to D \\ b)))$ is equivalent to", "$$(x) :((x') \\to D \\ b), \\text{ is equivalent to } :((Ex)(x' \\to D \\ b), \\text{ in all universes.})$$", "$$(x) :((x') \\to D \\ Ex), \\text{ is equivalent to } :b \\ D \\ (x') \\to x, \\text{ in all universes.})$$", "(aif):", "Two brancbles concerning the conditional are compared as colorifirs of", "can be deduced from (iii)' (iv)' and (aif) that as (aif) was deduced from (iii)-(a). (viii) can be contradicted by a similar dialogue into cases; or, contradict, if", "consequently false. But so does $b \\to (x') \\to x$.", "so that, $(x') \\to (x \\to D \\ x)$ becomes true if $x'$ is interpreted as denoting that object or not; because true. Case S: O is false if $b$. Then $b \\to (x') \\to x$ becomes true or false. Because true: and also, $b \\to (x') \\to x$ becomes true for every object $x$ so that, $(x') \\to (x \\to D \\ x)$ is false. Then $b \\to (x') \\to x$ is true in truth value under either assignment of truth value to $b$, and such assignment is false in all universes; or, in other words, that, $(x') \\to (x \\to D \\ x)$, and $b \\to (x') \\to x$ come to be equal (aif). What we want to show is that, $(x') \\to (x \\to D \\ x)$, $b \\to (x') \\to x$", "$$(aif) : ((x') \\to (x \\to D \\ Ex), \\text{ is equivalent to } :b \\ (Ex)(x' \\to D \\ b), \\text{ in all universes.})$$", "$$(aif) : ((x') \\to (x \\to D \\ Ex), \\text{ is equivalent to } :b \\ (x') \\to x, \\text{ in all universes.})$$", "The following are two of four branches leading us to continue a disnitiier to", "($x$ is not $req$), is true.", "Assuming, for, $((Ex)(x \\to D \\ x) \\to x$ is not $req$), is false, unless $((Ex)(x \\to D \\ x) \\to x$ is not $req$), is true, otherwise, $((x \\to D \\ x) \\to x$ is not $req$), is

89.

(xi) $'(x)(p \cdot Fx)'$ is equivalent to $'p \cdot (x)Fx'$ in all non-empty universes.

(xii) $'(\exists x)(p \vee Fx)'$ is equivalent to $'p \vee (\exists x)Fx'$ in all non-empty universes.

Let us first examine (xi), in non-empty universes. Case 1: O is assigned to 'p'. Then 'p \cdot Fx' comes out false for each object x. Then, the universe not being empty, there are objects x for which 'p \cdot Fx' comes out false; so '(x)(p \cdot Fx)' fails. But so does 'p \cdot (x)Fx'. Case 2: l is assigned to 'p'. Reasoning similar to that of Case 2 of (vii) serves here.

Law (xii) can be justified by similar reasoning, or by deduction from (iii), (iv), and (xi).

It is curious to note that the equivalences in (xi) and (xii) fail for the empty
universe. For that universe, every universal quantification is valid and every exis-
tential one inconsistent; for, '(x)Fx' amounts merely to '$F$', which is valid in the
empty universe, and '(∃x)Fx' amounts merely to '$\neg F$', which is inconsistent in the
empty universe. In the empty universe, then, '(x)(p \cdot Fx)' comes out true regardless
of which truth value is assigned to 'p', whereas 'p \cdot (x)Fx' comes out false when O
is assigned to 'p'. Again '(∃x)(p \vee Fx)' comes out false regardless of which truth
value is assigned to 'p', whereas 'p \vee (∃x)Fx' comes out true when l is assigned to
'b'.

Because of the exceptional behavior of the empty universe in connection with (xi)-(xii) and certain other useful principles, it is convenient in quantification theory to gear our techniques to proving validity, equivalence, etc. for all non-empty universes rather than for all universes. Hereafter the word valid, without qualifying adverb, is to be taken to mean 'valid in all non-empty universes'; and similarly for 'inconsistent', 'equivalent', and 'implies'.

If we ever care to determine whether a schema which has been found valid in the above sense is valid also in the empty universe, we can always do so by an easy separate operation: put '1' indiscriminately for all universal quantifications and '0' for all existential ones, and reduce by the methods of Part I. If we end up with '1' or a valid truth-functional schema, the original was valid in the empty universe; otherwise not. Note incidentally that any open schema is valid in the empty universe; for, it is valid if its closure is valid, and its closure is a universal quantification.

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(53)

it is valid if its closure is valid, and its closure is a universal quantification. It is also true that any open formula is valid in the empty universe; hence not. Moreover, it is also true that any open formula is valid in the empty universe: a valid truth-functional formula, the original one, is valid in the empty universe; it is still existentially one, and reduces to the method of but I. It we end up with it, the operation: but, if, indeterminately, for all universal quantifications and 0, it has sense if valid in the empty universe, we can easily do so if we allow some conditions for the definition of the empty universe in connection with it.

It we allow to determine whether a scheme which we mean long valid in the similar way for 'invariant', 'adjevant', and 'ambitess'.

The following sharp, is to be taken to mean, valid in all non-empty universes; the empty universe rather than for all universes. Hence the word valid, without any restriction to our technique to proving validity, adjevance, etc. for all non-

universes. In the empty universes, then, for (x)(b' > x'), comes out true regardless of the empty universe, and (Ex)(x' > x'), means respectively to: "x", which is inconsistent in the first one, inconsistent; for, (x)(x' > x'), means respectively to: "x", which is valid in the empty universe. For that universe, every universal quantification is valid and every ex-

It is convenient to note that the adjevance in (xi) and (xii) follows the empty universe is satisfied to 'b', whereas 'b' (Ex)(x' > x'), comes out true when it is satisfied to 'a' in the same way as 'b', whereas 'b' (Ex)(x' > x'), comes out false regardless of which truth value 'a' or 'b' is assigned to. Again, (Ex)(b > x'), comes out false regardless of which truth value 'a' or 'b' is assigned to. This is satisfied to 'b', whereas 'b' (x)(b > x'), comes out false. In the empty universe, then, for (x)(b' > x'), comes out true regardless of the empty universe, and (Ex)(x' > x'), means respectively to: "x", which is inconsistent in the first one, inconsistent; for, (x)(x' > x'), means respectively to: "x", which is valid in the empty universe. For that universe, every universal quantification is valid and every ex-

It follows (xi) can be justified by reasoning, or by deduction from (iii).

Case S of (aii) serves here.

is so does b, (x)(x' > x'), Case S: it is satisfied to 'b'. Reasoning similar to that of the objects x for which 'b' > x', comes out false: so, (x)(b' > x'), yields.

Now 'b' > x', comes out false for each object x. Then, the universe not being empty, it has itself examine (xi'), in non-empty universes. Case I: O is assigned to 'b'.

(xii) : (Ex)(b > x') is adjevant to 'b' (Ex)(x' > x'), in all non-empty universes.

(xi) : (x)(b > x') is adjevant to 'b' : (x)(x' > x'), in all non-empty universes.

33.

90.

§30. SUBSTITUTION

By substitution in a valid schema we get new schemata which, being merely
special cases of the old, are valid likewise. By substitution in equivalent schemata,
similarly, we get schemata which are equivalent to each other; and correspondingly
for implication.

E.g., substituting '(Ey)Fy' for 'p' in the schemata which appeared in the
equivalence (x) of §29, we may conclude that '(x)(Fx ⊃ (Ey)Fy)' is equivalent to
'(3x)Fx ⊃ (Ey)Fy'.

Note further that '(3x)Fx ⊃ (Ey)Fy' is obviously valid, since there is none
but a notational difference between use of 'x' and of 'y' as the variable of quanti-
fication. From the equivalence just previously noted, then, we may infer that the
schema:

$$
(1) \qquad (x)(Fx \supset (Ey)Fy)
$$

is valid. Hence also the open schema:

$$
(2) \qquad Fx \supset (Ey)Fy
$$

is valid, by the definition of validity of open schemata (§29); for, the closure of
(2) is (1).

By substitution in (1) or (2) we may obtain further valid schemata, e. g.:

$$
(3) \qquad (Ez)Gxz \supset (Ey)(Ez)Gyz.
$$

The process of substitution involved calls, however, for careful formulation.
Limiting our consideration for the present to substitution upon monadic letters, i.e.,
term letters followed by single variables, we may visualize the process as follows:
what is substituted for a monadic term letter is an incomplete schema, with some holes
left for variables; and those holes are then filled with the particular variable which
was attached to the term letter. (3) may be thought of as obtained from (2) by put-
ting '(Ez)G_z' for the 'F' of 'Fx', and then moving the 'x' into the hole; and put-
ting the same '(Ez)G_z' also for the 'F' of 'Fy', and then moving the 'y' into the
hole. It is essential, of course, that the same incomplete schema supplant all occur-
ences of a term letter; the holes, however, will be variously filled depending on
what variables are attached to the various occurrences of the supplanted term letter.

The incomplete schema put for a term letter may, unlike the above example, have
several holes; e. g., '(Ez)G_z'. Substitution of this for 'F' in (2) yields:

$$
(Ez)Gxzx \supset (Ey)(Ez)Gyzy.
$$

The two examples thus far considered have this in common: the schemata which
supplant 'Fx' and 'Fy' are alike except that the one exhibits 'x' wherever the other
exhibits 'y'. This, however, is not necessary; we might substitute 'G_x' for 'F' in
(2), obtaining:

$$
(4) \qquad Gxx \supset (Ey)Gyx.
$$

It was in order to allow for such cases that care was taken to formulate the concept
of substitution in terms of schemata with holes.

Actually the formulation is at present too broad, however, and needs a

(30

(30)

Vitality of the formulation is at present too broad, however, and needs a
substitution in terms of symbols with holes.

The same in order to allow for some cases that one may wish to formulate the concept of a symbol in terms of symbols with holes.

Oxx D (Ex)Oxx.

(1)

Operating:

(S) symbols 'A', 'B', 'C', 'D' etc. however, is not necessarily; we might substitute 'O_x' for 'A' in
'mainland' 'A_x' and 'B_x' etc. since except that one symbol 'x' wherever the other
the two examples same in common: the symbols which

(Es)Oxx D (Ex)(Es)Oxx:

(2)

General rules: a. E. e. (Es)O_x's. Superscript of this for 'A' in (S) yields:

The incomplete scheme for a term letter may substitute the above example, save

the variables are attached to the various consequences of the exponentiated term letter.
Since of a term letter: the hole, powered, will be automatically filled by substituting on
the same: (Es)O_x's, also for the 'B' of 'A_x' and then leaving the 'A' into the
letter (Es)O_s, for the 'B' of 'A_x' and then leaving the 'A' into the hole; and but-
noted to the term letter. (C) may be found of the form obtained from (S) by put-
ing in variables; and those holes are then filled with the barbiture variables which
is substituted for a monadic term letter is an incomplete scheme with some holes
from letters followed by single variables; we may utilise the process as follows:
limiting our consideration for the present to substitution above monadic letters, i.e.,
the process of substitution involves calls, powers, for complete termination.

(3)

(Es)Oxx D (Ex)(Es)Oxx:

Ex substitution in (1) on (S) we may obtain further valid schemes, e. g.:

(S) valid for the definition of validity of open schemes (SS); for the closure of

the valid rule for the definition of validity of open schemes (SS); for the closure of

x D (Ex)x

(S)

is valid. Hence also the open scheme:

(1)

(x)(Ex)D (Ex)x

schemata:
From the equivalence that previous noted, then, we may infer that the
is a notational difference between use of 'x' and 'A' as the variable of distinct
Note further that (Ex)x = (Ex)x, is optionally valid, since there is none

(Ex)x > (Ex)x.

Equivalence (x) of SSS, we may conclude that: (x)(x' D (Ex)x), is equivalent to
E. E., suppositing: (Ex)x, for 'b' in the schemata which occurred in the

for implication.

Inference, we get schemes which are equivalent to each other; and correspondingly

special cases of the cf., are valid likewise. By supposition in equivalent schemes,
Equ supposition in a valid scheme we get new schemes which, put merely

EFO. SUBSTITUTION

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91.

restriction which may be put figuratively thus: *Free variables must remain free, and bound ones must not shift their allegiance.* If the incomplete schema thought of as substituted for a term letter contains a free occurrence of a variable, that occurrence must remain free; we are forbidden, thus, to substitute 'Gy_' for 'F' in (2), since the result would be:

$$ (5) \qquad Gyx \supset (Ey)Gyy, $$

in which the free 'y' of 'Gy_' becomes bound. Again, the incomplete schema substituted for a term letter must not contain a quantifier capable of capturing a variable which was attached to that term letter; we are forbidden to substitute '(x)G_x' for 'F' in (2), since the result would be:

$$ (6) \qquad (x)Gxx \supset (Ey)(x)Gyx, $$

in which the free 'x' of (2) becomes bound. Also we are forbidden to substitute '(x)G_x' for 'F' in:

$$ (7) \qquad (x)Hx . Fy . \supset (\bar{E}) (Hx . Fx), $$

since the result would be:

$$ (8) \qquad (x)Hx . (x)Gyx . \supset (\bar{E}) (Hx . (x)Gxx), $$

wherein the 'x' of 'Fx' has shifted its allegiance; this 'x' was bound to the quantifier '(Ex)' in (7), but in (8) it has been captured by a new quantifier '(x)' intruded by the substitution.

The need of the restriction can be recognized by contrasting the meanings of (5) and (6) with that of (2). (2) says: 'If x is an F, something is an F.' Substitution of 'Gy_' for 'F' should, if it really yielded a special case of (2), yield this: 'If y is G to x then y is G to something.' But actually the result (5) of the substitution says something quite different, and groundless: 'If y is G to x then something is G to itself.' Again, substitution of '(x)G_x' for 'F' should, if it really yielded a special case of (2), yield this platitude: 'If x is G to everything then there is some one thing which is G to everything.' But actually the result (6) of the substitution says something quite different, and groundless: 'If everything is G to itself then there is some one thing which is G to everything.' Quantifiers have, in (5) and (6), become tangled in such a way that the substitution does not yield a genuine case of the original. The restriction serves to prohibit such spurious substitutions.

It is instructive to compare the legitimate (4) with the spurious (5). (4) being valid, 'Gxx' implies '(Ey)Gyx'; but (5) is not valid, and 'Gyx' does not imply '(Ey)Gyy'. From 'Gxx' -- say 'Herbert admires himself' -- it follows that somebody (viz. Herbert, at least) admires Herbert; but from 'y is G to x' -- say 'Amos is uncle of Herbert' -- it by no means follows that anyone is uncle to himself.

In much the same way in which the validity of (2) was arrived at from (x) of §29, we may argue from (ix) of §29 to the validity of:

$$ (9) \qquad (y)Fy \supset Fx. $$

Now it is interesting to see how our general restriction upon substitution operates when we shift our example from (2) to (9). By legitimate substitution we can proceed from (9) to:

$$ (y)Gyx \supset Gxx $$

(30)

["(20)", "(a) (a)C Gxx", "each from (d) to:", "a. We shift an example from (S) to (D). By legitimate apportionment we can produce it if it is interesting to see how our General restriction on apportionment operates", "(a) Aa C Ax.", "(d)", "b. We wish some from (ix) of BSO to the affinity of (S). In much the same way in which the affinity of (S) was arrived at from (x) of", "a. The affinity of Herbert, -- if he does not follow that someone is more to himself.", "c. (a) (a)Aa. From 'a, Gxx, -- say Herbert agrees himself; -- it follows that someone (a) (a)Aa, affix 'a, Gxx, implies 'a, (a)Aa, put (a) is not affix, and 'a, Gxx, does not imply that Herbert agrees himself; -- it follows that someone (a) (a)Aa, affix 'a, Gxx, implies 'a, Gxx, does not imply that Herbert agrees himself; -- it follows that someone is more to himself.", "d. It is inoperative to compare the legitimate (a) with the sample (a). (a)", "e. The sample is a genuine case of the original. The restriction serves to prohibit such a case in (c) and (d).", "f. The apportionment allows a something different and homogeneous: it everything then there is some one thing which is G to everything. But actually the result (c) really yields a special case of (S). Yield true apportionment: 'if x is G to everything something is G to itself. 'Vainly apportionment of 'a(x)Gxx, for 'a, should, if it is homogeneous allows something different and homogeneous: 'if x is G to x then 'a is G to something. But actually the result (c) of the apportionment of 'a, for 'a, is G, something is 'a, and something is 'a, 'a is G to x then 'a is G to something. But actually the result (c) of the apportionment of 'a, for 'a, is G, something is 'a, and something is 'a, 'a

92.

but not to:

(11)

$(y)Gyy \supset Gyx.$

'(y)Gyx' implies 'Gxx', but '(y)Gyy' does not imply 'Gyx'. From 'Everybody admires Herbert' it follows that Herbert admires himself, but from 'Everything is identical with itself' it by no means follows that everything is identical with Herbert. The situation in (10) and (11) is somewhat the reverse of the situation in (4) and (5). We can go from '(y)Gyx' to 'Gxx', creating a duplication of variables, and we can go from 'Gxx' to '(Ey)Gyx', destroying a duplication; but we cannot go from '(y)Gyy' to 'Gyx', destroying a duplication, nor from 'Gyx' to '(Ey)Gyy', creating a duplication.

Substitution for polyadic term letters admits of an analogous but more complicated description. What was thought of as put for a monadic letter was a schema with one or more undifferentiated holes in it, to be filled by the variable following the monadic letter in each case. What may be thought of as put for a polyadic letter is a schema with variously numbered holes; the holes numbered '1', then, are to be filled by the first of the variables following the polyadic letter, the holes numbered '2' are to be filled by the second of the variables, and so on. However, we shall have no occasion in these pages to make substitutions upon polyadic term letters; our techniques will take other lines.

Substitution upon statement letters also has to be considered. Such a substitution, viz. '(Ey)Fy' for 'p', occurred at the beginning of the section. In general we may substitute any schema for a statement letter so long as, in conformity with the restriction previously stated, all free variables of the substituted schema remain free in the result. This means that if 'p' lies within a quantification governed by '(x)' or '(Ex)', the schema substituted for 'p' must contain no free occurrences of 'x'. Thus the equivalences (vii)-(xii) of §29 can be used to confine a quantifier to one side of a compound only in case the variable in question has no free occurrences on the other side.

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§31. REDUCTION TO NULLITIES

93.

In view of (vi) and (xii) of §29, we need never acquiesce in an existential quantification of an alternation; for, '(Ex)(Fx v Gx)' reduces to '(Ex)Fx', and '(Ex)(Fx v p)' to '(Ex)Fx v p'. By substitution corresponding equivalences hold also when in place of 'Fx' and 'Gx' we have any complex schemata in which 'x' is a free variable, and in place of 'p' any schema in which 'x' is not free. By applying the same considerations in iteration, we can similarly distribute '(Ex)' through an alternation of three or more components - applying '(Ex)' individually to those components in which 'x' is free, and leaving other components unquantified. We can transform:

$$
(1) \qquad (\mathbf{Ex})(\mathbf{Fx} \lor \mathbf{Gx} \lor \mathbf{p} \lor \mathbf{Hx}),
$$

e.g., into:

$$
(2) \quad (\mathbf{Ex})\mathbf{Fx} \lor (\mathbf{Ex})\mathbf{Gx} \lor \mathbf{p} \lor (\mathbf{Ex})\mathbf{Hx},
$$

the successive stages being:

$(\mathbf{Ex})(\mathbf{Fx} \lor \mathbf{Gx} \lor \mathbf{p}) \lor (\mathbf{Ex})\mathbf{Hx} \quad \text{(by(vi) of §29),}$

$(\mathbf{Ex})(\mathbf{Fx} \lor \mathbf{Gx}) \lor \mathbf{p} \lor (\mathbf{Ex})\mathbf{Hx} \quad \text{(by(xii))},$

and thence (2) by (vi).

Analogously (v) and (xi) of §29 enable us to distribute '(x)' through a con-
junction of any number of components - applying '(x)' individually to those components
in which 'x' is free, and leaving other components unquantified. We can transform:

$$
(3) \qquad (x)(Fx \cdot Gx \cdot p \cdot Hx),
$$

e.g., into:

$$
(4) \quad (x)Fx . (x)Gx . p . (x)Hx.
$$

A quantification will be called _pure_ if the variable of the quantifier re-
appears in every component of the expression following the quantifier. Thus the
quantifications:

$(x)Fx, \quad (\mathbf{Ex})Fx, \quad (x)\mathbf{-Fx}, \quad (x)(\mathbf{Fx} \supset \mathbf{Gx} \ \mathbf{-Hx})$

are pure. On the other hand the quantifications (1), (3), and:

$(x)(Fx \supset p), \quad (\mathbf{Ex})(Fx \lor Gy), \quad (x)((\mathbf{Ey})(Fx \cdot Fy) \supset Gx)$

are not pure, since 'x' does not appear in 'p', 'Gy', or 'Fy'.

Consider, now, any impure existential quantification containing no shorter quantification within it. The variable of the quantifier is, let us suppose, 'x'. Thus our quantification consists of '(Ex)' applied to a truth function of such expressions as 'Fx', 'Gx', etc., together with a sprinkling of impurities such as 'p', 'Fy', etc. Let us put this truth function into alternational normal form, and then distribute the quantifier as illustrated in (1)-(2). Thus, if our original quantification is:

$$
(5) \quad (\mathbf{Ex})(\mathbf{Fx} \supset \mathbf{Gy} : \mathbf{Gx} \supset \mathbf{Hx} \lor \mathbf{Fy}),
$$

(31)

(21)

(22)

$$ (Ex)(Lx \ C \ GA : Ox \ C \ Hx \ a \ GA)' $$

determine the denominator as illustrated in (f)-(g).

Since, it can be original denominator, etc. Let us find this truth function into international term, and then equations as 'Lx', 'Hx', etc., together with a simplifying of denominator and so on. Determine the denominator within it. The ansды of the denominator is 'let us choose' 'Lx'. Consider now, any immune existentialization containing no shorter

are not true, since 'x' does not appear in 'b', 'GA', or 'HA'.

$$ (x)(LxCh), (Ex)(LxGA), (x)((EG)(Lx \ A) \ Lx) \ C \ Gx) $$

are true. On the other hand the denominator (f)' (g)' and:

$$ (x)(Lx'), (Ex)(Lx'), (x)=Lx', (x)(Hx \ C \ Gx \ -Hx) $$

determinations:

substitutes in every denominator of the expression following the denominator. Time the V disnification will be carried out if the variable of the denominator is:

$$ (x)Lx \cdot (x)C \cdot b \cdot (x)Hx $$

e.g., into:

$$ (x)(Lx \cdot Gx \cdot b \cdot Hx)' $$

in which 'x' is true, and leaving other components undetermined. We can transform:

the function of the denominator - supposing, (x), individually to those components which are distributed (a), and (x), through a continuous and in the place of 'b', any function in which 'x' is not true. By supposing the

and since (S) pA (A1).

$$ (Ex)(Lx \ a \ Gx) \ a \ b \ a \ (Ex)Hx \quad (pA(x1))' $$

$$ (Ex)(Lx \ a \ Gx \ a \ b) \ a \ (Ex)Hx \quad (pA(a1) \ ol \ #SD) $$

the successive steps being:

$$ (S) \quad (Ex)(Lx \ a \ (Ex)(Gx \ a \ b \ a \ (Ex)Hx) $$

e.g., into:

$$ (f) \quad (Ex)(Lx \ a \ Gx \ a \ b \ a \ Hx)' $$

in which 'x' is true, and leaving other components undetermined. We can transform: the following equations for those components a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, in which 'x' is the same as supposing (Ex), individually to those components a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, in which 'x' is not true. By supposing the given in place of 'b', any function in which 'x' is not true. By supposing the given in place of 'Lx', and 'Gx', we have an complex equation in which 'x' is the same as supposing (Ex)(Lx \ a \ b), to (Ex)(Lx \ a \ b). A supposition corresponding to this case holds also for supposition of an operation: let, (Ex)(Lx \ a \ Gx), reduce to (Ex)(Lx), and supposition of an operation: let, (Ex)(Lx \ a \ Gx \ a \ b), reduce to (Ex)(Lx \ a \ Gx), and so on.

REDUCTION TO MUTUATION

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95.

By the foregoing techniques, any monadic schema - i.e., any quantificational schema in which the term letters have only single variables attached - can be transformed into an equivalent in which all quantifications are pure. For, given any monadic schema, we may begin by transforming all its innermost quantifications -- i.e., all those which contain no shorter quantifications within them -- by the techniques set forth above. This done, our schema as a whole has given way to a schema in which all innermost quantifications are pure. But we have only monadic materials here; and obviously a pure quantification which is monadic can have no free variable, since each term letter in it has to be followed by the variable of the quantification. Each of these pure innermost quantifications thus constitutes, in relation to any longer quantification which may contain it, an impurity on a par with a 'p' or a 'q'. So we can proceed now to transform those next longer quantifications, by the same techniques which were applied to the innermost quantifications; and we continue thus until no impure quantifications remain.

E.g., consider the monadic schema:

$$ (11) \qquad (\mathbf{Ex})((y)(\mathbf{Fy} \dashv \mathbf{Gx}) \cdot \mathbf{Fx}). $$

Here there is one innermost quantification, $\mathbf{(y)(Fy} \dashv \mathbf{Gx)}$. When we transform the latter according to the procedure previously established, the whole becomes successively:

$$ (\mathbf{Ex})((y)(-\mathbf{Fy} \lor \mathbf{Gx}) \cdot \mathbf{Fx}), $$

$$ (\mathbf{Ex})((y)(-\mathbf{Fy} \lor \mathbf{Gx}) \cdot \mathbf{Fx}). $$

Now we can transform the whole in turn like an innermost quantification, since the $\mathbf{(y)-Fy}$ in it, containing no 'x', is an impurity to be handled simply as if it were say 'Hz' or 'p'. We obtain:

$$ (\mathbf{Ex})((y)(-\mathbf{Fy} \cdot \mathbf{Fx} \cdot \mathbf{v} \mathbf{Gx} \cdot \mathbf{Fx}), $$

$$ (\mathbf{Ex})((y)(-\mathbf{Fy} \cdot \mathbf{Fx}) \lor (\mathbf{Ex})(\mathbf{Gx} \cdot \mathbf{Fx}), $$

$$ (12) \qquad (y)(-\mathbf{Fy} \cdot (\mathbf{Ex})\mathbf{Fx} \cdot \mathbf{v} (\mathbf{Ex})(\mathbf{Gx} \cdot \mathbf{Fx}). $$

In this final result all quantifications are pure.

A schema which is not monadic cannot necessarily be thus transformed into an equivalent in which all quantifications are pure. Consider, e.g., $\mathbf{(x)(Fy} \cdot \mathbf{(Fy} \cdot \mathbf{Gxy)})$. The whole is not pure, because of the 'Fy' which occurs inside; yet our techniques avail us nothing here. The inner quantification $\mathbf{(Fy)(Fy} \cdot \mathbf{Gxy)}$ is already pure, and cannot be changed in any relevant way; nor yet, containing 'x' as it does, can this inner quantification be treated as an impurity and removed from the outer quantification.

The important thing about a pure quantification is that it can be translated into a mulity or the negation of a mulity, provided that it is monadic. Thus consider, to being with, any monadic pure universal quantification -- say $\mathbf{(x)(Fx} \dashv \mathbf{Gx} \cdot \mathbf{-Hx)}$. Get rid of any conditional or biconditional signs, by translating them into alternation, conjunction, and negation; thus we obtain $\mathbf{(x)(-Fx} \lor \mathbf{Gx} \cdot \mathbf{-Gx)}$. But this amounts, according to (1)-(3) of §17,

$$ \text{to } \mathbf{(x)} \text{ (FvGH)x, and hence to } \mathbf{v} \text{ (FvGH).} $$

In general, thus, it is evident that any monadic pure universal quantification can be translated into a mulity by the following quick steps: first, get rid of any conditional and

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96.

and biconditional signs; then simply drop the quantifier, delete the variable everywhere, and prefix '>-' to the resulting term schema.

Consider next any pure existential quantification -- say '(Ex)(Fx. -Gx.v Hx)'. This amounts, according to the distribution principle, to '(Ex) (FGvH)x', and hence to 'v(FGvH)'. The quick rule of thumb for translating any monadic pure existential quantification can be translated into the negation of a nullity is thus evident: get rid of any conditional and biconditional signs, and then simply drop the quantifier delete the variable everywhere, and prefix '->' to the resulting term schema.

So we have seen now that any monadic schema can be transformed into an equivalent in which all quantifications are pure, and that each such quantification, in turn, can be replaced by a nullity or the negation of a nullity. Starting e.g. with (11), we first transform it into (12) as above, and then we turn each quantification in (12) into a nullity or the negation of a nullity, thus ending up with a truth function of nullities:

$$ \rightarrow^{)}F . \rightarrow^{)}F . v \rightarrow^{)}GF. $$

(In view of the inconsistency of $\rightarrow^{)}F . \rightarrow^{)}F'$, indeed, we could reduce the whole to $\rightarrow^{)}GF'$ alone.)

We cannot always depend upon ending up thus with a truth function of nullities. For, if the monadic schema with which we start has some statement letters 'p', 'q', etc. in it, these may be expected to survive in the result. Similarly if it has parts such as 'Fx', 'Gy', etc. with no quantifications governing these variables, these expressions may be expected to survive in the result. However, a closed monadic schema -- one without free variables -- will not have any parts of the latter kind. We are in a position, therefore, to translate any monadic schema or its closure into a truth function of nullities, or of nullities and statement letters.

This provides a mechanical method of testing any monadic schema to see whether it is valid. First we transform the monadic schema, or its closure if it was open, into a truth function of nullities and perhaps statement letters, as above, and then we test the latter as in §§25-26 to see whether it is valid (at least in all non-empty universes).

We can then test monadic schemata likewise for consistency, equivalence, and implication. For, we can test for consistency by testing the negation for validity; we can test for equivalence by testing the biconditional for validity; and we can test for implication by testing the conditional for validity.

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§32. PROVING POLYADIC SCHEMATA

97.

We have a mechanical test now for the validity of monadic schemata. As for polyadic schemata -- those quantificational schemata in which some term letters have several variables attached -- there are many whose validity can be shown by substitution in valid monadic schemata. (3) and (4) of §30 are examples. From valid polyadic schemata thus reached, further ones can be reached in turn by applying universal quantifiers; for, the result of universally quantifying a valid open schema with respect to any if its free variables is valid. This follows from the fact that an open schema is valid if and only if its closures are valid (cf. § 29).

E.g., substitution in the valid monadic schema:

$$ (1) \qquad Gz \Dernote{\to} (Ex)Gx $$

yields the valid polyadic schema:

$$ (2) \qquad Fzy \Dernote{\to} (Ex)Fxy, $$

and universal quantification of this yields in turn the further valid polyadic schema:

$$ (3) \qquad (y)(Fzy \Dernote{\to} (Ex)Fxy). $$

Suppose, next, that we make a substitution in the valid monadic schema:

$$ (4) \qquad (y)(Gy \Dernote{\to} Hy) \Dernote{\to} (y)Gy \Dernote{\to} (y)Hy, $$

deriving the valid polyadic schema:

$$ (5) \qquad (y)(Fzy \Dernote{\to} (Ex)Fxy) \Dernote{\to} (y)Fzy \Dernote{\to} (y)(\exists x)Fxy. $$

Note that the valid schema (5) is a conditional with the valid schema (3) as antecedent. For any interpretation of 'F' and any choice of z, since (3) and (5) are true, the consequent:

$$ (6) \qquad (y)Fzy \Dernote{\to} (y)(\exists x)Fxy $$

must come out true; thus (6) also is valid. Here, then, we have found a further way of generating valid schemata from valid schemata: the consequent of a valid schema whose antecedent is valid is valid. This form of derivation, viz. derivation of the consequent of a conditional from the conditional together with its antecedent, is known as **modus ponens**.

Now we have a small but powerful battery of devices for obtaining valid polyadic schemata: **substitution** in valid monadic schemata, **universal quantification**, and **modus ponens**. Combined use of these devices has led to (6); by way of further illustration, let us go on from there. Universal quantification of (6) yields:

$$ (7) \qquad (z)((y)Fzy \Dernote{\to} (y)(\exists x)Fxy). $$

Substitution in the valid monadic schema:

$$ (8) \qquad (z)(Gz \Dernote{\to} p) \Dernote{\to} (Ex)Gx \Dernote{\to} p $$

Yields:

$$ (9) \qquad (z)((y)Fzy \Dernote{\to} (y)(\exists x)Fxy) \Dernote{\to} (\exists x)(y)Fxy \Dernote{\to} (y)(\exists x)Fxy. $$

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["(35)", "(a) ((λ)(λ)(λ)(λ)(λ)(λ)(λ)(λ)(λ)(λ)(

98.

Modus ponens leads from (9) and (7) to:

$$ (10) \qquad (\mathbf{Ex})(y)Fxy \dashv (y)(\mathbf{Ex})Fxy. $$

The validity of (10) shows that, although $\mathbf{(Ex)(y)Fxy}$ and $\mathbf{(y)(Ex)Fxy}$ are not equivalent (cf. §29), the former does imply the latter.

This method of proving valid polyadic schemata is complete, in the sense that every valid polyadic schema can be obtained from valid monadic schemata by substitution, quantification, and modus ponens. The proof of this completeness, however, is too elaborate to be included here.¹

Despite this completeness, it should be noted that our situation with respect to valid polyadic schemata differs in a very important way from our situation with respect to valid monadic or truth-functional schemata. We have a straightforward mechanical test whereby, given any monadic schema, we can decide whether or not it is valid; and similarly for any truth-functional schema. Such a method is called a decision procedure. But we have no general decision procedure for polyadic schemata. Given a polyadic schema at random, there is no unfailing mechanical routine to which we can subject it to find out whether it is valid. The best we can do is try to find a proof, in above style; but failure to find such a proof gives no assurance of non-validity, since a proof may exist to be found without our having the luck or ingenuity to hit upon it. Once found, a proof can indeed be checked quite mechanically and seen to be in fact a proof of the schema in question; but there is no unfailing method of finding a proof, if proof there be, and otherwise finding that there is none. For polyadic schemata we have, in short, a complete proof technique but no decision procedure; and it has been established, by Church,² that a decision procedure for this realm is forever and in principle impossible. That this is impossible is a sad and remarkable fact; but that such an impossibility should actually have been proved is a happy and yet more remarkable fact. Unfortunately we cannot embark upon the proof here.

What has been said of validity applies equally to inconsistency, equivalence, and implication: no general mechanical method is possible whereby to test polyadic schemata in these respects. We have, however, a complete method for proving inconsistency, equivalence, and implication of polyadic schemata; such a method is at hand in the foregoing method of proving validity. For, to prove a schema inconsistent we may prove its negation valid; to prove that one schema implies another we may prove the corresponding conditional valid; and to prove equivalence we may prove two reciprocal implications. The foregoing proof of validity of (10), e.g., was ipso facto a proof that $\mathbf{(Ex)(y)Fxy}$ implied $\mathbf{(y)(Ex)Fxy}$.

Gödel has shown (in "Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37, 1930, pp. 349-360), with much ingenuity, that a certain method of proof different from the present one is complete in the above sense; and I have shown (in "On the logic of quantification," Journal of Symbolic Logic, vol. 10, 1945, pp. 1-12) that whatever schemata can be proved by the method to which Gödel refers can also be proved by the present method.

²"A note on the Entscheidungs problem," Journal of Symbolic Logic, vol. 1, 1936, pp. 40-41, 101-102.

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100.

The rules governing the construction of proofs will now be stated. The justification of the rules, however, will be deferred to §35; it will there be established that the interpretation italicized above can be depended upon so long as proofs are constructed by the rules.

R1. Any schema or group of schemata may be set down as a line or group of lines at any stage in the course of a proof, provided that a new scope mark is initiated at that point. In the above example this rule is illustrated by the adoption of the premises (1) and (2), and again by the adoption -- within the scope of (1) and (2), as it happens -- of the further premiss (5).

R2. If the smallest scope in which a line L lies is that of a group of one or more premisses, P<sub>1</sub>, ..., P<sub>n</sub>, we may add as a subsequent line of proof, outside that scope but still within all other scopes in which L lies, the conditional whose consequent is L and whose antecedent is the conjunction of P<sub>1</sub>, ..., P<sub>n</sub> (or simply P<sub>1</sub>, if n = 1). In the above proof this rule is illustrated by (10), which has (9) as its consequent and, as antecedent, the premiss (5) whose scope ended with (9). It is illustrated also by (12), which has (11) as its consequent and, as antecedent, the conjunction of the premisses (1) and (2) whose scope ended with (11).

In stating the rest of the rules, this special terminology will be convenient: Inferring a line L' from a line L consists in writing L' as a line of the proof, later than L and within all scopes (at least) in which L lies, and appending to L' a numerical referring back to L. Thus, in the above example, (3) is said to be inferred from (1), and (4) from (2), and (6) from (3), and (7) from (4), and (9) from (8), and (11) from (10). Inferring a line L' from several lines L<sub>1</sub>, L<sub>2</sub>, ..., L<sub>n</sub>, likewise, consists in writing L' as a line of the proof, later than all of L<sub>1</sub>, L<sub>2</sub>, ..., and L<sub>n</sub>, and within all scopes (at least) in which any of L<sub>1</sub>, L<sub>2</sub>, ..., and L<sub>n</sub> lies, and appending numerals to L' referring back to L<sub>1</sub>, L<sub>2</sub>, ..., and L<sub>n</sub>. Thus (8) is said to be inferred from (5)-(7). This definition of "inference" is a broad one, covering both good and bad inference; it is the business of the ensuing rules to say what is good.

R3. From any given line or lines we may infer any line which is truth-functionally implied by the given line or by the conjunction of the given lines. It is this rule that led to (8) from (5)-(7); for, as is readily seen by test if not at a glance, (8) is truth-functionally implied by the conjunction of (5)-(7).

The remaining rules may conveniently be stated together, as follows:

If S and S' are schemata such that S' is like S except for containing free occurrences of some variable Z' at all places at least where S contains free occurrences of some variable Z, then:

R4. From the universal quantification of S with respect to Z we may infer S'.

R5. From S' we may infer the existential quantification of S with respect to Z.

R6. From the existential quantification of S with respect to Z we may infer S', provided that Z' is free in no line prior to this inferred S'.

R7. From S' we may infer the universal quantification of S with respect to Z, provided that Z' is not free in the line thus inferred, nor in any line previously established by R6, nor in any premiss whose scope continues.

It is R4 that leads from (2) to (4), in the foregoing example; for, (4) is like the '(y)(Gxy Hyy)' of (2) except for containing free occurrences of 'z' at all places

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101.

where $(y)(Gxy \supset Hyy)$ contains free occurrences of $x'$. The same rule leads from (3) to (6), and from (4) to (7). Note, however, that there is no need in general for $Z'$ to be foreign to S; thus the rule could lead also from $(x)Fxy$ to 'Fyyy', since, despite the '$y$' in 'Fxyx', it is still true that 'Fyyy' is like 'Fxyx' except for containing free '$y$' in place of every free $x'$. Note also that $Z'$ could, in particular, be the same as $Z$ (whereupon $S'$ would be the same as $S$). Thus the rule could lead from (3) not only to (6) but also to 'Fy \ Gzy'; all that happens in this case is the dropping of a universal quantifier.

It is R5 that leads from (8) to (9); for, (8) is like the 'Hwy' of (9) except for continuing a free occurrence of $w'$ at the one place where 'Hwy' contains a free '$y$'.

It is R6 that leads from (1) to (3). The '$z$' of (3) is a variable not hitherto appearing; and (3) is formed from (1) by dropping '(Ex)' and putting '$z$' for the free $x'$ in the ensuing $(y)(Fy \supset Gxy)$.

It is R7 that leads from (10) to (11). The legitimacy of this step, under the terms of R7, depended on the fact that $w'$ was not free in (11), nor in the line (3) which was established by R6, nor in either of the premisses (1)-(2). The fact that $w'$ is free in the premiss (5) was no hindrance, because the scope of (5) had meanwhile expired.

These are all the rules that govern the progressive writing down of lines to form a proof. However, in order that such an accumulation of lines constitute a whole *proof* and not a mere fragment, *the last line must meet these two additional requirements:* it must lie in the scope of none but initial premisses, and must contain no free occurrence of any variable which was first introduced by R6. Thus the top third of the above example, stopping with (4), would not by itself count as a proof of anything; for, the '$z$' which was introduced into (3) by R6 is still free in (4). Again the top three quarters, stopping with (9), would not count as a proof, because (9) lies in two scopes. But we could drop (12), or (12) and (11), and still have a proof.

It is well, after any line which enters by R6, to write 'R6' as a reminder; for we have to review all such lines in judging the legitimacy of any subsequent use of R7, and also in judging whether the last line of an alleged proof meets the requirements of last lines. Thus, at the end of line (3) in the above example the annotation '(1)' should be elaborated to read '(1) R6'.

### §34. REFLECTIONS ON THE RULES

The rules R1-7 show an outward symmetry: R1 introduces premisses; R2 gets us out from under premisses. R4 gets rid of a universal quantifier; R7 restores it. R6 gets rid of an existential quantifier; R5 restores it. R3, finally, provides a core of truth-functional reasoning, for use while obstructive quantifiers are out of the way.

Rules of proof similar to these in essential respects, but less simple in detail, were formulated both by Gentzen and by Jaśkowski, independently of each other, in 1934. The name "natural deduction" (naturliches Schliessen) is Gentzen's. The reason for the name is that this method of proof corresponds rather well to habitual, unformalized patterns of reasoning.

This may be seen by running through the illustrative proof in §33 again in using the vernacular and ignoring rules. What is to be proved, then, viz. (12), is that if

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102.

(Ex)(y)(Fy Gxy) and (x)(y)(Gxy Hyy) then (x)(Fx Ey)Hxy). So let us assume (1) that (Ex)(y)(Fy Gxy), and (2) that (x)(y)(Gxy Hyy), and undertake to show that (x)(Fx Ey)Hxy). Now (1) tells us that there is at least one thing x such that (y)(Fy Gxy). Let us consider, then, one such thing — call it z — and reason about it. So (y)(Fy Gzy); (3). Applying in particular to our chosen z what (2) says of everything x, we see further that (y)(Gzy Hyy); (4). Now what we want to show is that (x)(Fx Ey)Hxy); let us take, then, any unspecified object w such that Fw (5), and show of it that (Ey)Hwy. From (5) it follows, by the law (3), that Gzw, and from this in turn it follows, by the law (4), that Hww; (5). So w is H to something (viz., w itself). I. e., (Ey)Hwy; (9). It has been shown, then, for any unspecified object w, that Fw Ey)Hwy; (10). I. e., (x)(Fx Ey)Hxy); (11). So (12) is established.

R1-7, insofar as they reflect natural methods of reasoning, are of interest as a sharp analysis of what we had hitherto been only confusedly aware of. Moreover, insofar as R1-7 reflect natural methods, their justification (in the next section) is a justification of those methods. Any justification must itself appeal finally to natural standards, but, some such standards being far more dubious and less rigorous than others, justification has its scope.

Indeed, there are subtle restrictions in R6-7, and in the added rule of last lines, which would scarcely be apprehended at the uncritical level. Each of those restriction, moreover, is essential; by ignoring any one of them, false results could be obtained. The remainder of this section will be devoted to showing the need of the restrictions -- hence to showing that the rules minus the restrictions would be wrong. That the rules *with* the restrictions are *right* will not be shown until the following section.

By violating the requirement that the variable introduced by R6 be free in no prior line, we could get a wrong result as follows:

It is by no means true that '(x)(Ey)Fxy' implies '(Ex)Fxx'. If we take numbers as our universe, and construe 'Fxy' as 'x < y', clearly '(x)(Ey)Fxy' becomes true and yet '(Ex)Fxx' false. This fallacy is prevented by the restriction on R6 which requires us, in inferring from '(Ey)Fxy', to put for 'y' a variable which (unlike 'x') has not been free hitherto in the proof.

Another fallacy against which that restriction is aimed is seen in the following

It is by no means true that '(Ex)Fx . (Ex)Fx' implies '(Ex)(Fx . Gx)'; there are oysters, e.g., and things that fly, but no oysters fly.

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103.

The first restriction in R<sub>7</sub>, requiring that Z' not be free in the inferred quantification, is needed to avoid fallacies of this type:

It is not true that '(x)Fxx' implies '(Ez)(x)Fxz'. E. g., take 'Fxz' to mean that x and z are the same thing. Then '(x)Fxx' becomes true (everything is identical with itself); but '(Ez)(x)Fxz', which says there is some one object which is identical with every object, is -- in any universe of more than one object -- false.

The second restriction in R<sub>7</sub>, requiring that Z' not be free in any prior line established by R<sub>6</sub>, is needed to avoid fallacies of this type:

It is not true that '(Ex)Fx' implies '(x)Fx'.

Violation of this second restriction can lead to trouble even when the variable z' is not one which was introduced by R<sub>6</sub>, but merely happens to turn up free in a line justified by R<sub>6</sub>. An illustration is found in the following:

Actually it is not true that '(y)(Ex)Fxy' implies '(Ex)(y)Fxy'. E.g., '(y)(Ex)(x = y)' is true (where ' = ' expresses identity), but '(Ex)(y)(x = y)' is, in any universe of more than one object, false. (cf. §29).

It is important, therefore, to scan the lines marked 'R<sub>6</sub>', as well as the premises, before using R<sub>7</sub>.

The third restriction in R<sub>7</sub>, requiring that Z' not be free in any premiss in whose scope we are working, is needed to avoid fallacies of this type:

Certainly 'Fx' does not imply 'Fy'; Jones may be ill without Smith's being ill.

The "rule of last lines" requires that in the last line of a proof no free variable survive which was originally introduced by R<sub>6</sub>. This prevents us, e.g., from regarding the following:

as a finished proof; '(Ex)Fx' does not in fact imply 'Fy'. Variables introduced by

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(44)

The following block: (Ex)Ex, does not exist implicitly 'Ex'. Arithmetes introduced by
a tingeued block:

(S) A (Ex)Ex
(1) Hc

The following restriction:

The arithmetes which were explicitly introduced by Hc. This precedes me. e.g., from
a single line of last lines "require that in the last line of a block no tree
Ceratinly Ex, does not imply 'Ex'; those with ex ill without Smith's past ill.

(A) A (S)Ex
(S) Ex
(1) Hc

The above cases we are working. It is used to avoid tufificles of this fide:

The third restriction in Hl. requiring that S. not be free in such premises in
bundless, before using Hl.

It is important, therefore, to see this lines marked 'Hc', as well as the

more than one object, line, (ct. Sd). The line (where =, expresses identity), put (Ex)(A)(x = A), is in such a nature that it is not true that (A)(Ex)Ex, implies (Ex)(A)Ex. E.g., (A)(Ex)(x = A),

(2) (Ex)(A)Ex
(1) Hc
(2) (A)Ex
(3) Ex
(5) (Ex)Ex.
(1) Hc

The following restriction is found in the following:

S. is not one which was introduced by Hc, but merely because to find mb free in S
Atofation of this second restriction can lead to trouble even when the variables

It is not true that (Ex)Ex, implies (x)Ex.

(2) (x)Ex (S) (tufificone)
(5) Ex (1) Hc
(1) Hc

The following restriction in Hl. is needed to avoid tufificles of this fide:

The second restriction in Hl. requiring that S. not be free in such brorx free

with such object, is — in such a nature as were given one object — tufes.

 itself): put (Ex)(x)Ex, which same line is some one object which is identical
 such as the same thing. Then (x)Ex, becomes true (dividing it identified with
 it is not true that (x)Ex, implies (Ex)(x)Ex, E.g., true 'Ex, to mean that x

(1) (S)(x)Ex (3) (tufificone)
(1) (x)Ex (5) (tufificone)
(5) Ex (1) Hc
(1) Hc

The following restriction in Hl. requiring that S. not be free in the interred

The that restriction in Hl. requiring that S. not be free in the interred

103.

104.

R6 must be viewed as auxiliaries, eventually to be got rid of.

The other part of the rule of last lines, which stipulates that the last line of a proof must lie in the scope of none but initial premisses, is likewise really occasioned by R6. It prevents us from regarding the following as a finished proof:

last we be led to suppose, which is by no means true, that the conjunction of the premisses (1) and (3) implies (5). The stipulation is occasioned entirely, as it happens, by examples such as the above, in which a use of R6 intervenes between two premisses in whose scopes the last line lies. The rule of last lines could, without fear of fallacy, have been liberalized so as to withhold the title of proof from just such examples, and not from all cases where the last line lies in a non-initial scope. The rule as it stands is simpler, however, and its greater restrictiveness works no hardship in practice.

### §35. JUSTIFICATION OF THE RULES

Let the premisses in whose scope a given line lies be called, for brevity, the premisses of that line. What we want to show, in justification of R1-7, is that the last line of any proof by R1-7 is implied by the conjunction of the premisses of that line; or by the premiss of that line, if it has but one premiss; or is valid, if it has none.

The idiom 'implied by the conjunction of S<sub>1</sub>, S<sub>2</sub>, ..., and S<sub>n</sub>' may be accorded this convenient interpretation for its limiting cases: where n = 1 it will mean 'implied by S<sub>1</sub>', and where n = 0 it will mean 'valid'. With this understanding, the thesis italicized above may be put briefly: The last line of a proof by R1-7 is implied by the conjunction of the premisses of the line.

Three general principles which will figure later in the argument may conveniently be set down first.

(i) If S implies S', then S also implies the universal quantification of S' with respect to any free variable Z of S' which is not free in S.

For, if S implies S' then the conditional having S as antecedent and S' as consequent is valid. Then the universal quantification, with respect to Z, of this conditional is valid. But, by (ix) of §29, this quantification is equivalent to the conditional whose antecedent is S and whose consequent is the universal quantification of S' with respect to Z. So this conditional is valid; i.e., S implies the universal quantification of S' with respect to Z.

(ii) If S implies S', then the existential quantification of S, with respect to any free variable Z of S which is not free in S', implies S'.

This is established by similar reasoning, appealing to (x) of §29.

(iii) If the conjunction of S<sub>1</sub> and S<sub>2</sub> implies S', and Z is free in S<sub>2</sub> and not in S<sub>1</sub> nor S', and the existential quantification of S<sub>2</sub> with respect to Z is valid, then S<sub>1</sub> implies S'.

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["(32)", "(111) If the construction of 2f and 2S implies 2<sub>1</sub>, and 2 is free in S\nthen is established by similar reasoning, according to (x) of SBD.", "(11) If 2 implies 2<sub>1</sub>, then the existential quantification of 2<sub>1</sub>, with respect to 2<sub>1</sub>, with respect to any tree variables 2 of 2<sub>1</sub> which is not free in 2<sub>1</sub>, implies 2<sub>1</sub>.", "(111) If 2 implies 2<sub>1</sub>, then the existential quantification of 2<sub>1</sub>, with respect to 2<sub>1</sub>, with respect to any tree variables 2 of 2<sub>1</sub> which is not free in 2<sub>1</sub>, implies 2<sub>1</sub>.\n\n(1) If 2 implies 2<sub>1</sub>, then 2 also implies the universal quantification of 2<sub>1</sub>, with respect to any tree variables 2 of 2<sub>1</sub> which is not free in 2<sub>1</sub>.", "(1) If 2 implies 2<sub>1</sub>, then 2 also implies the universal quantification of 2<sub>1</sub>, with respect to any tree variables 2 of 2<sub>1</sub> which is not free in 2<sub>1</sub>.", "The term universal quantifier which will figure later in the argument may contain\nis set down first.", "The term universal quantifier which will figure later in the argument may contain", "The term universal quantifier which will figure later in the argument may contain\nis set down first.\n\nThe term universal quantifier which will figure later in the argument may contain\nis set down first.\n

105.

To see this, picture Z as 'x', S<sub>1</sub> as '--', and S<sub>2</sub> as '...x...'. By (ii), '(Ex)(--- ... x...)' implies S'. But, by (viii) of §29, '(Ex)(--- ... x...)' is equivalent to '--' (Ex)(...x...)', which, given the validity of '(Ex)(...x...)', is equivalent to '--' alone. So '--', or S<sub>1</sub>, implies S'.

Now to some temporary terminology which will be convenient in establishing the main thesis. Consider any use of R<sub>6</sub> in a proof. From one line, an existential quantification, it infers another line. Let us form a conditional having that existential quantification as antecedent and the inferred line as consequent. The conditionals which correspond in this way to all the uses of R<sub>6</sub>, prior to and including a given line of a proof, will be called the auxiliary conditionals of that line. A line of a proof will be called conditionally sound if it is implied by the conjunction of its premisses and auxiliary conditionals. (The limiting cases of the idiom 'implied by the conjunction of' are to be construed as already noted.)

A principal step toward establishing the main thesis, viz. that the last line of a proof is implied by the conjunction of its premisses, is substantiation of this preliminary law:

(I) Given any proof by R1-7 and any line L therein, if every line above L is conditionally sound then so is L.

To establish (I) we assume (what will be called the hypothesis) that every line above L is conditionally sound, and proceed to show that L is conditionally sound. Since L enters the proof through one or another of R1-7, there are seven possible cases to consider.

Case 1: L enters by R1. Then L is a premiss; hence it is one of its own premisses; hence, trivially, it is implied by the conjunction of its premisses, and is thus conditionally sound.

Case 2: L enters by R2. Then L is a conditional whose antecedent repeats a preceding premiss (or conjunction of premisses) P, and whose consequent repeats a line L' which lies in the scope of P. L itself lies outside that scope, but shares all other premisses of L', and of course all auxiliary conditionals of L'. Now L' is, by hypothesis, conditionally sound. Hence L' is implied by the conjunction of the auxiliary conditionals and premisses of L and the further premiss P. L, then, having P as antecedent and L' as consequent, is implied by that same conjunction minus P (cf. (II) of §13); so L is conditionally sound.

Case 3: L enters by R3. Then L is inferred from an earlier line which, or earlier lines the conjunction of which, truth-functionally implies L. But each such earlier line is, by hypothesis, implied by the conjunction of its own premisses and auxiliary conditionals. Therefore the conjunction of those lines (if there are several) is implied by the conjunction of all premisses and auxiliary conditionals of those lines. But, since L was "inferred" from those lines, all their premisses are among the premisses of L itself (cf. §33); and similarly, of course, for the auxiliary conditionals. Therefore L is implied by the conjunction of its own premisses and auxiliary conditionals; i. e., L is conditionally sound.

Case 4: L enters by R4. Then L is inferred from an earlier line L' which is related to L in the manner of '(y)Fy' to 'Fx', and therefore implies L (cf. §30).

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106.

L' is, by hypothesis, implied by the conjunction of its premisses and auxiliary conditionals; hence L is implied by that same conjunction. But since L was "inferred" from L', all premisses of L are premisses of L'; and similarly, of course, for the auxiliary conditionals. So L is implied by the conjunction of its own premisses and auxiliary conditionals, and is thus conditionally sound.

Case 5: L enters by R5. Then there is an earlier line L' which is related to L in the manner of 'Fx' to '(Ey)Fy', and therefore implies L. The argument continues as in Case 4.

Case 6: L enters by R6. Then L is inferred from a line L' such that the conditional C whose antecedent is L' and whose consequent is L is an auxiliary conditional of L. L is truth-functionally implied by L' and C. But L' is, by hypothesis, implied by the conjunction of its premisses and auxiliary conditionals; and all those premisses and auxiliary conditionals of L' are included among those of L itself. So L is implied by the conjunction of its own premisses and auxiliary conditionals (including C); i.e., L is conditionally sound.

Case 7: L enters by R7. Then L is a universal quantification with respect to some variable Z, and is inferred from an earlier line L' which is like L except that it lacks the initial quantifier and shows, in place of Z, free occurrences of some variable Z' which is not free in any premiss or auxiliary conditional of L' (cf. the restrictions in R7). Now the conjunction of those premisses and auxiliary conditionals of L' implies L', by hypothesis. By (i), then, since Z' is not free in that conjunction, that conjunction likewise implies the universal quantification of L' with respect to Z'. But, since Z' is not free in L (cf. the restriction in R7), the quantification of L' with respect to Z' is equivalent to L -- being the same as L except for an alphabetically different rendering of a bound variable. So the conjunction of the premisses and auxiliary conditionals of L' implies L. But all the premisses and auxiliary conditionals of L' are among those of L. So L is implied by the conjunction of its own premisses and auxiliary conditionals; i.e., L is conditionally sound.

So (I) is now established. From it we easily infer that

(II) Every line of any proof by R1-7 is conditionally sound. For, the first line of a proof is necessarily a premiss and therefore conditionally sound by the trivial argument of Case 1 above. By (I) it then follows that the second line is likewise conditionally sound. Then, by (I), so is the third line; and so on to the end.

We are now ready to establish our main thesis:

(III) The last line of any proof by R1-7 is implied by the conjunction of its premisses.

Let the last line be L. By (II), L is conditionally sound. If L has no auxiliary conditionals (i.e., if R6 was not used in the proof), then conditional soundness of L means simply that L is implied by the conjunction of its premisses, which is what we wanted to show. So it remains only to consider the case where L has auxiliary conditionals. Let them be C<sub>1</sub>, C<sub>2</sub>, ..., C<sub>n</sub> (n > 0), in the order of occurrence of those uses of R6 to which they correspond. Let the premisses of L be P<sub>1</sub>, P<sub>2</sub>, ..., P<sub>m</sub> (where m may be 0). These are initial premisses, since, by the rule of last lines, all premisses of last lines of proofs must be initial. By (II), L is implied by the

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107.

conjunction of $P_1, ..., P_m, C_1, ..., \text{ and } C_n$. Now let us examine $C_n$. It may be re-presented as:

$$ (\mathbf{Ex})(\dots\mathbf{x}\dots) \supset \dots\mathbf{w}\dots $$

(if we imagine 'x' and 'w' to be the appropriate variables). '(Ex)(...x...) here represents some existential quantification occurring as a line in the proof, and '...w...' represents the result of putting 'w' for the free occurrences of 'x' in the schema '...x...'. In view of the restriction in R6, 'w' is free in no earlier line of the proof. So 'w' is not free in '(Ex)(...x...)'. Also it is free in none of $C_1, C_2, ..., \text{ and } C_{n-1}$, since these correspond to earlier uses of R6 and hence consist of antecedents and consequents which appeared as earlier lines of the proof. Similarly 'w' is free in none of $P_1, ..., P_m$, these being initial premisses. Also 'w' is not free in L, by the rule of last lines. Furthermore, the existential quantification of $C_n$ with respect to 'w' is valid, being obtainable by substitution in the monadic schema '(Ew)(Ex)Fx \supset Fw)' whose validity may be checked by the method of §31. From these various observations it follows by (iii), then, that we can ignore $C_n$ and conclude that the conjunction of $P_1, ..., P_m, C_1, ..., \text{ and } C_{n-1}$ implies L. By repeating the above reasoning we can get rid in turn of $C_{n-1}$, then of $C_{n-2}$, etc., concluding finally that L is implied by the conjunction merely of $P_1, ..., P_m$.

So (III) is established. Stated more fully: *The last line of any proof by R1-7 is implied by the premiss or the conjunction of the premises. If any, in whose scope it lies; or, if there are none, it is valid.*

## §36. STRATEGY

What was said of the system of §32, viz. that every valid schema is demonstrable, may be said also of the system R1-7. Every valid schema can be reached as the last line, with no scope mark alongside it, of a proof by R1-7. Also every implication can be established in a proof by R1-7. The rules R1-7 are, in short, *complete*. This assertion must, like the similar one in §32, be left unsubstantiated here,¹ and no more use will be made of it.

R1-7 have the advantage over the method of §32 that on the whole proofs by R1-7 are easier to discover. Strategies readily suggest themselves. If we want to end up with a quantification, it is natural to try first to prove it with the quantifier dropped and the variable perhaps changed, and afterward hope to get the quantification by R5 or R7. If we want a conjunction, it is natural to try to prove its parts and then bring them together by R3. Proofs of desired results are thus likely to be invented backwards. But forward strategy is obvious too, and may be pursued concomitantly with backward strategy until the results meet in the middle and form a proof. Thus, given a quantification as a line in a proof, it is natural to use R4 or R6 on it. Given an conditional, it is natural to try somehow to establish its antecedent as a line then derive the consequent by R3.

Thus, suppose we want to prove that '(x)(Fx \supset Gx)'. (Ex)(Fx . Hx)' implies '(Ex)(Gx . Hx)'; i.e., to derive '(Ex)(Gx . Hx)' from premises '(x)(Fx \supset Gx)' and

¹§99 §32, footnote 1. It is easy to show that whatever can be proved in the system which Gödel proves complete can also be proved by R1-7.

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110.

 subsidiary premis ' (x)Fx' gives 'Fy'. The resulting proof is this:

Here again, as an afterthought, uses of R₅ can be lumped and the proof shortened:

### §37. HARDER PROBLEMS

The predicament noted in §32 necessarily remains: there can be no unfailing mechanical method of deciding whether a given schema is valid, or whether it implies another. If it is valid, or if it does imply the other schema, a proof to that effect exists to be found; but there can be no unfailing method of finding proofs when they exist and finding that they do not exist when they do not.

If the strategies of §36 seemed infallible, let us remember that the examples there considered were monadic and so could even have been handled mechanically as explained in § 31.

Equally little is left to the imagination, indeed, when we apply those strategies to certain polyadic cases; viz., those cases where no rearranging of variables in their innermost contexts is required. The deduction of ' (y) (Ex) Fxy' from ' (Ex) (y) Fxy', e.g., issues quite automatically from the strategies, as the reader can quickly verify. Similarly for the deduction of ' (y) ((Ex) (Fx . Hyx) ∋ (Ex) (Gx . Hyx))' ('Whoever draws a circle draws a figure') from ' (x) (Fx ∋ Gx)' ('All circles are figures').

However, in some polyadic cases a shuffling of variables is called for which is not readily reducible to rules of thumb. Such a situation begins to arise, but still in an easily manageable form, when we try to deduce 'There is a philosopher who contradicts himself' from 'There is a philosopher whom all philosophers contradict'. The problem here is, given the premis ' (Ex) (Fx . (y) (Fy ∋ Gyx))', to deduce ' (Ex) (Fx . Gxx)' . Backward strategy suggests that we aim for 'Fz . Gzz'; hence for 'Fz' and 'Gzz'. Turning then to forward strategy, we naturally apply R₆ to our premis and get 'Fz' and '(y) (Fy ∋ Gyz)' . Forward strategy then dictates application of R₄ to '(y) (Fy ∋ Gyz)' . But how shall we pick the variable which is to supplant 'y' ? We pick it as 'z', getting 'Fz ∋ Gzz', and our proof is complete.

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111.

It took very little guessing to pick the variable for (4) in such a way as to make things come out right. Sometimes, though, enough alternatives are open to render the happy choice of variables harder to discover. It may even happen that we have to use the same quantification several times in a proof, making different substitutions of variables each time. Under such circumstances, ingenuity begins to play a part.

Preparatory to one illustration of such a situation, let us acquaint ourselves with symmetry, transitivity, reflexivity, and related concepts -- these being worth noting also in their own right. A relative term is called symmetrical, asymmetrical, transitive, intransitive, totally reflexive, reflexive or irreflexive according as it fulfills:

<table><tr><td>(x)(y)(Fxy &rarr; Fyx)</td><td>(symmetry)</td></tr><tr><td>(x)(y)(Fxy &rarr; -Fyx)</td><td>(asymmetry)</td></tr><tr><td>(x)(y)(z)(Fxy . Fyz . &rarr; Fxz)</td><td>(transitivity)</td></tr><tr><td>(x)(y)(z)(Fxy . Fyz . &rarr; -Fxz)</td><td>(intransitivity)</td></tr><tr><td>(x)Fxx</td><td>(total reflexivity)</td></tr><tr><td>(x)(y)(Fxy &rarr; . Fxx . Fyy)</td><td>(reflexivity)</td></tr><tr><td>(x) -Fxx</td><td>(irreflexivity)</td></tr></table>

The relative term 'compatriot' is symmetrical, in that if x is a compatriot of y then y is a compatriot of x. It is also transitive, if we disallow multiple nationality; for then if x is a compatriot of y and y of z, x will also be a compatriot of z. It is also reflexive, if we consider a person a compatriot of himself -- as indeed we must if 'compatriot of' means 'having same nationality as'. But it is not totally reflexive, if we think of our universe as containing any things devoid of nationality. Examples of total reflexivity are rare and trivial; 'identical' and 'coexistent' are two such.

The relative term 'north' is again transitive, but it is asymmetrical and irreflexive; 'x is north of y' excludes 'y is north of x', and nothing is north of itself. The relative term 'mother' is intransitive, asymmetrical, and irreflexive.

The relative term 'loves' lacks all seven properties. Where x loves y, y may or may not love x; thus 'loves' is neither symmetrical nor asymmetrical. Where x loves y and y loves z, x may or may not love z; thus 'loves' is neither transitive nor intransitive. And, since some love themselves while others (even among those who love or are loved) do not love themselves, 'loves' is neither reflexive nor irreflexive.

As an example now of a proof calling for some ingenuity in the manipulation of variables, let us prove that symmetry and transitivity imply convexity. Our problem is to deduce '(x)(y)(Fxy &rarr; . Fxx . Fyy)' from the premises '(x)(y)(Fxy &rarr; Fyx)' and '(x)(y)(z)(Fxy . Fyz . &rarr; Fxz)'. The backward strategy is to aim for 'Fuw &rarr; Fuu . Fww', planning afterward to apply R7 twice to it. To get 'Fuw &rarr; . Fuu . Fww', in turn, the backward strategy is to assume 'Fuw' as subsidiary premiss and try to deduce both 'Fuu' and 'Fww'. So the problem reduces to getting 'Fuu' and 'Fww' from 'Fuw' and

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on the ground that this would naturally be understood by all concerned.¹

In everyday discourse most logical inference is enthymatic. We are constantly sparing ourselves the reiteration of known facts, trusting the listener to supply them where needed for the logical completion of an argument. But when we want to analyze and appraise a logical inference which someone has propounded, we have to take such suppressed premisses into account. At this point two problems demand solution simultaneously: the problem of filling in the details of a logical proof leading from premisses to desired conclusion, and the problem of making out the premisses so that such a proof can be constructed. Solution of either problem presupposes solution of the other; we cannot set up the proof without adequate premisses, and we cannot know what added premisses will be needed until we know how the proof is to run.

Sometimes, as in the syllogistic example above, the form of logical inference intended by the speaker suggests itself to us immediately because of its naturalness and simplicity; in such a case there is no difficulty in identifying the tacit premiss which the speaker had in mind. Sometimes, on the other hand, the form of inference itself may not be quite evident, but the relevant tacit premisses are already somehow in the air because of recently shared experiences; such a case differs in no practical way from the case where all premisses are explicit, and there is no problem beyond the normal one of working out the details of the proof.

Sometimes, finally, neither the intended form of inference nor the intended tacit premisses are initially evident; and in this case the best we can do is try to solve both problems concurrently. Thus we may start a tentative proof on the basis of the explicit premisses, and then, on coming to an impasse, we may invent a plausible tacit premiss which would advance us toward the desired conclusion. Alternating thus between steps of proof and supplementation of premisses, we may, with luck, achieve our goal. Of course, the tacit premisses thus invoked must always be statements which can be presumed to be believed true by all parties at the outset; for it is only under such circumstances that a proof using those tacit premisses would give reason for belief in the conclusion. If we were to invoke as a tacit premiss some statement which was (from the point of view of concerned parties) as much in need of proof as the conclusion itself, we should be guilty of what is known as circular reasoning, or begging the question, or pettitio principii; any added conviction that might accrue to the conclusion

1. Traditionally "enthymeme" meant, more specifically, a syllogism with suppressed premiss -- like the above example; but it is natural, now that logic has so far outstripped the syllogism, to refer to a logical inference of any form as an enthymeme when some premisses are left tacit.

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through such a proof would be purely deceptive. Deciding whether a statement is believed true by all parties at the outset is a task of applied psychology, but in most cases it offers no difficulty, there commonly being a wide gulf between the moot issues of an actual argument and the common fund of platitudes.

As an example of the kind of problem discussed in the foregoing paragraph, consider the explicit

**Promisses:** All natives of Ajo have a cephalic index in excess of 96,
All women who have a cephalic index in excess of 96 have Pima blood

and the

**Conclusion:** All persons both of whose parents are natives of Ajo have Pima blood.

Let us put these statements into logical notation, but for the present let us use obvious contractions instead of schematic letters 'F', 'G', etc., for we must keep the meanings of the words in mind in order to be able to think of relevant platitudes for use as tacit premises. The following, then, are the results of translation:

**Promisses:**
(x)(x is nat $\supset$ x has ceph),
(x)(x is wom . x has ceph . $\supset$ x has P bl)

**Conclusion:**
(x)( x is pers . (y)(y par x $\supset$ y is nat)
. $\supset$ x has P bl).

Since the conclusion is a universally quantified conditional, the natural strategy is to get it by $\exists$ from a result of R3.

Thus we may expect our finished proof to conform to the adjoining outline. Lines (3) and (4) here are reserved for eventual tacit premises. (The number of tacit premises to allow for in this outline, and the number of intermediate lines of proof to allow for, may be supposed to have been fixed upon by guess subject to eventual correction; actually they were fixed upon by peeking.)

(1) (x)(x is nat $\supset$ x has ceph)
(2) (x)(x is wom . x has ceph . $\supset$ x has P bl)
(3)
(4)
(5) z is pers . (y)(y par z $\supset$ y is nat)
...
(15) z has P bl
(16) z is pers . (y)(y par z $\supset$ y is nat)
. $\supset$ z has P bl (15)
(17) (x)((x is pers . (y)(y par x $\supset$ y is nat)
. $\supset$ x has P bl) (16)

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Now the question is how to derive (16). Let us first look back to (1) and (2) and see what we can hope to do with them. The only reference to Pima blood in (1) and (2) makes it dependent on cephalic index, and having the required cephalic index is conditional in turn on being a native; so we cannot hope, from (1) and (2), to infer Pima blood for anyone except natives of Ajo. But (5) does not make z a native of Ajo; it merely makes z's parents natives of Ajo, allowing z still to have been born elsewhere. At best we might hope to prove something about Pima blood for z's parents, those being natives, but not for z himself -- pending some tacit promiss. At this point a properly platitudinous tacit promiss suggests itself: z (or anyone) will have Pima blood if either of his parents does. So we introduce this tacit promiss as (3):

(3) (x)(y)(x par y . x has P bl . y has P bl).

Now, in order to reach our desired result (15), all we have to do is somehow get a couple of preliminary lines of this sort: 'w par z', 'w has P bl'. For, (3) itself yields:

(14) 'w par z . w has P bl . z has P bl (5)

by two applications of R4;

and (15) would follow by R3 from (14) together with 'w'par z' and
'w has P bl.'

So it remains to find something w satisfying 'w par z' and 'w
has P bl'. According to the line of reasoning which has brought us
thus far, we expect w to be a parent of z. But if we are to use (2)
to attribute Pima blood to w, we shall need to know further that w
is a woman. This suggests another tacit promiss of properly plati-
titudinous kind: everyone has a woman for a parent.

(4) (x)(x is pers ␣ (Ey)(y is wom . y par x)).

Application of this generality to z involves filling in a couple of
further lines of proof as follows:

(6) z is pers ␣ (Ey)(y is wom . y par z)) (4)

(7) (Ey)(y is wom . y par z) (6), (5).

Since we want w to be the woman in question, the next step is ob-
viously a case of R6:

(8) w is wom . w par z (7), R6

We had wanted to get 'w par z' and 'w has P bl'. One of those is
already contained in (8); we now need to derive only:

(13) w has P bl.

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The channel of course is (2), which, by R4, gives:

(9) w is wom . w has ceph . D w has P bl (2)

Since we already have 'w is wom' (in (8)), all we shall need
an order to get (13) from (9) is 'w has ceph'. But the way
to get this from (8), (5), and (1) is obvious:

(10) w par z D w is nat (5)

(11) w is nat D w has ceph (1)

(12) w has ceph (8), (10), (11)

So the proof is now complete; the reader has only to put all
the pieces in place.

Then the suppressed premisses of an enthymeme are hard
to recover, certainly we are under no obligation to the pro-
pounder of the argument to rack our brains over the matter;
we have every right to ask him how he figures, or to dismiss
the matter altogether. It is only when the propounder is
out of reach, and we have some interest of our own in sus-
taining his argument, that a real search for tacit premisses
is worth bothering with. Study of the details of the above
complex example, however, can be rewarding also apart from
the special matter of enthymemes; for it illustrates also
the kind of thinking that is called for generally in devis-
ing proofs.

839. VIRTUES OF LOGICAL NOTATION

An illustration of how logical notation circumvents
tricky features of ordinary language is seen by comparing the
two statements:

A lady is present, A Scout is reverent.

These are alike in grammatical form, yet their likeliest mean-
ings are logically altogether dissimilar:

$(\exists x)(x \text{ is a lady} . x \text{ is present})$

$(x)(x \text{ is a Scout} \supset (x \text{ is reverent}))$

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A striking further instance is afforded by the four statements:

John can outrun every man on the team,
John can outrun any man on the team,
John cannot outrun every man on the team,
John cannot outrun any man on the team.

The first two of these are clearly equivalent, and mean simply:

(1) (x) (x is a man on the team) John can outrun x).

It would be natural to suppose, accordingly, that the other two are al
equivalent to each other, being merely the negations of the first two.
But the caprice of English is otherwise. 'John cannot outrun every ma
on the team' is indeed a proper negation of the first two statements;
it can be quite correctly rendered as the negation of the quantifica-
tion (1), in the fashion:

-(x) (x is a man on the team) John can outrun x),

or, rewriting '-(x)' as '(Ex)-' and '(p) q)' as 'pq',

(2) (Ex) (x is a man on the team. John cannot outrun x),

But the case is quite otherwise with 'John cannot outrun any man on the team'; this is not the negation of (1) at all, but has rather the meaning:

(3) (x) (x is a man on the team) John cannot outrun x).

Another misleading feature of ordinary language, obviated in
logical notation, is the grammatical role of substantive which is
accorded to such words as 'nothing', 'nobody', 'something', 'every-
thing'. The statements:

London is big and noisy,
Something is big and noisy,

e.g., look as if they were quite parallel constructions. Yet the anal-
ogy is misleading. For, the first is just a short way of saying:

London is big and London is noisy

whereas the second one is *not* just a short way of saying:

Something is big and something is noisy.

On the contrary, this last says only that there are big things and

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there are noisy things, without requiring that the two classes in-
clude any things in common; it goes into logical notation as:

(4) (Ex)(x is big) . (Ex)(x is noisy),

which is the conjunction of the two separate truths '(Ex)(x is big)'
and '(Ex)(x is noisy)'. On the other hand 'Something is big and
noisy' says that there is some one and the same thing (at least one)
which is at once big and noisy; it goes into symbols as:

(5) (Ex)(x is big) . x is noisy).

(5) is not equivalent to (4), but is stronger. This becomes especially evident if, changing the example, we use 'small' instead of 'noisy'; thus revised, (5) becomes false but (4) remains true.

In the logical notation of (4) and (5) no single expression survives that could be said to correspond exactly to the substantive 'something'; the original outward resemblance to the 'London' examples is dissipated. No force of analogy remains that could tempt us to confuse (4) with (5).

Parallel to the above illustration concerning 'something', similar ones can be devised concerning 'nothing' and 'everything'; see the exercises.

The following is a still simpler illustration of how the spurious substantive 'nothing' differs from genuine names. Statements like 'London is noisy', 'Fenway Park is noisy', and 'Jimmy Durant is noisy', in which genuine names occur, imply that there are noisy things; but the parallel statement using the spurious substantive 'nothing', viz., 'Nothing is noisy', implies no such consequence.

Lewis Carroll exploits the misloading substantial status of
'nobody' in Through the Looking Glass:

"I see nobody on the road," said Alice.

"I only wish I had such eyes," the King remarked in a
fretful tone. "To be able to see Nobody! And at that distance
too! Why, it's as much as I can do to see real people by this
light!"

1. The use of the same variable in both quantifications is, of course, an insignificant notational accident; we could just as well write (4) in the fashion '(Ex)(x is big) . (Ey)(y is noisy)'.

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["120", "Heidegger's 'Only the existent is to be studied, and otherwise -- nothing' becomes:", "$(x)(x \\text{ is to be studied} \\supset x \\text{ exists}).$", "There survives in these translations no substantive like 'nothing' or 'nobody' which might seem to name some strange void object, tempting us to embark on mistaken lines of reasoning like those noted above.", "The reader, I grant, would not have been tempted to embark on such lines of reasoning anyway. Nevertheless these examples illustrate how it is that logical notation, though not proposed as an everyday substitute for idiomatic English, can be of service as a tool of analysis.", "Besides clarity, there is another equally important reason for logical notation: economy of structural elements. Translation into logical notation makes a single standard idiom take the place of a multitude of idioms of ordinary language; $(x)(Fx \\supset Gx)$, for example, takes the place of:", "<table><tr><td>F are G,</td><td>Any F is a G,</td><td>Everything that is an F is a G,</td></tr><tr><td>All F are G,</td><td>No F are not G,</td><td>Anything that is an F is a G,</td></tr><tr><td>Each F is a G,</td><td>None but G is F,</td><td>If anything is an F it is a G,</td></tr><tr><td>Every F is a G,</td><td>Only G are F,</td><td>Whatever is an F is a G,</td></tr><tr><td></td><td></td><td>F are exclusively G.</td></tr></table>", "There is also a further economy, in this same example, in view of the fact that the structural elements $(x)'$ and '$D$' used in "$(x)(Fx \\supset Gx)$' are severally useful in rendering still further idioms of ordinary language. Thus "$(x)Fx$' answers to the words:", "<table><tr><td>All is F,</td><td>Everything is an F,</td></tr><tr><td>All things are F,</td><td>There is nothing but F;</td></tr></table>", "and '$Dq$' answers to a variety of idioms as noted in §15. In one way and another, thus, the notations of universal quantification and the conditional accomplish quite a lot. It is such an economy of structural elements that has enabled us to get by with so rudimentary a set of rules of inference: a few rules of truth-functional inference and four rules of quantifiers.", "\"We have been reflecting on the wide variety of idioms for which "$(x)'$ provides when used with '$D$' or alone. Further idioms are provided for by combining "$(x)'$ with '$-$', in the fashion "$(x)-Fx$'.", "<table><tr><td>There is no F,</td><td>Nothing is an F,</td></tr><tr><td>There are no F,</td><td>There is no such thing as an F.</td></tr><tr><td>F do not exist,</td><td></td></tr></table>", "A numerous further set of mutually synonymous idioms is represented by combined use of "$(x)'$, '$D$' and '$-$' in the fashion "$(x)(Fx \\supset -Gx)$'.", "<table><tr><td>No F is G,</td><td>Nothing is both an F and a G,</td></tr><tr><td>No F are G,</td><td>No things are both F and G,</td></tr><tr><td>There is no F that is a G,</td><td>Nothing that is an F is a G,</td></tr><tr><td>There are no F that are G,</td><td>No things that are F are G,</td></tr></table>", "(39)"]

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and even 'There is no F G' (e.g., 'There is no black swan'), 'There are no F G', 'F G do not exist'.

A similar capacity may be noted in the case of 'and' and '(Ex)'.
'(Ex)(Fx . Gx)' represents:

<table><tr><td>Some F is G,</td><td>Something is both F and G,</td></tr><tr><td>Some F are G,</td><td>Some things are both F and G,</td></tr><tr><td>F G exist,</td><td>There is an F that is G,</td></tr><tr><td>There is an F G,</td><td>There are F that are G.</td></tr><tr><td>There are F G,</td><td></td></tr></table>

We may also add 'An F is G', but in this case caution is needed, as noted earlier: the meaning can be either '(Ex)(Fx . Gx)' or '(x)(Fx ∨ Gx)', and we have to depend on context or the common sense of the concrete situation to decide which.

## 840. CANONS OF PARAPHRASING

In statements of the kind which analyze into quantifications, there is a special cue to grouping which, when it occurs, is always conclusive. The substance of it is the following self-evident principle: The matrix governed by a quantifier must reach out far enough to include any occurrence of a variable (corresponding to a pronoun, in ordinary language) that is supposed to refer back to that quantifier. E.g., the statement:

> If I want something badly then I manage to get it must not be interpreted as having the outward form of a conditional.

> I want something badly D I manage to get it.

If it were so interpreted, then the antecedent of this conditional would have to be construed as the existential quantification:

> (Fx)(I want x badly),

> but the 'it' at the end would be left out in the cold. What we must do rather is construe the whole statement as a quantification, so as to include the final 'it' within the reach of the quantifier. In fact, a moment's reflection shows that the whole thing must be construed as a universal quantification, despite the existential appearance of the word 'something'. The whole thing means:

> (x)(I want x badly D I manage to get x).

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Premisses: The guard searched all who entered the premises except those who were accompanied by members of the firm

Some of Fiorecchio's men entered the premises unaccompanied by anyone else,

The guard searched none of Fiorecchio's men;

Conclusion: Some of Fiorecchio's men were members of the firm.

The first premiss says in effect:

Every person that entered the premises and was not searched by the guard was accompanied by some member(s) of the firm.

This rephrasing, more readily translatable into logical notation than the original, is prompted by the second canon. Next, in view of the third canon, we inspect this result for its outermost structure, which obviously is: $((x)(... \rightarrow ...))$:

($x$)($x$ is a person that entered the premises and was not searched by the guard $\exists x$ was accompanied by some members of the firm)

The virtue of thus paraphrasing inward (third canon) is that the unparaphrased internal segments can now be handled each as a small independent problem. The clause 'x was accompanied by some members of the firm', e.g., regardless of context, becomes:

($Ey$)($x$ was accompanied by $y$ . $y$ was a member of the firm).

The other clause, 'x is a person that entered the premises and was not searched by the guard', needs little more attention; we have merely to make it an explicit conjunction of two matrices:

$x$ is a person that entered the premises . $x$ was not searched by the guard.

So the whole becomes:

($x$)($x$ is a person that entered the premises . $x$ was not searched by the guard $\exists x$ ($Ey$)($x$ was accompanied by $y$ . $y$ was a member of the firm)).

Care must be taken, as here, to insert parentheses to indicate intended grouping.

Finally, writing 'Fx' for 'x is a person that entered the premises', 'Gx' for 'x was searched by the guard', 'Hxy' for 'x was

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accompanied by y', and 'Ky' for 'y was a member of the firm', we
have:

$(x) (Fx \cdot \neg Gx \rightarrow (Ey)(Hxy \cdot Ky))$

as the final rendering of the first premiss into logical notation.
We are safe in writing 'Fx' for the whole of 'x is a person that
entered the premises', without further analysis, because we know no
further analysis of it will be needed for the proof (fourth canon).
We are assured of this by the fact that 'entered' never occurs in
premisses or conclusion except as applied to persons entering the
Premises. Similarly we were able to leave 'x was searched by the
guard' unanalyzed, because 'searched' never occurs except with 'by
the guard'. On the other hand it behooved us to break up 'x was
accompanied by some members of the firm', since accompaniment and
membership in the firm are appealed to also outside this combination
in the course of the premisses and conclusion.

Turning to the second premiss, and writing 'Lx' for 'x was one
of Fiorecchio's men', we get this as the obvious outward structure
(third canon):

$(Ex)(Lx \cdot Fx) \rightarrow x \text{ was unaccompanied by anyone else).}$

It remains to paraphrase the component clause 'x was unaccompanied
by anyone else'. Clearly the intended meaning is:

Anyone accompanying x was one of Fiorecchio's men
(second canon), which becomes:

$(y)(Hxy \rightarrow Ly),$

so that the second premiss as a whole becomes:

$(Ex)(Lx \quad Fx) \cdot (y)(Hxy \rightarrow Ly)).$

To paraphrase the third premiss we need observe only the first
canon, and to paraphrase the conclusion we need hardly observe even
that much. We get, respectively:

$(x)(Lx \rightarrow \neg Ex), \qquad (Ex)(Lx \cdot Kx).$

The proof, finally, proceeds as follows.

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Everyone who pays his dues receives the Bulletin.

In this example 'everyone' is used instead of 'everything' only because of a habit of language, and not because the speaker feels any need of hedging his statement against such absurd objects as subhuman payments of dues. It would be pedantic to paraphrase the statement as:

(x) (x is a person) x pays his dues ␣ x receives the Bulletin) and quite proper to write simply:

(x) (x pays his dues ␣ x receives the Bulletin).

On similar grounds it may be felt that 'x is a place' could be deleted from the preceding example; anything that I can get to by plane is _ipso facto_ a place.

On the other hand the clause 'x is a person' could not, by such considerations, be deleted from the translation of the earlier example 'Everyone in the room speaks English'; being in the room is no assurance of being a person, and consequently the additional personal restriction is needed (unless we limit the universe to persons).

There are some minor irregularities of idiom that are obvious enough. The 'ever' of 'whoever', 'wherever', and 'whenever' (and likewise 'whatever') is sometimes left tacit, as in 'who hesitates is lost', 'I want to go where you go', 'when it ruins it pours'. Such use of 'who', 'where', etc. is not to be confused with the normal use as a relative pronoun meaning 'the person which', 'the place which', etc. (discussed below, §29). In practice the cases involving an implicit 'ever' are readily enough spotted.

Another irregularity to look out for is the non-temporal use of the time-words 'always', 'whenever', 'sometimes', 'never'. E.g., the statement:

The sum of the angles of a triangle is always equal to two right angles

really means:

The sum of the angles of any triangle is equal to two right angles,

and goes into logical notation:

(x) (x is a triangle ␣ the sum of the angles of x is equal to two right angles)

without any reference to time at all.

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We turn next to some constructions, having to do with time, that
have no parallels for places or persons. Consider the statement:

I knew him before he lost his fortune.

'Before' here appears in the guise of a statement connective, for
all the world like 'and' or 'or' or 'D'. But the whole is better
analyzed as follows (understanding 'a time' here and henceforward
as short for 'a period of time'):

(Ex)(x is a time . I know him during x . (Ey)(y is a time.
x is before y . he lost his fortune during y)).

This analysis does not get rid of the word 'before', but it gets it
into a position where it explicitly connects time-references (as in
'x is before y') rather than statements. The whole turns out not
to be a compound of simple statements 'I know him' and 'he lost his
fortune' at all, but a compound rather involving matrices 'I know
him during x' and 'he lost his fortune during y' (along with other
matrices).

We might indeed have stopped with the shorter analysis:

(Ey)(y is a time . I know him before y . he lost his for-
tune during y).

The first analysis has the virtue of treating the clauses 'I knew
him' and 'he lost his fortune' in uniform fashion, connecting each
with time by means of the same adverb 'during'; but either analysis
is correct, and in practice either would be chosen or none at all
depending on the exigencies of what we want to prove. Cf. the
fourth canon of §40: Expose no more structure than you think you
will need for the proof.

The word 'after', or 'since', being the opposite of 'before',
admits of course of treatment analogous to 'before'. Another word,
intermediate in meaning between 'before' and 'after', is 'while':

I knew him while he was with Sunnyrinse.

The translation might seem to be:

(Ex)(x is a time . I knew him during x . he was with
Sunnyrinse during x),

but there are difficulties with it. Does 'during x' mean merely
'within x', or does it mean 'throughout x'? If the former, then
the translation is inadequate. For, if I knew him *at all*, and if
he was *ever* with Sunnyrinse, then there would be *some* period of
time x (e.g. the Christian Era) such that I know him *within x* and
he was *with* Sunnyrinse *within x*; thus the proposed "translation"
would be a true statement even though the original 'I know him while

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he was with Sunnyrinse' in its normal sense might well be false. If
on the other hand we take 'during x' to mean 'throughout x', then
this difficulty does not arise and our translation is perfectly good

(1) (Ex)(x is a time . I know him throughout x . he was with Sunnyrinse throughout x).

The reader may think at first that this gives a meaning to 'I
knew him while he was with Sunnyrinse' which is inappropriately
strong, viz. that I began knowing him the moment he took up with
Sunnyrinse and stopped knowing him the moment he turned elsewhere.
But this is a misreading of (1). (1) says only that there is some
period of time x, which may be very short, such that I knew him
throughout x (as well as earlier and later perhaps) and such that
he was with Sunnyrinse throughout x (as well as earlier and later
perhaps). (1) is thus true provided merely that my knowing him
overlapped his Sunnyrinse career at all; and this presumably suits
the intended meaning of the original words. An alternative trans-
lation which achieves the same effect is:

(Ex)(x is a moment . I knew him at x . he was with Sunnyrinse at x).

There may of course be many such moments; 'E' means merely 'there
is at least one'.

There is a use of 'once', as in:

Once you can do this problem you have mastered the theory,

which comes out rather like the 'while' example except that it calls
for universal quantification instead of existential:

(x)( x is a time . you can do this problem in x .) you are master of the theory in x).

This example offers a good occasion for reminding ourselves that we
intend 'a time' to mean, not a quantity of time, like five hours,
but an epoch or dated period of time, like the afternoon and early
evening of the nineteenth of October, 1945; so that in 'you can do
this problem in x' the idea is not how fast you can do it, but when.

Note further that in this example, unlike the preceding one, we
can say simply 'in x' instead of 'throughout x'. This is because
we have a universal quantification; all times come in for considera-
tion, short ones included, and consequently the Christian Era which
precipitated the previous discussion is innocuous here.

Frequently essential time relationships are expressed merely
in the form of the verbs, as in:

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We saw Stromboli but it was not erupting.

This statement would lose its main point if rendered as a mere con-
junction of 'We saw Stromboli' and 'Stromboli was not erupting'.
What we want is rather:

(Ex)(x is a time . we saw Stromboli in x . Stromboli did not erupt in x).

Here again we are able to say simply 'in x' instead of through-
out x'; this time not because there is universal quantification, but
because of the denial in the last part. The reader will find it
instructive to see to the bottom of this by his own lights.

The foregoing examples give an inkling of the sort of subtlety
that is involved in everyday remarks about time. Translation of
such idioms into logical notation brings these things out into the
open; and if translation is sometimes hard, the difficulty is due
not to logical notation but to the hidden complexity of the idea
itself. And it is very clear that the attempt to put such idioms
into logical notation is doomed to failure unless we fall in square-
ly with the first canon of paraphrasing (§40) and rethink. This is
brought out forcibly again by the seemingly quite innocent example:

Tai always eats with chopsticks.

This is not properly interpretable as:

(x)(x is a time $\supset$ Tai eats with chopsticks in x);

nor (which is equivalent):

(x)(x is a moment $\supset$ Tai is eating with chopsticks at x),

for these imply that Tai is always eating. What we want is rather:

(x)(x is a time and Tai eats in x $\supset$ Tai uses chopsticks at x).

(41.)

PART IV. IDENTITY AND SINGULAR TERMS

§42. Non-existence.

There is no such thing as Pegasus. When I say 'thing', moreover I mean entity of any description, physical or mental, concrete or abstract, insofar as there are physical, mental, concrete, or abstract entities at all. There is nothing, no entity of any description, that is Pegasus.

Let it not be protested that there is such a thing but that it exists in the mind; for what exists in the mind is not Pegasus, but the Pegasus-idea. Thus, to change the example for a moment, take the Taj Mahal: the Taj-Mahal-idea and the Taj Mahal are two different things, the one existing in the mind and the other in India. Pegasus, if in addition to there being such a thing as the Pegasus-idea there were such a thing as Pegasus, would be distinct from the Pegasus-idea, and would, like the Taj Mahal, occupy portions of the space and time of the actual world. But it just happens in the case of Pegasus, unlike the Pegasus-idea, the Taj-Mahal idea, and the Taj Mahal, that there is no such thing.

This is not the place to try to say what an idea is, or what existence in the mind means. Perhaps, from the point of view of experimental psychology, an idea should be explained somehow as a propensity to certain patterns of reaction to words or other stimuli of specified kinds; and perhaps "existing in the mind" then means simply "being an idea". But precision on such matters is immaterial to present purposes. The present point is purely negative: Pegasus is not to be said to exist as a mental entity. Whatever our special theory of the Pegasus-idea or other mental entities may be, there is no such thing as Pegasus himself.

I do not mean to suggest that everything that there is must occupy space and time in the physical world: that there are not also abstract objects, e.g. the number 7, out of space and out of time. This is a separate question, which will be touched on in another place.

Also, it is clear from foregoing remarks that I do not mean here to deny in particular that there are mental entities. But the meaning in particular of the word 'Pegasus' happens to be such that, if the word did name an object, that particular object would be a physical object in space and time. Nothing but a physical object in space and time could ever be horse-shaped or winged. The word 'Pegasus' is like 'Taj Mahal' and 'Bucephalus' in this respect, and unlike 'Pegasus-idea' and '7'. But the word 'Pegasus' fares differently from 'Taj Mahal' and 'Bucephalus' in the following respect: whereas there is actually something in space and time such as the word 'Taj Mahal' purports to name (viz., at least from the 17th century onward), and whereas there is (tenselessly) presumably something in space and time such as the word 'Bucephalus' purports to name (viz., at a succession of positions in the Near and Middle East in the fourth century B.C.), on the other hand, there happens to be

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nothing such as the word 'Pegasus' purports to name, near or remote, past, present, or future. (Factual statements about the whole of time usually call for some caution and suspension of judgment, but when in particular it comes to denying winged horses there is little need for misgivings.)

I spoke above of the meaning of the word 'Pegasus'. The fact that there is no such thing as Pegasus does not make the word meaningless. The word 'Pegasus' has to have meaning, or be meaningful, in order that we even be able meaningfully to deny that there is any such thing as Pegasus. But the meaning of a word is not to be confounded with the object (if any) of which the word is a name. Many words, like 'and', and even many nouns, like 'sake', are quite meaningful without even purporting to be names at all.

Precise and satisfactory formulation of the notion of meaning is an unsolved problem of semantics. Perhaps the meaning of a word is the associated idea, in some sense of 'idea'; or perhaps it is the system of implicit rules in conformity with which the word is used. 'Meaning', for me, like 'idea, remains vague. The point now relevant in connection with meaning, like the previous point in connection with idea, is purely negative; viz., that the meaning of a word is not to be confounded with the object, if any, of which the word is a name. The named object, if such there be, may be of any sort, depending on what name we happen to be considering; it may even itself happen to be a meaning (whatever meanings may be); but it is not to be confused with the meaning of the word that names it. And many meaningful words, e.g., 'and', 'sake', and 'Pegasus', name nothing.

We may pass quickly over the relativistic point of view according to which Pegasus exists in the universe of Greek mythology and not in the universe of modern science. This is a perverse way of saying merely that Greeks believed Pegasus to exist and that (if we may trust modern science thus far) they were wrong. Myths which affirm the existence of Pegasus and recount his exploits have aesthetic value and anthropological significance; moreover they have internal structures upon which our regular logical techniques can be brought to bear; but it does happen that the myths are literally false, and it is sheer obscurantism to phrase the matter otherwise. There is really only one universe, and there is not, never was, and never will be any such thing as Pegasus.

Another opposing view which had best not detain us long, if only because of the mazes of metaphysical controversy in which it would involve us if we were to tarry, is the view that concrete individuals are of two kinds: those which are actualized and those which are possible but not actualized. Pegasus is of the latter kind, according to this view; so that there is such a thing as Pegasus, and the proper content of the vulgar denial of Pegasus is more correctly expressed in the fashion 'Pegasus is not actualized'. The universe, in the broader sense, becomes rather over-populated under this view but

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the comfort of it is supposed to be that there comes to be something Pegasus about which we may be said to be talking when we rightly say (in lieu of 'There is no such thing as Pegasus') 'Pegasus is not actualized'. The mysterious business of attributing something, non-existence, to Pegasus, when there is no such thing as Pegasus to attribute it to, is thereby dispelled.

But this comfort is short-lived, for we can change our example, and, instead of the word 'Pegasus', use 'the round square', which describes nothing that could be reckoned even as an unactualized possible. We thus end up still with the same mysterious business, if mysterious it be, of attributing non-existence where there is nothing to attribute it to; the only difference is that it is now a question of the round square instead of Pegasus. Our stuffing the universe with a weird lot of unactualized possibilities seems to have come to naught. We could go on, I suppose, and add a realm of unactualized impossibles such as the round square, but this surely stretches common sense to the breaking point.

In my view the better course is to lay all these ghosts, the unactualized possibilities included, and accept the universe as it really (tenselessly) is. This course involves simply acknowledging that we may attribute non-existence where there is nothing to attribute it to. Nor is this such a mysterious business after all, for we have seen that the word 'Pegasus' still has meaning, or is meaningful, even if it names nothing; the meaningfulness of the statement 'There is no such thing as Pegasus' is not dependent on there being any such thing as Pegasus for 'Pegasus' to name.

It may be felt that the realm of unactualized possibilities has uses apart from the purpose noted above. Perhaps this is true; one would have to become clearer on what sort of thing is meant by unactualized possibilities before judging. Perhaps they are no different from the things which I have provisionally tolerated above under the vague head of "ideas"; or perhaps they are no different from the things which I have provisionally tolerated under the vague head of "meanings"; or perhaps all three are the same. It is not necessary for present purposes to decide these things. All it is really necessary to insist on is that the particular word 'Pegasus' (like 'Taj Mahal') be recognized as meaning nothing if not an actual concrete object (and in fact nothing). Entities of more shadowy kinds, if admitted, can still be named, but in some distinguishing fashion -- 'the Pegasus-possibility', or 'the Pegasus-idea', or 'the meaning of 'Pegasus'''. If we can get together on this much by way of convention, then everyone can be left to his favorite metaphysics so far as anything further is concerned. The essential message to be carried over from this section into succeeding ones is simply this: Some meaningful words which are proper names from a grammatical point of view, notably 'Pegasus', do not name anything.

### §43. Singular terms vs. general terms.

The grammarian's expression 'proper name' is logically misloading, since many words which are proper names in the grammatical sense do

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not name anything; witness 'Pegasus'. A better designation, for
purposes of logic, is singular term. A singular term is any ex-
pression which purports to name one and only one object. Singular
terms comprise simple singular terms like 'Socrates', 'Pegasus',
'earth', or '7', together with various complex phrases like '3 / 4',
'the prime number between 5 and 11', 'the author of Waverly'. (The
concept of proper name in grammar is somewhat narrower, comprehend-
ing only the simple singular terms and perhaps not all even of
these; however, this distinction need not concern us.)

Terms divide into singular terms, as above, and general terms.
What have been called "terms" in Parts II-III, and represented by
'F', 'G', etc., are general terms. Generality is not to be con-
fused with ambiguity. The term 'Jones' is ambiguous in that it
might be used, in different contexts, to name any of various persons.
but it is still a singular term, in that it purports, in any particu-
lar context, to name one and only one person. The same is true even
of pronouns such as 'I' and 'you'; these again are singular terms,
but merely happen to be highly ambiguous pending determination through
the context or other circumstances attending any given use of them.
(Cf. § 15.) The same may be said of 'the man', or more clearly 'the
President', 'the cellar'; these phrases (unlike 'man', 'president',
and 'cellar' themselves) are singular terms, but the one and only one
object to which they purport to refer in any given use depends on
attendant circumstances for its determination.

Besides the classification of terms into singular and general,
there is a cross-classification into concrete and abstract. Con-
crete terms are those which purport to refer to individuals, phy-
sical objects, events; abstract terms are those which purport to
refer to abstract objects, e.g. to numbers, classes, qualities.
Thus some singular terms, e.g. 'Socrates', 'Pegasus', 'earth', 'the
author of Waverly', are concrete, while other singular terms, e.g.
'7', '3 / 4', 'kindliness', are abstract. Again some general terms,
e.g. 'man', 'house', 'red house', are concrete (since each man or
house is a concrete individual), while others, e.g. 'prime number',
'zoological species', 'virtue', are abstract (since each number is
itself an abstract object, if anything, and similarly for each
species and each virtue).

Thus the distinction between singular and general is indepen-
dent of the distinction between concrete and abstract. But there
are singular abstract terms which correspond in a certain intimate
way to general concrete ones. 'Kindliness', e.g., is a singular
abstract term corresponding to the general concrete term 'kindly
person'; 'redness' is a singular abstract term corresponding to
the general concrete term 'red thing'; and 'mankind' and 'humanity'
are singular abstract terms corresponding to the general concrete
term 'man'. In each such correspondence the singular abstract term
purports to name a property (or perhaps a class) which is shared

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by (or embraces) all and only those individuals to which the corresponding general concrete term applies. Despite this correspondence, however, the singular abstract term differs from the general concrete term in an important way: it purports to name one and only one object, abstract object though it be, while the general term does not purport to name at all. (The general term does indeed "denote" or "refer to" each of many things, viz, each red thing, or each man, but this kind of reference is not called naming; "naming", at least as I shall use the word, is limited to the case where the named object purports to be unique.)

Occasionally, as language is ordinarily used, a word like 'man' which is normally a general concrete term may be used as a singular abstract term; e.g., 'Man is a zoological species'. But from a logical point of view it is well to think of such examples as rephrased using a distinctively singular abstract term (thus 'Mankind is a zoological species').

The division of terms into concrete and abstract is a distinction only in the kinds of objects referred to. The distinction between singular and general terms is more vital from a logical point of view. General terms figured explicitly in the analyses of Parts II-III, and techniques for manipulating them were developed at some length; rules for manipulating singular terms, on the other hand, have yet to be taken up in succeeding sections.

But let us get this distinction between general and singular terms more clearly in mind. The distinction is, as we have seen, quite independent of the distinction between abstract and concrete. Thus far, however, it has been drawn only in a very vague way: a term is singular if it purports to name an object (one and only one), and otherwise general. Note the key word 'purports'; it separates the question off from such questions of fact as the existence of Socrates and Pegasus. Whether a word purports to name one and only one object is a question of the logical structure of language, and is not contingent on facts of existence.

In terms of logical structure, what it means to say that the singular term "purports to name one and only one object" is just this: The singular term belongs in positions of the kind in which it would also be coherent to use variables 'x', 'y', etc. (or, in ordinary language, pronouns). Contexts like:

Socrates is wise, Kindliness is a virtue,
Bellerephon captured Pegasus, $7 = 3 + 4,$

etc. are parallel in form to matrices:

$x$ is wise, Bellerephon captured $x$, $x$ is a virtue, $x = 3 + 4$

such as may occur in meaningful statements having the form of quantifications. $(Ex)(x \text{ is wise})$, etc. The terms 'Socrates', 'Pegasus',

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'kindliness', and '7', are, in short, substitutable for variables
in matrices without violence to grammar; and it is this that makes
them singular terms. Whether there is in fact such an object as
Socrates (which, tenselessly, there is) or Pegasus (which there is
not) or kindliness or 7 (on which philosophers disagree) is of course
a separate question.

General terms, in contrast to singular ones, do not occur in
positions appropriate to variables. Typical positions of the gen-
eral term 'man' are seen in 'Socrates is a man', 'All men are mor-
tal'; it would not make sense to write 'Socrates is an x' or 'All
x are mortal', or to imbed such mock-matrices in quantifications
in the fashion:

(1) (Ex)(Socrates is an x), (Ex)(all x are mortal).

The 'x' of a matrix may refer to objects of any kind, but it is sup-
posed to refer to them one at a time; and then the quantifier '(x)'
or '(Ex)', applied to the matrix, says that what the matrix says
of x is true of all or some of the objects taken thus one at a time.

There are indeed legitimate quantifications some what resembling
(1) but phrased in terms of class-membership, as follows:

(2) (Ex)(Socrates is a member of x),

(3) (Ex)(every member of x is mortal);

this latter in turn becomes:

(4) (Ex)(y)(y is a member of x ⊃ y is mortal).

But the matrices in the quantifications (2) and (3) do not, like
those in (1), attempt to parallel 'Socrates is a man' and 'All men
are mortal'; rather they parallel the statements:

(5) Socrates is a member of mankind (= class of all men),

(6) Every member of mankind is mortal,

wherein 'mankind' is a straightforward singular term (an abstract
one, purporting to name a class). As an alternative to the class
version (2)-(3) we might also appeal to attributes:

(7) (Ex)(Socrates has x),

(8) (Ex)(everything having x is mortal);

the matrices in these quantifications are parallel to the statements:

(9) Socrates has humanity, Everything having humanity is mortal.

wherein 'humanity' is a singular abstract term purporting to name an
attribute. The statements (2)-(9) exhibit variables and singular
terms in like positions, as is normal; the mock-quantifications (1),

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on the other hand, which use variables in positions appropriate to general terms, are nonsense.

It may be felt by now that logic is being chopped rather fine, and that there would have been no harm in admitting the quantifications (1) as meaningful, simply according to the meanings (2)-(3) or (7)-(8). Actually there is an important philosophical reason for not doing this. As soon as we elevate general terms to the status of singular abstract terms, as we should be doing if we were to let variables occupy the positions of general terms as in (1), we run into the philosophical question whether there are abstract entities for singular abstract terms to be names of; for, as we shall see in §44, the special rules of inference governing the interplay between a singular term and a variable depend on a promise to the effect that there is the object which the singular term purports to name. Most of our logic can be kept clear of this question of there being abstract entities if we adhere to the point of view that words like 'man', in contexts like 'Socrates is a man' and 'All men are mortal', occupy positions which are inaccessible to variables -- just as inaccessible as the positions occupied by 'man' or 'and'.

If for any perverse reason we should want to rephrase 'Socrates is a man' or 'All men are mortal' in such a way as to involve ourselves in the question of abstract entities, the formulations (5)-(6) and (9) are at our disposal. These versions do contain singular terms which purport to name abstract entities. But we keep the record straight by reserving recognizably singular terms like 'mankind' and 'humanity' for such naming of abstract entities. Then, so long as we are minded to proceed independently of the question of the existence of abstract objects, we can also use idioms like (5)-(6) and (9) in favor of 'Socrates is a man' and 'All men are mortal'.

This is a way, as remarked, of keeping most of our logic clear of the question of there being abstract entities. After a certain point, however, the question will still have to be faced even independently of any use of singular abstract terms like 'mankind' or 'humanity'. For, there is also another channel, more fundamental than that of singular terms, through which reference to objects, abstract or otherwise, regularly occurs; viz., variables. An existential quantification says that such-and-such is the case for some object x; and if in particular the "such-and-such" happens to be something which could hold only for abstract objects x, then whoever affirms the quantification has committed himself to the view that there are abstract objects. The quantifications:

$(Ex)(x \text{ has many members})$,

$(Ex)(x \text{ is prime})$,

e.g., affirm that there are many-membered things and prime things; but many-membered things are classes and prime things are numbers, and classes and numbers are abstract. The truth of these quantifications depends squarely on the questions of there being abstract

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entities, despite the complete absence of singular terms; and the
same may be said, for that matter, of the quantifications (2)-(4),
(7), and (8).

Singular abstract terms ultimately play an inessential role
in the issue over existence of abstract objects, particularly since,
as we shall see in §47, singular terms of all kinds are in prin-
ciple dispensable altogether. Whether we are to commit ourselves
to recognizing abstract objects depends finally on what we do with
our variables; it depends on whether, e.g., we propound quantifi-
cations such as these last examples. But it is only in special sit-
uations, particularly in mathematics, that this becomes an issue.

Meanwhile the essential point, of this section as of the pre-
ceding one, is negative. The essential point of this section is
that general terms are not to be forced into the mold of singular
abstract terms — i.e., are not to be viewed as significantly re-
placeable by variables. The question of the existence of abstract
objects can thus be held in abeyance until it becomes really rele-
vant, viz. in connection with the foundations of mathematics; and
meanwhile we can continue to logicize over 'Socrates is a man' and
'All men are mortal' without embroilment in metaphysical issues.

§44. <u>Singular Inference</u>

The following are three simple examples of inference involving
the singular term 'Socrates':

<table>
  <tr>
    <td>All men are mortal,</td>
    <td>No men fly,</td>
    <td>Socrates is a Great</td>
  </tr>
  <tr>
    <td>Socrates is a man;</td>
    <td>Socrates is a man;</td>
    <td>Socrates is wise;</td>
  </tr>
  <tr>
    <td>...</td>
    <td>...</td>
    <td>...</td>
  </tr>
  <tr>
    <td>Socrates is mortal.</td>
    <td>Socrates does not fly.</td>
    <td>Some Greeks are wise.</td>
  </tr>
</table>

Statements like 'Socrates is a man' and 'Socrates does not fly',
known traditionally as singular statements, were arbitrarily inter-
preted in traditional logic as categorical statements of types A
and B. Thus 'Socrates is a man' was treated as if it read 'Every-
thing that is Socrates is a man', and 'Socrates does not fly' was
treated as if it read 'Nothing that is Socrates flies'. In this
way the three inferences exhibited above were assimilated to syllo-
gisms: AAA in the first figure, EAE in the first figure, and AAI in
the third.

It will be recalled that the mood last mentioned is one that
needed, for its validity, a supplementary existential assumption
'-xG' (cf. §22). This limitation could have been overcome by

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construing 'Socrates is wise' rather as a particular (I), 'Some-
thing that is Socrates is wise', while adhering to the universal (A)
interpretation of 'Socrates is a Greek', the syllogism would then
become AII in the third figure.

Actually the universal and the particular have equal rights as
interpretations of singular statements. Since there is just one
thing that is Socrates, the statements 'Everything that is Socrates
is wise' (A) and 'Something that is Socrates is wise' (I) both amount
merely to saying that Socrates is wise. Similarly 'Nothing that is
Socrates flies' (E) and 'Something that is Socrates does not fly'
(O) both amount merely to saying that Socrates does not fly.

The most natural course, however, once we have adopted the
point of view of quantification theory, is simply to abandon all
interpretation of singular statements as categoricals, whether uni-
versal or particular. Let us return to the first of the three in-
ferences exhibited above, translating its first premiss in terms
of quantification theory:

(x)(x is a man ∃ x is mortal),
Socrates is a man;
∴ Socrates is mortal.

It now appears artificial indeed to construe the second premiss and
conclusion as categorical statements on a par with the first premiss.
The second premiss and conclusion resemble rather the clauses 'x is
a man' and 'x is mortal' which appear inside the first premiss. In
fact, if by something akin to R4 we could infer:

Socrates is a man ∃ Socrates is mortal

from the first premiss, then we could use R3 on this result toge-
ther with the second premiss, gaining our desired conclusion 'So-
crates is mortal'.

Similar remarks apply to the second of the three inferences ori-
ginally exhibited:

(x)(x is a man ∃ x does not fly),
Socrates is a man;
∴ Socrates does not fly.

The third of the original three inferences; viz:

Socrates is a Greek,
Socrates is wise;
∴ (Ex)(x is a Greek , x is wise),

would be taken care of if, having combined the promises in the
fashion:

Socrates is a Greek . Socrates is wise

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by conjunction, we could pass from this conjunction to our desired
conclusion by something akin to R5.

The extensions of R4 and R5 which seem to be called for are
very simple: allow Z', in the original versions of R4-5 (§33), to
be a singular term. These extensions work well in the Socrates
examples, and furthermore their justification is obvious: whatever
is true of everything is true in particular of Socrates, and what-
ever is true of Socrates is true of something. The analogous jus-
tification when the singular term concerned happens to be 'Pegasus'
instead of 'Socrates', however, is less clear. We should hesitate
to argue that whatever is true of everything is true of Pegasus
and whatever is true of Pegasus is true of something, considering
that there is nothing Pegasus for anything to be true of. Perhaps
we can gain more light by some experimenting: let us see what kinds
of results the above extensions of R4 and R5 would lead to in con-
nection with 'Pegasus'. From the truth '(x) (x ≡ x)', R4 as extended
would enable us to infer:

(1) Pegasus = Pegasus.

From

(2) Pegasus is a horse. Pegasus has wings,
again, R5 as extended would enable us to infer the falsehood:

(Ex) (x is a horse. x has wings).

General acceptance of the contemplated extensions of R4 and R5
would therefore compel us to regard (1) and (2) themselves as re-
spectively true and false.

Except in a few statements like:

(3) There is no such thing as Pegasus,

a singular term tends to be used only when the speaker believes it
names something. (He may of course assume this in deliberate con-
tradiction of fact, and with the listener's connivance, as in story-
telling.) Hence it is hard to decide whether (1) and (2) should in
fact be regarded as true or as false; there would seem to be room
for an arbitrary fiat on this point. Perhaps, then, we may con-
veniently adhere to the proposed extensions of R4-5 by ruling that
(1) and (2) are indeed to be construed respectively as true and
false; and similarly for analogous examples. However, such an
attempt to sustain the extensions of R4-5 encounters difficulties
when we come to (3); for this is one statement whose truth has been
insisted on ever since the present chapter began, and yet our

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Schematically:

<table><tr><td>(x)Fx,</td><td>Fu,</td></tr><tr><td>a exists;</td><td>a exists;</td></tr><tr><td>∴ Fu.</td><td>∴ (Ex)Fx.</td></tr></table>

The earlier problem of whether to regard statements like (1) and (2) as true or as false becomes an idle question now which can be left forever in abeyance. Inferences of the sort that had seemed to bear on the question are contingent now on a contrary-to-fact premiss 'Pegasus exists', thus throwing no light on the actual question of the truth or falsehood of those statements. The truth or falsehood of statements involving a singular term, say, failing the truth of the associated singular existence statement, in general be viewed as indeterminate: not decided by usage, as noted earlier, and not in need of deciding. Indeterminacy as between truth and falsehood is already familiar as an adjunct of vagueness or ambiguity in statements; and what we have before us now is merely a more radical form of the same phenomenon. Statements involving a singular term are in general simply waste statements, omitted from further consideration and evaluation, failing the singular existence premiss.

But an exception to all this is to be noted in the existence premiss itself, e.g. 'Pegasus exists', 'There is such a thing as Pegasus', which remains determinate in point of truth or falsehood even failing the existence premiss; indeed false, since it is simply the existence premiss. The exception extends also of course to (3), which is the denial of the existence premiss, and to various other logical contexts containing the existence statement as a component. Existence (along with non-existence and other logical derivatives) occupies a very special position: it is the one kind of thing that can be affirmed in connection with a proper name, with full determinacy in point of truth or falsehood, even when the proper name is not really a name of anything.¹

Exception must also be made of compounds which, though containing statements like (1) or (2) as components, may still be determinately true or false because of decisive further components. A trivial example is:

(4) Pegasus is a winged horse and 5 < 3,
which is false because of '5 < 3'.

1. Kant was thinking of somewhat this sort of exceptional behavior on the part of existence and derivative concepts when he urged (though in terms not suited to present purposes) that "existence is not a predicate".

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§45. Identity.

We have encountered the notation 'x = y' of identity repeatedly in the course of examples, but we have yet to study identity explicitly as one of the ideas of logic. Identity is such a simple and fundamental idea that it is hard to explain otherwise than through mere synonyms. To say that x and y are identical is to say that they are the same thing. Everything is identical with itself and with nothing else.

Despite its being so elementary a notion, identity has been the object of persistent confusions. An example of such confusion is the argument whereby Wittgenstein believed to show that the idea of identity is useless and dispensable: "To say of an object that it is identical with itself is empty, and to say of it that it is identical with something else is false." Essentially the same idea is set forth less succinctly by Hume:

> In that proposition, an object is the same with itself, if the idea expressed by the word, object, were no ways distinguish'd from that meant by itself; we really should mean nothing.... One single object conveys the idea of unity, not that of identity.

> On the other hand, a multiplicity of objects can never convey this idea, however resembling they may be suppos'd. (Treatise of Human Nature, Book I, Part IV, Section II).

This kind of confusion is cleared up by reflecting that there are really not just two kinds of cases to consider, one trivial and the other false, but three:

Cicero = Cicero, Cicero = Catiline, Cicero = Tully.

The first of these is trivial and the second false, but the third is neither trivial nor false. The third is informative, because it joins two different terms; and at the same time it is true, because the two terms are names of the same object. Wittgenstein, not distinguishing carefully between objects and their names. He supposed that every true statement of identity had to exhibit the two-place predicate '=' between repetitions of the same name in the manner 'Cicero = Cicero'; he failed to recognize that for truth of the statement of identity it is necessary only that '=' appear between names of the same object, and that the names may (and in useful cases will) themselves be different. For it is not the names that are affirmed to be identical, it is the things named. Cicero is identical with Tully (same man), even though the name 'Cicero' is different from the name 'Tully'; just as the Nile is longer than the Tuscaloosahatchie, even though the name 'Nile' is shorter than the name 'Tuscaloosahatchie'. To say anything about given objects

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we apply the appropriate verb or predicate to *names of the objects*; and there is no reason to expect or desire that what is thereby said of the objects may happen incidentally to be true also of the names themselves.

Confusion between identity of objects and identity of names is observable in the minds of occasional mathematicians who hold that that an equation like '5 + 3 = 2 + 6' relates two numbers which are "equal", in some sense, but not identical. Sometimes this confusion evolves into the idea that identity is a relation between signs rather than objects; thus Leibniz: "Two terms are the same (eadem) if one can be substituted for the other without destroying the truth of the context (salva veritate)." Clearly it is not the terms, but the objects named by the terms, that are the same.

Another difficulty commonly associated with identity dates from the early Greek philosopher Heraclitus, who said "You cannot step into the same river twice, for fresh waters are ever flowing in upon you." The same difficulty reappears in the case of the human body itself, since its substance is continually changing. In general, the problem is this: how can a thing that changes its substance be said to remain identical with itself?

Actually the key to this difficulty is to be sought not in the idea of identity but in the ideas of thing and time. A physical thing -- whether a river or a human body or a stone -- is, at any one moment, a sum of simultaneous momentary states of spatially scattered atoms or other small physical constituents. Now just as the thing at a moment is a sum of these spatially small parts, so we may think of the thing over a period as a sum of the temporally small parts which are its successive momentary states. Combining these conceptions, we see the thing as extended in time and in space alike; the thing becomes a sum of momentary states of particles, or briefly particle-moments, scattered over a stretch of time as well as space. All this applies as well to the river (or human body) as to the stone. There is only a difference of detail in the two cases: in the case of the stone the constituent particles moments pair off fairly completely, from one date to another, as momentary states of the same particles, whereas in the case of the river (or human body) there is more heterogeneity in this respect. The river (or human body) will regularly contain some momentary states of a particle and exclude other momentary states of the same particle, whereas with the stone (barring small peripheral changes or ultimate destruction) this is not the case. Here we have a distinction reminiscent of the distinction in traditional philosophy between "modes" and "substances". But things of both kinds are physical things in one and the same sense: sums of particle-moments. And each thing is identical with itself; we can step into the same river twice. Naturally we come into contact with different particle-

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moments the second time, and this would be so even if fresh waters
were not ever flowing in upon us, but in any case the varied par-
ticle-moments belong to one and the same spatio-temporal collection
known as the river in question.

So clarification of Heraclitus' problem, the problem of change,
is best achieved by abstracting from grammatical tense and treat-
ing time as spread out analogously to a dimension of space: a
fourth dimension. The issue, properly conceived, does not chal-
lenge the logical idea of identity at all; any object, momentary
or enduring or changing or eternal, is (tenselessly) identical
with itself and with nothing else.

The utility of identity is not limited to cases like 'Cicero =
Tully' involving simple proper names; more useful cases arise when
instead of one or both proper names we have a complex name of a
descriptive kind, e.g. 'the author of De Senectute' (cf. next sec-
tion);

Cicero = the author of De Senectute.

Examples of the same sort abound in arithmetic; e.g., '7 = 3 + 4',
wherein '7' is a simple name and '3 + 4' a complex name of one and
the same (abstract) object, viz. the number 7. Also variables may
of course occur, as in the laws of algebra:

$(x)(x + 0 = x) \dots (x)(y)(x + y = y + x)$

Even in application to mere variables the predicate of identity
has it in uses. A trivial statement involving such use of it is:

(I) (x)(x = x),

which attests to the total reflexivity of identity (cf. §37). Be-
cause of its brevity and truth, this statement has been useful as
an example occasionally in earlier pages. But there are also less
trivial uses of the identity sign between variables, as in:

$(x)(y)(x \text{ is a god}) \dots y \text{ is a god} \dots \nV. x = y$

$(\exists x)(x \text{ is a god}) \dots (y)(y \text{ is a god}) \dots \nV. x = y)$

$(\exists x)(\exists y)(x \text{ is a god}) \dots y \text{ is a god} \dots x \neq y$

$(x)(y)(z)(x \text{ is a god}) \dots y \text{ is a god}) \dots z \text{ is a god}) \dots \nV. x = y$

$\nV. x = z \dots \nV. y = z)$

(The notation $'x \neq y'$ is a convenient abbreviation for $'-(x = y)'$.)
as the reader can verify on a little reflection, these four formu-
las are equivalent respectively to the following verbal statements:

There is one god at most,

There is exactly one god,

There are at least two gods,

There are two gods at most.

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The statement (I) above is a logical truth of a kind not provable by the rules of proof thus far set forth. (I) is a logical truth belonging to a new department of logic, the theory of identity. Further logical truths belonging to the theory of identity are:

$(x)(y)(x \text{ is green} \quad x = y \quad \supset \quad y \text{ is green})$

and all other statements like it -- all statements of the form:

$(\text{II}) \quad (x)(y)(Fx \quad . \quad x = y \quad \supset \quad Fy).$

There are also various additional logical truths belonging to the theory of identity; but it happens that every one of them can be proved, by the technique of proof hitherto developed, provided merely that we allow ourselves this further liberty: Set down (I), or any statement or matrix of the form (II), as a line of proof anywhere at will.

The following is a simple example of such a proof:

(1) $(x)(y)(\underline{z} \supset x \quad . \quad x = y \quad \supset \quad \underline{z} \supset y)$

(II)

(2) $(w)(x)(y)(\underline{w} \supset x \quad . \quad x = y \quad \supset \quad \underline{w} \supset y)$

(I)

The first line here is a matrix of the form (II), wherein 'w = x' and 'w = y' answer respectively to 'Fx' and 'Fy'. There is no scope-mark, because no premiss. Line (2), proved as it is outside the scope of any premiss, is logically true; it is another of the logical truths belonging to the theory of identity. In fact it is the familiar law which says that things equal to the same thing are equal to each other. It is the law of *transitivity* of identity; cf. § 37.

A less simple example of such a proof appears below, wherein it is proved that order in an identity is immaterial:

$(x)(y)(x = y \quad \supset \quad y = x).$

This is the law of *symmetry* of identity; cf. § 37. Note the elliptical usage in proving line (3): we cite (I) (viz. $(x)(x=x)'$) as if it had been written in as a line of the proof, omitting the formality of actually writing it in.

(1) $(x)(y)(x = x \quad . \quad x = y \quad \supset \quad y = x)$

(II)

(2) $z = z \quad . \quad z = w \quad \supset \quad w = z$

(I)

(3) $z = z$

(I)

(4) $z = w \quad \supset \quad w = z$

(2), (3)

(5) $(x)(y)(x = y \quad \supset \quad y = x)$

(4)

Note that R4 is applied twice in succession in passing from (1) to (2); and similarly for R7, in passing from (4) to (5).

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The statement (I) and the statements and matrices of the form (II) may be called the *axioms of identity*. An axiom, according to traditional usage, is any self-evident truth which is adopted without proof, as a basis for proving other things. From the point of view of the proof techniques of the present book, an axiom may be defined as an expression which we allow ourselves to insert as a line of proof anywhere at will, neither proving it nor giving it a scope-mark as we would a promiss. Speaking of (I) and the statements and matrices of the form (II) as "axioms" is thus just another way of announcing adoption of the rule which was italicized three paragraphs back.

In practice it is convenient to add a supplementary rule obeying the symmetry of identity, and used always tacitly: *Disregard the order in identities, thus treating 'x = y' at will as if it were 'y ≠ x'. This rule is dispensable, but it shortens proofs.*

### §46. Descriptions.

Singular terms of the special type 'the author of *Waverly*', 'the prime number between 5 and 11', 'the brightest student at Harvard', are called *singular descriptions* or *briefly descriptions*. A description may be thought of as consisting of a matrix e.g. 'x wrote Waverly' governed by the prefix 'the object x such that'. The notation customarily used for this prefix in modern logic is '(1x)', containing an inverted Greek iota. Thus the description 'the author of Waverly' becomes:

(1x)(x wrote Waverly).

Similarly 'the prime number between 5 and 11' becomes:

(7x)(x is prime. 5 < x < 11),

and 'the brightest student at Harvard' becomes:

(1x)(x goes to Harvard . (y)(y goes to Harvard D y is not brighter than x)).

So the general form of notation for a description is '(1x)Fx', read 'the object x such that Fx'.

The description prefix '(1x)'

applies to matrices as quantifiers do, and in fact it is grammatically exactly like a quantifier except that where a quantifier produces statements a description prefix produces singular terms.

Though the word 'the' in the singular is the characteristic mark of a description in ordinary language, many idioms of ordinary

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language which lack this mark are best analyzed in terms of descriptions. The words 'what', 'who', 'where', 'when' are often used with the meanings 'the object which', 'the person which', 'at the place at which', 'at the moment at which', so that the characteristic 'the' of description comes into evidence on paraphrasing. E.g., the statements:

He got what he went after. He died where he was born
may be construed in the fashion:

He got (1x)(he went after x),
He died at (1x)(x is a place . he was born at x).

Also the possessive pronoun commonly conceals a description;
thus 'Smith's house' means 'the house which belongs to Smith', i.e.:

(1x)(x is a house . x belongs to Smith),
and 'my mother' means 'the object which bore me', i.e.:

(1x)(x bore me).

The machinery adopted for inference in connection with singular terms applies in particular to descriptions. Two simple illustrations of such inference are as follows:

f (1) (1x)Fx exists
{ (2) (1x)Fx = (1x)Fx (I), (1)

| ( (1) Tom admires (1x)Fx
| ( (2) (1x)Fx exists
| (3) (Ex)(Tom admires x) (1), (2)

But the singular existence premiss for a description, viz.:

(1x)Fx exists, or: There is such a thing as (1x)Fx,
proves to admit of some further analysis. The description '(x)Fx'
is supposed to name the one and only object x such that Fx; to say
that this object exists, then, is to say that

There is one and only one object x such that Fx,
i.e.,

(Ex)(Fx . F nothing but x),
i.e.,

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(6) (nx)(x is prime .. 5 < x < 11) is prime . 5 < (nx)
(x is prime . 5 < x < 11) < 11.

The general form illustrated by these two examples is:

'F' is true of (nx)Fx,

i.e.:

(III) F(nx)Fx.

An illustration of another sort of axiom needed is:

Nobody but the author of *Waverly* wrote *Waverly*,

i.e.:

(7) (y)(y wrote Waverly ∃ y = (nx)(x wrote Waverly)),

or in general:

(IV) (y)(Fy ∃ y = (nx)Fx).

Compare (III) and (IV) with (4). What follows '(Ex)' in (4)
says, in effect, 'x is the one and only object such that Fx';
and (III) and (IV) say the same thing of (nx)Fx:

(nx)Fx is the one and only object x such that Fx.

The *axioms of description* are the statements or matrices having
either of the forms (III) and (IV) -- e.g. the statements (5)-(7).
Axioms of description may, like axioms of identity, be inserted
at will as lines of proof.

It may seem that (III) and (IV) should be assumed only on con-
dition of (4), for surely we do not want to say that (nx)Fx is
the one and only object x such that Fx (which is what (III) and
(IV) say) unless there is one and only one object x such that Fx
(which is what (4) says). Actually, though, this point need not
worry us; for, any use of R4 or R5 in application to descriptions
is already conditional upon (4), and we are powerless to draw any
embarrassing consequences from (III) and (IV) without help of R4
and R5. For this reason we can adopt (III) and (IV) themselves
with no strings attached.

A result is that some proofs based on axioms of description
can be carried out without appeal to the singular existence state-
ment for the description. Thus, suppose we are given the

Premiss: The broker who hired John hired only honors graduates
and want the

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Conclusion: John was an honors graduate.

It turns out that though we need the singular existence statement for 'John' ('John exists') as an added premiss, we do not need a singular existence premiss for 'the broker who hired John'. The proof is as shown below, where 'Fx' stands for 'x is a broker', 'Gxy' for 'x hired y', 'Hy' for 'y was an honors student', and 'a' for 'John'. Note that 'G (7x)(Fx and Gxy) y', in (1), means 'the broker who hired John hired y'.

### §47. Dispensability of singular terms.

Recourse to 'a exists', 'John exists', as a premiss for R4 and R5 has put us in the position of countenancing a new unreduced element of logical structure, 'exists', supplementary to the other recognized structural elements 'D', 'and', 'not', '(x)', '(Ex)', 'ég', '(Ax)'. When applied to '(Ax)Fx', the verb 'exists' can indeed be analyzed out, as seen early in the foregoing section; as applied to singular terms other than descriptions, however, 'exists' remains unreduced.

A good expedient, if we are intent on consolidating our logical theory and eliminating 'exists', is simply to identify all singular terms with descriptions. This is easily done. We may merely suppress every simple singular term, say 'a' ('Socrates', 'Pegasus'), in favor of '(Ax)Fx', fashioning 'Fx' in such a way as to attribute to x whatever we think of as the distinguishing feature of the supposed object a.

The name 'Homer' lends itself especially clearly to this procedure. Homer is known to all of us as the author of the *Odyssey* and the *Iliad* and thus only; so in place of the simple singular term 'Homer' we can write:

(Ax)(x wrote The Odyssey ... x wrote The Iliad)

Without any real change in meaning.

In the case of any simple singular term there ordinarily will be some such distinguishing characteristic which marks off the alleged object in the speaker's mind, though the particular characteristic selected may vary from one speaker to another. 'Pegasus',

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tures of our language. Since 'exists' had remained rather anoma-
lous in connection with simple singular terms (see especially §44)
but was satisfactorily analyzable in connection with descriptions,
this is worth knowing.

Such reducibility of singular terms to descriptions holds not
only for simple singular terms, like 'Socrates', but also for
whatever complex singular terms there may be over and above des-
criptions themselves; e.g., '3 + 4'. The sign '+' is known as a
two-place functor; the characteristic of a functor is that it is ap-
plies to singular terms (two at a time, in the present case) to
produce a singular term.

Now instead of using
this two-place functor we could use a corresponding three-place rela-
tive term, let us say 'F', where 'Fxyz' is interpreted as holding
if and only if y and z are numbers and x is their sum. For,
'3 + 4' can then be rendered on this new basis as a description
'(1x)Fx34'. The surviving simple singular terms within this, viz.
'3' and '4', can themselves be supplanted in turn by descriptions
in view of the previous considerations. Thus it is that descrip-
tions can be made to serve the purposes of singular terms of all
kinds by a superficial change of language.

Furthermore descriptions, it turns out, can be dispensed with
completely in turn! This fact, first pointed out by Bertrand
Russell, is seen as follows. We have already observed (§ 16) that
the idiom '(1x)Fx' or its equivalent is in practice used only when
the user believes or pretends that there is one and only one object
x such that Fx. But, if this much is already acceptable, the nota-
tion '(1x)Fx' itself becomes superfluous; wherever it is used we
could instead use simply 'x', provided merely that we quantify the
whole context in the fashion '(Ex)' and insert a further restric-
tive clause of such kind as to determine x as the one and only
object such that Fx. This restrictive clause is:

Fx . F-nothing-but-x,

i.e.:

Fx . (y)(Fy ⊢ y = x).

In short, if '... (1x)Fx...' represents a statement involving the
description '(1x)Fx', we can accomplish the purpose of that state-
ment without use of the description by saying instead:

(Ex)(Fx . (y)(Fy ⊢ y = x) ... x ...)).

Instead of saying:

The woman who took my order this morning had red hair,

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we may say:

(Ex) (x is a woman. x took my order this morning.

(y) (y is a woman. y took my order this morning.)

y = x) . x has red hair).

It is not always convenient to rephrase our observations in this
way, but it is theoretically important that it can always be done;
for, this means that descriptions, and therefore singular terms
generally, are theoretically dispensable altogether. It is theo-
retically possible to drop '(x)' and 'exists' from the list of
the structural elements of logic, to drop the axioms of descrip-
tion and the extensions of R4 and R5 from our machinery of logical
inference, and to drop singular terms and functors and description
prefixes from our list of the parts of speech of logical grammar.

The parts of speech then come to comprise merely general terms,
variables, quantifiers, truth-function signs, and parentheses. The
complete disappearance of singular terms does not mean that our
language has lost all means of talking about objects; on the contrary,
the foregoing considerations show that the extrusion of singular
terms is unaccompanied by any diminution in the power of the langu-
age. What the disappearance of singular terms does mean is that all
reference to objects of any kind, concrete or abstract, is narrowed
down now to one specific channel: variables of quantification. We
can still say anything we want about any one object or all objects,
but we say it always through the idioms of quantification: 'There
is an object x such that ...' and 'Every object x is such that ...'.

The importance of this elimination of singular terms is that it
shows the variable to be the key to extra-linguistic reference. The
objects whose existence is implied in our discourse are finally just
the objects which must, for the truth of our assertions, be acknow-
ledged as "values of variables" — i.e., be reckoned into the to-
tality of objects over which our variables of quantification range.
To be is to be a value of a variable. There are no ultimate philo-
sophical problems concerning terms and their references, but only
concerning variables and their values; and there are no ultimate
philosophical problems concerning existence except insofar as exis-
tence is expressed by the quantifier '(Ex)'. Except when we are
concerned with philosophical issues of linguistic reference and
existence, on the other hand, there is no point of depriving our-
selves of the convenience of singular terms; and accordingly the
techniques of inference developed in §44 and 46 are not to be
thought of as abandoned.

Finally a word should be said of term matrices. This topic
has been withheld in order not to clutter the foregoing discussion.

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Term matrices are to singular terms what statement matrices (or
briefly matrices, hitherto so-called) are to statements. Just as
a statement matrix can be turned into a statement by putting singu-
lar terms for variables, so a term matrix can be turned into a
singular term by putting singular terms for variables. Examples
of term matrices are the variables themselves, also 'x + 4',
'x - y', 'the author of y' (i.e., '(nx)(x wrote y)'). A descrip-
tion, as this last example illustrates, should no longer be classed
merely as a kind of singular term (as was done in §48), but rather
as a kind of singular term or term matrix; it may be a term matrix
or a term according as it has a free variable or not. What has
been said of the reduction of complex singular terms to descriptions,
and of the eliminability in turn of descriptions themselves, applies
as well to term matrices as to terms; the fact that a free variable
is being carried along alters nothing in the reasoning. Thus singu-
lar terms and term matrices are all in theory dispensable, with the
sole exception of the variables themselves.

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