1.

1.

# I. Introduction.

An important part of the study of logic is the development and analysis of systems. A system has been defined as the knowledge of how statements stand together, preliminary to achieving a set or conjunct of postulates, axioms or code ascriptives which will allow the deduction of the remainder of the propositions or ascriptives contained in the system.

The differences in the approach to this study, between the literature and the school of thought exhibited in the lectures are, however, of a different kind than the divergences treated in the first paper. The points at issue between pre-assertional and propositional logic, are, to a large degree, of a metaphysical nature, and the answers to the metaphysical problems, in propositional and sentential logic, determine the formulation and the analysis of the "logical primitive." In the various approaches to the study of systems, there is, on the other hand, a large measure of agreement on the metaphysical issues. Points of divergence arise regarding precise formulation and nomenclature, of considerable importance, because exactness is a determining

1. In this introduction the word "set", "conjunct", "axiom", "postulate", "code ascriptive" will be used interchangeably pending definition.

apply bending definition.

"broad", "code scriptive" will be used interchange-

I. In this introduction the word "set", "connect", "axiom"

considerable importance, because exactness is a determining
raise regarding precise communication and nomenclature, of
agreement on the metachronous issues. Points of divergence
of systems, there is, on the other hand, a large measure of
"logical primitive". In the various abscapes to the same
logic, determine the communication and the analysis of the
metachronous problems, in probostitional and sentential
large degree, of a metachronous nature, and the answers to
between pre-saiential and probostitional logic, are to a
divergences treated in the first paper. The points at issue
in the texts are, however, of a different kind than the
tween the literature and the school of thought expounded

The differences in the approach to this study, pe-
sonal.

minder of the abstractions or abstractions contained in the
code scriptives which will allow the deduction of the re-
sary to deduce a set or context of postulates, axioms or
as the knowledge of how statements stand together, preemin-
ent and analysis of systems. A system has been defined

An important part of the study of logic is the devel-

I. *Introduction*.

I.

I.

2.

factor in a study of such scope.

There exist, nevertheless, wide areas of disagreement. Whether the system is implied by a set of postulates or a conjunct of ascriptives, matters not only in respect to clearness, but vitally influences the notion of a generalization of the code. Our examination of pre-codes will illustrate that point.

Nevertheless, the nature of the subject is such, the agreement as to the object of codifications so wide, that, contrary to the previous paper, the various theories will be developed concurrently and only their points of divergence brought into focus. Throughout, the metaphysical implications of such terms as "proposition" and "ascriptive", will be assumed as covered in the previous paper. This can, moreover, be done safely since none of the theories presented in this paper lean on such theories as the existential import of propositions.

port of propositions.

in this paper few on any theories as the existential im-
moreover, be done safely since none of the theories presented
will be assumed as covered in the previous paper. This can,
implications of any term as "proposition" and "axiomatic",
divergence brought into loona. Throughout, the metaphysical
will be developed concomitantly and only their points of
that, contrary to the previous paper, the various theories
the agreement as to the object of codifications so wide,
Nevertheless, the nature of the subject is one,
illustrate that point.

simplification of the code. Our examination of pre-codes will
be clearmeas, put vitally influences the notion of a gener-
on a context of axiomatics, matters not only in respect
ment. Whether the system is implied by a set of postulates
There exist, nevertheless, wide areas of disagree-
factor in a study of any scope.

2.

A

3.

## II. Systems and Codes.

1

A system has been defined as any body of propositions arranged according to a sequence of logical deductions, so that every proposition, after a certain one, is a logical consequence of some, or all, of the preceding propositions. This body of propositions that implies the rest, has been called the code by the lecturer, though this term is not used in any of the texts listed in the bibliography. The necessity of the code is given by the limitations of the human mind. A Being, such as God, whose knowledge is instantaneous, would have no recourse to a code. Yet the average person, who can not grasp the totality of the propositions (or ascriptives, see below) contained in the system without deduction, requires a body of propositions from which such a deduction can be effected. Few classical sciences and no empirical science, have in fact been completely systematized, and the degree of systematization is usually indicated by the necessity which attends conclusions. Mathematics, particularly geometry, have been the most widely systematized, resulting in Peirce's definition of "Mathematics as the science that draws necessary conclusions." Whatever the merits of this definition - and against it might be set Russell's epitaph as the science "where we

1. Young - Fundamental Concepts of Algebra, p. 2.

I. Young - Fundamental Concepts of Algebra, p. 5.

might be set forth as the science "where we
whatever the merits of this definition - and against it
maths as the science that draws necessarily conclusions."
systematized, resulting in being a definition of "Math-
ematics, partially Geometry, have been the most widely
nearly indicated by the necessity which attends conclusions,
directly systematized, and the degree of systematization is
science and no empirical science, have in fact been com-
which and a deduction can be effected. New classical
without deduction, requires a body of propositions from
tions (or axiomatics, see below) contained in the system
age person, who can not grasp the totality of the probab-
ity, would have no recourse to a code. Yet the aver-
human mind. A Being, and as God, whose knowledge is in-
necessary of the code is given by the limitations of the
used in any of the texts stated in the bibliography. The
called the code by the lecturer, though this form is not
this body of propositions that implies the rest, has been
consequence of some, or all, of the preceding propositions.
so that every proposition, after a certain one, is a logical
tions arranged according to a sequence of logical propositions,
A system has been defined as any body of probab-
If

II. Systems and Codes.

3.

4.

never know what we are talking about, nor whether what we
are saying is true", - the literature almost exclusively
deals with mathematical systems and so will this paper.
The applicability of codifying to all fields, however, will
be discussed under the heading of "generalizing the notion
of a code."

Historically it had been assumed that all systems,
such as Euclidian geometry, were based on a set of proposi-
tions, self-evidently true, the axioms, from which all other
propositions, the theorems, followed deductively. Mathemat-
icians have proved, in the meantime, however, that the
Euclidian axioms can not be considered universally self-
evident. It is possible, for example, to conceive a world
entirely enclosed within a large sphere, with the tempera-
ture decreasing point by point, reaching zero at the boun-
dary. In such a world, every inhabitant would conceive him-
self in a universe of infinite extent, for as he approached
the boundary, the falling temperature would decrease his
size, causing his steps to be smaller and smaller. The
Geometry of such a world would have many axioms, the self-
evidence of which would vary considerably from Euclid's.
It can be proved, f.e., that the shortest distance between
two points in that case would be the arc of a circle, which

1. Poincare - Science and Hypothesis, p. 49.

I. Boinville - Science and Hypothesis, p. 49.

two points in that case would be the arc of a circle, which
It can be proved, i.e., that the shortest distance between
evidence of which would vary considerably from Euclid's.
Geometry of such a world would have many axioms, the self-
size, causing his steps to be smaller and smaller. The
the pomperly, the falling temperature would decrease his
self in a universe of infinite extent, for as he abridged
qarly. In such a world, evenly inaptant would conceive him-
time decreasing point dy point, reaching zero at the pom-
fully encased within a large sphere, with the tempe-
rent. It is possible for example, to conceive a world
Enclidian axioms can not be considered universally self-
to have proved, in the meantime, however, that the
propositions, the theorems, followed deductively. Mathemat-
tions, a self-evidently true, the axioms, from which all other
and so Enclidian geometry, were based on a set of proposit-
Historically it had been assumed that all axioms,
of a code."

be discussed under the heading of "Generalizing the notion
The applicability of codifying to all types, however, will
does with mathematical systems and so will this paper.
are saying is true", - the literature almost exclaimingly
never know what we are talking about, nor whether what we

4.

5.

cuts the bounding sphere orthogonally. ¹ This is the basis of non-Euclidean mathematics.

Moreover, from the point-of-view, of modern logic, "self-evidence" can not be conceived as a predicate of a proposition, but as a dyadic relation between the person making the statement, and the proposition.

² Professor Royce has classified the various assumptions underlying the use of the term axiom. It can be used either in the sense of inner necessity or self-evidence, subjective necessity, as an unproved hypothesis from which, however, other propositions can be deduced, as a proposition universally assented to, and lastly as a proposition resulting from an individual's psycho-genetic constitution. The inner necessity theory is discarded both because of the reasons above stated, as well as because of the impossibility of defining a criterion for certainty. Descartes and Spinoza, for example, despite the fact that their whole philosophy was based on innate ideas, could offer no better criterion than inner certainty. The fact that Descartes invoked the intervention of God as a guarantor of objective reality, has no logical significance in this connection. According

1. Young, op. cit., p. 17.

2. Axioms - Hastings Encyclopedia of Arts and Sciences.

2. **Axioma** - A Handbook Encyclopedia of Arts and Sciences.
   J. Young, op. cit., p. 17.

has no logical significance in this connection. According
intervention of God as a guarantee of objective reality,
than inner certainty. The fact that Descartes invoked the
was based on innate ideas, could offer no better criterion
for example, despite the fact that their whole philosophy
of defining a criterion for certainty. Descartes and Spinoza,
reasons spoke stated, as well as because of the impossibility
inner necessity theory is disallowed both because of the
ing from an individual's balanced-genetic constitution. The
universally asserted to, and fairly as a proposition result-
however, other propositions can be deduced, as a proposition
subjective necessity, as an improved hypothesis from which,
either in the sense of inner necessity or self-evidence,
tions underlying the use of the term axiom. It can be need
Professor Royce has classified the various assemp-
S

making the statement, and the proposition.
proposition, put as a dyadic relation between the bemon
"self-evidence" can not be conceived as a predicate of a
Moreover, from the bound-ol-view of modern logic,
of non-Enlighten mathematics.
out the poming sphere orthogonally. This is the basis
f

c.

[y]

6.

to Professor Royce, this criterion can only qualify in the
sense of "subjective certainty", not self-evidence and there
is no necessary connection between the statement "This is
true" and the objective reality of the subject.

The third sense in which axiom has been used, as an
unproved hypothesis, will be discussed fully below. In the
Aristotelean sense, by which a first principle must inhere
in all scientific deductions as a part of a universal first
principle, the discovery of a valid system, though its
axioms, might have been assumed as hypotheses to start with,
imparts to those axioms a certain, highly mediated "self-
evidence". For practical purposes, however, this self-evi-
dence is equivalent to the assumption of unproved hypothesis
as a starting point.

The fourth usage of universal assent defining an
axiom, implies its own contradiction, according to Professor
Royce. For one would not invoke universal assent as an argu-
ment, except against a person not assenting. One dissent
will therefore shake the validity of the axiom, forcing re-
course to such limitations on universality as "assent of the
wise", "assent of the capable", etc.

The last meaning of an axiom, imposed by a self-evi-
dence dependent on psycho-genetic factors, is thereby made so
vague as to lose all usefulness. To begin with, the sub-
conscious operations of the mind very frequently obscure our

conjunction operations of the mind very frequently opaceme on
agene as to lose all meaningfulness. To begin with, the and-
gence dependent on batocho-genetic factors, is therelby made so
wise as "assent of the cabaple", etc.

come to such limitations on universality as "assent of the
will therefore abate the validity of the axiom, forcing re-
ment, except against a person not assenting. One disassent
Bolee. For one would not inoke universal assent as an argu-
sion implies its own contradiction, according to Professor
The fourth made of universal assent defining an
as a starting point.

gence is equivalent to the assumption of improved hypothesis
evidence". For practical purposes, however, this self-evi-
dence to those axioms a certain, highly mediated "self-
axioms, might have been assumed as hypothesis to start with,
principle, the discovery of a valid system, through its
in all scientific deductions as a part of a universal that
Virtuteleesn aena, dy which a first principle must inhere
improved hypothesis, will be disallowed fully below. In the
The third aena in which axiom has been made, as an
true, and the objective reality of the subject.

is no necessary connection between the statement "This is
aena of "subjective certainty", not self-evidence and there
to Professor Bolee, this criterion can only qualify in the

6.

[A]

7.

real meanings and thus this is a proper field for the psy-
chiatrist, rather than the philosopher.

This leaves the definition of axiom, alluded to before.
The axiom is conceived as an unproved proposition, composed
of undefined terms and assumed relations, which are required
to be intuitively understood, from which to deduce the re-
maining propositions contained in the system. To be sure,
this precludes the existence, or at least the necessary self-
evidence, of a universal first principle and a self-contained
system of the universe. But, on the other hand, it allows
the formulation of systems, in the Hegelian sense, in which
the axioms and theorems mutually imply each other and in
which the axioms of one system can theoretically be the
theorems of another and a high degree of coherence thereby
achieved. We will come back to this when the generalized
notion of a code is discussed.

The arguments of Professor Royce have been developed
at such length because they contain the theoretical basis for
the modern conception of axioms. Since "self-evidence" has
been discarded as a criterion, the last formulation of Pro-
fessor Royce's is the general usage of an axiom. (Professor
Huntington's concept of an axiom as a statement of fact,
opposed to postulates as defining required conditions, will

1. Young - Monographs on Modern Mathematics, p. 172.

J. Young - Monographs on Modern Mathematics, p. IVs.

obposed to postulates as defining required conditions, will
Huntington's concept of an axiom as a statement of fact.
Professor Royce's is the General scheme of an axiom. (Professor
been discussed as a criterion, the last formulation of Pro-
the modern conception of axioms. Since "self-evidence" has
at and length because they contain the theoretical data for
The arguments of Professor Royce have been developed
notion of a code is discussed.

suggested. We will come back to this when the generalized
theorem of another and a high degree of conference thereto
which the axioms of one system can theoretically be the
the axioms and theorems mutually imply each other and in
the formulation of axioms, in the Hegelian sense, in which
system of the universe. But on the other hand, it affords
evidence, of a universal fact, which is true and a self-contained
this precedes the existence, or at least the necessarily self-
mining propositions contained in the system. To be ame,
to be intuitively understood, from which to deduce the re-
of undefined terms and assumed relations, which are required
The axiom is conceived as an improved proposition, composed
This leaves the definition of axiom, affixed to before.

qualified, rather than the builbooth.
This leaves the definition of axiom, affixed to before.

The meanings and thus this is a proper field for the bay-

8.

be discussed below). The axiom then contains certain undefined terms and relations and is itself an unproved-deductively-proposition. In order to extend the area of deductive reasoning and to achieve the greatest simplicity, it is useful to limit the number of terms and relations required to be intuitively understood. The choice of the terms and relations is, however, entirely arbitrary and limited only by their appropriateness to the system, the considerations set forth above and the proviso of intersignificance. (Thus the term point would preclude the notion of perpendicularity).

This raises a question as to the concept of simplicity. By simplicity, both in the literature and in lectures, is meant logical simplicity. From a logical point of view, every term is equivalent, or determined only by its logical implications, regardless of how many psychological difficulties it presents. Thus a triadic relation is logically simpler than a quadric relation, (See below comparison of Huntington's and Veblen's code).

The objects of codification are the same in the literature and in the ascriptival logic as presented during lectures. Several differences, some in nomenclature, and some of fundamental importance, are apparent, however. To begin

1. Young - Fundamental Concepts, p. 163.

["J. Young - Fundamental Concepts, b. 193.", "of fundamental importance, are absent, however. To begin\nfrom time. several differences, some in nomenclature, and some\nare and in the abstract logic as presented giving the\nidea and in the abstract relation it presents. Thus a logical\nrelation is a logical implication, regardless of how many beliove-\npoint of view, every term is equivalent, or determined only\nfrom the abstract relation it presents. Thus a logical relation\nis a logical implication, regardless of how many beliove-\npoint of view, every term is equivalent, or determined only\nfrom the abstract

9.

with, ascriptival logic does not utilize propositions.

Starting with an undefined term and an undefined, intersig-
nificant relation, it develops an ascriptive, a pure hypo-
thesis, exhibiting fully the argument before endorsement
occurs. For the sake of simplicity, moreover, ascriptival
logic admits only those ascriptives that can be positively
endorsed. Ascriptives are particularly useful in codifica-
tion. Since the code implies the system, but subjectively
requires a deduction of the system from the code, it is im-
perative that the constituent elements of the code (be they
propositions or sentences or ascriptives) exhibit their
arguments with maximum clarity. Since no semantic necessi-
ties are imposed on ascriptives by metaphysical doctrines
such as existential import, the possibilities of an argument
are most fully presented. Moreover, even a propositional
logician like Stebbing (though her theory can not be called
so in the strict sense as was discussed in the first paper),
maintains that the pure logician is not concerned when he
looks for axioms, with their truth or falsity, but only with
testing all their consequences, the real criterion of the
validity of the axioms being the validity of the theorems.

A more fundamental divergence occurs in the very con-
cept of codification. It was already mentioned that what

1. Stebbing - An Introduction to Logic, p. 164.

f. Stepping - An Introduction to Logic, p. 164.

The term 'axioma' has already been mentioned that what
A more fundamental divergence axioma in the very con-
cept of codification. It was already mentioned that what
f. Stepping - An Introduction to Logic, p. 164.

3.

10.

1

Young calls his "set of assumptions", the lecturer calls a code. The basic distinction, however, consists of the fact that the code is not a *set*, but a conjunct of all its ascriptives which implies the conjunct of all ascriptives constituting the system, and, being itself of the system, is implied by it. The notion of a conjunct as opposed to a set, is of fundamental importance in developing a formulation for the pre-code. (See below). Several other terms must be discussed. The ascriptives contained in the code are called code ascriptives. Throughout a strict distinction is maintained between the meaning of the expression and the words denoting it. This is achieved by utilizing the device of single quotes. Thus the highly complex ascriptive, which is the conjunct of all code ascriptives, is called the code, while the same expression in single quotes and then denot-ing merely the constituent words, is the code expression.

Definitions play an important role in making code- ascriptives or axioms psychologically simpler. Russell offers the following definition of "definitions".² "The definition must be in terms of other expressions already defined. Which are the terms to be previously defined depends

1. Young - Fundamental Concepts of Algebra, p. 222.

2. Quoted in Stebbing, op. cit., p. 441.

S. Graded in stepping, ob. cit., b. 44f.
J. Young - Fundamental Concepts of Virgina, b. 555.

defined. Which are the terms to be broadly defined depends
definition must be in terms of other expressions already
offers the following definition of "definitions". "The
sacriſives or axioms formally ſimper. Hsueſſ
Definitions play an important role in making code-

ing merely the constituent words, is the code expression.
while the same expression in single quotes and then denot-
ing the constituent of all code sacerſtives, is called the code,
or single quotes. Thus the highly complex sacerſtive, which
words denoting it. This is achieved by utilizing the device
is maintained between the meaning of the expression and the
called code sacerſtives. Throughout a strict diſtinction
be disallowed. The sacerſtives contained in the code are
to the pre-code. (See below). Several other terms must
set, is of fundamental importance in developing a formulation
implied by it. The notion of a constituent as opposed to a
constituting the atom, and being itself of the atom, is
sacerſtives which implies the constituent of all sacerſtives
that the code is not a set, put a constituent of all its
a code. The basic diſtinction, however, consists of the
Young calls his "set of assumptions", the lexicographical call

10.

11.

to a large extent on the person making the definition." To
this it might be added that the definition might also be in
terms of notions and relations left undefined to serve as a
starting point. The definition thus substitutes a term for
an expression. This illustrates that the lecturer's choice
of terms expresses more clearly the nature of the operation,
by calling it abbreviation.

It must be noted, as Russell does by implications,
that abbreviation is not a question of fact, but of conven-
tion and that the only requirement imposed is, that if the
abbreviandum is in terms of a subject-matter having certain
conventional meanings, the abbrevion must be the term com-
monly associated with the abbreviandum in order to be mean-
ingful. Thus, a definition which conventionally describes
a line-segment could not as an abbrevion utilize the word
"circle." The word "blah" would be logically acceptable,
but meaningless in terms of the subject-matter.

The difference between definitions and primitive
code-ascriptives or axioms is difficult to fix. Indeed
Young calls his set of assumptions "hidden definitions."

Three properties are usually sought for in a code:
consistency, independence and completeness (or categoricalness).

1. Young - Fundamental Concepts of Geometry, p. 53.

J. Yonnel - Fundamental Concepts of Geometry, p. 22.

certainty, independence and completeness (or categoricity).

Yonnel calls a set of assumptions "biggen definitions." A code-ascription or axiom is sufficient to fix. Indeed the difference between definitions and primitives put meaningless in terms of the subject-matter.

"Circle." The word "plan" would be logically acceptable, a five-segment, could not be an appreciation utilize the word itself. Thus, a definition which conventionally describes commonly associated with the appreciation in order to be meaningful meaning, the appreciation what be the term conventional appellation is in terms of a subject-matter having certain tion and that the only requirement imposed is, that it be that appreciation is not a question of fact, put of conven-

It must be noted, as Unself goes by implications,

by calling it appreciation.

of terms expresses more clearly the nature of the operation, an expression. This illustrates that the lecturer's choice starting point. The definition thus substantiates a term for terms of notions and relations left undefined to serve as a this it might be added that the definition might also be in to a large extent on the basis making the definition." To

ix.

12.

Both the literature in the bibliography and the lecturer,
maintain that the only required property is consistency,
and that independence and completeness are sometimes desir-
able, but not a logical necessity.

Independence is the predicate of a postulate or code
ascriptive that is not implied by any of the preceding code-
ascriptives, nor their conjunct. Independence is proved by
finding a concrete application which satisfies all axioms
or code-ascriptives except the axiom or code-ascriptive in
question. Independence is useful for a distinguishing be-
tween theorems and axioms but has no necessary significance.
Moreover, complete independence, despite the above proof,
may still be lacking, for a certain amount of implication
or overlapping, by which a part of the postulate is implied
by the preceding or their conjunct, is frequently unavoidable.

Completeness is the predicate of a code that implies
all the remaining ascriptives and theorems that belong to
the whole subject-matter. The test for completeness, that
any two concrete applications of a code are simply isomorphic,
is beyond the scope of this paper. Completeness also is a
property, not absolutely requisite. Indeed, sometimes a
part-code to analyze special portions of a system, is even
desirable.

Consistency is the one essential property of a code.
Without it no deduction would be possible, no system could

Without it no deduction would be possible, no system could
consistency is the one essential property of a code.

barb-code to analyze specialist portions of a system, is even
property, not absolutely redundant. Indeed, sometimes a
is beyond the scope of this paper. Completeness also is a
and two concrete applications of a code are simply incommensible,
the whole subject-matter. The test for completeness, that
all the remaining axioms and theorems that belong to
Completeness is the predicate of a code that implies
by the preceding or their context, is frequently unambiguous.

or overabbing, by which a part of the boatnate is implied
with a still be packing, for a certain amount of implication
Moreover, complete independence, despite the above proof,
tween theorems and axioms put has no necessary significance.

dismantle. Independence is meant for a distinguishing per-
or code-sacriptives except the axiom or code-sacriptive in
finding a concrete application which satisfies all axioms
sacriptives, not their context. Independence is proved by
sacriptive that is not implied by any of the preceding code-
Independence is the predicate of a boatnate or code
apse, put not a logical necessity.

and that independence and completeness are sometimes gear-
maintain that the only required property is consistency,
Both the literature in the bibliography and the literature,

13.

be constructed. Tests for consistency and a general discussion of the notion will be developed in relation to the pre-code. A basic distinction between traditional and ascriptival logic must be mentioned, however. Traditional logic defines consistency as the predicate of a postualte from which no contradictory arguments can be deduced. Ascriptival logic, on the other hand, can never face the problem of consistency in this form. Ascriptives have to be affirmed or denied, no other endorsement is possible. The problem of consistency consequently precedes the problem of endorsement. If this is the case, consistency in ascriptival logic depends on the internal structure of the argument, resides in the combination of predicates and relations, and can be defined as an ascriptive that involves no internal contradiction.

contradiction.

can be defined as an inscription that involves no *inferential*
action in the computation of predicates and relations, and
logic depends on the internal structure of the argument, re-
endorsement. It this is the case, consistency in inscriptional
problem of consistency consequently precedes the problem of
attested or denied, no other endorsement is possible. The
tem of consistency in this form. Inscription have to be
inscriptival logic, on the other hand, can never face the hop-
from which no contradictory arguments can be degreed. As-
logic defines consistency as the predicate of a bivalent
inscriptival logic must be mentioned, however. Traditional
brie-code. A basic distinction between traditional and as-
ciation of the notion will be developed in relation to the
be constructed. Tests for consistency and a general dia-

13.

14.

# Generalization of the Notion of a Code. 1

## a. Concrete Applications of Codes.

In 1904 Oswald Veblen developed a code for geometry utilizing only the notion of a point and the relation between. In 1911, Veblen wrote another code adding the notion of congruence to the terms above. In order to make his code psychologically simpler, Veblen divided his second code into several sections based on code-ascriptives dealing with a certain subject matter, such as assumptions of order, of points on a line, etc.

Huntington in 1913 developed a code based on the notion of sphere and the relation of inclusion. Faugère complexity of code ascriptives resulted and yet Huntington's code is logically simpler than Veblen's as inclusion, a diadic relation, is simpler than the triadic relation, between. As a general, though not necessary, rule, it can be stated that the more sparing the code concepts, the more complex, though logically simple, the code ascriptives will become.

Hilbert in an attempt to achieve a maximum of psychological simplicity, chose a wide range of code-concepts,

1. This section is based on the articles by Veblen and Huntington in Young - Monographs on Modern Mathematics as well as lectures. Points of divergence are so indicated.

created.

 points of divergence are so in-
Huntington in Young - Monographs on Modern Mathematics
f. This section is based on the articles by Veblen and
as well as rectangles.

epological simplicity, choose a wide range of code-concepts,
Hilbert in an attempt to achieve a maximum of ba-
become.

complex, though logically simple, the code scribed will
 stated that the more abating the code concebts, the more
tween. As a General, though not necessarily, rule, it can be
disadic relation, is simpler than the triadic relation, pe-
code is logically simpler than Veblen's as inclination, a
complexity of code scribedives resulted and yet Huntington's
notion of sphere and the relation of inclination. A guest-
 Huntington in 1913 developed a code based on the
points on a line, etc.

certain subject matter, and as assumptions of order, of
several sections based on code-scribedives dealing with a
epologically simpler, Veblen divided his second code into
gruence to the terms spake. In order to make his code bal-
In 1911, Veblen wrote another code adding the notion of com-
utilizing only the notion of a point and the relation between.

In 1904 Oaswaj Veblen developed a code for Geometry

a. Concrete Abstractions of Codes.

Generalisation of the Notion of a Code.

I

14.

15.

the notion of point, plane, line and the relations of be-
tween and congruence.

Several examples from Veblen's first code will be
presented to illustrate the nature of a code ascriptive and
to serve as a connecting link to our discussion of the gener-
alized notion of the code.

Assumption I: (Code Ascriptive I): If points A, B, C are
in the order ABC, they are distinct.

Assumption II. If points A, B, C are in the order (ABC)
they are not in the order (BCA).

These two code ascriptives, or in Veblen's terms
assumptions, can be combined into a quantified code ascrip-
tive in ascriptival logic.

(A, B, C): punctual <sup>A</sup>, punctual <sup>B</sup>, punctual <sup>C</sup>.

not identical A. not identical B. not identical C.

between ABC implying not between BAC.

Analysis reveals that this code ascriptive is com-
posed of two elements, the geometrical, represented by the
terms point and between, and the logical, expressed symboli-
cally by $\lambda$.

## II. Generalization of the Notion of a Code.

The first code-ascriptive, being composed of a logi-
cal part, represented symbolically by $\lambda$ and the notion of
point and between can be written thus $\lambda, \check{\lambda}$ (pt, between).

point and between can be written thus $y'N$ (bt. between).
cat bar', represented analytically $\lambda y$ and the notion of
The first code-scribative, being composed of a logi-
II. Generalization of the Notion of a Code.

caffly $\lambda y$.
terms point and between, and the logical, expressed analy-
boag of two elements, the geometrical, represented $\lambda$ the
Analysis reveals that this code scribedive is com-
$\lambda$ between ABC implying not between BAC.

not identical A. not identical B. not identical C.
(A, B, C): $\lambda$ mutual A, $\lambda$ mutual B, $\lambda$ mutual C.
tive in scripttval logic.

assumptions, can be composed into a quantified code scribed-
these two code scribedives, or in Aeplen's terms
they are not in the order (BCA).

Assumption II. It points A, B, C are in the order (ABC)
in the order ABC, they are distinct.

Assumption I: (Code Acribttive I): It points A, B, C are
sized notion of the code.

to serve as a connecting link to our discussion of the General-
presented to illustrate the nature of a code scribedive and
Several examples from Aeplen's first code will be
tween and convergence.

the notion of point, plane, fine and the relations of pe-

A

16.

The conjunct of the 12 code ascriptives, which constitute Veblen's first code will be written as $\lambda_0^\Upsilon$ (pt, between).

The act of logizing is thus independent of the notion of point and between which arises only out of the actual representation in a concrete system of the logical part as applied to a predicate and a triadic relation. The generalization of the notion of the code is made possible by conceiving of the code as a single, highly complex, quantified ascriptive derived as a conjunct of its constituent code-ascriptives.

Since the notions of point and between arise only through application to a concrete system, generalization will consist of substituting a quantified ascriptive composed of schemata, and thereby allowing the full examination of $\lambda_1$,

thus:

$$ [Q_2 R_3]^{\lambda_0} \{Q_2 R_3\} $$

This representation is called the pre-code. The argument that the code only arises after the code is given and should therefore, properly called post-code, is countered by stating that the time a function arises is not the determining factor but its logical precedence.

The fundamental importance of the code as a conjunct of the assumptions rather than a set of assumptions is now revealed. Both Huntington and Young attempted the generalization of a code. Yet since they were forced to generalize their individual

1. See Young - Concepts of Algebra, p. 41.
   Young - Monographs on Modern Mathematics, p. 186.

Yong - Monograph on Modern Mathematics, b. 186.
f. 2 see Yong - Concepts of Algebra, b. 41.

Yet since they were forced to generalize their individual
Huntington and Yong attempted the Generalization of a code.
f rather than a set of assumptions is now revealed. Both
The fundamental importance of the code as a conjoint or
mining factor put its logical precedence.
by stating that the time a function arises is not the deter-
and should therefore, probably, called host-code, is counter-
argument that the code only arises after the code is given
This representation is called the host-code. The
themselves:

```
{ (Q, R) (Q', R') } (Q, R)
```

Theorem: If $\mathfrak{R}$ is a field of characteristic $p$ and $\mathfrak{A}$ is a set of axiomatics, a quantified formula of a concrete system, Generalization will continue since the notions of point and between arise only through derived as a conjoint of its constituent code-suffixes. The code as a single, highly complex, quantified suffixive of the notion of the code is made possible by conceiving of it as a predicate and a triadic relation. The Generalization station in a concrete system of the logical part as applied to point and between which arises only out of the actual represeem- The set of logizing is thus independent of the notion of Aepin's first code will be written as $\mathfrak{A}$ (bt. between). The conjunction of the IS code suffixes, which constitute

18.

17.

axioms, no such clearly apparent pattern of logical structure appeared. Each axiom of their pre-code would have to be individually examined to determine its utility for a code. [See below]
This resulted in Huntington's definition of the assumptions of his precode as postulates or conditions, each of which had to be satisfied as a prerequisite to the construction of a code.

Ascriptival logic as represented in the quantified ascriptive of the pre-code, makes no demands. A code is obtained by instatement of the pre-code. This raises again the question as to how the consistency of the pre-code is established. Two considerations must be discussed. Classical logic asserts that to call a proposition consistent is equivalent to affirming it. The fallacy of this is particularly apparent when applied to pre-codes. If a consistent pre-code is instantiated by an empirical code, this code must also be consistent. But consistency in the empirical world does not guarantee existence. (An ascriptive about the existence of a 500 year old man, would be consistent, but also EN).

Though this consideration delimits the validity of inference from consistency, it tells nothing about a criterion for consistency. Yet an absolute criterion for consistency can not be given. To prove consistency, concepts would be utilized that are in turn assumed consistent. A

would be utilized that are in turn assumed consistent. A
consistency can not be given. A broad consistency, conceived
as a consequence from consistency, it tells nothing about a con-
tradiction for consistency. Yet an absolute criterion for con-
ference from consistency, it tells nothing about a crir-
on of a kind).

The existence of a 200 year old man, would be consistent,
world does not guarantee existence. (An ascriptive about
what also be consistent. But consistency in the empirical
bre-code is intuited by an empirical code, this code
familiar abherent when applied to pre-codes. It a consistent
infallible logic asserts that to call a proposition consistent is
eatified. Two considerations must be discussed. Classif-
the creation as to how the consistency of the pre-code is
tried by the determination of the pre-codes. This relates again
ascriptive of the pre-code, makes no demands. The
quantified
ascribital logic as represented in the quantified
of a code.

and to be ascribed as a prudence to the construction
of his precode as postulates or conditions, each of which
This resulted in Huntington's definition of the assumptions
be individually examined to determine its utility for a code.
per code.

The above is a free translation,

of the above as clearly apparent pattern of logical attri-
sioms, no and clearly apparent to the reader.

14.

18.

test for consistency, with these reservations, would be the
deducing of the pre-code as a theorem from a code of logic
that is assumed consistent.

A more general test for consistency is the instan-
tiation of the pre-code. Three possibilities of instantia-
tion, logical, mathematical and empirical, exists. Thus if
the pre-code is instantiated by a code in Euclidian geometry,
and we grant that Euclidian geometry is consistent, we have
a proof for the consistency of the pre-code.

In instantiation in an empirical system, the element
of simply passing the burden of proof to a higher order of
logic or another set of assumptions, is compounded by the
following epistemological assumptions:

(1) The empirical world is consistent.

(2) Therefore this part (say astronomy) of the
physical world is consistent.

(3) And our propositions about this part are true.

(4) Therefore $\lambda_0 \mathbf{Q}_{\text{astron}} \mathbf{R}_{\text{astron}}$ is true.

(5) Therefore $\lambda_0$ of $[\neg \mathbf{Q} \mathbf{R}] \wedge \{\mathbf{Q} \mathbf{R}\}$ is consistent.

This raises the question of the use of a pre-code.

By generalizing the notion of a code, both in the ascriptival
as well as in the propositional form, the possibility of
instantiation in any number of different sciences is given.
The realization that two different sciences have the same
logical structure (that is, utilize the same pre-code) is

["logical structure (that is, utilize the same brie-code) is\nthe realization that two different sciences have the same\ninstantiation in any number of different sciences is given.\nAs well as in the propositional form, the belpdillity of\nEl Generalizing the notion of a code, both in the scriptivat\nThis relates the question of the use of a brie-code.", "(2) Therefore $\\Upsilon$ of $\\mathfrak{C} \\otimes \\mathfrak{R} \\otimes \\mathfrak{Y}$ {OR} is consistent.", "(4) Therefore $\\Upsilon$ of $\\mathfrak{C}$ $\\mathfrak{R}$ $\\mathfrak{Y}$ $\\mathfrak{R}$ $\\mathfrak{R}$

20.

## Conclusions

The study of systems then is a study of a certain unity in the totality of nature. The unity is no longer given by self-evident propositions, but by studying all the implications and consequences of a conjunct of originally undefined assumptions or code-ascriptives. The theoretical frame-work within which various logicians have attempted to deal with this subject is relatively uniform. Yet ascriptival logic has achieved preciser formulation, a clearer definition of its object, and added at least one concept, that of the conjunct of ascriptives, which makes its pre-code considerably more serviceable and psychologically simpler than e. Huntington's or Young's.

Logic then has ceased to be a mere study of the meanings or possibilities of meanings of statements. It has been seen to be at the root of whatever man deduces out of the totality of a system by means of a code. Moreover, by generalizing the notion of a code and calling attention to the fact that logical similarity implies translatability, logic has re-established a unity based on the inner necessity of the totality of a system and opened new vistas for expanding our knowledge.

for expanding our knowledge.

necessarily of the totality of a system and opened new ways
of thinking, logic has re-applied a truth based on the inner
action to the fact that logical similarity implies translation
over, by generalizing the notion of a code and calling atten-
out of the totality of a system by means of a code. More-
over, it has been seen to be at the root of whatever man deduces
meanings or meanings of statements. It

logic then has ceased to be a mere study of the
similarity theme. Huntington's or Young's code considerably more serviceable and more interestingly

that of the context of sentences, which makes its pre-
delinition of its object and added at least one concept,
flexible logic has achieved greater tomulation, a clearer
to deal with this subject, is relatively uniform. Yet as

frame-work within which various logicians have attempted
undertaken assumptions or code-ascriptions. The theoretical
implications and consequences of a context of originally
Given by self-evident propositions, but by studying all the
mity in the totality of nature. The mity is no longer

**Conclusions**

50

1.

I.

I. Introduction.

An important part of the study of logic is the devel-
opment and analysis of systems. A system has been defined
as the knowledge of how statements stand together, prelimin-
ary to achieving a set or conjunct of postulates, axioms or
code ascriptives which will allow the deduction of the re-
mainder of the propositions or ascriptives contained in the
system.

The differences in the approach to this study, be-
tween the literature and the school of thought exhibited
in the lectures are, however, of a different kind than the
divergences treated in the first paper. The points at issue
between pre-assertional and propositional logic, are, to a
large degree, of a metaphysical nature, and the answers to
the metaphysical problems, in propositional and sentential
logic, determine the formulation and the analysis of the
"logical primitive." In the various approaches to the study
of systems, there is, on the other hand, a large measure of
agreement on the metaphysical issues. Points of divergence
arise regarding precise formulation and nomenclature, of
considerable importance, because exactness is a determining

2.

There exist, nevertheless, <u>fundamental</u> areas of disagree-
ment. Whether the system is implied by a set of postulates
or a conjunct of ascriptives, matters not only in respect
to clearness, but vitally influences the notion of a gener-
alization of the code. Our examination of pre-codes will
illustrate that point.

Nevertheless, the nature of the subject is such,
the agreement as to the object of codifications so wide,
that, contrary to the previous paper, the various theories
will be developed concurrently and only their points of
divergence brought into focus. Throughout, the metaphysical
implications of such terms as "proposition" and "ascriptive",
will be assumed as covered in the previous paper. This can,
moreover, be done safely since none of the theories presented
in this paper lean on such theories as the existential im-
port of propositions.