# THE PARADOXES OF LOGIC

TRANSLATION BY JÉRÔME VERSTRYNGE

ABSTRACT. In order to facilitate historical research in logic, we provide an English translation of a French article written by Chaïm Perelman, titled 'Les paradoxes de la logique', published in *Mind* in April 1936.

## The paradoxes of logic

[Off-printed from MIND: a Quarterly Review of Psychology and Philosophy. Vol. XLV., N.S., No. 178.]

by Ch. Perelman

## THE PARADOXES OF LOGIC<fnref n="1" />

1. At the end of the last century, logic and set theory were disturbed by the appearance of strange antinomies,<fnref n="2" /> which seemed to require the modification of the very bases of logic. This enterprise, in which many scientists were interested, resulted in the creation of Lord Russell's theory of types and the success of Mr. Brouwer's intuitionist conception of mathematics.

Let us consider these paradoxical hypotheses which lead to antinomies. We find ourselves in the presence of a paradoxical hypothesis when one manages to deduce from it a proposition *p* such that, by laying it down, one finds itself obliged to affirm not-*p* and, on the other hand, if one supposes the truth of not-*p*, one is led to affirm *p*. It is this continual swing between *p* and not-*p*, excluding all possibilities of avoiding a contradiction, which has led logicians who have dealt with this question to doubt the validity of some rules of classical logic.

Note that this see-saw is necessary for there to be a paradox. Indeed, suppose that the affirmation of *p* leads us to not-*p*, but one cannot deduce *p* from not-*p*: one will not have a paradox. For, if the reciprocal implication of *p* and not-*p* lays down the equivalence of two contradictory propositions, i.e.

<footnote n="1">¹This article was prompted by research in the mathematical logic seminar of the University of Brussels, and the support of its director, Professor Barzin, played a large part in its realization. I ask my master to accept all my thanks here.</footnote>

<footnote n="2">²For a presentation of the main antinomies, see in particular: B. Russell and A. Whitehead, *Principia Mathematica*, I, Cambridge, 1910; A. Fraenkel, *Zekn Vorlesungen über die Orundlegung der Mengenlehre*, Leipzig, 1927.</footnote>

<footnote>R. Camap, *Die Antinomien und die Unvollständigkeit der Mathematik*, Monatshefte für Mathematik und Physik, 41st v., 2nd notebook, pp. 263-284, Leipzig, 1934.</footnote>
a true antinomy, the fact that *p* implies not-*p*, without the contrary being true, is easily explained by the hypothesis that *p* is false. This is why only the hypotheses where one arrives at the equivalence of a proposition and its negation will be studied more closely by us.

2. We will focus, in what follows, on the analysis of the paradoxes of the liar, of the barber, of the class of classes that are not a member of themselves (to which one can reduce the antinomies of the predicate not attributable to itself and the concept of "heterologisch"), and that of relations.

(a) He who says "I am lying now", affirms that he is lying by this proposition. If he lies, he does not lie; if he doesn't lie, he lies. Moreover, to say that one is lying by affirming this proposition is to say "this proposition is false". If it is false, then it is true; if it is true, it is false. These two antinomies are logically identical and only differ in their expression.

(b) The barber's antinomy assumes that the men shaved by the only barber in the village are all those who do not shave themselves. Does the barber shave himself or not? If he shaves himself, he does not shave himself, because he only shaves those who do not shave themselves. If he does not shave himself, he shaves himself, the barber shaving everyone who does not shave themselves.

(c) One can distinguish two kinds of classes, those that satisfy the function of which they are the extension, and those that do not satisfy it; in other words, the classes that are members of themselves, and those that are not. The class of classes that are not members of themselves must also either be members of themselves or not be members of themselves. Let's suppose that they are a member of themselves; they are then part of the class of classes that are not a member of themselves and are therefore not a member of themselves. On the other hand, if they are not a member of themselves, they are part of the other class, the class of classes that are members of themselves and they are therefore a member of themselves.

(d) Whenever any relation *R* does not have the relation *R* with *S* it has the relation *T* with *S* and vice versa. Let us replace *R* with *T*. It follows that, whenever *T* does not have the relation *T* with *S*, it has the relation *T* with *S*, and vice versa.

3. Here is the problem laid down by these antinomies: one is assuming that the hypotheses from which one starts are perfectly valid for classical logic, and that, despite their legitimacy, one manages to deduce contradictory consequences from these. One concludes that one must restrict the rules of classical logic which tolerates these hypotheses leading us to inextricable antinomies; one modifies them; one imposes additional conditions on the construction of valid propositions. One seeks to remedy the antinomies by this or that modification of the rules of classical logic, which, unfortunately, usually entails the obligation to sacrifice other logical or mathematical propositions, those which are precious, become innocent victims of the
purification work undertaken without pity. If one wants to save these pro-
posals, whose legitimacy or fruitfulness no one had previously contested, one
is left with more or less ingenious and rarely convincing expedients.

Considering the uncertainty thus introduced into logic and mathematics,
it seems to us that one can lift it by showing that these are not the rules
of classical logic, which must be held responsible for the paradoxes, but the
transgressions of these rules, committed by laying down hypotheses which
lead to antinomies.

4. If one had laid down as a premise of reasoning, the assertion *p* is equivalent to not-*p* (*p* ≡ ~ *p*) one could have obviously deduced all sorts of contradictory consequences, and in particular, by laying down *p*, one could have deduced not-*p*, and from not-*p* one would easily have deduced *p*. No one, however, would have accused the rules of logic of being responsible for these contradictions, which one would have found, not paradoxical, but entirely natural. It would have been the same if one had affirmed that *p* is equivalent to not-*p*, not at once, but with the help of two simultaneous assertions, like "*p* is equivalent to *q*" and "*q* is equivalent to not-*p*". We will see that all the paradoxes exposed in paragraph 2, can be reduced, basically, to the affirmation that *p* is equivalent to not-*p*, the latter being introduced in a more or less indirect way, being in itself a source of astonishment.

Let us try to formalize the antinomy "this proposition is false". To assert
that "proposition *p* is false" (1) amounts to asserting not-*p*. Let's give
proposition (1) the name *q*; hence, not-*p* is equivalent to *q*. To say "this
proposition is false" is to replace in proposition (1) *p* by *q*, therefore one
affirms that non-*q* is equivalent to *q*. One thus sees that the first antinomy
is about laying down a contradiction in substance, and one should not be
surprised if the consequences one draws from it are also contradictory.

It could be objected to me that the assertion "this proposition is false"
does not constitute the premise of the antinomy, but that it is already the
contradictory conclusion, deduced however from perfectly valid premises.
Let's ask ourselves what these premises are. The only premise that one
could indicate is the equivalence between not-p and q, where one would
replace p with q. Now for this replacement operation to be possible, one
must have affirmed the equivalence of not-p and q whatever the value of
p, or (p) . ~ p ≡ q. But this proposition is false. Indeed, for a formal
equivalence to be true, it must be verified by all the values of the variable,
which is not the case here; because there is one value, at least, for which the
equivalence is false, and that is the value q. Moreover, this equating of a
function with a constant will immediately arouse everyone's distrust. If the
proposition ~ q ≡ q is false, it is because it was deduced from an equally
false premise, which contained in its seed, the proposition (p) . ~ p ≡ q.
We see here the use of a very ingenious process to hide the contradiction:

We see here the use of a very ingenious process to hide the contradiction:
it is that of the generalization, in the equivalence between *q* and non-*q*, of
the latter term. One thus manages to hide, amid the infinity of possible
values of $p$ in $\sim p \equiv q$, the case obtained by the replacement of $p$ by $q$, where the contradiction bursts out to anyone's astonishment.

It is easy to show that the expression $(x) \sim aRx \equiv aRb$ is as false as $(p) \sim p \equiv q$, and for the same reason. The falsity of this equivalence is moreover so apparent that no logician would have found the slightest difficulty in denouncing it.

The most ingenious paradoxes hide their faults better. First of all, so as not to arouse the suspicion that one feels when faced with an equivalence where a member contains an extra variable, it is the same variable that, in the considered paradoxes, one will put in the different variable places. This is how, by generalizing $(x) \sim aRx \equiv aRb$, one will not come to the expression $(y) \cdot (x) \cdot yRx \equiv yRb$, but to the equivalence $(x) \sim xRx \equiv xRb$. This last formal equivalence, even though it seems more correct, is as false as the previous ones, because it is not verified in the case where one considers $b$ as the argument of the function; but since, in general, in paradoxes, the equivalence is not formalized, and since such is true in all cases except one, one does not think of this case where the equivalence's failure is exposed, and thus one is surprised to deduce the antinomy $\sim bRb \equiv bRb$. The expression $(x) \sim xRx = xRb$ constitutes the common form to the barber and the class of classes that are not members of themselves paradoxes. To obtain the barber's paradox, one only has to replace $R$ by "being shaved by", $b$ by "the village barber", and consider that only men of the village are possible values of $x$. To obtain the second paradox, it is enough to replace $R$ by the relation $\epsilon$ (being a member of), $h$ by the class of classes that are not members of themselves, and restrict the domain of variation $x$ to classes.

The last paradox only differs from the previous ones in that one considers the relationship itself to be variable. We immediately see that the equivalence $\sim aRS \equiv aTS$ is false in the case where one replaces $R$ with $T$, and that one, therefore, does not have the right to assert the formal equivalence $(R) \sim aRS \equiv aTS$. To hide the falsity of this expression, and to avoid making a constant equivalent to a propositional function, one will replace $a$ by $R$, hence $(R) \sim RRS \equiv RTS$, an equivalence as false as the previous one.

Let us note here that if the construction of the paradox does not require a reflexive relation, it nevertheless requires, for the hypothesis to have an appearance of validity, that one of the members of the equivalence, by which the antinomy is laid down, to contain the same variable in two different places.

5. The analysis that we have just made allows us to conclude that, in order to avoid the paradoxes of logic, it is not necessary to modify its fundamental rules; it is enough to observe them carefully<fnref n="3" />.

<footnote n="3">³In a subsequent article, we will bring precision to the slight modification of the definition of the class, that the idea we have developed above begs.</footnote>
The antinomies arise, indeed, from the illegitimate affirmation (which
could easily be detected through a formalization) of a formal equivalence.
It is always illegitimate to make equivalent, for all its values, a function to
its negation, even if one modifies a member of the equivalence by putting an
argument of the function in place of a variable. The contradiction then arises
when one makes the two members symmetrical again, by putting the same
argument in place of the variable in the other member of the equivalence.

If, as I believe I have shown, the considered hypotheses, leading to antinomies, violate the rules of logic, it is not necessary to amend these using the theory of types, to avoid contradictions. I think, about the beneficial return, that a great number of mathematicians, realizing the ease with which one can avoid the antinomies of set theory, will no longer find the necessity to redesign mathematical principles, of the kind undertaken by Mr. Brouwer and his school.

CH. PERELMAN.