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FUTURE LOGIC

© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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CHAPTER 32. DOUBLE PARADOXES.

1. Definition.

2. The Liar Paradox.

3. The Barber Paradox.

1. Definition.

We have seen that logical propositions of the form ‘if P, then nonP’ (which equals to ‘nonP’) or ‘if nonP, then P’ (which equals to ‘P’), are perfectly legal. They signify that the antecedent is self-contradictory and logically impossible, and that the consequent is self-evident and logically necessary. As propositions in themselves, they are in no way antinomial; it is one of their constituents which is absurd.

Although either of those propositions, occurring alone, is formally quite acceptable and capable of truth, they can never be both true: they are irreconcilable contraries and their conjunction is formally impossible. For if they were ever both true, then both P and nonP would be implied true.

We must therefore distinguish between single paradox, which has (more precisely than previously suggested) the form ‘if P, then nonP; but if nonP, not-then P; whence nonP’, or the form ‘if nonP, then P; but if P, not-then nonP; whence P’ — and double paradox, which has the form ‘if P, then nonP, and if nonP, then P’.

Single paradox is, to repeat, within the bounds of logic, whereas double paradox is beyond those bounds. The former may well be true; the latter always signifies an error of reasoning. Yet, one might interject, double paradox occurs often enough in practise! However, that does not make it right, anymore than the occurrence of other kinds of error in practise make them true.

Double paradox is made possible, as we shall see, by a hidden misuse of concepts. It is sophistry par excellence, in that we get the superficial illusion of a meaningful statement yielding results contrary to reason. But upon further scrutiny, we can detect that some fallacy was involved, such as ambiguity or equivocation, which means that in fact the seeming contradiction never occurred.

Logic demands that either or both of the hypothetical propositions which constituted the double paradox, or paradox upon paradox, be false. Whereas single paradox is resolved, by concluding the consequent categorically, without denying the antecedent-consequent connection — double paradox is dissolved, by showing that one or both of the single paradoxes involved are untrue, nonexistent. Note well the difference in problem solution: resolution ‘explains’ the single paradox, whereas dissolution ‘explains away’ the double paradox.

The double paradox serves to show that we are making a mistake of some kind; the fact that we have come to a contradiction, is our index and proof enough that we have made a wrong assumption of sorts. Our ability to intuit logical connections correctly is not put in doubt, because the initial judgment was too rushed, without pondering the terms involved. Once the concepts involved are clarified, it is the rational faculty itself which pronounces the judgment against its previous impression of connection.

It must be understood that every double paradox (as indeed every single paradox), is teaching us something. Such events must not be regarded as threats to reason, which put logic as a whole in doubt; but simply as lessons. They are sources of information, they reveal to us certain logical rules of concept formation, which we would otherwise not have noticed. They show us the outer limits of linguistic propriety.

We shall consider two classical examples of double paradox to illustrate the ways they are dissolved. Each one requires special treatment. They are excellent exercises.

2. The Liar Paradox.

An ancient example of double paradox is the well-known ‘Liar Paradox’, discovered by Eubulides, a 4th cent. BCE Greek of the Megarian School. It goes: ‘does a man who says that he is now lying speak truly?’ The implications seem to be that if he is lying, he speaks truly, and if he is not lying, he speaks truly.

Here, the conceptual mistake underlying the difficulty is that the proposition is defined by reference to itself. The liar paradox is how we discover that such concepts are not allowed.

The word ‘now’ (which defines the proposition itself as its own subject) is being used with reference to something which is not yet in existence, whose seeming existence is only made possible by it. Thus, in fact, the word is empty of specific referents in the case at hand. The word ‘now’ is indeed usually meaningful, in that in other situations it has precise referents; but in this case it is used before we have anything to point to as a subject of discourse. It looks and sounds like a word, but it is no more than that.

A more modern and clearer version of this paradox is ‘this proposition is false’, because it brings out the indicative function of the word ‘now’ in the word ‘this’.

The word ‘this’ accompanies our pointings and presupposes that there is something to point to already there. It cannot create a referent for itself out of nothing. This is the useful lesson taught us by the liar paradox. We may well use the word ‘this’ to point to another word ‘this’; but not to itself. Thus, I can say to you ‘this “this”, which is in the proposition “this proposition is false”‘, without difficulty, because my ‘this’ has a referent, albeit an empty symbol; but the original ‘this’ is meaningless.

Furthermore, the implications of this version seem to be that ‘if the proposition is true, it is false, and if it is false, it is true’. However, upon closer inspection we see that the expression ‘the proposition’ or ‘it’ has a different meaning in antecedents and consequent.

If, for the sake of argument, we understand those implications as: if this proposition is false, then this proposition is true; and if this proposition is true, then this proposition is false — taking the ‘this’ in the sense of self-reference by every thesis — then we see that the theses do not in fact have one and the same subject, and are only presumed to be in contradiction.

They are not formally so, any more than, for any P1 and P2, ‘P1 is true’ and ‘P2 is false’ are in contradiction. The implications are not logically required, and thus the two paradoxes are dissolved. There is no self-contradiction, neither in ‘this proposition is false’ nor of course in ‘this proposition is true’; they are simply meaningless, because the indicatives they use are without reference.

Let us, alternatively, try to read these implications as: if ‘this proposition is false’ is true, then that proposition is false; and if that proposition is false, then that proposition is true’ — taking the first ‘this’ as self-reference and the ‘thats’ thereafter as all pointing us backwards to the original proposition and not to the later theses themselves. In other words, we mean: if ‘this proposition is false’ is true, then ‘this proposition is false’ is false, and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true.

Here, the subjects of the theses are one and the same, but the implications no longer seem called for, as is made clear if we substitute the symbol P for ‘this proposition is false’. The flavor of paradox has disappeared: it only existed so long as ‘this proposition is false’ seemed to be implied by or to imply ‘this proposition is true’; as soon as the subject is unified, both the paradoxes break down.

We cannot avoid the issue by formulating the liar paradox as a generality. The proposition ‘I always lie’ can simply be countered by ‘you lie sometimes (as in the case ‘I always lie’), but sometimes you speak truly’; it only gives rise to double paradox in indicative form. Likewise, the proposition ‘all propositions are false’ can be countered by ‘some, some not’, without difficulty.

However, note well, both the said general propositions are indeed self-contradictory; they do produce single paradoxes. It follows that both are false: one cannot claim to ‘always lie’, nor that ‘there are no true propositions’. This is ordinary logical inference, and quite legitimate, since there are logical alternatives.

With regard to those alternatives. The proposition ‘I never lie’ is not in itself inconsistent, except for the person who said ‘I always lie’ intentionally. The proposition ‘all propositions are true’ is likewise not inconsistent in itself, but is inconsistent with the logical knowledge that some propositions are inconsistent, and therefore it is false; so in this case only the contingent ‘some propositions are true, some false’ can be upheld.

3. The Barber Paradox.

The Barber Paradox may be stated as: ‘If a barber shaves everyone in his town who does not shave himself, does he or does he not shave himself? If he does, he does not; if he does not, he does’.

This double paradox arises through confusion of the expressions ‘does not shave himself’ and ‘is shaved by someone other than himself’.

We can divide the people in any town into three broad groups: (a) people who do not shave themselves, but are shaved by others; (b) people who do not shave themselves, and are not shaved by others; (c) people who shave themselves, and are not shaved by others. The given premise is that our barber shaves all the people who fall in group (a). It is tacitly suggested, but not formally implied, that no one is in group (b), so that no one grows a beard or is not in need of shaving. But, in any case, the premise in fact tells us nothing about group (c).

Next, let us subdivide each of the preceding groups into two subgroups: (i) people who shave others, and (ii) people who do not shave others. It is clear that each of the six resulting combinations is logically acceptable, since who shaves me has no bearing on whom I can shave. Obviously, only group (i) concerns barbers, and our premise may be taken to mean that our barber is the only barber in town.

Now, we can deal with the question posed. Our barber cannot fall in group (a)(i), because he is not shaved by others. He might fall in group (b)(i), if he were allowed to grow a beard or he was hairless; but let us suppose not, for the sake of argument. This still allows him to fall in group (c)(i), meaning that he shaves himself (rather than being shaved by others), though he shaves others too.

Thus, there is no double paradox. The double paradox only arose because we wrongly assumed that ‘he shaves all those who do not shave themselves’ excludes ‘he shaves some (such as himself) who do shave themselves’. But ‘X shaves Y’ does not formally contradict ‘X shaves nonY’; there is no basis for assuming that the copula ‘to shave’ is obvertible, so that ‘X shaves Y’ implies ‘X does not shave nonY’.

If the premise was restated as ‘he shaves all those and only those who do not shave themselves’ (so as to exclude ‘he shaves himself’), we would still have an out by saying ‘he does not shave at all’. If the premise was further expanded and restricted by insisting that ‘he somehow shaves or is shaved’, it would simply be self-contradictory (in the way of a single paradox).

Further embellishments could be made to the above, such as considering people who shave in other towns, or making distinctions between always, sometimes/sometimes-not, and never. But I think the point is made. The lesson learned from the barber ‘paradox’ is that without clear categorizations, equivocations can emerge (such as that between ‘shaves’ and ‘is shaved’), which give the illusion of double paradox.

See also, regarding the Liar Paradox: Ruminations, chapter 5.1, and A Fortiori Logic, appendix 7.4.

2016-08-05T10:23:04+00:00