Paradoxes and Their Resolutions

1.    Double paradox

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 antinomic; 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 practice! However, that does not make it right, anymore than the occurrence of other kinds of error in practice 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.

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.    More on the Liar paradox

Once we grasp that the meaning of words is their intention, singly and collectively – the solution of the liar paradox becomes very obvious. Self-reference is meaningless, because – an intention cannot intend itself, for it does not yet exist; an intention can only intend something that already exists, e.g. another intention directed at some third thing.

In view of this, the proposition “this proposition is false” is meaningless, and so is the proposition “this proposition is true”. Both may freely be declared equally true and false, or neither true nor false – it makes no difference in their case, because the words “this proposition” refer to nothing at all[1].

Although the words used in these sentences are separately meaningful, and the grammatical structure of the sentences is legitimate – the words’ collective lack of content implies their collective logical value to be nil. Self-reference is syntactically cogent, but semantically incoherent. It is like circular argument, up in the air, leading nowhere specific.

Regarding the exclusive proposition “Only this proposition is true”, it implies both: “This proposition is true” and “All other propositions are false” – i.e. it is equivalent to the exceptive proposition “All propositions but this one are false”. The latter is often claimed by some philosopher; e.g. by those who say “all is illusion (except this fact)”.

My point here is that such statements do not only involve the fallacy of self-reference (i.e. “this proposition”). Such statements additionally involve a reference to “all others” which is open to criticism, because:

• To claim knowledge of “all other propositions” is a claim to omniscience, a pretense that one knows everything there is to know, or ever will be. And generally, such statements are made without giving a credible justification, though in contradiction to all prior findings of experience and reason.
• Surely, some other propositions are in fact regarded and admitted as true by such philosophers. They are generally rather talkative, even verbose – they do not consistently only say that one statement and refuse to say anything else.
• And of course, formally, if “this” is meaningless (as previously shown), then “all others”, which means “any other than this” is also meaningless!

The liar paradox, by the way, is attributed to the ancient Greeks, either Eubulides of Miletus (4^th Cent. BCE) or the earlier Epimenides of Crete (6^th Cent. BCE). I do not know if its resolution was evident to these early logicians, but a (European?) 14^th Cent. CE anonymous text reportedly explained that the Liar’s statement is neither true nor false but simply meaningless. Thus, this explanation is historically much earlier than modern logic (Russell et alia, though these late logicians certainly clarified the matter).[2]

4.    The utility of paradoxes

A (single) paradoxical proposition has the form “if P, then notP” or “if notP, then P”, where P is any form of proposition. It is important to understand that such propositions are logically quite legitimate within discourse: a (single) paradox is not a contradiction. On the other hand, a double paradox, i.e. a claim that both “if P, then notP” and “if notP, then P” are true in a given case of P, is indeed a contradiction.

The law of non-contradiction states that the conjunction “P and notP” is logically impossible; i.e. contradictory propositions cannot both be true. Likewise, the law of the excluded middle states that “notP and not-notP” is logically unacceptable. The reason for these laws is that such situations of antinomy put us in a cognitive quandary – we are left with no way out of the logical difficulty, no solution to the inherent problem.

On the other hand, single paradox poses no such threat to rational thought. It leaves us with a logical way out – namely, denial of the antecedent (as self-contradictory) and affirmation of the consequent (as self-evident). The proposition “if P, then notP” logically implies “notP”, and the proposition “if notP, then P” logically implies “P”. Thus, barring double paradox, a proposition that implies its own negation is necessarily false, and a proposition that is implied by its own negation is necessarily true.

It follows, by the way, that the conjunction of these two hypothetical propositions, i.e. double paradox, is a breach of the law of non-contradiction, since it results in the compound conclusion that “P and notP are both true”. Double paradox also breaches the law of the excluded middle, since it equally implies “P and notP are both false”.

These various inferences may be proved and elucidated in a variety of ways:

• Since a hypothetical proposition like “if x, then y” means “x and not y is impossible” – it follows that “if P, then notP” means “P and not notP are impossible” (i.e. P is impossible), and “if notP, then P” means “notP and not P are impossible” (i.e. notP is impossible). Note this explanation well.

We know that the negation of P is the same as notP, and the negation of notP equals P, thanks to the laws of non-contradiction and of the excluded middle. Also, by the law of identity, repeating the name of an object does not double up the object: it remains one and the same; therefore, the conjunction “P and P” is equivalent to “P” and the conjunction “notP and notP” is equivalent to “notP”.

Notice that the meaning of “if P, then notP” is “(P and not notP) is impossible”. Thus, although this implies “notP is true”, it does not follow that “if notP is true, P implies notP”. Similarly, mutadis mutandis, for “if notP, then P”. We are here concerned with strict implication (logical necessity), not with so-called material implication.

The reason why this strict position is necessary is that in practice, truth and falsehood are contextual – most of what we believe true today might tomorrow turn out to be false, and vice-versa. On the other hand, logical necessity or impossibility refer to a much stronger relation, which in principle once established should not vary with changes in knowledge context: it applies to all conceivable contexts.

• Since a hypothetical proposition like “if x, then y” can be recast as “if x, then (x and y)” - it follows that “if P, then notP” equals “if P, then (P and notP)”, and “if notP, then P” equals “if notP, then (notP and P)”. In this perspective, a self-contradictory proposition implies a contradiction; since contradiction is logically impermissible, it follows that such a proposition must be false and its contradictory must be true. This can be expressed by way of apodosis, in which the laws of thought provide the categorical minor premise, making it possible for us to exceptionally draw a categorical conclusion from a hypothetical premise.

• We can also treat these inferences by way of dilemma, combining the given “if P, then notP” with “if notP, then notP” (the latter from the law of identity); or likewise, “if notP, then P” with “if P, then P”. This gives us, constructively:

Paradox sometimes has remote outcomes. For instance, suppose Q implies P, and P implies notP (which as we saw can be rewritten as P implies both P and notP). Combining these propositions in a syllogism we obtain the conclusion “if Q, then P and notP”. The latter is also a paradoxical proposition, whose conclusion is “notQ”, even though the contradiction in the consequent does not directly concern the antecedent. Similarly, non-exclusion of the middle may appear in the form “if Q, then neither P nor notP”. Such propositions are also encountered in practice.

It is interesting that these forms, “Q implies (P and notP), therefore Q is false” and “Q implies (not P and not notP), therefore Q is false”, are the arguments implicit in our application of the corresponding laws of thought.

When we come across an antinomy in knowledge, we dialectically seek to rid ourselves of it by finding and repairing some earlier error(s) of observation or reasoning. Thus, paradoxical argument is not only a derivative of the laws of thought, but more broadly the very way in which we regularly apply them in practice.

That is, the dialectical process we use following discovery of a contradiction or an excluded middle (or for that matter a breach of the law of identity) means that we believe that:

Every apparent occurrence of antinomy is in reality an illusion.

It is an illusion due to paradox, i.e. it means that some of the premise(s) that led to this apparently contradictory or middle-excluding conclusion are in error and in need of correction. The antinomy is never categorical, but hypothetical; it is a sign of and dependent on some wrong previous supposition or assumption. The apparent antinomy serves knowledge by revealing some flaw in its totality, and encouraging us to review our past thinking.

Contradiction and paradox are closely related, but not the same thing. Paradox (i.e. single not double paradox) is not equivalent to antinomy. We may look upon them as cognitive difficulties of different degrees. In this perspective, whereas categorical antinomy would be a dead-end, blocking any further thought––paradox is a milder (more hypothetical) degree of contradiction, one open to resolution.

We see from all the preceding (and from other observations below) the crucial role that paradox plays in logic. The logic of paradoxical propositions does not merely concern some far out special cases like the liar paradox. It is an essential tool in the enterprise of knowledge, helping us to establish the fundaments of thought and generally keeping our thinking free of logical impurities.

Understanding of the paradoxical forms is not a discovery of modern logic[3], although relatively recent (dating perhaps from 14^th Cent. CE Scholastic logic).

From Future Logic 32, and Ruminations 1 & 5.