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Wittgenstein on Logic and Contradiction 

If you’ve had to study analytic philosophy, you’re probably aware of the special status “contradiction” has in the academic field of logic. 

Classical logicians treated contradictions with the principle of explosion: from a contradiction, anything and/or everything follows: If a contradiction is true, then you can say anything is true. And that would ruin everything. Human society would fall into chaos. The philosophers had to do something.

To this end, philosophers in a capacity labeled “logician” spent many hours laboring over supposedly difficult contradictions to resolve like the Liar Paradox (is “this sentence is a lie” a true statement?1??). They found this task hard and and thus constructed elaborate systems of mathematical-looking symbols in attempts to get around the problem.

Some decided they had succeeded. Others that they had failed and embraced a “truth” in pure, actual contradictions and they thus fell into trivialism (‘all propositions of all kinds are true!’), strong paraconsistency (‘contradictions may be true!’), dialetheism (‘some contradictions are true!’), polylogism (‘truth is relative to race, culture, nationality, or class!’) and similarly silly ideas that still amazingly persist in current academic philosophy.

Wittgenstein answers this supposedly difficult “Liar Paradox” and at the same time shows the pointless nature of classical logician’s projects, their obsession with contradictions, and their failed principle of explosion. I’ll let him speak for himself:

Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think… Because the thing works like this: if a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter. It is just a useless language-game, and why should anyone be excited? … Suppose I convince Rhees of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 x 2 = 369.’ Well, we should not call this ‘multiplication,’ that is all.

LFM 21-22

We exclude contradictions from language; we have no clear-cut use for them, and we don’t want to use them.

RPP II §290

To understand how Wittgenstein can see clearly to dismiss these supposed problems, we must recall the central theme of almost all Wittgenstein’s work: natural language and its relation to human thought.

In order to be able to draw a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought). It will therefore only be in language that the limit can be drawn, and what lies on the other side of the limit will simply be nonsense.

TLP Pref.

Thought can never be of anything illogical, since, if it were, we should have to think illogically. … It used to be said that God could create anything except what would be contrary to the laws of logic. – The truth is that we could not say what an ‘illogical’ world would look like. … It is as impossible to represent in language anything that ‘contradicts logic’ as it is in geometry to represent by its coordinates a figure that contradicts the laws of space or to give the coordinates of a point that does not exist.

TLP 3.03-3.032

The limits of my language mean the limits of my world. … Logic pervades the world: the limits of the world are also its limits. So we cannot say in logic, ‘The world has this in it, and this, but not that.’ For that would appear to presuppose that we were excluding certain possibilities, and this cannot be the case, since it would require that logic should go beyond the limits of the world; for only in that way could it view those limits from the other side as well. We cannot think what we cannot think; so what we cannot think we cannot say either.

TLP 5.6-5.61

In giving explanations I have already to use language full-blown … but then how can these explanations satisfy us? - Well, your very questions were framed in this language; they had to be expressed in this language, if there was anything to ask! One might think: if philosophy speaks of the use of the word “philosophy” there must be a second-order philosophy. But it is not so: it is, rather, like the case of orthography, which deals with the word “orthography” among others without then being second-order.

PI I 120-121

It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into mathematics.

RFM 5.2

As the Wittgensteinian linguist Noam Chomsky put it:

Factual beliefs and common-sense expectations also play a role in determining that a thing is categorizable and hence namable. Consider Wittgenstein’s disappearing chair. In his terms, we have no “rules saying whether one may use the word ‘chair’ to include this kind of thing” (PI, p.38). Or to put it differently, we keep certain factual assumptions about the behavior of objects fixed when we categorize them and thus take them as eligible for naming or description.

Chomsky. Reflections on Language (1975)

An actual, literal contradiction as such violates the presuppositions—the factual assumptions about the behavior of objects—of any language meant to be taken literally. If a language game does not presuppose non-contradiction, it is useless if meant to be taken literally. That is all one really needs to say. The claim that “some pure contradictions are literally true” is nonsense. It is itself a performative contradiction. It is nonsense. One does not need to construct elaborate systems of mathematical symbols to figure this out (and doing so doesn’t really help). 

Of course, contradictions can be useful in poetry, as literary devices, in mysticism, in religion or what have you. Furthermore, in programming and mathematics, the principle of explosion is often an impractical way to deal with contradictions (thus weak paraconsistency is sometimes useful). But this is not the issue at hand.