On the Liars Paradox

I recently came across Arthur Priors argument, his so-called resolution of the Liar’s Paradox.  I adamantly disagree with this mans entire argument and resolution of the paradox.  This post aims to discredit the argument.

Statements, for the most part, can be true or false. But they can’t be both – at least not in the common applications of logical reasoning. Much like a question, as a matter of fact. Is a question true or false? It cant be either. Nor can general exclamations. I assert that any statement which is self-contradictory simply does not have a truth value, and it is logically meaningless.  Any other statement which leads to a contradiction is false.

There is fundamental difference between proofs by contradiction and Arthur Priors perspective. In indirect proofs, a statement does have a truth value… but the value is often times unknown. By assumption (of the negation of the actual truth value) we can arrive at a contradiction, which thus proves the assumption false.

But the assumed proposition (and the concluded proposition) in indirect proofs cannot be both true and false simultaneously. The actual logical statement must be either true or false. Only the false assumption can be self-contradictory, but that false statement is not true and can never be true, for it would lead to contradiction otherwise. For a statement to actually be true and false is nonsense.

Let me rephrase what I mean. In the case of the indirect proofs, a false statement is posited as a hypothetical truth. This produces a contradiction, yes, but this contradiction is hypothetical. The implication is that a single non-contradictory statement emerges from the mess.  Dual possibilities converge to a single possibility.

This is not what is happening in the Liars Paradox. Here, the statement is self-contradictory, but not hypothetical. Its actual. Arthur Prior would have you believe that a self-contradictory statement can exist in actuality without dilemma. Furthermore, he essentially purports that “self-contradiction” and “false” are equivalent states for a single proposition. Arthur’s conclusion is that the falsity of the statement can be asserted, but with no regard that, through the statement itself, false implies true in turn.

Simply put, self-contradictory statements and entities cannot exist in actuality… only hypothetically. When in the course of hypothetical reasoning we arrive at a contradiction, we must necessarily trace our logic back to whatever assumed proposition was made, and negate it – this is the entire point of indirect proof and proof by contradiction. You cannot negate something that is, only assumptions about what might be. Arthur Priors treatment of a self-contradictory statement as actual instead of as hypothetical is nonsense to me.

If his reasoning is indeed valid then proof by contradiction is no longer a valid technique, as it may be assumed from here on that contradiction is not grounds for calling absurdity on an argument.

In proof by contradictions, the contradictory statements exists hypothetically… but then it is necessarily debunked and replaced by a new line of reasoning that does not lead to contradiction. The contradiction is removed from possibility.  However, Arthur Prior allows the continued existence of the contradictory statement.  The contradiction persists and imposes a truth value.

Arthur Prior is right about one thing. That all logical statements assert themselves as true implicitly. I agree for the most part. But bare in mind that this is true for only logical statements, not illogical or logically devoid ones.   Statements must also provide meaningful information. If the sole purpose of a statement is to declare itself false then nothing informative is contributed.  Nothing is added to an argument.  The statement is no more true or false than had you not said anything at all… which, obviously by the nature of remaining silent, has no truth value. Such statements undermine (worse, they contradict) the entire point of using language in the first place.

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