Thursday, November 1, 2012

[PL 211] Is the liar both true and false?

In this interesting Rationally Speaking podcast, Graham Priest says that one of the major motivations behind paraconsistent logic is the view that there are true contradictions. Here's what Priest says about this in the SEP:
If there are true contradictions (dialetheias), i.e., there are sentences, A, such that both A and ¬A are true, then some inferences of the form {A , ¬A} ⊨ B must fail. For only true, and not arbitrary, conclusions follow validly from true premises. Hence logic has to be paraconsistent. One candidate for a dialetheia is the liar paradox. Consider the sentence: ‘This sentence is not true’. There are two options: either the sentence is true or it is not. Suppose it is true. Then what it says is the case. Hence the sentence is not true. Suppose, on the other hand, it is not true. This is what it says. Hence the sentence is true. In either case it is both true and not true.
Is the liar paradox an instance of a true contradiction? Is the sentence 'This sentence is not true' both true and false at the same time?

Priest reasons as follows:
  1. If 'This sentence is not true' is true, then 'This sentence is not true' is false.
  2. If 'This sentence is not true' is false, then 'This sentence is not true' is true.
Because of (1) and (2) Priest says that 'This sentence is true' is both true and false at the same time. Two questions:
  • Clearly, 'This sentence is not true', (1), and (2) express different propositions. For one thing, (1) and (2) are conditionals and 'This sentence is not true' isn't. But we want to know if 'This sentence is not true' is both true and false at the same time, not whether (1) or (2) are. So do (1) and (2) show that 'This sentence is not true' is both true and false at the same time?
  • Priest needs to assume that "There are two options: either the sentence is true or it is not" in order to show that 'This sentence is not true' is both true and false. Shouldn't there be three options? Is Priest being inconsistent or is he making some sort of reductio? If the latter, what does this reductio look like?

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