Consider the following sentence: “The self-referential statement expressed by this sentence is not true.” Taking the phrase “this sentence” to refer, self-referentially, to that very sentence, the most obvious meaning of that sentence is that it is not the case (is not true) that the self-referential statement expressed by that sentence is not true. But that is just to say that the statement expressed by that sentence is true, which is the negation of the obvious meaning of that sentence.
......Since the statement expressed by that sentence is both that such and such is the case and that it is not the case, which is self-contradictory, it may well follow that the statement in question is false, as suggested by Dale Jacquette (2007: ‘Denying the Liar’, Polish Journal of Philosophy 1, 91–8). But other philosophers – amongst whom I would once (five years ago) have counted myself – think that because an assertion that such and such is the case is clearly different in meaning to an assertion that it is not the case, such Liar sentences do not express any proposition at all, but are rather meaningless nonsense.
......However, I argued recently that Liar statements are in fact as true as not, and that the Liar paradox is, in that sense, a typical semantic paradox (for details see my The Liar Paradox, and my On the Cause of the Unsatisfied Paradox, in the April and June issues of this year's The Reasoner respectively); whereas, the problem above seems to be unique to the Liar paradox, e.g. it does not arise with Yablo’s paradox, in which there is no self-reference. So, I am wondering how else we might address this part of the Liar paradox.
......Could the problem be due to substitution failure? Perhaps replacing “this sentence”, in the sentence in question, with a near-copy of the sentence itself – the only difference being that ‘that’ replaces ‘this – changed the proposition expressed by that sentence to its negation. Similar failures can occur with propositional attitude reports, e.g. consider the difference between “Lois believes that Clark is thirsty” and “Lois believes that Superman is thirsty”; for details see Jennifer Saul (2007: Simple Sentences, Substitution, and Intuitions, OUP). But then, Liar sentences are sentences of a very different kind; they need only involve a self-referential name, e.g. ‘L’, plus ‘is’, ‘not’ and ‘true’.
......Another possibility is that Liar statements are identical to their negations. As a rule, the negation of a proposition is a different proposition, of course; but, propositions are either true or else false, as a rule, whereas we are now looking at propositions that are as true as not. Now, an elementary part of language is the subject-predicate description, “S is P” (e.g. “that salmon is pink”), and so a simple model of truth might use strips of paper with “S is P” on one side of the strip and “S is not P” on the reverse side, for all S and P in some simple language: All the strips with non-fictional S get stuck onto the things of the world, with “S is P” uppermost if S is P and “S is not P” uppermost if S is not P. We might extend that model to include cases where S is as P as not by giving the strips a twist in the middle before sticking them down, and by including non-fictional strips with no worldly referent, such as “100 is a round number that is also a square.” And then we might think of our Liar sentence as being like a Möbius strip, the twist due to its being as true as not, and the joining of its ends being due to its being self-referential.
......In any case, this peculiarity of the Liar paradox gives us an easy answer to the Revenge problem for this resolution of the Liar paradox, which is as follows: If “what I am saying is not true” is as true as not, then what about “what I am now saying is not only not true, it is not even as true as not”? Were that about as true as not, what it said would seem false. But, what it said was that it was not at all true that what was said was not at all true, so it said not only that what was said was not at all true, but also that it was to some extent true. So if it was as true as not, then although it would indeed seem to have been false – false that what was said was not even as true as not – it should also, and to the same extent, appear true – true that what was said was to some extent true – whence it should seem to have been as true as not after all.
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