Let's have some fun, which may not lead anywhere. While the meaning of a sequence of characters is a contingent matter, the meaning in Contemporary English of a sequence of characters is not a contingent matter, because "Contemporary English" rigidly refers to a particular dialect of English. I shall suppose abstracta, like Contemporary English, to be necessary beings. I will now use a modified diagonalization procedure which neatly simplifies things. I shall use 'the Goedel number of s' as an abbreviation for some complex expression that I shan't bother to give, but I shall assume that Goedel numbers are all positive. (I will use single quotes to introduce abbreviatory marks; abbreviations are to be substituted for both in unquoted contexts and in double-quoted contexts. Thus, if 'x' is an abbreviation for "the dog", then each time I write "the x runs" and x, that should be taken as short for "the the dog runs" and the dog.) I shall use 'the modified Goedel number of s' as an abbreviation for "the Goedel number of the sequence of characters formed from s by replacing the first contiguous sequence of arabic digits with 0 if this Goedel number is equal to the number expressed by that sequence and zero otherwise or if there are no arabic digits in that sequence". Now, let 'n' abbreviate the arabic expression of the Goedel number of:
The sequence of characters whose modified Goedel number is 0 expresses in Contemporary English an impossible proposition.Now, I will use 'S' be an abbreviation for the above sequence of characters (including the period) but with "0" replaced by "n". Observe that n is the modified Goedel number of "S". I could, if I so wanted, expand out "modified Goedel number", and calculate n according to some Goedel numbering scheme, and use no abbreviations, but I am too lazy.
Now, let us reason. If S, then the proposition expressed by "S" is impossible, and so it is impossible that S, and hence it isn't the case that S.[note 1] Since the inference "If p, then not p; therefore not p" is valid, it follows that it is not the case that S. But then the sequence of characters whose modified Goedel number is n does not express an impossible proposition in Contemporary English. Suppose that that sequence does express a proposition—after all, it seems that every sequence obtained from the block-quoted sequence displayed above by replacing "0" with a different arabic sequence expresses a proposition. So, the sequence expresses a possible proposition in Contemporary English.
Thus, there is a possible world where S. Let w be such a world. Then, at w, S. So, at w, the proposition expressed in Contemporary English by the sequence with modified Goedel number n is impossible. But "S" expresses in Contemporary English the very same proposition at every world, since "S" has no indexical elements and "Contemporary English" refers rigidly, and S is the only possible sequence with modified Goedel number n. Thus, at w, the proposition expressed by "S" is impossible. But if it is impossible at w, then at w, it is not the case that S. Hence, at w, it is and is not the case that S, which is a contradiction, given that w is a possible world.
This argument shows that we can run a liar argument using modal properties of propositions instead of truth, as long as we accept the following disquotation schema:
- The proposition expressed by "..." is impossible iff it is impossible that ....
- What proposition an indexical-free sentence type expresses in Contemporary English is not a contingent matter.
- It makes sense to talk of a sentence type having a meaning.
- The rule of inference "If p, then not p; hence, not-p" is correct.
- From impossibility one can infer non-actuality.
Actually, it should be no surprise that one can generate liar paradoxes using an impossibility predicate that satisfies (1), since one can directly use an impossibility predicate that satisfies (1) to define a predicate coextensive with truth in w, where w rigidly designates a world:
- p is true in w if and only if the conjunction of p with the proposition that w is actual is not impossible.
- Sentence (7) expresses a proposition that is not true in @.
It seems, then, that the liar paradox is not so much a phenomenon of truth, as of disquotation, and other predicates that have disquotation schemas generate liar paradoxes as well. It would be interesting to come up with a general characterization of the sorts of disquotation schemas that generate liar paradoxes. For instance the schema:
- x believes the proposition expressed by "..." iff x believes that ...