Saturday, August 1, 2009

A modal liar

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:

  1. The proposition expressed by "..." is impossible iff it is impossible that ....
Of course, the liar argument makes a whole bunch of other assumptions, including:
  1. What proposition an indexical-free sentence type expresses in Contemporary English is not a contingent matter.
  2. It makes sense to talk of a sentence type having a meaning.
  3. The rule of inference "If p, then not p; hence, not-p" is correct.
  4. From impossibility one can infer non-actuality.
I myself deny (2).

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:

  1. p is true in w if and only if the conjunction of p with the proposition that w is actual is not impossible.
And of course we don't need truth, but just truth in @ (where "@" is a name of the actual world), to generate the liar paradox:
  1. 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:

  1. x believes the proposition expressed by "..." iff x believes that ...
does not appear to generate a paradox.

4 comments:

MC said...

...therefore naturalism is false!

Ted Parent said...

I was searching for "modal liar" and came upon this page. I'm wondering if anyone has proposed the following as a "modal liar":

(*) This statement is not necessary.

It seems we can show (*) is true at @ iff (*) is false at @, assuming that accessibility is reflexive.

Suppose first that (*) is false at @. Then, (*) is necessary--hence, at every world W accessible from here, (*) is true at W. So, if our world accesses itself, (*) is true at @. Contradiction.

So suppose (*) is true at @. (The reductio in this case takes a tad more work.) If so, then (*) is not necessary. Hence, there is an accessible world W such that (*) is false at W. However, if (*) is false at W, then it is true at W that (*) is necessary. Hence, at every world accessible from W, (*) is true. Assuming reflexivity of access, it thus follows (*) is true at W. But ex hypothesi, W is a world where (*) is false. Contradiction.

Do you have thoughts on this? If you prefer, please feel free to email me at parentt@vt.edu. Thanks! Ted

Alexander R Pruss said...

Very nice. I don't know what exactly I was thinking in my post, but I think I had some worries about self-reference that I was trying to work through.

I guess my version was basically:

(**) This statement is impossible.

Suppose (**) is true at an accessible w. Then (**) is impossible at w, and hence false at w (reflexivity). So, (**) is true at every accessible w. So, (**) is necessarily true. But then it is actually true. (Reflexivity.) So, it is actually impossible, and hence it is actually both true and impossible. Contradiction.


What I was thinking in my convoluted post may be starting to come back to me now. I may have been worried that the same sentence type might express a different proposition at different worlds.

If that's so, then one cannot go from "(*) is false at w" to "(*) is necessary at at w" (and similar inferences fail for (**)). For (*) is false at some world w iff the proposition that (*) expresses at w is false at w. But maybe the proposition that (*) expresses at w is not the proposition that (*) is not necessary?

So I felt it necessary to rigidly fix a particular dialect to ensure that meanings can't shift between worlds. Maybe that was an unnecessary worry.

Ted Parent said...

Yes, well done.
Whether we should worry about the contingency of language depends, I think, on whether 'necessarily' expresses an operator on propositions (like '~') or a predicate defined on sentences. In the latter case, Nec('a = a') seems dubious; it suggests that 'a = a' expresses a necessary proposition in every world. But I think you're safe if 'necessarily' expresses the proposition-operator (which is the standard view).