Friday, February 11, 2011

The Liar Paradox and conjunction introduction

Let F be some property had by and only by the sequence of symbols:

  1. The sequence of symbols with property F is not a sentence expressing a truth, and 2+2=4.
Then, (1) is nonsense. (For if it makes sense, then it is true if and only if it is not true.) Since sequences of symbols that are nonsense don't express propositions, and hence don't express truths:
  1. The sequence of symbols with property F is not a sentence expressing a truth.
And we know:
  1. 2+2=4.
Hence, (1), appearances to the contrary notwithstanding, is not a conjunction of (2) and (3). For if it were a conjunction of (2) and (3), it would make sense and be true.

In my previous post, I tried to create space for the idea that in natural language not every pair of sentences can be conjoined. The above argument extends this to sufficiently rich artificial languages, since the above case could be formulated in an artificial language.

We can keep classical logic rules such as:

  1. If c is the conjunction of a and b, and a and b are both true, then c is true,
as long as we recognize that in many natural languages and some artificial languages there can be sentences a and b that have no genuine conjunction. They may have a syntactic conjunction—a sequence of symbols formed out of the symbols in a and b and the conjunction symbol—but the preceding post shows that a syntactic conjunction is not the same as a conjunction simpliciter. And (4) must be understood in terms of genuine conjunction, not merely syntactic conjunction. This modification of classical logic does, of course, screw up the meta-theory. It doesn't affect soundness results, but completeness results may be in trouble.


Heath White said...

This argument works even if we'd rather say "some propositions have no truth value" than "propositions have truth values essentially and sentences with no truth values are nonsense." For we can modify the first steps of the argument to

"Then, (1) [expresses a proposition which] has no truth value. For if it did..."

Strawson's view of "meaningful" was something like "having the capacity to have a truth value" and he assumed that this capacity was not always exercised. I sometimes think this would not be a bad way to go.

Alexander R Pruss said...

That might work, but I at least want to keep this claim:
If (1) expresses a proposition, the proposition it expresses is a conjunction of (2) and (3).

Keith Brian Johnson said...

Hmm. Curious. The propositions expressed by (2) and (3) can still be conjoined--just write (2) in French or write (3) with Roman numerals. What apparently won't work is mechanically conjoining the *strings of symbols* in (2) and (3) and expecting the conjunction-string to express in English a proposition. This makes some sense, since property F was defined as a property of a string of symbols. But if there can be such a property F, it would seem to play havoc with the principle that exactly translated sentences express the same propositions.

Alexander R Pruss said...

Yes, that's interesting.

One can get around this by making F be a property that a sequence of symbols has if and only if it expresses the conjunction of the two conjuncts of (1). But that's tricky--expresses *in what language*?