Paradox Lost
I've learned that my book, Paradox Lost, will be on sale for only 9.99 on Thursday, May 2! (e-book only, one day only) Go to https://www.palgrave.com/us and click on the "Daily Deal" in the upper left. This is your chance to get the solutions to 10 paradoxes for cheap. :)
In honor of this, I am going to discuss my solution to the Liar Paradox.
The Paradox
Consider sentence L: "This sentence is false." Is L true or false? Well, if it's true, then it's true that that sentence is false, so it's false. But if it's false, then it's false that the sentence is false, so it isn't false, so it's true. More explicitly, we can derive a contradiction from just two obvious premises:
In general, if a sentence says that a is F, then it is true if and only if a is F.
Sentence L says that L is false.
Therefore, L is true if and only if L is false.
Conclusion 3 is a contradiction. The inference form is modus ponens, so not very controversial. Premise 1 seems to be just a correct account of the meaning of “true”. What could be wrong?
Some Unsatisfying Approaches
You might think "Maybe there is a third truth-value besides 'True' and 'False', say, 'Indeterminate'. And L is indeterminate." Reply: Change the sentence to "This sentence is false or indeterminate." Then the sentence is true iff it is false or indeterminate.
Graham Priest says the example shows that some sentences are both true and false, that some contradictions are true, etc. I reject this because I think it's built into the meaning of "contradiction" that contradictions are never true. (I.e., if you think that "not P" can obtain at the same time that "P" obtains, you've misunderstood the word "not".)
Maybe there is something wrong with self-reference in general? But there are many other self-referential sentences that are fine, e.g., "This sentence is in English."
Some people say our concept of truth is inconsistent or otherwise illegitimate. That is too radical for me. It seems that there are many true sentences!
My Solution
In the argument above, it must be that premise 2 is false: L doesn’t say that L is false. Why? Because L doesn’t say anything, i.e., it lacks propositional content. My account:
The conventions of our language create a set of rules for interpreting sentences – i.e., for assigning a propositional content to a sentence. (These rules are not propositions; they are more like imperatives: they tell us to assign certain propositions as the meaning of certain sentences.)
In most cases, the rules are fine -- they can be followed so that we coherently interpret sentences. But the set of rules is contradictory (logically cannot be followed) as applied to certain cases.
In such a case, the sentence lacks a propositional content.
L is such a case. The rules of our language, as applied to L, require that L be assigned, as its meaning, the proposition that obtains just when that very proposition doesn't obtain. There cannot be a proposition that is true when it is not true, so there is no proposition to serve as the meaning of L.
Because it fails to express a proposition, L is neither true nor false.
Q: What about sentence L': "This sentence is false OR fails to express a proposition"?
A: L' also fails to express a proposition. The rules of the language require L' to be assigned, as its meaning, the proposition that obtains when that very proposition is false, and also obtains when that proposition isn't assigned as the meaning of L'. But there is no such proposition.
For more, see ch. 2 of Paradox Lost. And buy it on Thursday. :) :)