You Are Underestimating the Liar Paradox

Many people who don’t work in the philosophy of logic think the Liar Paradox isn’t terribly important. Call these people the Confidents. They think that the philosophers who continue to work on the paradox are excellent philosophers but are making some false assumption that they are either blind to or not evaluating properly — an assumption that the Confidents have spotted and that generates virtually the whole contemporary debate about the paradox. Moreover, the Confidents usually endorse some variant of a particular solution to the paradox, one that is in accord with common sense. In my experience, just about every smart person thinks of and endorses some version of this solution at some point in their thinking about the paradox. Let us call this solution the Commonsensical Solution, CS. This paper addresses not philosophers of logic but Confidents. I will prove that CS has grave difficulties that should prevent anyone from thinking that it shows that the paradox is not too hard to solve.

1. The Paradox

As we all know, the following string of symbols, call it ‘A’, presents a problem.

A isn’t true

Assuming A is true leads to a contradiction. That contradiction is not true. So, the assumption is not true. A isn’t true. Oops.

Of course, if the linguistic strong of symbols A refers to anything at all, it refers to itself. As we all know, self-reference is often not a problem. For instance, the last sentence of the only paragraph in §1 of this paper that starts with ‘Of course’ contains more than five words.

What makes the Liar Paradox prima facie stunning is the apparent absence of reasonable responses to it. Consider the following five L (for ‘liar’) claims:

• A = ‘A isn’t true’. (True in virtue of a linguistic stipulation for the proper name ‘A’. Alternatively, we could use a self-referring definite description such as one of the form ‘The first indented sentence on such-and-such page’.)
• If A = ‘A isn’t true’ and A is true, then ‘A isn’t true’ is true. (Cf. if D = ‘Dogs bark’ and D is true, then ‘Dogs bark’ is true.)
• If A = ‘A isn’t true’ and ‘A isn’t true’ is true, then A is true. (Cf. if D = ‘Dogs bark’ and ‘Dogs bark’ is true, then D is true.)
• If ‘A isn’t true’ is true, then A isn’t true. (Cf. if ‘Dogs bark’ is true, then dogs bark.)
• If A isn’t true, then ‘A isn’t true’ is true. (Cf. if dogs bark, then ‘Dogs bark’ is true.)

What makes the Liar Paradox stunning is the apparent lack of even remotely plausible commonsensical responses to it. For starters, it painfully obvious that there exists an interpretation W such that: from just the W-interpreted liar Ls there is a derivation of a contradiction using just familiar elementary rules of inference from sentential logic, and those elementary inference rules are truth-preserving when applied to the W-interpreted liar Ls. (In admitting this claim, we can be generous and accept that there are alternative “literal” meanings of the Ls such that there is no such derivation. Hence, the claim that W exists is quite accommodating!) Interpretation W will have the four conditionals be material conditionals and avoid equivocation. And yet, under that interpretation, the liar Ls all seem obviously true. The first W-interpreted L is true via successful linguistic stipulation — and as noted above we needn’t use linguistic stipulation to generate the first L, as we could use a definite description instead. The second and third look to be elementary truths of first-order predicate logic with identity. And the fourth and fifth are instances of the two halves of the T-schema which, as the ‘dogs bark’ illustrations show, are awfully intuitive when applied to simple sentences of the form ‘X is not true’ or ‘The first indented sentence on such-and-such page is not true’.

It seems that we are faced with a stark choice: either say W doesn’t exist (so there is no way to interpret the Ls in order to derive a contradiction from them), say that the simplest rules of sentential logic aren’t truth-preserving, or say that one of the Ls is not true. How can one possibly find a commonsensical solution to this predicament?

2. The Commonsensical Solution, CS

CS doesn’t answer that question directly. It doesn’t even address the above reasoning. Instead, it says that the reasoning in the previous section equivocates because it wasn’t decided whether ‘A’ is the name of a sentence token, a sentence type, or a proposition.

For instance, if we take ‘A’ from the previous section to name a sentence token, then CS says it’s not true. Suppose I produce another token of that sentence type in order to say something about token A:

A is not true

According to CS this new token, B, is true since it correctly remarks that the first token, A, wasn’t true. A and B refer to the very same sentence token but only one succeeds in being true. Hence the sentence type had by A and B has some true tokens and some untrue tokens. Alternatively, for friends of propositions: some tokens of the same sentence type express a true proposition while others do not.

3. The Other-Token Problem

CS holds that token B is true and token A is not true. But how could that be? (Alternatively: how can B express a truth while A does not?) After all, token B is exactly like token A: the same words, the same meanings (no matter how you construe ‘meaning’), the same order, the same sentential structure, etc. How on earth could one be true while the other is not true?

Of course, A is self-referential while B is not, but that hardly means there is any difference in truth conditions. For instance, tokens S1 and S2,

S1 is less than fifty words long

S1 is less than fifty words long

have the same truth conditions despite only S1 being self-referential. And self-reference plus occurrences of ‘is true’ can’t be a bar to being true, as the true sentence ‘All English sentences that are true are true’ demonstrates: it is self-referential, contains ‘is true’, and is true. This is the Other-Token Problem for CS: since A and B are linguistically the same, there is no way that B can be true while A isn’t true, as CS claims.

4. Can CS Be Saved from the Other-Token Problem?

The advocate of CS responds to the Other-Token Problem by claiming that since token B is true while A is not true (and it’s not the case that either B isn’t true or A is true; she accepts the law of non-contradiction), A and B must have different truth conditions due to some hidden linguistic properties. Sure, there is nothing in A or B that signals any truth-conditional difference, but any decent philosopher of language should be accustomed to subtle contextual factors that make a difference to truth conditions.

For instance, two tokens of ‘Jo is tall’ may differ in truth conditions due to subtle contextual factors. More to the point, a token of that sentence type fails to have any truth condition absent the contextual factors. ‘Jo is tall’ has no truth condition until it links up with something like a contrast class (Jo is tall in comparison to what class of objects?). Similarly, and roughly put, perhaps a sentence token that includes a use of ‘true’ has no truth condition until it links up with something Z — and token B but not A has the required link, which is why B ends up true even though A does not.

Hence, the advocate of CS is saying that there will be two apparently contextually unproblematic tokens that are of the same sentence type but that do not have the same truth condition. If that upsets current thoughts about context dependence, so be it. Or so she says.

In order to make the task faced by the advocate of CS vivid, consider these four sentence tokens, S1-S4:

S1 is not true.

S2 is longer than fifty words in length.

S3 is in English.

S1 is not true.

CS says that the self-referential S3 and non-self-referential S4 are true, S2 is self-referential and false, and S1 has no truth condition. The task the advocate of CS faces is to find out why S1 fails to have a truth condition while S2-S4 do. Her provisional hypothesis, arrived at through thinking about how ‘tall’ works, is that the words in S1 have their usual meanings, but its occurrence of ‘true’ reaches out to the world to get the sentence’s meaning completed, so to speak, and since the completion never happens, it doesn’t get to have a truth condition. It’s easy to note that the S1/S2-S4 difference in truth condition possession must come from the use of ‘true’ in S1, but that is to merely locate the solution, not reveal it.

5. The Revenge Problem

Unfortunately, like many attempted solutions to the Liar Paradox, CS appears to completely fail for some clever paradoxical sentences. Hence, even if we found some way around the Other-Token Problem, CS appears to be in big trouble.

Consider sentence token X immediately below:

No token of X’s type is true.

If X is true then what it says is true: no token of X’s type is true. But X itself is a token of that type and it is true. Contradiction. Hence, X is untrue.

If you like CS, then this reasoning probably strikes you as right: X is another bizarre sentence token, so it’s no surprise that it’s not true.

Now consider token Y immediately below and bear with me while I repeat some of the above reasoning:

No token of X’s type is true.

Suppose for a moment that token Y is true. Since Y is true, what it says is true: no token of X’s type is true. And yet, Y is true and is a token of X’s type. Hence, it’s not the case that no token of X’s type is true, as we just found one (i.e., Y) that is true. We have reached a contradiction, based on the assumption that Y is true. Thus, Y is untrue.

Things get interesting when we realize that Y is an entirely arbitrary token of X’s type. And we proved, in the immediately preceding paragraph — call it the proof paragraph — that Y is untrue. In fact, we could use that paragraph-proof for any token of X’s type, any one at all! That is, whenever you give me a token T of X’s type, I can just slap down the proof paragraph to show that T is untrue (all I have to do is replace ‘Y’ with a name for T in the paragraph-proof). Hence, we have an important result: for any token of X’s type, there is a proof that it isn’t true.

In other words, we can conclude from all these proofs that no token of X’s type is true — since for each token there is a corresponding proof that it is untrue. As a consequence, we have just proved that

No token of X’s type is true.

Call that immediately preceding token string S. S is true, as was just proven. But S is a token of X’s type. Thus, at least one token of X’s type — namely, S — is true. But that contradicts S, which we proved to be true! This is the Revenge Problem.

Here is a way to make the reasoning explicit, pretending Z is a completely arbitrary token of X’s type:

1. Z is true. (Assumption for reductio.)

2. If Z is true, then Z says that no token of X’s type is true. (There is no assumption that this is all that Z says. In other words, we are allowing that token Z may have multiple truth conditions.)

3. From (1), (2), and modus ponens: Z says that no token of X’s type is true.

4. From (1), (3), and conjunction-introduction: Z is true and Z says that no token of X’s type is true.

5. If Z is true and Z says that no token of X’s type is true, then no token of X’s type is true. (This is an instance of the schema ‘If token P is true and says that ____, then _____’.)

6. From (4) (5), and modus ponens: no token of X’s type is true.

7. Z is a token of X’s type (empirical fact).

8. From (1) & (7): it’s not the case that no token of X’s type is true.

9. Hence, the assumption in (1) has led to the contradiction found in (6) and (8).

10. The contradiction found in (6) and (8) isn’t true. (At least that particular contradiction isn’t true.)

11. From (1), (9), & (10): Z is not true.

12. A generalization of (1)-(11): for any token of X’s type, there is a proof (modelled on (1)-(11)) that that token is not true. (All we have to do is take (1)-(11) and interchange for the occurrences of ‘Z’ occurrences of a name for the new token.)

13. If for any token of X’s type, there is a proof that that token isn’t true, then no token of X’s type is true. (Hopefully this principle is obviously correct — although I suppose nothing is really obvious here.)

14. From (12), (13), and modus ponens: no token of X’s type is true.

15. From (1)-(14): the sentence token after the colon in (14) has been proven.

16. If the sentence token after the colon in (14) has been proven, then it is true. (This is an instance of the general principle ‘For any sentence token, if that sentence token has been proven, then it is true’.)

17. By (15), (16), and modus ponens: the sentence token after the colon in (14) is true.

18. The sentence token after the colon in (14) is a token of X’s type (empirical fact).

19. From (17) & (18) it’s not the case that no token of X’s type is true.

(14) contradicts (19). It isn’t obvious how the advocate of CS can respond to this argument. Since CS doesn’t embrace true contradictions, the only premises it can challenge are (2), (5), (7), (12), (13), (16), and (18). Can it do so reasonably (see Gaifman 2010 and Simmons 2015 for extended treatment)?

It’s difficult to see how CS could reject either (7) or (18). Perhaps there is a way of precisifying ‘sentence type’ so that one of those is not true, but all we need for the proof to go through is a precisification that makes them true.

In order for (2) to not be true, Z has to be true even though it doesn’t say that no token of X’s type is true. But if Z is true, how could it not say that? That sounds miraculous.

In order for (5) to not be true, Z has to be true and say that no token of X’s type is true, but for some reason it’s not the case that no token of X’s type is true. This would be bizarre: if token P is true and says that blah, then surely blah. However, maybe we’re supposed to believe that although Z’s being true has something to do with its saying something true, that true thing it says is not that no token of X’s type is true. That is, perhaps it says multiple things, at least one of them is true — which is why Z is true — but the true thing it says is not the thing we were thinking of, saying that no token of X’s type is true. But what on earth would that other thing be?

In order for (12) to not be true, there has to be a token of X for which we cannot use (1)-(11) to show it’s not true. But if we can give this token a name, why can’t we plug that name into the (1)-(11) template?

In order for (13) to not be true, this has to happen: for any token T of X’s type, there is a proof that T isn’t true, but even so some token of X’s type is true. That just can’t happen: if there is a proof that T isn’t true, then it’s not the case that T is true.

In order for (16) to not be true, there has to be a sentence token T such that T has been proven but it isn’t true. That just can’t happen: if there is a proof that T is true, then it’s not the case that T isn’t true.

Of course, unless there are true contradictions there has to be a mistake somewhere in (1)-(19) since (14) contradicts (19). Even if one rejects CS one is confronted with finding the error in that “proof”. Hence, (1)-(19) presents a problem for more than the advocate of CS. That’s fine, but my point here is that it presents a serious problem for CS.

Let me be clear. I’m not arguing that CS is false or that the problems above don’t apply to any other approaches to the paradox. All I’ve proven is this: CS has grave difficulties that should prevent anyone from thinking that it shows that the paradox is not too hard to solve. That is, the Confidents are wrong.

6. References

Gaifman, Haim 2010. “Vagueness, Tolerance and Contextual Logic”, Synthese 174 (1): 5–46.

Simmons, Keith 2015. “Paradox, Repetition, Revenge”, Topoi 34 (1): 121–131.