Phil 2511: Paradoxes

Lecture 9 The Liar Paradox: Introduction

1. The Liar paradox, which arises from the assertion `What I am now saying is false’, is a semantical paradox. What is `semantical'? It is common to divide linguistics into three fields: syntax, semantics and pragmatics. Syntax (grammar) is the study of the rules for combining linguistic expressions so as to produce well-formed sentences. Semantics is the study of meaning. Pragmatics is the study of the use of language in real situations. The relation of semantics to pragmatics is controversial – for a recent discussion: Kent Bach, `The Semantics-Pragmatics Distinction: What It Is and Why It Matters’, http://online.sfu.edu/~kbach/semprag.html Semantical paradoxes are those that concern meaning and thought.

Giuseppe Peano` in `Addition E', Rivista di Mathematica 8 (1906), pp. 143-157, says that these paradoxes belong to `linguistics', and he distinguishes them from those that belong to mathematics. This division of paradoxes into two distinct groups was also advocated by Frank Ramsey in `The Foundations of mathematics' (1925). We shall need to question whether there really is a principled distinction between the two groups. For a discussion (and an answer in the negative), see Graham Priest, Beyond the Limits of Thought (Cambridge, Cambridge University Press, 1995), pp.155-162.

2. Some members of the family of semantical paradoxes:

Epimenides the Cretan. In verses 12-13 of Chapter 1 of his Epistle to Titus, St. Paul writes: `One of them, a prophet of their own, said “The Cretans are always liars, evil beasts, slothful bellies.” This testimony is true’. This allegation about Cretans is supposed to have been made by Epimenides, a native of Cnossus, which is the capital of Crete. So here we have a Cretan saying that all Cretans are liars. On this paradox and its connection with the standard Liar, see Alan Ross Anderson’s beautiful introduction to R.L. Martin (ed.), The Paradox of the Liar.

The Liar (and its strengthened variants). The standard Liar (attributed to Eubulides) is `This statement is false’. The strengthened version is `This statement is either false or neither true nor false’, or, more simply `This statement is not true’.

Loops. E.g. Jourdain's card paradox. There can, of course, be much larger loops.

Chains - infinite sequences of sentences where each refers to the next, e.g.

(ß₁) ß₂ is false

(ß₂) ß₃ is true

(ß₃) ß₄ is false

(ß₄) ß₅ is true

.

.

.

(ß_2k-1) ß_2k is false

(ß_2k) ß_2k+1 is true

.

.

For discussion of these types of paradox, see Tyler Burge, `The Liar Paradox: Tangles and Chains’, Philosophical Studies, 1981. For a demonstration of how to make loops out of chains, see Laurence Goldstein, Circular Queue Paradoxes -- the Missing Link', Analysis 59 (1999), pp.284-290.

Empirical variants. E.g. Arthur Prior's example, where the policeman testifies: `Everything the prisoner says is false’, and the prisoner says `Something which the policeman testifies is true’. And the following (adapted from Saul Kripke, `Outline of a Theory of Truth', The Journal of Philosophy 72 (1975) 690-716; reprinted in Robert L. Martin (ed.), Recent Essays on Truth and the Liar Paradox (Oxford: Oxford University Press, 1984), pp.53-81.))

Charles: Over half of Diana's statements about Camillagate are false.

Diana: Everything Charles says about Camillagate is true

This is a tangle (since the second statement is talking about the first, and the first is talking about the second) but there is no paradox. However, if Charles's statement were the only one he ever made about Camillagate, and Diana's statements about Camillagate, apart from the above mentioned one, were divided 50-50 between the true and the false, then, under those circumstances, paradox would arise. So paradoxes are not exclusively the artifacts of logicians. Ordinary (if unusual) circumstances can render ordinary utterances paradoxical.

Truth variants, e.g., the simplest of which is `This statement is true'.

The statement below is false.

The above statement is false.

(John Buridan, the great 14^th Century philosopher has an example very similar to this. In Buridan’s eighth sophism, Socrates says `What Plato is saying is false’, while Plato says `What Socrates says is false’.) In all of these, we can consistently assign a truth-value, but it seems an arbitrary matter whether to assign `true’ or `false’. So here we have indeterminacy, not inconsistency.

3. Concepts that occur in discussions of the Liar:

Truth (Tarski Convention T: S is true iff p, where S is a name of p. This leads to contradiction when S = L = `L is not true'.

Groundedness. When someone claims that Nelson Mandela is black, one can check whether this statement is true by checking Nelson Mandela’s colour. His being black grounds the truth of the statement that he is black. A formal account of grounding is given by Kripke in his `Outline of a Theory of Truth’ (op. cit.), but here is Kripke’s informal account:

`In general, if a sentence … asserts that (all, some, most etc.) of the sentences of a certain class C are true, its truth value can be ascertained if the truth values of the sentences in the class C are ascertained. If some of these sentences themselves involve the notion of truth, their truth value in turn must be ascertained by looking at other sentences, and so on. If ultimately this process terminates in sentences not mentioning the concept of truth, so that the truth value of the original can be ascertained, we call the original sentence grounded; otherwise, ungrounded’ (p.57 of R.L. Martin (ed.)).

4. Two other concepts that seem to recur in discussions of paradox are reference and negation. Are these important? Self-reference of some kind seems to figure in the semantical paradoxes, and the more general notion of reflexiveness embraces also the logical paradoxes. But note Yablo's paradox, in which there is an infinite sequence of S-sentences, of which the following is an arbitrary member:

(S_n) For all k>n, S_k is untrue.

There does not seem to be self-reference here, because each sentence seems to be referring only to other sentences. But the claim is controversial, and some authors have argued that Yablo’s Paradox is covertly self-referential. The discussion has been quite technical, but one way to see why these authors might be right is to re-cast the paradox , with each sentence in the sequence saying `All of the sentences following this one are untrue’. In this formulation, self-reference is present, and the question to be asked is whether it is essentially present (overtly or covertly) in every variant of Yablo’s Paradox.

Negation may be important, since without negation, you can't have contradiction. On the other hand, there are some paradoxes, like Curry's and the `truth-teller’ versions above, which are negation-free.

5. Some possible types of solution.

i) Paradoxical statements are semantically unacceptable, e.g. they commit category mistakes, or are perniciously vague, or incorporate an unacceptable form of self-reference etc.

ii) Paradoxical statements have one truth value (not both) and the argument apparently showing that they have the other one is faulty

iii) The paradox reasoning contains an informal fallacy, e.g. the fallacy of equivocation, e.g. the paradox statement is ambiguous

iv) One of the concepts employed in framing the paradox (e.g. truth) is incoherent

v) Paradoxical statements are neither true nor false

vi) Contrary to appearances, paradoxical sentences fail to express statements

vii) Paradoxical statements change their truth value in the course of the paradox reasoning

viii) The utterance of a paradoxical sentence is pragmatically self-defeating

ix) The Liar is no paradox – just a demonstration that some sentences can be both true and false.

These approaches are not all mutually exclusive. For example, you might argue that, if a paradoxical statement is perniciously vague ((i)) or fails to express a statement ((vi)) then it is neither true nor false ((v)).

6. Back to Epimenides.

We could say (paradoxically) that the Epimenidean Liar paradox is not a paradox. Epimenides denounced his countrymen, saying `All Cretans are liars’, meaning by this that everything ever said by any Cretan is false. Obviously what he said cannot be true, for if it were then it itself, being a statement by a Cretan, would have to be false. It can, however, be false just in case some Cretan at some time said something true. But although it does not engender contradiction in the way paradoxes generally do, Epimenides’ statement does have a very disturbing peculiarity of its own. Since it can only be false, what follows is that some true statement must have been made by a Cretan. We seem to have demonstrated this by pure logic. But surely it is a contingent matter whether any other statement was made by a Cretan – how could the existence of such a statement be guaranteed just by the fact of some other statement’s having been made by Epimenides? If we suppose that (as is logically possible) that remark of Epimenides’ was the only one he ever made, then his making it would entail the existence of some other person who made a true statement. How can the existence of one extremely untalkative person require the existence of another person? It cannot.

There is, of course, no reason to believe that Epimenides made just that one statement in his lifetime. An adult who made just one statement in his whole lifetime would be a strange individual indeed. However, one can conceive of a community of such adults – we shall call them Cretins – each of whom makes one and only the one statement `All Cretins are liars’. As we have already seen, such a statement can only be false, but it entails that some other Cretin uttered a truth. Yet, since all the other Cretins say only the falsehood `All Cretins are liars’, we seem to have shown, by pure logic alone, that a community of Cretins could not exist. Again, this seems much too powerful a result, because there is nothing logically impossible about there being a bunch of people each of whom says `All Cretins are Liars’, and nothing else.

A large community of Cretins is, perhaps, a little far-fetched, so let us suppose that there is just a very small community consisting of just two Cretins. Each of them makes just one utterance which is the same as the utterance made by the other, and what each says is that the statements made by himself and his fellow Cretin are false. Each Cretin utters the same type-sentence, but let us distinguish the tokens each utters by calling the first Cretin’s statement `S1’ and the second `S2’. So, if these Cretins express themselves rather formally, here is what each says:

S1: S1 is false and S2 is false

S2: S1 is false and S2 is false

It is easy enough to see that this constitutes a paradox. If S1 is true, then its first conjunct (which states that it is false) is true, i.e., contrary to assumption, S1 is false. But if S1 is false, then its first conjunct (which states that it is false) is true, so the second conjunct (`S2 is false’) must be the one that is false, i.e., S2 must be true. But that cannot be, because the second conjunct of S2 says that it is false. Hence the assumption that S1 is false also cannot be sustained. Hence no truth-value can be consistently assigned to S1 and S2. In the general case, a paradox arises for any n statements each of which states that all n of them are false.

Nobody can rationally deny the possibility of a two-Cretin community or wish out of existence the associated tokens S1 and S2. But does not the very existence of these tokens engender contradiction! No, it does not. The reasoning to contradiction presupposes that the tokens in question had just one of two truth-values. So, assuming that we do not want to embrace inconsistency, a way out of our difficulty is to deny that either of S1 and S2 has a classical truth-value. But is this a satisfactory response? Is this denial just an ad hoc means of escaping our difficulties? And could we not invent a strengthened version that would plunge us straight back into paradox?

[Note: The word `cretin’ is a medical term that means a person who is deformed and of very low intelligence because of a disease of the thyroid gland. But the word is also used offensively to mean a very stupid person. The word has nothing to do with Cretans (see Anderson, op. cit.).]