Thursday, 24 February 2011

A Liar Paradox of Material Implication

Here I present a new objection to the material or "hook" analysis of indicative conditionals - the thesis that an indicative conditional 'If A then C' has the truth-conditions of the so-called material conditional - based on Liar-like reasoning. This objection seems invulnerable to any Grice-Lewis-Jackson-inspired pragmatic rejoinder.

(1) If (1) is true, (1) is false.

Let us call (1)'s antecedent 'A', and its consequent 'C'. I propose that the following sentence is intuitively true, or true based on intuitive and unproblematic reasoning:

(S) On the assumption that (1) is neither true nor false, A and C are false.

The reasoning is: Assume that (1) is neither true nor false. Then A is false, since it says that (1) is true, and C is false since it says that (1) is false.

If we accept the hook analysis, however, this reasoning is not secure. For if A and C are false, and the hook analysis is right, then (1) must be true. But if (1) is true, then A must be true (since it says that (1) is true), and C must be false (since it says that (1) is false). But then, by the hook analysis, (1) must be false, since it has true antecedent and false consequent. But if (1) is false, then A must be false since it says that (1) is true, but then by the hook analysis (1) must be true. But then...

The point is, the hook analysis treats (1) as a truth-functional compound, and this places it squarely in the Liar family, making our straightforward argument to (S) veer into paradox. Yet (S), and our argument for it, seem clearly correct. Therefore we should reject the hook analysis.

There are already plenty of intuitions around which seem to cast doubt on the hook analysis. This one has more bite, I submit, since it cannot be explained away with the customary sorts of pragmatic story. Take a case like 'If I die tonight, I will be alive tomorrow'. The typical proponent of the hook analysis will maintain that, given that I will not die tonight, this sentence is true but not assertable, since I should assert something stronger,1 or something robust with respect to the antecedent,2 etc. But (S) is an example of an intuitively true sentence which comes out as paradoxical (i.e. leads to paradox) if we apply the hook analysis to (1). It is hard to see how any Grice-Lewis-Jackson-inspired pragmatic story could account for our asserting, or treating as true, sentences which are "really" paradoxical.

That's the objection. While not exceedingly complicated, it is quite easy to misunderstand, so I shall conclude with a few clarifications. Firstly, the argument is not: when we apply the hook analysis to (1) we get Liar-like paradox, and since Liar-like paradox is undesirable, we should reject the hook analysis for (1). It is irrelevant to my objection whether Liar-like paradox is good, bad or indifferent. It is also irrelevant whether there is (or could be) a solution to these paradoxes. The point is simply that, intuitively, we do not get into Liar-like paradox with (1) and (S), and so the hook analysis seems to deliver the wrong answer on this point. For a truth-functional analysis of English conjunctions, on the other hand, generation of Liar-like paradox would be the intuitively right answer for certain sentences (e.g. 'This sentence is false and this sentence is false').

Secondly, I put (S) in the form 'On the assumption that X, Y', because if I had used the conditional form, the objection would have become messy through having to avoid begging the question against the hook analysis.

Finally, I am not maintaining that (1) is in no way paradoxical. It is paradoxical. To illustrate:

Suppose (1) is true. Then by (1) and modus ponens, it is false. Therefore, by conditional proof, if (1) is true then it is false. But that is just what (1) says, so it is true, but then by modus ponens (this time not within the scope of any assumption), it is false. Paradox.

What should be said about this and similar arguments, I regard as an open problem. Some thoughts: rejecting unrestricted conditional proof seems like a promising avenue, since several authors have done this for independent reasons.3 However, it is hard to shake the feeling that if (1) is true, it is false. Perhaps the object of this feeling could be accounted for as a 'Mackie conditional' or 'telescoped argument',4 and thus kept semantically distinguished from (1) read as an ordinary conditional. But if these telescoped arguments turn out to be truth-apt in some sense, and to sustain modus ponens, we would seem to be back where we started.

In sum, (1) does not appear to be a (full) member of the Liar family. An important difference can be expressed thus: while there are arguments involving the assumption that (1) is true which lead to paradox (and not just within the scope of the assumption), the bare assumptions that (1) false, or that it is neither true nor false, do not intuitively yield any such arguments (as they do with Liar-like sentences). Hence, we can and should accept (S) as straightforwardly true. And this means rejecting the hook analysis.

Tristan Haze
The University of Sydney


References

Beall and Murzi. draft. 'Two flavours of curry paradox'.
[draft available on the authors' websites, where it is listed as under review]

Bennett, Jonathan. 2003. A Philosophical Guide to Conditionals. Clarendon Press, Oxford.

Grice, Herbert Paul. 1975. Logic and Conversation, in The Logic of Grammar, D. Davidson and G. Harman (eds.), Encino, California, Dickenson, pp. 64-75. Reprinted in Grice (1989).

Jackson, Frank. 1979. 'On assertion and indicative conditionals.' in The Philosophical Review 88, 565-589.

King, Peter. 2004. "Peter Abelard" in The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2008/entries/abelard/>.

Lewis, David. 1976. 'Probabilities of conditionals and conditional probabilities.' in Philosophical Review, 85(3):297315. Reprinted with Postscript in Philosophical
Papers, Volume II, pp. 133-152.

Mackie, J.L. 1962. Counterfactuals and causal laws, in R.J. Butler, (ed.), Analytical Philosophy, 1st series, Blackwell.

Thomason, Richmond H. 1970. 'A Fitch-style formulation of conditional logic' in Logique et Analyse, 52:397412.

1 cf. Grice (1975), Lewis (1976).
2 cf. Jackson (1979).
3 According to King (2004), Abelard rejected something like conditional proof. More recently, cf. Thomason (1970), Bennett (2003), and Beall and Murzi (draft).
4 cf. Mackie (1962).

Welcome!

Here, until further notice, I will occasionally post short and often critical articles on various topics in philosophy, mainly the philosophies of logic and language. For now (at least) these will be written by me, but anyone who wishes to contribute can email an article to tristanhaze at <the domain-name of Google's email service> dot com, and I will consider it. (Needless to say, this is not a peer-reviewed journal.)

My name is Tristan Gr√łtvedt Haze, and I am a PhD student at The University of Sydney.