Tuesday 12 March 2013

On the Truth-Functional Account of Indicative Conditionals

The "if/⊃"-question has an interesting history. It had evidently been considered (in essentials) by the Stoics, and by some mediaevals (Abelard especially). By the 19th century, many logicians endorsed the view that '⊃' (or whatever symbol was used) could be read as 'if...then'. This continued through the early years of the 20th century, but conscientious objectors came into view. This is socially and historically interesting, in that (as we shall see) the essential matter of the controversy had lain dormant in logic books for years beforehand, without being much discussed. It is as though logic had started to come to life again: gradually, more people were moved to think critically (but without complete dismissal) about what they read in logic books.

In the English-speaking world, MacColl was one of the earlier dissenters, though his criticisms were partly obscured by his own unpopular doctrines and procedures.

In 1908, in a short polemic against Russell, MacColl wrote: 'For nearly thirty years I have been vainly trying to convince [logicians] that this assumed invariable equivalence between a conditional (or an implication) and a disjunctive is an error'. (This is a reference to the Or-to-If Argument, which we will consider in a future post.) Russell's reply was made easy by the fact that MacColl had, in his objection, overlooked the former's distinction between propositions and propositional functions. After correcting this, Russell addressed the main issue swiftly, writing 'I say that p implies q if either p is false or q is true. This is not to be regarded as a proposition, but as a definition', and admitting happily that this definition does not give 'implies' its usual meaning. But this does not square well with the justification of the 'Definition of Implication' given in Principia.

More successful criticisms came later from Strawson. By the time of Quine's (1953) review of Strawson's Introduction to Logical Theory, the former was able to treat the semantic divergence between '⊃' and 'if...then' as rather old news:

The well-known failure of the ordinary statement operators 'or', 'if-then', 'and', and 'not' to confirm in all cases to the precepts of truth- functional logic is well expounded by Mr. Strawson. Because 'and' and 'not' deviate less radically than the others, I have found it pedagogically helpful (in Elementary Logic) to treat the translation of ordinary language into logical form, at the truth-functional level, as funnelled through 'and' and 'not'; and Mr. Strawson follows suit.
And later:
Mr. Strawson is good on '⊃' and 'if-then'. He rightly observes the divergences between the two, and stresses that 'p⊃q' is more accurately read as 'not (p and not q)' than 'if p then q'.
This state of affairs did not last. A series of post-1960 events has changed things irrevocably, so that Quine's comments above seem to come from a bygone era when things were much simpler. In my own view, the Quine-Strawson view was basically right, but one cannot make a respectable case for that today without discussing the post-1960 events. Therefore I shall now give a summary of the events, followed by a series of critical comments.

The resurgence of '': a potted history

Phase 1: In his William James Lectures at Harvard in 1967, Grice makes public his theory of implicature and conversational maxims. People are impressed by this idea: 'John is poor but honest' has the same truth-conditions as 'John is poor and honest', but the former (in some contexts) strikes people as objectionable and unassertable, even when the latter may be both true and assertable, the difference being that the former can carry an implicature that poor people aren't honest. Secondly, the maxim of 'Assert the Stronger' is developed; if someone asks where John is, and I know he's at the library, it's not proper to respond that he is either in the library or at the pub. Similarly, Grice argues, sentences like 'if snow is green then I am king' are true (just because snow isn't green), but unassertable, since we should assert the stronger: that snow isn't green. (The work is published in Grice (1975).)

Phase 2: Meanwhile, other philosophers had been continuing to develop more sophisticated accounts of the truth-conditions of conditionals. Among these is the possible worlds account of Stalnaker (1968), who, following Adams (1965) (who himself wasn't interested in the question of truth-conditions), conjectured that the probability (in some sense) of a conditional 'If A then C' is the probability of 'C' given 'A'. That is: P(If A then C) = P(C/A) = P(C & A)/P(A) (where P(A) is positive).

Phase 3: David Lewis proves his triviality results in Lewis (1976), to the effect that 'there is no way to interpret a conditional connective so that, with sufficient generality, the probabilities [of truth] of conditionals will equal the appropriate conditional probabilities'. He considers the possibility of accommodating this with a theory on which conditionals do not have truth-values (i.e. are not truth-apt): 'Why not? We are surely free to institute a new sentence form, without truth conditions, to be used for making it known that certain of one's conditional subjective probabilities are close to 1. But then it should be no surprise if we turn out to have such a device already.' He writes: 'I have no conclusive objection to the hypothesis ... . I have an inconclusive objection, however: the hypothesis requires too much of a fresh start. ... [W]hat about compound sentences that have ... conditionals as constituents? We think we know how the truth conditions of compound sentences of various kinds are determined by the truth conditions of constituent sentences, but this knowledge would be useless if any of those subsentences lacked truth conditions.' This boosts Grice's proposal, which Lewis has come to endorse: 'It turns out that a quantitative hypothesis based on Grice's ideas gives us just what we want: the rule that assertability goes by conditional subjective probability.' And so the truth-conditions of indicative conditionals are identified with those of '⊃'-statements. And for sophisticated reasons.

(To complete the story, though this is less important for what follows: in a postscript to his (1973) in his Philosophical Papers, Volume II, Lewis admitted that in 'special cases', assertability and conditional probability diverge. Secondly, he abandoned the 'Assert the Stronger' explanation of apparent counterexamples to the '⊃'-analysis, due to apparent counterexamples to the 'Assert the Stronger' maxim itself, in favour of an ingenious alternative theory devised by Frank Jackson: one may assert 'if A then C', even when one is in a position to assert the stronger 'C', if one wants to give information which is robust with respect to 'A' (which could have low probability): information which, even if 'A' turned out true, would still hold. For more details on how this theory works, see Lewis's postscript and Jackson (1979).)

Thus the Grice-inspired Lewis-Jackson version of the '⊃' analysis is today regarded as a serious proposal, even if it is not widely accepted. Some other major accounts on the market deny truth-aptness, either completely (cf. Edgington 1991, 1995) or in certain cases, such as when the antecedent is false (cf. McDermott 1996). All these accounts have in common that they are error-theoretic with respect to many or most competent speakers: the '⊃' analysis implies that competent speakers often get a conditional's truth-value wrong, while accounts which partially or totally deny truth-aptness have it that competent speakers often mistakenly ascribe truth-values to sentences which have none.

Comments on the resurgence

Comment on Phase 1: Note a fundamental difference between the cases of 'but' and 'or' on the one hand, and the case of 'if' on the other: people do not generally judge it false to say that a poor and honest person is poor but honest, but rather wrong in some other sense. This is even more pronounced in the case of 'or'. In that case, we can see perfectly well that the misleading statement about John is true. By contrast, competent speakers will confidently classify a sentence like 'If grass is blue, it isn't blue' as not true. Thus it seems any view which says that for every '⊃' sentence, there is a corresponding 'if' sentence with the same truth-conditions, will inescapably be an error theory with respect to competent speakers.

Comment on Phase 2: The notion that assertability or probability of conditionals goes by conditional probability may seem initially appealing, but apparent counterexamples abound: sentences such as 'If 6 is greater than 5, then 7 is greater than 6' and 'If Gödel's proof really was valid, the sun will thankfully rise again' do not seem at all assertable or probable. They seem like bits of nonsense. Furthermore, the idea that assertability can be quantified, and that it equals any sort of probability, seems odd; if I attach a probability of only .5 to some proposition P, why would I assert it? Such a proposition seems not assertable at all in a normative sense - and therefore not 'half assertable' either, whatever that means. A common proposal in response to this is that assertability remains low until probability gets high, at which point it shoots up. This has been criticized by Dudman (1992), using lottery cases: someone who has a ticket in a lottery will usually not be prepared to assert that they won't win, even though they may realize that not winning is very highly probable indeed.

Comment on Phase 3: Lewis, wanting to maintain that assertability of conditionals goes by conditional probability ('A = CP' for short), uses his triviality results to argue in effect that, since we can't give any truth-conditional analysis of conditionals such that probability of truth will equal conditional probability, any truth-conditional account will (by A = CP) have to explain divergences between assertability and probability of truth, so why not at least start with something simple like the '⊃' analysis? The quite different course of denying truth-aptness remains open, but - says Lewis - that requires too much of a fresh start.

The first thing to note about this line of argument is that, for reasons given in the previous comment, A = CP is really not independently attractive, once you consider certain examples. So perhaps no 'divergences' need explaining at all, and philosophers can go on looking for a non-gappy truth-conditional account of conditionals which is more plausible than the '⊃' analysis.

The second thing to note is that the logical space between giving a truth-conditional analysis of conditionals and denying truth-aptness remains largely unexplored. Consider the case of subject-predicate statements about explanatorily basic things possessing explanatorily basic properties: this is a class of truth-apt statements for which no non-circular truth-conditional analysis can be given - what we might call an 'analytically basic' class of statements. A view on which conditionals are analytically basic - an antitheory of conditionals - can happily avoid the error-theoretic consequences of prevailing views, although it could be retorted that such a view is error-theoretic with respect to analytic philosophers. Surely the response to that is: when faced with a choice between a set of accounts which are error theoretic with respect to (almost) all competent speakers, and an error theory with respect to some philosophers, one of whom also believed in other universes inhabited by donkeys which speak, the latter should at least be examined properly. (This, of course, would go beyond the scope of the present inquiry.)

There is a different family of accounts, known as "support" theories, which are not strikingly error-theoretic. Such accounts are for the most part out of favour today, but a highly sophisticated one has been developed by my teacher Adrian Heathcote, in unpublished work. In my view, all such accounts - if they purport to be reductive - will face circularity problems. (A defence of this view is beyond our scope here.) However, even if they don't succeed as reductive analyses, the key ideas behind "support" theories seem important for understanding the logic and context-sensitivity of conditionals.

In a post coming soon, I will discuss the Or-to-If Argument. This is a simple, initially-compelling deductive argument-form which, if valid, would suggest that '⊃' can be read as 'if...then'.

Adams, Ernest W. 1965. 'The Logic of Conditionals', Inquiry 8, pp. 167-197. Adams, Ernest W. 1975. The Logic of Conditionals, Dordrecht, Reidel.

Dudman, V.H. 1992. ‘Probability and Assertion’, Analysis, 52:204-11.

Edgington, Dorothy. 1991. 'Do Conditionals Have Truth-Conditions?' in Jackson. ed. (1991, pp. 176-201).

Edgington, Dorothy. 1995. 'On Conditionals', Mind 104.414., (Apr. 1995), pp. 235-329.

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).

Grice, Herbert Paul. 1989. Studies in the Way of Words, Cambridge MA, Harvard University Press.

Jackson, Frank. 1979. 'On assertion and indicative conditionals.' in The Philosophical Review 88, 565-589. Reprinted in Jackson, ed. (1991, pp. 111-135).
Lewis, David. 1976. 'Probabilities of conditionals and conditional probabilities.' in Philosophical Review, 85(3):297–315. Reprinted with Postscript in Philosophical Papers, Volume II, pp. 133-152.

Lewis, David. 1986. Philosophical Papers, Volume II. Oxford University Press, Oxford.

McDermott, Michael. 1996. 'On the truth conditions of certain “If”-sentences' in The Philosophical Review, Vol. 105, No. 1 (Jan., 1996), pp. 1-37.

Quine, W.V.O. 1953. 'Mr. Strawson on Logical Theory' in Mind, New Series, Vol. 62, No.248 (Oct., 1953), pp. 433-451.

Russell, Bertrand. 1908. '"If" and "Imply", A Reply to Mr. MacColl' in Mind, New Series, Vol. 17, No. 66 (Apr., 1908).

Stalnaker, Robert: 'A Theory of Conditionals', Studies in Logical Theory: American philosophical quarterly monograph, Oxford, Blackwell 1968, pp. 98-112.

Whitehead, Alfred North and Russell, Bertrand. 1910. Principia Mathematica, Vol. 1. Cambridge: Cambridge University Press. Second edition 1925.


  1. Could you give more informations about "support" theories? I'm curious to know more about this approach.

    1. The classic source for support theories is Nelson Goodman's 1947 paper 'The Problem of Counterfactual Conditionals'. As the title suggests, the approach Goodman develops (but criticizes too) is about counterfactuals rather than indicatives. (There's currently a free copy available here: http://thatmarcusfamily.org/philosophy/Course_Websites/Readings/Goodman%20-%20Counterfactual%20Conditionals.pdf)

      The more recent theory of Adrian Heathcote alluded to above provides a unified treatment of indicatives and subjunctives - unfortunately I'm not in a position to give you more information about it. (You could try emailing him - he might send you a draft of the unpublished book where he develops the theory.)

    2. P.S. See also chapter 20 ('Support Theories') of Jonathan Bennett's 2003 book A Philosophical Guide to Conditionals.