‘Two Recent Theories of Conditionals’ vs. Two Recent Theories of Conditionals

The tide is beginning to turn against counterintuitive theories of indicative conditionals which either deny them truth-values or give them apparently wrong ones, but a deductive argument in Gibbard’s 1981 paper ‘Two Recent Theories of Conditionals’ appears to show that those unhappy options are the only viable ones. Here I summarise some fascinating recent technical work on an escape route and argue that Gibbard’s reason for not taking that route stemmed from a (forgivable) failure of theoretical imagination and a too-narrow view of the motivation for granting truth-values to indicatives.

Introduction

Indicative conditionals seem to have truth-values. Just as ‘I will not eat a grapefruit tomorrow’ and ‘You are a horse’ are true and false respectively, so it seems that ‘If I have breakfast tomorrow, it won’t be a grapefruit’ and ‘If tomorrow someone tells you you’re a horse, you’ll become a horse’ are true and false respectively.

It also seems that an indicative conditional does not always have the same truth-value as the corresponding material conditional (which is true in all cases except when the antecedent is true and the consequent false). For example, both ‘If you die tonight, you’ll be alive tomorrow’ and ‘If you die tonight, the French Government will collapse tomorrow’ seem false - the first due to the nature of life and death, the second due to the way the world is organized - even though ‘You will die tonight ⊃ you’ll be alive tomorrow’ and ‘You will die tonight ⊃ the French Government will collapse tomorrow’ are both true provided that you don’t die tonight.

An ingenious deductive argument from Allan Gibbard’s 1981 paper ‘Two Recent Theories of Conditionals’ appears to show that these two seemings cannot both be right. Gibbard’s collapse argument is so called because threatens to collapse any truth-conditions that an indicative conditional might have down to those of the corresponding material conditional.

But Gibbard’s argument does not by itself demonstrate collapse, and the larger context of Gibbard (1981) shows that he was aware of that fact. Indeed, he identified an escape route - one which takes some noticing, and may not be noticed by many who encounter this much-discussed argument outside the context of Gibbard’s paper. However, upon identifying the escape route Gibbard gave what may seem like a compelling reason not to take it. Having also rejected the material conditional account of indicatives, Gibbard ends up adopting the NTV thesis - the view that indicative conditionals lack truth-values.

Subsequently, theories which take the escape route Gibbard identified have been pursued in earnest anyway, with impressive results. After a period in which the NTV thesis was beginning to look like the dominant view, the tide is finally beginning to turn.

The purpose of this article is to contribute to turning the tide by confronting and neutralizing Gibbard’s reason for not taking the escape route, and along the way to provide a high-level summary of some recent relevant work (some of which can be highly technical). We will see that by drawing on this work we can uphold, in a principled way, the intuitive view that indicative conditionals do indeed have truth-values, and ones which can differ from those of the corresponding material conditionals.

1. Gibbard’s Collapse Argument

I begin with a reader-friendly reconstruction of Gibbard’s collapse argument.

Assumptions:

If Implies Hook: An indicative conditional ‘If A then C’ always implies ‘A ⊃ C’, i.e. indicatives are at least as strong as material conditionals.

Conditional Conjunction Elimination: All indicative conditionals of the form ‘If (A & C) then C’ are logical truths.^[1]

Import-Export: In any arbitrary context, all pairs of indicative conditionals of the forms ‘If A then (if B then C)’ and ‘If (A & B) then C’ are logically equivalent.

Equivalent Antecedents: In any arbitrary context, all pairs of indicative conditionals which share the same consequent, and whose antecedents are logically equivalent, are themselves logically equivalent.

Reasoning:

Consider any arbitrary indicative conditional ‘If A then C’ in an arbitrary context and its corresponding ‘A ⊃ C’.

By Conditional Conjunction Elimination, ‘If (A & C) then C’ is a logical truth.

By Equivalent Antecedents, ‘If ((A ⊃ C) & A) then C’ is then also a logical truth, since ‘A & C’ is logically equivalent to ‘(A ⊃ C) & A’ by propositional logic.

By Import-Export, ‘If (A ⊃ C) then (if A then C)’ is then also a logical truth. (Here ‘(A ⊃ C)’ plays the ‘A’ role in Import-Export as stated above, ‘A’ plays the ‘B’ role, and ‘C’ plays the ‘C’ role.)

By If Implies Hook, ‘(A ⊃ C) ⊃ (if A then C)’ is then also a logical truth, since the implications of logical truths are logical truths.

By If Implies Hook again, ‘(If A then C) ⊃ (A ⊃ C)’ is a logical truth.

By propositional logic applied to the last two sentences, ‘(If A then C) ≡ (A ⊃ C)’ is a logical truth. Hence, any arbitrary indicative conditional in any arbitrary context is logically equivalent to its corresponding material conditional. QED.

If we accept the reasoning and want to maintain that indicatives have truth-values that don’t always agree with the corresponding material conditional, we need to reject one of the assumptions of the argument - either one of the explicit ones listed above, or some auxiliary assumption.

2. The State of the Art of Resisting Collapse

Some have suspected Import-Export. For instance, a detailed axiomatic analysis of Gibbard’s proof leads Fitelson to conclude as follows:

The only axioms that seem plausibly deniable (to me — in the context of a sentential logic containing only conditionals and conjunctions) are [...] the import-export laws, and they seem to be the most suspect of the bunch. I find it difficult to see how any of the other axioms could (plausibly) be denied (but I won’t argue for that claim here). (Fitelson (2013), p. 184.)

However, Import-Export has proven difficult to reject. It strikes many as plausible, and counterexamples have been elusive. Edgington, for instance, finds them plausible in the abstract, and suggests that any example one tries seems to obey the principle:

Here are two sentence forms instances of which are, intuitively, equivalent:

(i) If (A&B), C.

(ii) If A, then if B, C.

(Following Vann McGee (1985) I'll call the principle that (i) and (ii) are equivalent the Import-Export Principle, or “Import-Export” for short.) Try any example: “If Mary comes then if John doesn't have to leave early we will play Bridge”; “If Mary comes and John doesn't have to leave early we will play Bridge”. “If they were outside and it rained, they got wet”; “If they were outside, then if it rained, they got wet”. (Edgington (2014), Sec. 2.5.)

There is one notable attempt at a counterexample in the literature, due to Kaufmann (2005, pp. 213 - 214). In Fitelson’s (2016) presentation:

Suppose that the probability that a given match ignites if struck is low, and consider a situation in which it is very likely that the match is not struck but instead is tossed into a campfire, where it ignites without being struck. Now, consider the following two indicative conditionals.

(a) If the match will ignite, then it will ignite if struck.

(b) If the match is struck and it will ignite, then it will ignite.

It seems like it is possible to understand (a) and (b) in such a way that (a) expresses a logical truth and (b) does not, suggesting that they may not be equivalent, making for a counterexample to Import-Export. But this has been challenged. Khoo and Mandelkern (forthcoming) write:

However, we suspect the intuitive grip of this example rests on an equivocation in ‘will’ between a broadly dispositional meaning and a temporal meaning. We can disambiguate these readings by replacing ‘will ignite’ with ‘is ignitable’, to select for the dispositional meaning, and by replacing ‘will ignite’ with ‘will ignite at t’, to select for the temporal meaning. (We also replace ‘struck’ with ‘struck at t0’, to thoroughly regiment the readings.) We suspect that the reading on which (a) and (b) strike us as inequivalent is:

(a’) If the match is ignitable, then it will ignite at t if struck at t0.

(b’) If the match is struck at t0 and it will ignite at t, then it will ignite at t.

(b’) does indeed strike us as a logical truth, while (a’) certainly does not. But this pair is of course no longer a counterexample to the pattern we are exploring; we would only get a counterexample if we were to disambiguate (a) and (b) in a uniform way. But no matter how we do this, the resulting sentences strike us as equivalent. (Khoo & Mandelkern (forthcoming), pp. 8 - 9 in online version).

In view of the fact that even the most suspect of Gibbard’s explicitly stated principles has proven difficult to reject, it is not surprising that some have rejected auxiliary assumptions not directly appealed to in the derivation. According to Kratzer (1986, 2012, p. 105 in latter) - whose syntactically distinctive theory of indicatives was inspired by Lewis (1975) - the problem with Gibbard’s argument is that it relies on the assumption that indicative conditionals are propositions formed by an operator, ‘if’, which takes two propositions and yields a proposition. If instead we follow Kratzer and treat ‘if’ as a restrictor, and regard ordinary indicative conditionals as containing an unvoiced necessity operator restricted by ‘if’, the conclusion of Gibbard’s argument no longer leads to the result that indicative conditionals, if they have truth-conditions at all, are truth-functional. For Gibbard’s conclusion is that if indicative conditionals are propositions in which a two-place propositional operator is applied to two propositions, then their truth-conditions collapse to those of the material conditional.

However, as Khoo (2013) has shown in detail, an analogous argument can be given directly in terms of the semantic values of sentence-schemas, without assuming that ‘if’ is a two-place propositional operator. But it turns out that Kratzer’s theory is nevertheless able, in another way, to block both the original and the analogous argument. Kratzer’s theory predicts subtle counterexamples to the principle that whenever an indicative conditional is true, so is the corresponding material conditional, thus invalidating the If Implies Hook assumption of Gibbard’s argument. So too does Gillies’ (2009) theory, on which ‘if’ is a two-place operator, but one which is able to shift the index and context^[2] against which the consequent of an indicative conditional is evaluated (in the course of the evaluation of the conditional containing it).

On Khoo’s analysis, Kratzer’s alternative view of the syntax of indicative conditionals is orthogonal to the collapse issue. Both her theory, on which ‘if’ is a restrictor, and Gillies’ theory, on which ‘if’ is a “shifty” two-place propositional operator, avoid Gibbard’s conclusion. But in consequence of how they avoid Gibbard’s conclusion - by invalidating If Implies Hook - both theories predict counterexamples to modus ponens construed as a semantic thesis according to which ‘C’ is true whenever ‘A’ and ‘If A then C’ are both true.

Completely invalidating modus ponens would be a serious issue and would naturally cast doubt on these theories. But, like McGee’s (1985) independently-motivated counterexample to modus ponens, the main conditionals in the predicted counterexamples feature indicative conditionals in their consequents. That the predicted counterexamples are in this way similar to independently-motivated ones suggests that they are not mere artefacts of faulty theories. Furthermore, while modus ponens construed as a semantic thesis as explained above turns out to be invalid on these theories, modus ponens as a practical inference rule remains unaffected, insofar as asserting or supposing something has the effect of restricting the range of possibilities against which conditionals are evaluated to ones in which that thing holds. In this way, both theories are compatible with modus ponens being “dynamically valid” (for details see Khoo (2013)). It seems reasonable to suppose that this is all the modus ponens we need.

Although the whole of this intricate story could not have been imagined by Gibbard, he certainly was aware in the abstract that theories which, like Kratzer’s and Gillies’, allow embedded indicative conditionals’ semantic values to differ from the semantic values they would get if taken alone, have the resources to avoid his conclusion.

This possibility, now realised in detail by existing theories, was the very escape route that Gibbard identified and gave reason not to take. The assumption that a given indicative conditional sentence in a given context always gets the same semantic value, regardless of whether it is embedded in a larger conditional, is thus an auxiliary assumption of Gibbard’s proof.

3. Why Gibbard Wouldn’t Take the Escape Route

Gibbard’s identification of this auxiliary assumption and his argument against rejecting it are contained in the following passage:

One other possibility remains: that → always represents a propositional function, but that what that function is depends not only on the utterer's epistemic state, but on the place of the connective in the sentence. In a → (b → c), for instance, we might suppose that the two different arrows represent two different propositional functions. Nothing we have seen rules that out.

The pursuit of such a theory, though, has now lost its advantage. A theory of indicative conditionals as propositions was supposed to give, at no extra cost, a general theory of sentences with indicative conditional components: simply add the theory of conditionals to our extant theory of the ways truth-conditions of sentences depend on the truth-conditions of their components. The alternative was to develop a new theory to account for each way indicative conditionals might be embedded in longer sentences, and that seemed costly. Now it turns out that for each way indicative conditionals might be embedded in longer sentences, a propositional theory will have to account for their propositional content, and do so in a way that is sensitive to the place of each indicative conditional in its sentence. In a → (b → c), the right and left arrows must be treated separately. What must be done with the left and right arrow in (a → b) → c or with the arrows in a & (b → c) and a ∨ (b → c) we do not yet know. Thus, for instance, no account of sentences of the form (a → b) → c will fall out of a simple general account of indicative conditionals as propositions; rather the account of indicative conditionals itself will have to confront separately the way left-embedded arrows work. A propositional theory would not save labor; instead it would demand all the labor that would have to be done without it. (Gibbard (1981), pp. 236-237)

The way Gibbard puts it, the assumption at issue is that ‘→ is a fixed propositional function’ (Gibbard (1981, p. 236)), but for present purposes it is the ‘fixed’ part that is relevant, and in view of the possibility of a Kratzerian treatment of the syntax of indicatives, we should separate the ‘fixed’ part out and state it in a way that does not presuppose that ‘→’ is syntactically a two-place propositional operator. Hence our statement of it at the end of the previous section: a given indicative conditional sentence in a given context always gets the same semantic value, regardless of whether it is embedded in a larger conditional. Or in other words again, the assumption is that in a given context, there is no more than one indicative conditional with one set of truth-conditions per pair of antecedent and consequent. Henceforth let’s call this the fixity assumption.

4. The Escape Route is Open

I will give a four-pronged argument against Gibbard’s defense of the fixity assumption. If it is successful, we are left free to abandon the fixity assumption and thus to resist the collapse of indicative conditionals into material conditionals while maintaining a truth-conditional approach to indivatives.

Prong 1. Gibbard’s description of the extra work we must do if we abandon the fixity assumption in the pursuit of truth-conditions for indicatives overplays the amount of extra work required, due to what appears to be a (forgivable) failure of theoretical imagination on his part.

Gibbard says that if we give up fixity, then ‘for each way indicative conditionals might be embedded in longer sentences, a propositional theory will have to account for their propositional content, and do so in a way that is sensitive to the place of each indicative conditional in its sentence’. This may be strictly correct, but it doesn’t follow that such a theory has to confront each form of embedding separately, or that this sensitivity to place cannot come about in an elegant, systematic way.

Indeed, the sensitivity to place of indicatives-inside-indicatives that we need in order to block Gibbard’s collapse argument does come about in an elegant, systematic way on both of the theories we have been discussing. On Kratzer’s theory, it stems from the fact that ‘if’ restricts a modal and that such restriction may occur more than once in a single sentence. On Gillies’, it stems from the fact that ‘if’ shifts index and context, and that such shifting may occur multiple times in a single sentence.

Thus, when Gibbard says that ‘no account of sentences of the form (a → b) → c will fall out of a simple general account of indicative conditionals as propositions; rather the account of indicative conditionals itself will have to confront separately the way left-embedded arrows work’, this - provided that Kratzer’s and Gillies’ theories qualify as ‘simple’ - is simply false. An account of sentences of that form does fall out of both accounts.

Krazter’s and Gillies’ theories deliver, in an elegant way, different semantic values for conditionals depending on where they are in a sentence. And it seems to me that there is a good sense in which these theories are such that we can ‘simply add the theory of conditionals to our extant theory of the ways truth-conditions of sentences depend on the truth-conditions of their components’.

Prong 2. Following Gibbard in embracing the NTV thesis creates special work of its own, which does not have to be done if we hold that they have truth-values.

For one thing, there is an irony in his complaint that if we give up the fixity assumption ‘we do not know’ what to do with the arrow in a sentence of the form ‘a ∨ (b → c)’. In keeping with what we saw in the previous prong, the fixity-denying theories we now have do not encounter any special difficulty in handling such sentences, and we do not have to consider such forms of embedding on a one-by-one basis. Now we may observe further that, if anything, it is the NTV route that leads to issues with such a form; if we deny truth-values to indicative conditionals, we don’t know what to do with the wedge in such a sentence. That is, we face the extra work of making sense of, or denying sense to, embeddings of allegedly truth-valueless sentences in what appear to be truth-functional contexts. (See, however, Edgington (1995) for a classic defense of the view that such embeddings are not problematic after all.)

That is one sort of extra work the NTV theorist seems to be saddled with. And there is another, quite different sort. Namely, the work of explaining what is going on when people appear to ascribe truth-values to indicatives. A truth-value-granting view of indicatives such as Kratzer’s or Gillies’ lets us take these ascriptions at face value, and to allow that they are often correct. An NTV view must either reinterpret these ascriptions so that they aren’t all incorrect, or explain why people so often say these incorrect things. So if we want to avoid extra work, it may be that we do better to uphold truth-value-granting theories like Kratzer’s and Gillies’.

Prong 3. It’s not all about extra work! The issue is whether we should or should not respond to the collapse argument by denying that indicatives have truth-values. To proceed as though this issue turns just on whether we save labor by maintaining that indicatives have truth-values is too narrow. Labor-saving patently isn’t the only reason why we might want to maintain that indicatives have truth-values. A distinct and arguably very important reason is that they seem to have truth-values! (How compelling you find this will depend on your philosophical orientation, but if you think that what pre-theoretically seems to be the case is an important guide in philosophy, it should count for quite a bit.)

Prong 4. Gibbard’s argument against abandoning the fixity assumption obscures the fact that, when you think about it, it makes sense to expect the assumption to be false. Rejecting the assumption is presented by Gibbard as a last resort. But rejecting the fixity assumption is not, on reflection, some intuitively unpalatable thing which we get forced into doing just so that we can uphold a prejudice.

There are well-developed, intuitively motivated views which enable us to think of indicative conditionals, schematically, as saying something like ‘In all relevant possibilities in which the antecedent holds, the consequent holds’. And it is quite natural to think that what is known to be true, or what is being supposed to be true, can affect what possibilities are relevant. Furthermore, it is quite natural to think of the antecedents of conditionals, for example, as introducing a supposition. Putting these last two things together, it is quite natural to think that the possibilities relevant for the ‘if B then C’ in ‘If A, then if B then C’ may differ from the possibilities relevant for an unembedded ‘If B then C’. In particular, it is natural to think that only A-possibilities will be relevant to the embedded conditional, while not-A-possibilities may still be relevant to the unembedded one.

So, the negation of the fixity assumption is something which has quite a bit of plausibility. At the very least, it seems plausible from within the general way of looking at indicatives which the collapse argument is supposed to threaten. Namely, a perspective according to which indicatives have truth-values and in some sense deal with ranges of relevant possibilities. And obviously, such a perspective has much to recommend it besides helping us to resist Gibbard’s argument.

5. Conclusion

Starting from the intuitiveness of the view that indicative conditionals have truth-values which can differ from those of the corresponding material conditional, we looked at how Gibbard’s collapse argument threatens that view, and how Import-Export, flagged as suspicious by Fitelson, is hard to fault. Drawing on work by Khoo, we then saw that both Kratzer’s and Gillies’ independently-motivated theories of indicative conditionals block Gibbard’s argument at the cost of invalidating modus ponens construed as a general semantic thesis, but that the predicted counterexamples coincide with McGee’s independently-motivated ones and leave modus ponens unscathed as a form of dynamically valid inference. We then looked at Gibbard’s argument against truth-value-granting theories of indicatives which, like Kratzer’s and Gillies’, reject the fixity assumption, and saw that the threat is not serious. Gibbard’s refusal to abandon fixity in pursuit of truth-conditions for indicatives stemmed from a failure of theoretical imagination and a too-narrow view of the motivations for non-material, truth-value-granting accounts of indicatives. The prospects for such accounts appear to be brightening.

References

Edgington, Dorothy (1995). On conditionals. Mind 104 (414):235-329.

Edgington, Dorothy (2014). Indicative Conditionals. In The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), ed. Edward N. Zalta. https://plato.stanford.edu/archives/win2014/entries/conditionals/

Fitelson, Branden (2013). Gibbard's Collapse Theorem for the Indicative Conditional: An Axiomatic Approach. In Automated Reasoning and Mathematics: Essays in Memory of William W. McCune, M.P. Bonacina and M. Stickel (eds.), Springer.

Fitelson, Branden (2016). Two new(ish) triviality results for the indicative conditional. Lecture Notes. http://fitelson.org/triviality_handout.pdf

Gibbard, Allan (1981). Two Recent Theories of Conditionals. In William Harper, Robert C. Stalnaker & Glenn Pearce (eds.), Ifs. Reidel. pp. 211-247.

Gillies, Anthony S. (2009). On truth-conditions for if (but not quite only if ). Philosophical Review 118 (3):325-349.

Kaufmann, Stefan (2005). Conditional predictions. Linguistics and Philosophy 28 (2):181 - 231.

Khoo, Justin (2013). A note on Gibbard's proof. Philosophical Studies 166 (S1):153-164.

Khoo, Justin & Mandelkern, Matthew (forthcoming). Triviality results and the relationship between logical and natural languages. Mind.

Kratzer, A. (1986). Conditionals. Chicago Linguistics Society, 22(2), 1–15.

Kratzer, A. (2012). Collected papers on modals and conditionals. Oxford: Oxford University Press.

Lewis, D. (1975). Adverbs of quantification. In: E. L. Keenan (Ed.). Formal semantics of natural

language. Cambridge, MA: Cambridge University Press.

---

[1] Gibbard leaves the notion of ‘logical truth’ unexplicated in his proof, but the arguments in the present article do not turn on any particular understanding of it.

[2] Contexts, whatever they are, should be thought of as determining ranges or sets of possibilities relevant to the evaluation of conditionals in that context. Cf. Gillies (2009), p. 329 (incl. f.n. 5). Note also that the use of ‘possibilities’ here should not be taken to imply that the possibilities in question are all metaphysical possibilities.

Transitivity and Conditionals

Followup:

Sufficient Conditions for Some Inference-Patterns Involving Conditionals

Let us assume that a conditional A > C is true iff in all relevant scenarios the corresponding material conditional is true.

Let's leave it completely open here what makes a scenario relevant for a conditional. Let's also leave it open what scenarios are like.

(That something like the above is true for counterfactual or subjunctive conditionals seems more widely accepted than that something like it is true for indicatives, so the following will be most widely acceptable as an observation about the logic of counterfactuals. I think it probably applies to indicatives too. That it holds on the assumption of the above schematic semantics seems to me to be almost beyond dispute.)

In their 2008 paper 'Counterfactuals and Context', Brogaard and Salerno attempt to block a famous counterexample to transitivity for counterfactuals (cf. Lewis , p. 33) with the proposal that to have conditionals for which different scenarios are relevant figuring in the same argument is illicit.

But an inference from A > B and B > C to A > C will be truth-preserving as long as the set of relevant scenarios for the second is a subset of that for the first, and the set of relevant scenarios for the third is a subset of that for the second. (Note I don't say 'proper subset': they could all be the same set, but that's a special case.)

Illustration: If I had spoken to a cat then I would have spoken to an animal. If I had spoken to an animal I would have been happy. Therefore, if I had spoken to a cat then I would have been happy. (It is natural to think of the set of relevant scenarios for the first sentence as larger than that for the second. This could be further brought out by adding something like 'no matter what' to the first sentence.)

(This post builds on this.)

References

Brogaard, Berit & Salerno, Joe (2008). Counterfactuals and context. Analysis 68 (297):39–46.

Lewis, David K. (1973). Counterfactuals. Blackwell Publishers.

A Problem for the Simple Theory of Counterfactuals

In a recent blog post called 'The Simple Theory of Counterfactuals', Terrance Tomkow argues extensively for a theory of counterfactual conditionals along broadly Lewisian lines, explicitly restricted to counterfactuals with nomologically possible antecedents. The theory, Tomkow says, was first proposed by Jonathan Bennett in 1984, but later abandoned. Lewis held a more complicated theory.

Tomkow argues successfully, in my opinion, against Bennett's reasons (given in his Philosophical Guide to Conditionals) for rejecting his own theory. (Tomkow tells me, in a private communication, that Bennett has agreed with these arguments of Tomkow's, also in a private communication.) There is much else of value in the post as well. However, I cannot agree with Tomkow that the theory as he states it, even with its restriction, is correct.

The Simple Theory, or the Bennett-Tomkow Theory, is this:

THE SIMPLE THEORY
A > C iff  C is true at the legal A-worlds that most resemble @ at TA.

('A > C' is a shematization of 'counterfactual statements of the form: If ANTECEDENT had been the case then CONSEQUENT would have been the case.'

'@' denotes the actual world. 'Tp' denotes the time that the proposition 'p' is about. 'Legal' worlds are nomologically possible worlds.

The restriction of the this theory is then given as follows: 'To keep things simple, we will only deal with cases where A is false at @ but nomologically possible.')

Now, before giving the objection which is the main point of the present post, I want to note a simpler but less powerful objection. Some counterfactuals with nomologically possible antecedents are categorical - that is, require that all A-worlds are C-worlds. For example 'If I had met a bachelor this morning, I would have met an unmarried man this morning', in the context of a language-lesson. I argue for this here. The Simple Theory seems to assign the wrong meaning here, since it says that such a counterfactual is true iff C is true at the legal A-worlds that most resemble @ at TA, and these won't be all A-worlds, as intuitively required by the counterfactual. This objection is less powerful than the one I am about to give, because it can be easily avoided by simply restricting the theory to non-categorical counterfactuals.

Now the more powerful objection. This is inspired by my cartoon understanding of the confirmation of relativity, but let's just treat it as a fiction. Einstein asserted a law in paper N which actually holds, and which, together with the facts of some experimental setup E, predicts that some light will bend.

Now, it seems to me we can evaluate counterfactuals where the relevant closest A-worlds are worlds where the law doesn't hold, for example ones with the antecedent '~L' (where L is the law in question). Tomkow seems to agree, saying in a comment that 'we do need an account of counterfactuals with contra-legal anteced[e]nts'. So far, no problem for the Simple Theory.

My idea is that there are counterfactuals whose antecedents are legal, but where the similarity relation is contextually understood in such a way that the closest relevant A-worlds are counter-legal. So, with the following counterfactual:

(H) If Einstein had been wrong in paper N, this light would not have bent.

both what Einstein wrote and the experimental setup may be held fixed during evaluation (i.e. match in these respects required for close similarity), while the actual laws of nature are not held fixed. The antecedent itself is legal, however, since there are legal worlds where Einstein is wrong in paper N, but where he writes something else.

I will now try to make this more precise, and spell the objection out.

For a given counterfactual and contextual understanding of it, call the 'focus set' the set of A-worlds at which C is required, by the counterfactual, to be true. (This of course assumes that a theory with broadly Lewisian/strict-implication outlines is basically right.)

The special property (H) was designed to have is thus: having a legal antecedent, yet being legitimately and naturally understandable such that its focus set contains counter-legal worlds.

If there are counterfactuals with that property, that's a problem for the Simple Theory as stated, since it says that 'A > C iff C is true at the legal [my emphasis] A-worlds that most resemble @ at TA'.

Their having legal antecedents puts them in the scope of the Simple Theory as stated, but the presence of counter-legal worlds in their focus sets (on the relevant understandings of them) conflicts with it.

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.

A Modification to Lewis's Theory of Counterfactuals

I propose a modification to Lewis's (1973) theory of counterfactuals, which has come to be treated by many as the standard semantics for counterfactuals. Lewis's theory is that a counterfactual conditional with antecedent A and consequent C is true iff all the most similar A-worlds (worlds at which A is true) are C-worlds. Lewis admits that what matters for similarity varies a lot from sentence to sentence, and from context to context.

What I propose is that similarity sometimes plays no part at all, and that whether it does also varies with sentence and context. When it plays no part, the truth of the counterfactual in question requires that all A-worlds are C-worlds. (To state the modified theory elegantly, we could speak of 'all relevant A-worlds', defining 'relevant' using 'most similar' but adding that sometimes all A-worlds will be relevant.)

The argument for this modification involves what could be called categorical counterfactuals. Consider the following sentence, uttered in the context of teaching someone how to use the word 'bachelor':

(A) If I had spoken to a bachelor this morning, I would have spoken to an unmarried man this morning.

Intuitively, the truth of this hinges on the fact that bachelors are necessarily unmarried men. Lewis's analysis, without my proposed modification, although it gives the right truth-value, gives the wrong truth-condition and thus distorts the meaning of (A); it is false to say that the truth-condition for this sentence is that all the most similar A-worlds are C-worlds - on any understanding of similarity.

The modified theory handles (A) much better: this is one of those cases where similarity plays no part, and so (A) is true iff all worlds where I spoke to a bachelor this morning are worlds where I spoke to an unmarried man this morning. This seems right.

(A note on the structure of Lewis's theory as formally developed with systems of spheres: this can remain as is, but in the case of categorical counterfactual conditionals the “innermost” sphere will contain all worlds, and so it would be misleading to call the worlds in this sphere 'the most similar A-worlds'.)

Reference

Lewis, D. 1973. Counterfactuals. Basil Blackwell: Oxford.