Saturday, 2 November 2019

‘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.
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.
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.
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.
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.
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.
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.


  1. This is very interesting. I've always thought If Implies Hook is the assumption to reject (I think arguments for it often implicitly make the mistake of assuming that if the indicative conditional's behavior is partly described by some lines on the material conditional truth table that it is fully described by the whole material conditional truth table), but I seem to have missed Kratzer's work completely.

    It's probably only loosely connected, but the fixity issue reminds me a bit of Lewis Carroll's Nemo and Outis (or A Logical Paradox, which just uses different names); I played around with all sorts of different ways to that problem years ago and came to the tentative conclusion that Outis (or Uncle Jim) will always get the better of the argument, even varying lots of different assumptions for how conditionals work, as long as we assume that conditionals remain the same in all contexts.

    1. Thanks for the reference. Carroll was very penetrating when it comes to conditionals. In fact he anticipates the 'Sly Pete' example of Gibbard (which later appeared in the cleaned up form of the 'Top Gate' example in Bennett's book) with an example involving barbers. (I learned this from Adam Rieger's 2013 article 'Conditionals are material: the positive arguments' published in Synthese.)

      Kratzer's work may have been neglected in philosophy for a time in part because she's a linguist, but I think that this has largely been gotten over more recently.

  2. I would like to inform about some papers devoted to the entailment problem.

    In the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment", Journal of Mathematics and System Science , vol. 5, 60-71 (2015): ; arXiv:1603.06621 , among others, a new solution of the well-known problem of relevant logics was given, i.e. atomic entailment was defined and Atomic Logic was formulated (the abstracts related to this paper: T. Stepien, „Logic based on atomic entailment”, Bulletin of the Section of Logic, vol. 14 (2), 65 – 71 (1985) ; T. J. Stepien and L. T. Stepien, „Atomic Entailment and Classical Entailment”, The Bulletin of Symbolic Logic, vol. 17, 317 – 318 (2011))

    In the next paper: T. J. Stepien and L. T. Stepien, „The Formalization of The Arithmetic System on The Ground of The Atomic Logic”, Journal of Mathematics and System Science, vol. 5, 364 – 368 (2015): ; arXiv:1603.09334, it has been proved that the classical Arithmetic can be based on the Atomic Logic (presented in the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment"). The abstract of this paper: T. J. Stepien and L. T. Stepien, "The formalization of the arithmetic system on the ground of the atomic logic" , The Bulletin of Symbolic Logic 22 , No. 3, 434 - 435 (2016).

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