‘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.

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[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.

The Or-to-If Argument

This is a sequel or appendix to 'The Truth-Functional Account of Indicative Conditionals'.

Before we look at this argument, it needs to be made clear exactly how, if valid, it would support the view that '⊃' can be taken as an abbreviation of 'If...then'. Canonically, the argument proceeds from a truth-functional disjunction of the form '~p ∨ q' to a conditional of the form 'If p then q'. And the truth-function associated with 'p ⊃ q' in the propositional calculus is equivalent (or identical, extensionally speaking) with the truth-function associated with '~p ∨ q'. Secondly, it is widely accepted that 'If p then q' implies '~p ∨ q'. (This has been questioned on subtle grammatical/syntactic grounds, but we will not discuss that here.) Thus it seems that if the Or-to-If Argument is valid,'⊃'-statements can be taken as logically equivalent to indicative conditionals.

Historical preliminary
 
The origin of the idea that one can infer a conditional from a disjunction appears to be unknown. There has been speculation that it originated with Stalnaker. Priest (2001, p. 17) says the Or-to-If Argument was 'given by' Faris (1968) - and it was, but not for the first time. While one of those authors might have made the first use of the inference form as an explicit argument for a truth-functional reading of 'if' after the issue had become controversial in our era, the form itself has a long and venerable history. We find it on p.64 of Cohen and Nagel (1934):
 

Equivalence of Compound Propositions
 
. . .
 

Consider next the alternative proposition Either a triangle is not isosceles or its base angles are equal. To assert it means to assert that at least one of the alternants is true. If, therefore, one of the alternants were false, the other would have to be true. Hence we may infer from the alternative above the following hypothetical If a triangle is isosceles, its base angles are equal.

This textbook, which was popular in its day, also contains quite extensive discussion of the relation between 'formal' and 'material' implication - including a resolution of the 'paradox' attending to the latter (there is no paradox, since the term 'implication' is just given a special technical use in the propositional caclulus). Curiously, this 'paradox' is not related to hypotheticals (conditionals). In fact, hypotheticals are not discussed in the chapter on 'the calculus of propositions' at all, but in two more old-fashioned chapters near the beginning called 'The Analysis of Propositions' and 'Relations between Propositions'. (No doubt this has partly to do with the dominance of the denotational approach to the propositional calculus at that time.) It is in the latter that the Or-to-If Argument and its conclusion appear as a bland lesson.

Even C.I. Lewis, who famously raised the 'paradoxes of material implication' in his 1918 Survey of Symbolic Logic (and articles written earlier), had no problem with '⊃' being read as 'If...then'. He appeared to regard the latter as ambiguous between an "extensional" and an "intensional" reading. A curious passage on p.225 reads [and bear in mind that Lewis was using the notation of the algebraic tradition]:

we can now prove that we have a right to interchange the joint assertion of p and q with p × q, "If p, then q", with p ⊂ q, etc. We can demonstrate that if p and q are members of the class K, then p ⊂ q is member of K, and that "If p, then q", is equivalent to p ⊂ q. And we can demonstrate that this is true not merely as a matter of interpretation but by the necessary laws of the system itself. We can thus prove that writing the logical relations involved in the theorems—"Either ... or ...," "Both ... and ...," "If ... , then ..."—in terms of +,×,⊂, etc., is a valid procedure.

In this case, the "proof" does not proceed from Or to If , but by the previously "established" theorem '(1 a) is equivalent to (a = 1)', together with the rather Tarskian postulate 'For any proposition p, p = (p = 1)', and a tacit use of something like Conditional Proof (which, we shall see, is crucial in the Or-to-If Argument). Today this reasoning would be regarded as metalinguistic, not 'by the necessary laws of the system itself'.

Earlier, we find the Or-to-If Argument given in support of the very first definition in Principia Mathematica, 'Definition of Implication':
 

*1 01. p ⊃ q . = . ~ p ∨ q Df.

. . .
 

According to the above definition, when 'p ⊃ q' holds, then either p is false or q is true; hence if p is true, q must be true. Thus the above definition preserves the essential characteristic of implication . . .

This was then taken to be authoritative in Hankin (1924), a widely-cited legal article on 'Alternative and Hypothetical Pleadings', with the groan-inducing remark:

"If A, then B" is equivalent to the statement "either A is false or B is true". To persons not engaged in the study of logic this may at first appear absurd; yet it can be proved.

In Boole (1847) p.54, the supposed equivalence - except with the negation in the conditional instead of the disjunction - is baldly stated:

To express the conditional Proposition, If X be true, Y is not true. The equation is obviously

                                                              xy=0, (37);

this is equivalent to (33), and in fact the disjunctive Proposition, Either X is not true, or Y is not true, and the conditional Proposition, If X is true, Y is not true, are equivalent.

Earlier still, according to Ashworth (1968), 'The Spanish scholastic, Petrus Fonseca ... [wrote] that the name 'hypothetical' most properly applies to conditional propositions, but can also be used of disjunctions, because they imply a conditional.' Ashworth tells us that Abelard discussed the point in his Dialectica.

It is known that Abelard learnt about the theory of hypothetical syllogisms from Boethius, whose De Hypotheticis Syllogismis, written during the years 516–22, contains what seems to be a related but distinct idea:

[1.3.2] Fiunt uero propositiones hypotheticae etiam per disiunctionem ita:

Aut hoc aut illud est.

Nec eadem uideri debet haec propositio quae superior, quae sic enuntiatur:

Si hoc est, illud non est.

haec enim non est per disiunctionem sed per negationem.

This may be translated as:

[1.3.2] But propositions become hypothetical also through disjunction, thus:

Either this is, or that is.

Neither should the proposition pronounced as follows:

If this is, then that is not.
 
seem the same as the one above. For this one is not through disjunction but through negation.

(Thanks to P.V. Spade for this translation.) Boethius intends exclusive disjunction. To help corroborate the suggestion that this can be seen as a precursor to the Or-to-If Argument: Lagerlund (2010), discusing Boethius's work on hypothetical syllogisms, makes the following suggestion (without specific reference to the text):

Boethius also treats ‘P or Q’ as hypothetical, apparently because he thinks that disjunction can be translated in terms of a conditional sentence

Criticism of the argument
 
Here is the Or-to-If argument in schematic form:
 
1. ~A ∨ B. (Premise)
2. A. (Hyp)
3. B. (1, 2, Disj. Elim)
4. If A then B. (2 - 3, Cond. Proof)
 
Consider the following instance:

1. ~grass is green ∨ grass isn't green. (Premise) 
2. Grass is green. (Hyp)
3. Grass isn't green. (1, 2, Disj. Elim.)
4. If grass is green then grass isn't green. (2 - 3, Cond. Proof)
 
I think there is something wrong with this argument, and I suspect most unindoctrinated people who comprehend it would agree. If a demon somehow convinced me of the truth of '(~grass is green ∨ grass isn't green)', and if I were rational, I would conclude that grass isn't green. In that situation, it would not appear rational (valid, truth-preserving) to conclude further that if grass is green, then grass isn't green. Of course, a defender of '⊃' as 'if' will argue that I have been deceived by appearances on this point. I have tried to undermine the motivation for this in the post on the truth-functional account of indicative conditionals. However, the question remains: what should we say is wrong with the argument?
 
The fallacy occurs, I think, in the step of discharging the hypothesis and deriving a conditional. That is, in the application of the rule of Conditional Proof (roughly speaking, the natural language analogue of the Deduction Theorem for the propositional calculus - I say 'roughly' because DT is strictly a metatheorem, not a proof-rule). Notice that, together with (2) (whose scope it appears in), (3) is an absurdity; it can't be that grass is and isn't green. Accordingly, I propose that CP becomes unavailable once an absurdity has been derived within the scope of the supposition. (Here I count as an 'absurdity' anything which, when conjoined with the supposition, yields an absurdity in an ordinary sense.) That CP is unavailable in such circumstances should not be surprising; if it were not so, all sound reductio arguments could be used to establish bizarre conditionals.

(This constraint is arguably insufficient to make Conditional Proof valid. There will remain the problem of Strengthening the Antecedent, and perhaps others. For a more thorough treatment of this matter, see Thomason (1970) (thanks to Adrian Heathcote for the reference). According to King (2004), Abelard rejected something like Conditional Proof. Given his interest in the semantics of conditionals, it is conceivable that his reasons were closely related to ours.)

Essentially the same point can be seen from another side, if we change the premise to something we actually believe, such as: ~grass is blue ∨ grass isn't blue. Coming to step (2), in this case the hypothesis that grass is blue, if we really want to assume this hypothesis for the sake of argument, then we can hardly use the above disjunction in the ensuing reasoning, unless we are trying for a simple reductio of the proposition that grass is blue. And that would be epistemically queer, since it is hard to see how we could rationally be more sure of the disjunction than the "conclusion" that grass is not blue.
 
What I think all this shows is that the Or-to-If Argument form is not generlly valid, as it would have to be if '⊃' could be read as 'If...then'. Therefore '⊃' cannot be read as 'If...then'. There is, of course, much more to say, in particular concerning the wide range of cases in which one seemingly can argue from Or to If; it seems that while '⊃'-sentences aren't conditionals, assurance of the truth of a '⊃ '-sentence can in many cases serve as a basis for a conditional. The common talk about ordinary conditionals differing from '⊃'-sentences in asserting some kind of natural "connection" between antecedent and consequent is, for this reason, highly suspect.

For a differently orientated discussion of the Or-to-If Argument which culminates in the same verdict - that it is not valid - see Bennett (2003).

References

Ashworth, E.J. 1968. 'Propositional logic in the sixteenth and early seventeenth centuries' in Notre Dame Journal of Formal Logic, Vol. 9, No. 2, 179-192.

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

Boethius, Anicius Manlius Severinus. 516–22. De Hypotheticis Syllogismis.
Original Latin available at Peter King's website:
<http://individual.utoronto.ca/pking/resources.html>
Published in Italian:
(Istituto di Filosoofia dell'Università di Parma, Logicalia 1).
ed. Obertello, L. Brescia, 1968.
 
Boole, George. 1847. The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Cambridge: MacMillan, Barclay and MacMillan. London: George Bell.

Cohen, Morris R. and Nagel, Ernest. 1934. An Introduction to Logic and Scientific Method, London: Routledge & Kegan Paul Ltd.

Faris, J.A. 1969. 'Interderivability of "⊃" and "if"' in Logic and Philosophy: Selected Readings, ch. 7, Iseminger, G., ed., Appleton-Century-Crofts, New York.

Hankin, Gregory. 1924. 'Alternative and Hypothetical Pleadings' in The Yale Law Journal, Vol. 33, No. 4 (Feb., 1924), pp. 365-382.

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

Lagerlund, Henrik, "Medieval Theories of the Syllogism", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2012/entries/medieval-syllogism/>.

Lewis, Clarence Irving. 1918. A Survey of Symbolic Logic, University of California Press, Berkeley.

Priest, Graham. 2001. An Introduction to Non-Classical Logic, Cambridge University Press.

Thomason, Richmond H. 1970. 'A Fitch-style formulation of conditional logic' in Logique et Analyse, 52:397–412.

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

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.