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.