Monday 24 June 2013

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).


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:

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 = <>.

Lagerlund, Henrik, "Medieval Theories of the Syllogism", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <>.

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.


  1. 1. ~grass is green ∨ grass isn't green. (Premise)

    is equavelent to say that grass isn't green. And assuming grass isnt green it follows that, if grass is green then it is not green. I see no problem with that since you didnot and cannot discharge the assumption. And given the assumption your conclusion follows. There is nothing wrong with or to if instence

    1. 'And assuming grass isnt green it follows that, if grass is green then it is not green.' - I deny this, since I argue that conditional proof is subject to the restriction I propose in the post.

      'I see no problem with that since you didnot and cannot discharge the assumption.' - I don't understand this as a reason. The assumption in the (in my opinion invalid) argument in question is indeed not discharged before the disjunction elimination step, but then it is discharged, and a conditional is derived - invalidly, as I have argued.

      'And given the assumption your conclusion follows.' - What are you calling the assumption here? 'Grass isn't green' follows from what I have called the premise. But, as I have argued, the conditional doesn't follow. You are of course free to contradict me without argument, but be clear that that's what you're doing.

      If we change 'green' for 'blue' in this example, the premise is true. So by your lights, it's true that if grass is blue, it isn't blue. I take it you accept the truth-functional account of indicative conditionals. I argue further against that in the prescursor post.

  2. With regards to the criticism it seems that the main issue arises because A and B essentially mean the same thing:

    ~grass is green ∨ grass isn't green.

    but "grass isn't green" just is ~"grass is green", so I suspect there's a problem with using A and B to represent these two atomic sentences rather than use A and ~A, which results in:

    (~A v ~A) instead of (~A v B)

    which will have none of the problems your argument point out. Perhaps another example might raise a similar result but as of now the example doesn't seem convincing.

    1. Thanks for commenting.

      I don't think what you're saying here is right. First of all, why do you say that with (~A v ~A) there are none of the problems my argument points out? Starting with that we may argue:

      (~Grass is green v ~Grass is green) (premise)
      ~~Grass is green (hyp)
      ~Grass is green (disj. elim)
      If ~~Grass is green then ~Grass is green (cond. proof)

      And then that conclusion can also be made more manifestly absurd by eliminating the double negation.

      Secondly and more generally, if the argument form were generally valid, how could there be a problem with any uniform substitution?

      To be sure, valid argument forms are of course such that if you substitute different things into the same placeholder the result may be invalid, but, by definition, they are never such that if you uniformly substitute the same things into different placeholders the result may be invalid.

  3. ~grass is green ∨ grass isn't green
    The premise is equivalent to saying that grass isn't green.
    Then the statement:
    if grass is green, then grass isn't green
    if false, then true
    given the original premise.
    The conditional proof holds under the premise that grass isn't green.

    1. 'grass is green ∨ grass isn't green
      The premise is equivalent to saying that grass isn't green.'

      I agree.

      'Then the statement:
      if grass is green, then grass isn't green
      if false, then true
      given the original premise.'

      I agree.

      'The conditional proof holds under the premise that grass isn't green.'

      I disagree. I don't think all conditionals with false antecedents and true consequents are true. For example, I think it's false that if I die tonight I will be alive tomorrow. But assuming I won't die tonight, that conditional has false antecedent and true consequent.

      You seem to just be assuming the material conditional analysis of the material conditional. But that is what the or-to-if argument sets out to establish, and here I am taking issue with that argument on intuitive grounds.

  4. Thanks for the nice write up.

    Might you kindly post a reference or two to work in which the implication from 'if p then q' to '~p ∨ q' "has been questioned on subtle grammatical/syntactic grounds"? I for one am interested to have a look.


    1. Tristan Haze2 November 2015 at 15:15
      Pardon the late reply.

      Unfortunately, the main work I had in mind was and remains unpublished. It was a manuscript by Adrian Heathcote - you could perhaps try contacting him and asking for it. One of the things he argued was that truth-functional propositions, properly analyzed, have to mention, rather than use, sentences. Whereas conditionals, according to Heathcote, do not (in general) *mention* their antecedents or consequents, so you cannot infer the truth function from the conditional. There may be other views delivering the same result, but I have always found this unconvincing, as I see no reason to think that Heathcote is right that truth-functional propositions have to *refer to* rather than just *contain* the sentences/propositions being operated on. (I think Heathcote has too narrow conception of what logical forms there can be - this also shows up in his treatment of identity statements, which he has - again erroneously in my view - argued to be metalinguistic.)

    2. Interesting. Thanks for the lead and the info, Tristan!