Issue #01. April 1995. Pp. 96-99.
When Is If?
Copyright (C) by SORITES and M. G. Yoes, Jr.
When Is If?
M. G. Yoes, Jr.
Consider Adam's example:
(1) If it rained, it did not rain hard.
(2) It did rain hard.
Therefore,
(3) It did not rain.
Modus Tollens, it might seem, has absurdly led us to call this intuitively invalid argument valid. A counterexample to Modus Tollens? What has gone wrong?
Indeed, if we accept
(4) If it did rain hard, it rained
as logically true, then Modus Ponens leads from (2) and (4) to
(5) It rained
which together with (1) and Modus Ponens again implies
(6) It did not rain hard.
Should we say, then, that if we accept Modus Ponens and (4) as a logical truth, we are stuck with saying that the premise set {(1), (2)} of the argument is itself inconsistent? That the argument `If it rained, it did not rain hard; therefore, it did not rain hard' is valid'? Is Modus Ponens to be indicted as well?
No. This is a case of unusual symptoms but mistaken diagnosis. For the problem is not in the logic but in the representation of the logical form of (1) as a conditional, a problelm of surface grammar being a false clue. Here is another example of Russell's lesson that grammar can hide logical form. For (1) can be paraphrased as something like
(7) It may have rained, but it did not rain hard,
a mere conjunction. (1), then, on this reasonable representation, is a conjunction masquerading as conditional. The idea in (1) is not somehow that its not raining hard is conditioned on its having rained, but, as it were, to allow the possibility that it rained while denying that it did rain hard.
The matter can be put more cautiously. Perhaps (1) could be used to state a genuine conditional, but it would be a strange conditional indeed. If (1) is represented as a conditional, and we keep that assumption firmly in mind, then the surprise of these examples shifts from the validity of the argument to the peculiarity of the premise. If it rained, then it did not rain hard?
The analysis roughly is this. (1), and presumably some other classes of `if' statements, are not conditionals at all but conjunctions in disguise. The `if' in these statements functions as some sort of modal but with small scope: if it rained,...; that is, it may have rained, but.... . (Not that this analysis works for all `if's, for `if's are not univocal.)
On this analysis, of course, (3) does indeed follow from (1) and (2), though not by Modus Tollens since (1) is not a conditional. The premise set {(1), (2)} is likewise inconsistent on this analysis and the argument `(1), therefore ¬(2)' is valid. The argument `If it rained, it did not rain hard; it rained; therefore, it did not rain hard' is valid on this analysis, though not by Modus Ponens since again (1) is not a conditional.
An analysis that makes `If it rained, it did not rain hard; thus it did not rain hard' valid may seem counterintuitive; conditionals do not imply their consequents and the trained logical intuition sees a conditional behind every `if'. Still, recognizing that one quite normal reading of `If it rained, it did not rain hard' is `It may have rained, but it did not rain hard' may blunt the intuition.
This is not unlike Austin's exampleFootNote 74:
(8) There are biscuits on the sideboard if you want one.
If (8) is true and there are no biscuits on the sideboard, is it a fault in ancient and modern logic that no one would accept an inference to your not wanting a biscuit? No. Again we are better off saying that (8) is a hidden conjunction something like:
(9) There are biscuits on the sideboard and perhaps you want one.
Like any conjunction, of course, the whole will be false if either conjunct is false; so,
(10) There are no biscuits on the sideboard
implies that (9), and thus (8), are both false. But (10) does not imply that you do not want a biscuit. Likewise (2) does imply that (7) and thus (1) are both false. But (2) does not imply that it is false that it may have rained.
Construing (8) as a conjunction, despite its conditional disguise, allows us to infer ¬(8) from (10) since (10) gives ¬(9). But what if the right conjunct of (9) is false:
(11) ¬(perhaps you want a biscuit).
If (11) means
(12) You certainly do not want a biscuit
does ¬(8) follow? Isn't (12) consistent with (8)? Might it not be true that there are biscuits on the sideboard if you want one, while you certainly do not want one? Similarly, might it not be true that if it rained it did not rain hard, while it could not have rained? It is not merely a question of whether there is a likely conversational implicature that perhaps you do want a biscuit brought off by anyone who says there are biscuits on the sideboard if you want one. It is a question of truth conditions.
Yet there is no doubt that (8) is in some way incomplete. The implicit modality which the analysis in (9) brings out is necessarily read in so as to capture the weak `if'. There is ellipsis here. And perhaps `perhaps' is not quite the correct modality; the speaker may have to disambiguate for us. One reading which casts matters in a different light is
(13) There are biscuits on the sideboard and you may have one.
This statement is falsified, of course, by the assumption that you may not have a cookie. So if (8) is captured by (13) then (8) is likewise falsified, and the thesis that (8) is some sort of conjunction is confirmed. Moreover, it seems clear that
(14) There are biscuits on the sideboard if you want one; but you may not have one.
is more than conversationally at odds with itself, but in some way actually inconsistent.
The general point does not rest on the particular analysis of (8) as (9), but rather on the hypothesis that there exists a class of statements superficially of the form `A if B' which are best understood as conjunctions of the form `A and (Modal(B))'. What the modal actually is may strongly depend on context. Indeed, without a specific context (13) seems as good a candidate as (9) for a paraphrase of (8).
`If' may function as a mere modality, introducing doubt, uncertainty or whatever, in which case it is not functioning as a conditional, as a real iffy `if'. This mere modality of small scope is marked by the fact that it is hardly comfortable with a corresponding `then': If it rained, then it did not rain hard? If you want one, then there are biscuits on the sideboard?
When, then, is if? When it functions as an if. When is that? Standard formal logic answers: when it satisfies Modus Ponens, Modus Tollens, etc. Some `if's do not satisfy these formal properties, and therefore, according to this standard, are not conditionals at all. Thus do the formal properties define the conditional, as they define the other logical notions. The force of this definition is that it unmasks logical constructions in disguise.
M. G. Yoes, Jr.
Department of Philosophy
University of Houston
Houston, Texas 77204-3785
e-address: MGYoes@uh.edu