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Issue #17 -- October 2006. Pp. 56-67

Subcontraries and the Meaning of «If...Then»

Copyright © by Ronald A. Cordero and SORITES

Ronald A. Cordero

1. Conditionals Composed of Subcontraries

Paradox seems to arise when conditional statements have subcontrary statements as antecedent and consequent. I am going to argue here, however, that these conditionals can in fact function as well as ordinary conditionals and that a careful consideration of such cases can both resolve the apparent paradox and lead to a better understanding of the meaning of «if...then» in general.

Suppose, for a first example, that we know that our friend Jan never serves strawberries. If we learn that she had guests for dinner last night, it would be perfectly possible for us to say meaningfully, «Well, if she served fruit, she didn't serve strawberries.» We may not know whether or not she served fruit, but we know what kind of fruit she *didn't* serve if she served any fruit at all. Now, the statements «She served fruit,» and «She didn't serve strawberries,» fit the definition of subcontraries: they could be true simultaneously, but they could not be false at the same time. She could have served fruit but not strawberries but could not have served strawberries but not fruit. So we have a conditional statement composed of subcontraries that we might very well use in replying to a question such as «If she served fruit, was it strawberries?» And anyone who remembered our reply and later learned that Jan did in fact serve fruit could reason by the axiom (rule of inference) known as Modus Ponens (((p⊃q) ∧ p) ⇒ q ) to the conclusion that she did not serve strawberries.

Or suppose we learn, for a second example, that a certain figure drawn on the sidewalk was either a triangle or a square. We of course know that if it was a square, it was not a circle. So does it not follow that if the figure wasn't a triangle, it wasn't a circle?

1. T ∨ S Given

2. S ⊃ ¬C Analytic *a priori*

3. ¬T Assumed for CP

4. S 3,1; Disjunctive Syllogism

5. ¬C 4,2; Modus Ponens

----

6. ¬T ⊃ ¬C 3-5; CP

Line 6 represents a conditional composed of subcontraries that could presumably be used to transmit useful information. If someone learns the truth of (6) from a dependable source, and later learns that the figure was not a triangle, they will know that it was not a circle either -- even if they do not know what kind of figure it *was*.

It is worth noting that in the two examples considered so far, simultaneous falsity of antecedent and consequent is an *a priori* impossibility.

1. If Jan served fruit, she didn't serve strawberries.

2. If the figure wasn't a triangle, it wasn't a circle.

Because of what we mean by the terms «fruit» and «strawberries,» it cannot be that Jan failed to serve fruit but did serve strawberries. And because of the way we use the words «triangle» and «circle,» the figure cannot have been both. But not all cases of conditionals composed of subcontraries are like this. We can also encounter conditionals composed of statements that are *de facto* or *a posteriori* subcontraries. This occurs with subcontraries that are the contradictories of *de facto* contraries--statements which owing to some *a posteriori* fact of the situation cannot both be true but can both be false.

Suppose, for instance, that there is only room for one person to park in the drive. Then it cannot be true both that Jim is parked in the drive and that Mary is parked there too. But it can be the case that neither Jim nor Mary is parked there. It follows that the statements «Jim is parked in the drive,» and «Mary is parked in the drive,» are, in the circumstances given, *de facto* contraries. And this makes their respective contradictories *de facto* subcontraries:

Jim isn't parked in the drive.

Mary isn't parked in the drive.

In the circumstances specified, both can be simultaneously true, but both cannot be false at the same time.

Now imagine, for the sake of having an example parallel to those given earlier, that we learn that either Mary or Bill is parked in the drive. This means that if Mary isn't parked there, Bill is. But, of course, if Bill is parked there, then Jim (a third party) isn't -- because of the limited space. So...if Mary isn't parked in the drive, Jim isn't parked in the drive. This last conditional is composed of subcontraries but is potentially much harder to recognize as such than conditionals composed of *a priori* contraries. Anyone who knows what triangles and circles are can see the subcontrariety of antecedent and consequent in «If the figure wasn't a triangle, it wasn't a circle.» But just knowing the meanings of the terms involved will not enable one to see the subcontrariety of antecedent and consequent in «If Mary isn't parked in the drive, Jim isn't parked in the drive.» To see *that* subcontrariety, one would have to have *a posteriori* knowledge about the situation (viz., the knowledge that there is just room enough for one person to be parked there).

**2. Peculiarities**

Thus there are cases in which we can meaningfully assert and reason from conditional statements with subcontrary antecedents and consequents. Still, it cannot be denied that there is something strange about conditionals of this sort. In fact, they turn out to have certain features that are hard to characterize as anything other than paradoxical.

One unusual feature of conditionals composed of subcontraries is that falsity of the antecedent requires truth of the consequent. This follows, of course, simply from the nature of subcontraries. Two subcontraries cannot both be false, so it the antecedent and the consequent are subcontraries and the former is false, the latter must be true. Consider our three examples:

1. If Jan served fruit, she didn't serve strawberries.

2. If the figure wasn't a triangle, it wasn't a circle.

3. If Mary isn't parked in the drive, Jim isn't parked in the drive.

With (1) if the antecedent is false--if Jan did *not* serve fruit, she certainly did not serve strawberries. And as for (2), if the figure *was* a triangle, it can't very well have been a circle. Nor is the case regarding (3) much more complicated. If Mary *is* parked in the drive and the circumstances are as originally described, space limitations make it impossible for Jim to be parked there too.

Admittedly, this is not the way normal conditional statements behave. Ordinarily, knowing that p ⊃ q and learning that p is false does not enable us to conclude that q is true.

A second unusual feature of conditionals composed of subcontrary antecedents and consequents follows from the first: when such conditionals are true, they always have true consequents. As just seen, if the antecedent is false, the consequent is true. But if the conditional statement is true and the antecedent is true, the consequent must also be true (Modus Ponens). So whether the antecedent is true or false, the consequent is true. Consider the three examples once more:

1. If Jan served fruit, she didn't serve strawberries.

2. If the figure wasn't a triangle, it wasn't a circle.

3. If Mary isn't parked in the drive, Jim isn't parked in the drive.

If (1) is true, it is evident to anyone familiar with the meaning of «fruit» and «strawberries» that Jan must not have served strawberries: the true statement at hand asserts that she didn't serve any if she served fruit--and she obviously didn't serve any if she *didn't* serve fruit. So assuming (quite reasonably) that she either did or did not serve fruit, one can conclude that she didn't serve strawberries. Similarly, in the case of (2), the figure must have been a triangle or not. But then since being a triangle would have kept it from being a circle, and since not having been a triangle means it wasn't a circle, it can't have been a circle at all. The situation with (3) is essentially the same, in spite of the fact that the subcontraries involved are *de facto* rather than *a priori*. If Mary isn't parked in the drive and (3) is true, Jim is not parked there. But if Mary is parked in the drive, given the space limitations stipulated, it is also the case that Jim is not parked there. So granting that Mary either is or is not parked in the drive, Jim simply cannot be parked there.

Once again, this is obviously not a property shared by normal conditionals. Ordinarily, just knowing that p ⊃ q does not enable us to conclude that q .

Perhaps these two properties of conditionals with subcontrary antecedents and consequents could be considered merely unusual rather than paradoxical, since they do not prevent this sort of conditional from being employed in the way other conditionals are. There is, however, another feature of this kind of conditional that really does border on the paradoxical: if the consequent of such a conditional is false, the antecedent has to be *true*. This is clearly the case, since two subcontrary statements cannot both be false. But this is definitely not the way conditionals are supposed to behave. Once more, the three examples:

1. If Jan served fruit, she didn't serve strawberries.

2. If the figure wasn't a triangle, it wasn't a circle.

3. If Mary isn't parked in the drive, Jim isn't parked in the drive.

If (1) is true and it turns out that Jan did serve strawberries, then she obviously served fruit. Or (2), if the figure was in fact a circle, it certainly wasn't a triangle. And as for (3), discovering that Jim *is* parked in the drive would mean under the circumstances stipulated that Mary isn't parked there--which is what the antecedent says.

Yet according to Modus Tollens (((p ⊃ q) ∧ ¬q) ⇒ ¬p), falsity of the consequent is supposed to mean *falsity* for the antecedent--not truth. So it would seem that there is something seriously wrong with conditionals composed of subcontraries--unless in fact Modus Tollens does *not *apply to all conditionals.

Nor is the trouble limited to situations in which Modus Tollens would usually be thought to apply. With conditionals of the sort under consideration, it also turns out that Transposition ((p ⊃ q) ⇔ (¬q ⊃ ¬p)) cannot apply either. If p and q are subcontraries, a conditional of the form p ⊃ q simply cannot be equivalent to ¬q ⊃ ¬p , since subcontrary statements are not capable of being false at the same time.

1. If Jan served fruit, she didn't serve strawberries.

2. If the figure wasn't a triangle, it wasn't a circle.

3. If Mary isn't parked in the drive, Jim isn't parked in the drive.

How could (1) be equivalent to «If Jan served strawberries, she didn't serve fruit,»? How could (2) be equivalent to «If the figure was a circle, it was a triangle,»? Or how, given the limited space in the driveway, could (3) equate to «If Jim is parked in the drive, Mary is parked in the drive,»? In each case, if we suppose the original statement to be true, applying Transposition takes us from something true to something false. And that is paradoxical enough to cause concern. Either Transposition does not apply to conditionals composed of subcontraries or else there is something really amiss with such statements.

.......................................................................

**6. Conclusion**

If conditional statements are understood as claims to the effect that truth of one statement would permit a sound inference to the truth of another, the fact that there are conditional statements with subcontraries for antecedents and consequents poses no insuperable problem for logical analysis.

Department of Philosophy

The University of Wisconsin at Oshkosh

Oshkosh, WI 54901, USA

<cordero [at] uwosh.edu>

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