138 More on the material conditional
functional conditional has its actual truth-value fixed by the actual truth-values of its constituent sentences. By contrast, the truth-value of a possible-world condi- tional depends on what happens in other possible scenarios; so its truth-value can’t be fixed just by the this-worldly values of the antecedent and consequent.
Hence, quite uncontroversially, the truth-functional rendition can at most be used for conditionals like (2).
What is very controversial, though, is the range of ‘conditionals like (2)’. Cases like (3) are conventionally called subjunctive conditionals, for supposedly they are couched (according to some traditional grammars) in the subjunctive mood. (2), and all the conditionals in arguments A to G in §14.1 above, are called indicative conditionals (being framed in the indicative mood). The trouble is that the intended indicative/subjunctive division seems to carve up condition- als in an unhelpful way. For compare (3) with
(4) If Oswald doesn’t shoot Kennedy in Dallas, someone else will.
Someone who asserts (4) before the event – believing, perhaps, that a well- planned conspiracy is afoot – will after the event assent to (3). Likewise, some- one who denies (4) before the event – knowing that Oswald is a loner – will deny (3) after the event. Either way, it seems that exactly the same type of possible world assessment is relevant for (3) and (4): we need to consider what occurs in a world as like this one as possible given that Oswald doesn’t shoot Kennedy there. Plausibly, then, we might well want to treat (3) and (4) as belonging to the
same general class of possible-world conditionals. But, in traditional grammati-
cal terms, (4) is not subjunctive. So the supposed distinction between subjunctive and indicative conditionals divides the past interpretation of (3) from the future interpretation of (4) in a pretty unnatural manner.
We can make a related point in passing. Someone who asserts (2) leaves it open whether Oswald shot Kennedy; but someone who asserts (3) thereby implies that the antecedent is actually false, and that Oswald didn’t miss. For that reason, conditionals like (3) are often called counterfactual conditionals. But (4) is not a counterfactual in this sense: someone who asserts (4) again leaves it open whether Oswald will miss. So the counterfactual/non-counterfactual distinction again unnaturally divides (3) from (4).
We can’t pursue these matters further here. The point, to repeat, is that it isn’t obvious exactly where and how to draw the line between those possible-world conditionals like (3) which have to be set aside as definitely not apt for a truth- functional treatment, and those ‘indicative’ conditionals like (2) which are still in the running.
15.2 In support of the material conditional
We noted that the material conditional (like any truth-function) can be expressed with the unaugmented resources of PL; the two simplest ways of capturing the truth function using the original connectives are ¬(A∧¬C) and
15.3 Against identifying vernacular and material conditionals 139
look more or less like a conditional? Well, consider the following two lines of thought:
• If A then C rules out having A true while C is false; hence, borrowing the
PL symbols for brevity, if A then C implies the corresponding ¬(A∧¬C).
Suppose conversely that we believe that ¬(A∧¬C), i.e. that we think that
we don’t have A true while C is false. That seems to imply that if A is true, then C won’t be false, or in other words, if A then C. So, apparently, if A
then C implies and is implied by ¬(A∧¬C). Which is to say that the ordi-
nary language conditional is equivalent to the material conditional. • Suppose we hold that if A then C, and put that together with the law of
excluded middle (¬A∨ A). Then we either have ¬A, or we have A and so C. In other words, if A then C implies (¬A∨ C). Conversely, suppose we
hold (¬A∨ C); if A is in fact true, then – since that rules out the first
disjunct – we can conclude C must be true as well. So, apparently, if A then
C implies and is implied by (¬A∨ C). Which is again to say that the ordi-
nary language conditional is equivalent to the material conditional. These arguments are enough to lead many logicians to hold that at least some ordinary ‘if …, then …’ propositions are fundamentally truth-functional.
15.3 Against identifying vernacular and material conditionals
Do the arguments of the previous section really establish that the core meaning of indicative conditionals is truth-functional (i.e. that these ordinary language conditionals are material conditionals)?
Note that a material conditional, a proposition with the same truth-table as (¬A∨ C), can be true irrespective of any connection between the matters men-
tioned by A and C. All that’s required is A is false and/or C is true. You might hold that this simple observation is already enough to sabotage the claim that ordinary ‘if’s can be treated as equivalent to material conditionals. For don’t ‘if’s signal some kind of real connectedness between the topics of the antecedent and consequent?
No, not always. Suppose I remember that one of Jack and Jill has a birthday in April (I know that they threw a party last Easter, but I just can’t recall whose birthday it was celebrating). You say, a bit hesitantly, that you think that Jack has an October birthday. So I respond
(1) If Jack was born in October, then Jill was born in April.
Surely in the context that’s a perfectly acceptable conditional, though I don’t for a moment think that there is any causal or other connection between the facts about their birth months. All I’m saying is that either you are wrong about Jack, or else Jill was born in April: i.e., borrowing the PL symbols again, I’m just com- mitting myself to
(2) (¬Jack was born in October ∨ Jill was born in April).
140 More on the material conditional
Of course, when we assert conditionals, we often do think that there is (for example) a causal connection between the matters mentioned in the antecedent and the consequent (‘If you press that button, it will turn on the computer’). But equally, when we assert disjunctions, it is often because we think there is some mechanism ensuring that one disjunct or the other is true (‘Either the e-mail will be sent straight away or you’ll get a warning message that the e-mail is queued’). However, even if our ground for asserting that disjunction is our belief in an appropriate mechanism relating the disjuncts, what we actually say is true so long as one or other disjunct holds (exclusively, in our example). Likewise, our
reason for asserting a conditional may be a belief in some mechanism that
ensures that if the antecedent holds the consequent does too. It could still be the case that what we say is true just so long as the material conditional holds.
So far, the material conditional theory survives. But now compare the follow- ing (we’ll borrow ‘⊃’ to signal the material conditional in English):
(3) Wordsworth didn’t write Pride and Prejudice. So, (Wordsworth wrote
Pride and Prejudice ⊃ Wordsworth wasn’t a novelist).
(4) Wordsworth didn’t write Pride and Prejudice. So, if Wordsworth wrote Pride and Prejudice then he wasn’t a novelist.
Grant (3)’s premiss that Wordsworth didn’t write Pride and Prejudice. The con- clusion is of the form (A⊃ C) with a false antecedent, so the conclusion must be true too. Hence the inference in (3) is valid. That is no more puzzling than the validity of the inference from ¬A to (¬A∨ C); indeed, (3) in effect just is that
trivial inference. By contrast, the inference in (4) looks decidedly unhappy. Many will say: its conclusion looks unacceptable (being the author of Pride and Preju-
dice would surely be enough to make you a novelist!): so how can it validly
follow from the true premiss that Wordsworth didn’t write Pride and Prejudice? Generalizing, the inference pattern
(M) not-A; so (A⊃ C)
is unproblematically and trivially reliable. On the other hand, inferences of the type
(V) not-A; so if A then C
strike us as very often unacceptable (for indicative conditionals as well as sub- junctive ones). Equating the vernacular conditional in (V) with the material con- ditional in (M) has the highly implausible upshot that the two inference patterns should after all be exactly on a par.
Now we seem to be in a mess. On the one hand, we presented in the last sec- tion what seemed a good argument for saying that if A then C implies and is implied by ¬(A∧¬C), and then another good argument for saying that if A then C implies and is implied by (¬A∨ C), where ¬(A∧¬C) and (¬A∨ C) are
just two ways of expressing the material conditional. Yet here, on the other hand, we’ve emphasized what looks to be a very unwelcome upshot of treating ordinary ‘if’s as equivalent to material ‘⊃’s. What to do?
15.4 Robustness 141
15.4 Robustness
Go back to a remark made in passing in §14.2. I said: when we claim if A then
C, we are normally issuing a promissory note, committing ourselves to assent to C should it turn out that A true.
Now imagine that I currently hold not-A, and on that ground alone hold (¬A∨ C). Would I in this case be prepared to go on to issue a promissory note
committing myself to assent to C should it turn out that in fact A after all? No! If it turns out that A, I’d retract my current endorsement of not-A, and so retract the disjunction (¬A∨ C). Hence, if I accept (¬A∨ C) – or equivalently (A⊃ C)
– on the grounds that not-A, then I won’t be prepared to endorse the promissory conditional if A then C.
Let’s say, following Frank Jackson, that my assent to (¬A∨ C), or equiva-
lently (A⊃ C), is robust with respect to A if the discovery that A is true won’t lead me to withdraw my assent. Then the point we’ve just made comes to this: if my assent to (¬A∨ C), or equivalently (A⊃ C), isn’t robust with respect to A, I
won’t accept the force of the §15.2 argument from (¬A∨ C) to if A then C.
Exactly similarly, suppose I endorse something of the form ¬(A∧¬C) because
I believe ¬A. Again, I’ll take it back if I discover than in fact A holds. So again I won’t accept the force of the §15.2 argument from ¬(A∧¬C) to if A then C.
This observation suggests a new account of ordinary, indicative conditionals, an account which both recognizes the appeal of the arguments in §15.2 yet also recognizes the intuitive distinction between the inferences (M) and (V) that we stressed in §15.3. The suggestion is that ordinary conditionals are not just mate- rial conditionals, rather they are material conditionals presented as being robust (i.e. robust with respect to their antecedents).
Let’s explain that more carefully. Compare: (1) Jack came to the party.
(2) Even Jack came to the party.
These plainly differ in overall meaning. And yet it seems that the worldly state of affairs they report is one and the same, namely Jack’s presence at the party. The difference between (1) and (2) doesn’t seem to reside in what they tell us about the world. Rather, the word ‘even’ functions rather like a verbal highlighter – or a verbalized exclamation mark – typically used to signal the speaker’s feeling that Jack’s presence was surprising or out of the ordinary.
Similarly, compare the following:
(3) Jack came to the party and had a dreadful time. (4) Jack came to the party but had a dreadful time.
Again, the worldly state of affairs these two report seems to be the same. The difference between them is that the use of (4) signals, very roughly, that Jack’s going to the party without enjoying it was against expectation.
Likewise, it is now being suggested that the difference between these – (5) (Jack came to the party ⊃ Jack had a good time)
142 More on the material conditional
– is again not a difference in what they say about the world. Each is strictly speaking true so long as its antecedent is false and/or consequent true. Rather the difference arises because someone who uses ‘if’ signals that their claim is robust with respect to the antecedent. In other words, the speaker who uses (6) rather than (5) undertakes not to withdraw the conditional should it turn out that the antecedent is true and Jack did come to the party.
Go back to that example where I recall the Easter birthday party which shows that one of Jack and Jill has an April birthday. We argued that it could be natural to endorse
(7) If Jack was born in October, then Jill was born in April.
That’s nicely in keeping with the current theory, since (in the circumstances) although (7) may be no-more-than-material in propositional content, it is robust.
Now since, on this theory, the core content of if A then C is exactly (A⊃ C), it is no surprise that – quietly assuming robustness – we can construct plausible arguments as in §15.2, from (A⊃ C) to if A then C, as well as vice versa. But note too that, while we can accept the unproblematic inference
(M) not-A; so (A⊃ C)
we won’t robustly endorse (A ⊃ C) if our ground for believing the material conditional is simply the falsity of the antecedent. Hence, we won’t want to signal that the conclusion of argument (M) is being held robustly. In other words – according to our theory which equates ordinary-language conditionals with material conditionals signalled as being robust – we won’t want to say some- thing of the form
(V) not-A; so if A then C
Which accords exactly with the point of §15.3.
In summary, the ‘robust material conditional’ theory is one theory about ordi- nary ‘if … then …’ which neatly accommodates both the attractions of the §15.2
arguments for thinking that indicative conditionals are material conditionals, and also the §15.3 considerations that press against a pure equivalence.