Alineación del PEI y el POIA de la Gestión de Formación y Servicios Tecnológicos

The previous section’s theory about indicative conditionals is neat: but is it right? Is asserting a conditional always tantamount to giving a promissory note to accept the consequent should the antecedent turn out to be true? What about the following conditionals? –

(1) If George W. Bush is a great philosopher, then pigs can fly. (2) If George W. Bush is a great philosopher, then I’ll eat my hat. Or, to take a paradigm example, consider

(3) If George W. Bush is a great philosopher, then I’m a Dutchman. Here I assert the conditional inviting you to apply modus tollens, i.e. to infer from my assertion if A then C, together with the obvious truth of not-C, that

15.5 ‘Dutchman’ conditionals 143

not-A. And my grounds for asserting the conditional are none other than my

belief in not-A. I commit myself to no more than the corresponding (¬A∨ C).

Were I to discover that the antecedent A is in fact true – discover that Bush has a secret other life as the pseudonymous author of some much admired philosophi- cal works – then I’d have to take back my scornful conditionals.

In short, these ‘Dutchman’ conditionals are mere material conditionals which are not robustly asserted. What does that show?

You could take it as counting against the ‘robust material conditionals’ theory. But equally, you could take the opposing line. After all, ‘Dutchman’ conditionals do strike us pre-theoretically as exceptional cases, as being a jokey idiom. And our theory explains why: they are conspicuously non-robust, and so offend against one aspect of the usual rules governing the assertion of conditionals, which is why they seem non-standard. So, you might hold that far from being refuted by the behaviour of ‘Dutchman’ conditionals, our theory explains it.

But we can’t pursue this tangled issue any further here. We’ve already followed the twists and turns of arguments about the nature and variety of con- ditionals far enough. Two points, however, have clearly emerged.

First, it should now be clear why, when introducing truth-functional logic, we

didn’t attempt to deal with conditionals right from the start. Rendering ‘if …,

then …’ by a truth-function raises complex problems of a quite different order to the issues raised by treating ‘and’, ‘or and ‘not’ as truth-functions. To clutter the initial discussion of PL inferences with these problems would have been very dis- tracting; while to have introduced the material conditional straight off without much comment would have seemed at best mystifying, at worst a cheat.

Secondly, on the ‘robust material conditional’ theory, the rendition of indica- tive conditionals by ‘⊃’ captures that aspect of the content of ordinary indicative conditionals which has to do with matters of truth and falsehood, and hence the aspect which logic (the study of truth-preserving inferences) needs to focus on. And we can now see that, on any theory, there is some considerable plausibility in at least treating indicative conditionals as having a truth-functional core, even if there are other aspects of the use of vernacular conditionals that are missed if we try to insist on a bald equation between ‘if’ and ‘_⊃’.

The moral, then, is this:

If you want to be able to regiment arguments involving the vernacular

indicative conditional into a form that you can test using the truth-table test, then there is nothing for it but to use the material conditional ‘_{⊃’. If} you want to be able to apply the truth-table test, then the price is that you have to re-express the premisses and conclusion using ‘⊃’. The rendition preserves the validity/invalidity of arguments in at least some central cases. But arguably the translation is not always an accurate one (even once we’ve set aside the possible world conditionals for alternative treat- ment). Something – in some cases, you may think, too much – may be lost in translation.

144 More on the material conditional

Whatever you think about all the considerations in this chapter, the material conditional has to come with the warning use with care.

15.6 Summary

• Subjunctive’ or ‘counterfactual’ conditionals are possible-world condition- als, and are not truth-functional.

• Even restricting ourselves to indicative conditionals, there are various valid inferences involving the truth-functional conditional whose ordinary- language counterparts look dubious or positively invalid.

• The ‘robust material conditional’ theory gives one possible explanation of some of these divergences. (Note, however, the theory is a contentious one!)

• The material conditional must come with a Logical Health Warning.

Exercises 15

A (a) The pattern of inference C; so (A⊃ C) is trivially valid. What about the inference C; so if A then C? (Consider: given C is true, then it is true whether A is or not, so in particular it is true if A is. But also compare the inference Bush will win the election; so if there is a huge

financial scandal involving the Bush family then Bush will win the election.)

(b) The pattern of inference (B⊃ C); so ((A∧B)⊃ C) is trivially valid.

What about the inference if B then C; so if A and B then C? (Con- sider, ‘If you strike this match, it will light; so if you wet this match and strike it, it will light.’)

B Consider the variant language PLF whose only one- or two-place connec- tive is ‘_{⊃’, but which has – as well as the usual atoms, ‘P’, ‘Q’, etc. – the addi-} tional atomic wff ‘⊥’ which is governed by the semantic rule that it always takes the value F (so there is no option about how to evaluate this atom). (a) Show that negation, disjunction, and conjunction can be expressed in

PLF. (Hint: start by considering material conditionals relating two

propositions, one of which is ‘⊥’.)

(b) What does it mean to say that PLF is expressively adequate? Is it true? (c) You are familiar with the idea of a one-place truth-functional connec- tive (like negation) and a two-place connective (like conjunction): so what does it mean to talk of an n-place truth-functional connective? What, then, would be a 0-place truth-functional connective? Can we think of ‘⊥’ as a 0-place connective?

In previous chapters we defined what it is for PL (and PLC) inferences to be tautologically valid. We then introduced one way of testing for tautological validity, namely the brute force truth-table test, where we search through every assignment of values to the relevant atoms to see if there is one which makes the premisses true and the conclusion false. In this chapter, we introduce another way of testing for tautological validity, using ‘truth trees’. We begin with an informal exploration via examples: and for the moment we will concentrate on

PL inferences (we’ll extend the story to cover conditionals in Chapter 18).