126 PLC and the material conditional
These are plainly fallacious arguments. The premisses of F can be true and conclusion false; foolish Jack may have lost his money by betting on Pegasus, another hopeless horse. For the same reason, the premisses of G can be true and conclusion false.
It would be good to have a systematic way of evaluating such arguments involving conditionals. Now, on the face of it, ‘if …, then …’ seems to be a connective like ‘… and …’ and ‘… or …’, at least in respect of combining two propositions to form a new one. So the obvious question is: can we directly extend our techniques for evaluating arguments involving those other two-place connectives to deal with the conditional as well?
Here is a quick reminder how we dealt with arguments involving conjunction and disjunction.
• The strategy has two stages: (a) we translate the arguments into PL, and then (b) we evaluate the translated arguments, by running the truth-table test.
• Step (a) presupposes that the ordinary language connectives ‘and’ and ‘or’ are close in their core meanings to the corresponding connectives of PL. (If the translation were too loose, then a verdict on the PL rendition couldn’t be carried back into a verdict on the original argument.)
• Step (b) depends on the PL connectives being truth-functional. The truth- table test requires that, given any assignment of values to the relevant prop- ositional atoms, we can always work out whether or not that assignment makes the premisses and conclusion false.
So, if the approach of the previous chapters is to be extended to cover arguments involving the conditional, we need to find some way of rendering conditional statements with tolerable accuracy using a dyadic truth-function. Can we do the trick?
14.2 Introducing the material conditional
First, let’s introduce some entirely standard terminology:
Given a conditional if A then C, we refer to the ‘if’ clause A as the anteced-
ent of the conditional, and to the other clause C as the consequent.
An inference step of the form if A then C, A; hence C is standardly referred to by the label modus ponens. An inference step of the type if A
then C, not-C; hence not-A is called modus tollens.
Those two medieval Latin labels for the valid inference-moves in our examples A and B above are still in such common use that you just need to learn them.
Now, it is quite easy to see that there is only one truth-function which is a seri- ous candidate for capturing conditionals: this much is agreed on all sides. We’ll first give a longer argument which identifies this truth-function by exhaustion of cases, and then a second, shorter, argument for the same conclusion.
14.2 Introducing the material conditional 127
(1) Let’s for the moment informally use the symbol ‘⊃’ to represent a truth-function which is intended to be conditional-like. The issue is how to fill in the ‘?’s in its truth-table. To repeat, there must be no indeter- minate slots left in this table: if there were, we wouldn’t be able to apply the truth-table test to this conditional connective, and the whole point of the current exercise would be thwarted.
Step one Suppose we claim if A, then C. Then plainly we are ruling out the
case where the antecedent A holds and yet the consequent C doesn’t hold – i.e. we are ruling out the situation A⇒ T, C ⇒ F. Putting it the other way about, if we do have A⇒ T, C ⇒ F, then it would plainly be wrong to claim if A, then C. Hence, if ‘⊃’ is to be conditional-like, A ⇒ T, C ⇒ F
must likewise make (A⊃ C) false. Hence the entry on the second line of the table has to be F.
Step two A main point (maybe the main point) of
having a conditional construction in the language is to set up modus ponens inferences, from the prem- isses if A then C and A to the conclusion C. In assert- ing if A then C, I am as it were issuing a promissory
note, committing myself to assent to C should it turn out that A is true.
But if modus ponens inferences are ever to be sound, it must be possible for the two premisses if A then C and A to be true together. Hence the truth of A can’t always rule out the truth of if A then C.
Similarly, if ‘⊃’ is to be conditional-like, and so feature in useful modus pon- ens inferences, the truth of A can’t always rule out the truth of (A⊃ C). That means that the first line of the table for
‘⊃’ can’t be F too. Hence, the entry on the first line needs to be T.
Step three The table for (A⊃ C)
must therefore be completed in one of the ways (a) to (d).
We can now argue simply by elimina- tion:
• Column (d) can’t be right: for then
the full truth-table would be identical to the table for (A∧C), and ifs
aren’t ands.
• Column (b) can’t be right because that would make (A⊃ C) truth-function- ally equivalent to plain C; and conditionals aren’t always equivalent to their consequents (compare ‘If Jack bet on Eclipse, he lost his money’ and plain ‘Jack lost his money’).
• Column (c) won’t do either: for that column would make (A⊃ C) always equivalent to (C⊃ A), and real conditionals are not in general reversible
A C (A⊃ C) T T ? T F F F T ? F F ? A C (A⊃ C) T T ? T F ? F T ? F F ? A C (A⊃ C) T T T T F F F T T T F F F F T F T F a b c d
128 PLC and the material conditional
like that (compare ‘If Jo is a man, Jo is a human being’ and ‘If Jo is a human being, Jo is a man’).
• And so … cue drums and applause … the winner is column (a).
(2) Here’s a shorter argument for the same result. A proposition like If Jo is tall
and dark then Jo is tall is a necessary truth. Now suppose we translate it PL-
style, using ‘⊃’ for a truth-functional conditional. We get, say, ‘((P∧Q)⊃ P)’. This too should be a necessary truth, true in virtue of the truth-functional mean- ings of ‘∧’ and ‘⊃’; so it should in fact be a tautology. Consider, therefore, the following truth-table:
In the right-hand column we record the desired result that this conditional is a tautology. The middle columns give the values of the antecedent ‘(P∧Q)’ and the consequent ‘P’ of that conditional. And by inspection, to get from the assign- ments in the middle columns to the values in the final column, a truth-functional conditional (A⊃ C) must be true when A and C are both true; when A is false, and C true; and when A is false and C false. That fixes three lines of the table. Which leaves the remaining case when A is true, C is false: in that case (A⊃ C) must be false (otherwise all conditionals will be tautologous). So, again, …
The only candidate that even gets to the starting line as an approximately conditional-like truth-function is the material conditional, defined by:
Why ‘material conditional’? Let’s not worry about that. It’s one of those standard labels that has stuck around, long after the original connotation has been forgotten. (This truth-function is also sometimes known as the Philonian
conditional, after the ancient Stoic logician Philo, who explicitly defined it.)