## Thursday, February 09, 2006

### Are Disjunctions Truth-Functional?

A 'disjunction' is an either-or proposition. It has the form "Either A or B or ...", where the terms A, B, etc. are called 'disjuncts'. The simplest type of disjunction has only two disjuncts: Either A or B. Taking the "or" here in the usual inclusive sense, what this says is simply "Here are the possibilities (A,B); at least one of these is true."

Now, disjunctions are often considered to be 'truth-functional' compounds, i.e., their truth value is supposed to be determined entirely by the truth values of the disjuncts. Thus, "Either A or B" is true if A is true and B is false, B is true and A is false, or both A and B are true; and "Either A or B" is false if A and B are both false.

But it can be argued that disjunctions (or some of them at any rate) are not truth-functional. Consider the following argument patterns:
Either A or B
Not-A
Hence, B

If Not-A then B
Not-A
Hence, B
Since the second argument is obviously valid (modus ponens), if the first is to be valid, then "Either A or B" must say at least as much as the conditional "If Not-A then B". Conversely, since the first argument is obviously valid (disjunctive syllogism), if the second is to be valid, then "If Not-A then B" must say at least as much as the disjunction "Either A or B". Accordingly, it is natural to equate "Either A or B" with "If Not-A then B". Given that equation, disjunctions will be truth-functional if and only if the corresponding conditional is truthfunctional. It is arguably the case, however, that conditionals are not truth-functional.

The problem concerns what are known as the 'paradoxes of material implication'. ('Material implication' refers to the equation of "Either A or B" with "If Not-A then B", coupled with a truth-functional interpretation of each.) On the truth-functional interpretation of conditionals, a conditional is true whenever the antecedent is false and whenever the consequent is true. But this leads to counter-intuitive results. For example, it means that the following conditionals are true:
If the Eiffel Tower is in London, then the moon is made of cheese.
If the geocentric model of the solar system is correct, then water is wet.
If grass is green, then water is wet.
But many people would hestitate to call these conditionals 'true'. The problem is that the antecedent isn't relevant to the consequent. Strengthing the conditional to avoid such counter-intuitive cases, however, makes it no longer truth-functional.

For example, C.I. Lewis tried to get around the paradoxes of material implication by introducing what he called 'strict implication'. On his view
If p then q ≡ Nec(Either not-p or q),
where "Either not-p or q" is understood truth-functionally. Now, while this move doesn't avoid all of the counter-intuitive paradoxes, it does yield the result that all of the odd conditionals above are false. The result, however, is that "If p then q" is no longer truth-functional, for its truth value is no longer simply a function of the truth values of p and q. Rather, introducing the modal notion of 'necessity' makes the truth-value of "If p then q" be a function of the world-relative truth values of p and q across a domain of possible worlds.

So, the upshot is this: There are motivating reasons for interpreting (at least some) conditionals in a non-truth-functional way. If, therefore, (some) disjunctions are logically equivalent to any of those non-truth-functional conditionals, then those disjunctions are not truth-functional either. Furthermore, if all disjunctions are (by DeMorgan's laws) equivalent to conjunctions, then we get the further, and very surprising, result that (some) conjunctions are not truth-functional.