Monday, February 14, 2011

Indicative conditionals and material implication

I am partial to the view that "if p, then q" has the same truth conditions as "not-p or q", but there is pragmatic stuff going on. But after thinking hard about Field's remarks on the implications of the Montague Paradox for certain solutions to the liar paradox (in a Beall anthology), I've been drawn to think more about an odd but significant difference between "If p, then q" and "not-p or q".

Here is the less striking way to show the difference. Consider:

  1. (x)(Human(x) → Mortal(x))
  2. (x)(~Human(x) or Mortal(x))
Now, in (2) there is a symmetry between the disjuncts, where (1) has an asymmetry. Consider a particular case:
  1. Human(Seabiscuit) → Mortal(Seabiscuit)
  2. ~Human(Seabiscuit) or Mortal(Seabiscuit)
I am happy to grant that (3) and (4) are both true. But they are true for different reasons. I want to say that (3) is trivially true because of the falsity of its antecedent. Period. The fact that Seabiscuit is mortal is not explanatorily relevant to the truth of (3). But (4) is equally true because of the non-humanity of Seabiscuit as because of his mortality. There is a symmetry there.

Here is another way to see the difference. Some people think that sentences like "x is F" are nonsense (gloss: don't express a proposition) when x fails to be of a certain kind for which attribution of Fness makes sense. These people will, for instance, deny that "The chair is true-or-false" makes any sense. (I am inclined to think it makes perfect sense, but is false; chairs aren't propositions and don't express propositions, so they are neither true nor false.) Now, plausibly, if "q" doesn't make sense, neither does: "r or q". That's just nonsense. So, on such a view:

  1. This chair is true-or-false or this chair is not a proposition
is nonsense. And so is:
  1. This chair is not a proposition or this chair is true-or-false.
  1. If this chair is a proposition, this chair is true-or-false
makes perfect sense as an ordinary indicative conditional. It is trivially true because the antecedent is false. If this is right, then we can have meaningful conditionals whose consequents aren't meaningful, but not so for disjunctions. This may force a restriction on the use of modus ponens in subproofs.

For another illustration of this last point, go back to (1) and (2). The quantification seems to be unrestricted, including such entities as numbers and properties. But it is not clear that "Mortal(7)" makes sense. If not, and if disjunction requires both disjuncts to make sense, then (2) is in trouble. But (1) is just fine.

In a lot of programming languages, logical disjunction operators are actually asymmetrical. This means that if you do something like "f(x) || g(x)" ("||" being disjunction) in perl or C, the function g(x) is not evaluated when f(x) turns out to return truth. The disjunction operator shortcuts in this way. As a result, you can do things like "x == 0 || y/x == z" without worrying about the fact that the second disjunct is non-sense if x is zero, because the second disjunct is unevaluated when x is zero. But in human language, I suspect that there is no similar shortcutting in disjunction. Apart from implicatures, there is a symmetry in disjunction. But, I suggest, there is such a shortcutting in conditionals.

But I don't know exactly how far the shortcutting in conditionals goes. I am not sure I want to say:

  1. If the sky is green, then **##^^).
But at least to my ear it sounds better to say this than:
  1. The sky isn't green or **##^^).


Heath White said...

Here's a form of shortcutting disjunction: "Either he's unmarried or his wife lives out of state." Suppose he's unmarried; then "his wife" doesn't refer. Someone who knew this would balk (pick your favorite diagnosis) at the second disjunct baldly asserted, but not at the disjunction. (Note they would balk if you asserted "his wife lives out of state" in a conjunction.)

Alexander R Pruss said...

Nice example. So maybe disjunctions do shortcut, too? Do you think "His wife lives out of state or he's unmarried" is OK?

The example also neatly shows that in Russell's account of definite descriptions, the scope of a definite description can be a component. Thus, we wouldn't translate:

(E!x)(x is a's wife) and (x)(if x is a's wife, then (a is unmarried or x lives out of state)).

Instead we'd translate:

He's unmarried or [(E!x)(x is a's wife) and (x)(if x is a's wife, then x lives out of state)].

Heath White said...

I cannot decide whether it's semantically OK, or semantically not-OK but pragmatically OK.

You are right that the scope of the E! makes a big difference.

I notice, however, that if we take Russell's route, we need not "shortcut" anything. If the first disjunct is true (he's unmarried) then the second is false. The second disjunct is itself a conjunction, of which the first conjunct is false. If we think there is some shortcutting going on (i.e. that "his wife" is in some way nonsense when he's unmarried) then that's a strike against Russell's analysis.

Finally, I think both Russell and his competitors could explain why "If the sky is green, then **##)" is no good. :-)