T4.3 6 of 6
Consider the following argument:
(Argument
1)
I have no money.
_____________
If I have money, then I will spend it on moon rocks.
This reasoning appears poor: obviously the premise provide no reason to believe that I have an interest in purchasing moon rocks. But notice that it's valid if we take the conditional of the conclusion to be a material one! It has this form:
~M
M>R
Any truth value assignment making the premise true also makes the conclusion true. (Do the table, or think about any truth value assignment which makes the premise true. It makes 'M' false. So, this is a truth value assignment making the conclusion true.) So, this SL argument is valid.
What's gone wrong? Doesn't the fact that the English version of the argument is bad show that the (valid) symbolization given is not a proper symbolization? So that the "if...then..." of the English is not really translatable as the material conditional '>'?
I think the answer to these questions is "no" and that the problem with the above argument is different from invalidity.
One way to understand the problem is (again) in terms of conversational practicality. It would be silly to ask someone to make the inference from '~M' to 'M>R'. But this does not mean that the inference is an invalid one. Rather, at least in this case, the inference is an uninformative one. Because 'M>R' means the same as '~MvR', it would be typically be uninteresting to draw the conclusion given the premise. The following argument (replacing the conditional 'M>R' with the logically equivalent '~MvR') is also silly but plainly valid:
(Argument
2)
I have no money.
_____________
Either I have no money or I will spend it on moon rocks.
Argument 2 is just not worth giving. But it is valid. I take it that argument 1 is likewise valid but uninteresting.
Here's another way to think about argument 1. The conclusion
If I have money, then I will spend it on moon rocks
is in the indicative mood. But because it follows the premise
I have no money
we are inclined to think of it as a counterfactual. Thought of in this way, the conclusion cannot be symbolized with a horseshoe and it is clear why argument 1 is not valid. Only the symbolization
~M
M>R
is valid. That's different, that has no counterfactual. So there is no problem.
There is one more point which may make you more comfortable symbolizing English "if...then..." with the horseshoe. The horseshoe may be incorrect for hypothetical thinking ("What if I had money?") and so not correct for counterfactuals. But we have long known that. Instead, it's right for any English statement "If P, then Q" which can be parsed "If indeed it is the case that P, then it is in fact the case that Q (though if not-P, then all bets are off and no conclusion can be drawn)."
So, the horseshoe gives a decent translation of the English conditional so long as the English is about what is in fact the case rather than about the hypothetical. "If I had money, then I'd buy moon rocks" can be true because I've no money and all bets are off.
But the philosophy of language here gets rather deep. We needn't worry here. The best we can do for any English conditional is just to use the horseshoe for its symbolization. So, in what follows, do just that.
A nice and much more thorough introduction to these philosophy of logic issues can be found in chapter four of Richard Jeffrey's Formal Logic (only in the second edition, 1981).