Thursday, October 01, 2009

Material Conditionals in Natural Language

Bill Vallicella has a post up on an argument by Errol Harris on material implication. I don't have much to say about his objection to the argument; since I haven't read the work in question, I can only go on the excerpt given, and based on that, the Maverick Philosopher seems to be quite right. But he takes the example, "If the earth is flat, I am the Pope" and says of it:

I submit that this conditional is at once both a material conditional and a piece of ordinary language. Thus here we have an ordinary language example of a material conditional. So although the material conditional is a theoretical construct for logical purposes, it is exemplified in natural language.


But I think it's worth our time to press this somewhat. In order to be a material conditional, this conditional must meet two conditions:

(1) Its behavior for logical purposes must be describable using the entire material implication truth table.
(2) Its behavior for logical purposes must be entirely describable using the material implication truth table.

And by 'behavior' here I mean nothing more than the way the conditional is used in natural language reasoning, since we are discussing whether this can be both a material conditional and a piece of ordinary language. In order to be both, it must be a piece of ordinary language whose usage is entirely describable entirely by the right truth table.

But there is some reason to doubt that this is the case, because one might argue in the following way. I am not the pope whether the earth is flat or not. There is nothing about the earth being flat that would make it necessary for me to be pope, and there is nothing about my not being pope that makes it necessary that the earth is not flat. Therefore it would be logically consistent state of affairs if I were not the pope and the earth were flat. But if it is logically consistent for 'I am not the pope' and 'The earth is flat' to be true, then the truth table for the the conditional, 'If the earth is flat, I am the pope," is not the truth table of the material conditional, since it is inconsistent with that truth table for the truth of the antecedent and the falsehood of the consequent to be consistent with each other. To be sure, it happens to be a fact about the actual world that both antecedent and consequent are false, and this is the reason we are using the conditional in the first place. But it seems odd to say that whether or not this conditional is a material conditional depends on contingent facts about how the world is; and given that a true antecedent and false consequent are logically possible, it seems that we would have to say this in order to say that Condition (2) is not violated. The conditional has a modal behavior that would allow it to deviate, in principle if not in practice, from the behavior it would have to have as a material conditional. [ADDED LATER: Re-reading this, I don't find it to be adequately clear. The point boils down to this: If there is any modal or probabilistic component to the conditional, it is not a material conditional, since the material conditional truth table has no modal or probabilistic components. But the conditional as usually used doesn't rule out the combination of the truth of the antecedent and the falsehood of the consequent -- it merely relies on the fact that they both happen to be false, the consequent to a very high degree of certainty, and the antecedent to a degree of certainty that is being put on a level, for practical purposes, with the degree of certainty for the consequent. It is, in fact, what we would usually take it to be: a figure of speech classifying one possibility's likelihood of being false with another possibility that everyone is certain is false. Thus there is nothing about it that strictly rules out the compossibility of true antecedent and false consequent; it merely treats this as an extraordinarily unlikely possibility.]

Moreover, there is the difficult of showing that any piece of natural language exemplifies an entire truth table rather than only some fragment of it. That is, it's difficult to show that Condition (1) is met. In a formal language, where connections are defined in terms of truth tables, there is no problem. One can tell, simply by looking, that p → q has the truth table of a material conditional in standard propositional logic, because that is how → is defined in propositional logic. But natural language connections like 'if' are not defined in this way, and one might well hold that the conditional "If the earth is flat, I am the pope," was composed solely for the purpose of allowing modus tollens, and for no other reason. Can it then be represented by any part of the material implication truth table that does not deal with a false consequent? It isn't clear that it can. The behavior of 'if' in general isn't represented by the material implication truth table, because (to take just one example) common usage takes it in such a way that the counterpart of modus ponens in natural language is defeasible; so we can't take the right truth table to be built into 'if' itself. Therefore the missing parts of the truth table can't be supplied from the way 'if' is generally used. Perhaps there is some way around it, but it seems to be difficult to establish that any piece of natural language meets Condition (1).

This is not to say that there might not be cases of the material conditional being exemplified in natural language. But we run into the same problems here that Jennings has noted with regard to finding exclusive disjunction in natural language: matching natural language usage to truth tables is very tricky, because you have to use the entire truth table and that truth table has to be wholly sufficient for describing behavior for logical purposes. If either of these are violated, we don't have a material conditional exemplified in natural language but merely a conditional that is not a material conditional but sufficiently analogous to it that we can sometimes pretend that it is. And that's a different thing.