Material Implication, English, and Truth at a World

I was reading this post in the latest edition of the Philosophers’ Carnival on conditionals and free will…I don’t have anything to say about the main topic of the post so I thought I would comment here…

What caught my eye was someone in the comment section who said that material implication  ‘–>’ is not meant to capture ‘if…then’ in English rather it is merely a “convient way to combine negation and disjunction”…It is true that material implication is defined in terms of negation and disjunction but so is ‘&’ and in fact all of the truth functional connectives but that doesn’t mean that they do not attempt to capture the meaning of the English words.  The truth conditions for ‘–>’ are meant to capture at least on of the things that we mean when we say ‘if…then’.  Consider,

(1) if Santa is fat then Santa is jolly

The truth table definition of material implication says that this sentence is true when both antecedant and consequent are true and also when they are both false as well as wehn the atecedant is false and the consequent is true. The only time the sentence is false is when the antecedant is true and the consequent is false. So the claim is that these truth-conditions do not capture ‘if…then’ in English. Even the author of the post expresses some suprise that these turn out to be the truth-conditions for ‘if…then’.

But, what we mean when we say (1), and the reason why people intuitively accept Modus Ponens as a valid argument form, is that if it is true that santa is fat then it will also be true that Santa is jolly. The natural way to see if this sentence is true is by finding out whether Santa is fat or not and whether he is jolly or not. Now suppose that Santa in fact turns out to be neither fat nor jolly, does common sense expect (1) to be false? I don’t think so. (1) says that on the condition that Santa is fat he will be jolly as well, so if he is not in fact fat the condition doesn’t hold, but (1) could still be true because it might be true that if Santa were fat then he would be jolly. So the falsity of ‘Santa is fat’ is compatible with what the sentence says still being true counter-factually. But suppose that Santa turned out to be in fact fat but decidely NOT jolly. Then the sentence would be false because the relation that it asserts is shown not to hold. So it seems to me that the truth-table meaning of ‘–>’ does capture the meaning of ‘if…then’ in English.

It seems to me that the reason why the truth-conditons are suprising to people who see them for the first time is that it makes them realize that some sentence’s truth conditions depend on how the sentence behaves modally in opposition to the naive view that all sentences simply depend on how the world actually is for their truth. To put it a bit technically it forces them to realize that the sentence is made true in this world because it is true at some other possible world. It is NOT because these truth-conditions clash with what we take the meaning of ‘if…then’ to be…This, incidently, seems to me to be more evidence that there is something wrong with Williamson’s argument against the distinction.

These truth-conditions capture one of the meanings listed for the English conjunction ‘if”; that there are other uses of the conjunction doesn’t seem to matter.

5 thoughts on “Material Implication, English, and Truth at a World

  1. If Santa is not fat, then to interpret “If Santa is fat, then Santa is jolly” as being about what Santa would (probably? necessarily?) be (or have been), were he (or had he been) fat, is possible. And then its truth would depend upon what might be (or have been). But is that the material conditional? I thought that the material conditional came out true when the antecedent is false, however anything else is. (Of course, that is also a possible way to interpret the statement, much like “if pigs can fly,” or rather, since Santa is fictionally fat, the statement just says that Santa is fictionally jolly 🙂

  2. Hey, I am not really sure what the point you are making is…but let me take a stab at it. My claim is that to understand the truth conditions for matertial implication one needs to think modally which is suprising because our naive view is that the truth or falisty of atomic sentences (and so the the truth or falisty of molecular sentences made up of the atomic ones) depends only on the way that the world actually is (or is not).

  3. I was suggesting (inelegantly) that you don’t need to think modally to understand the truth-conditions of the material conditional. I think that although Modus ponens is often used in a context where we are uncertain, where we will be thinking about possibilities, it is useful precisely because in itself it is only about how things are, or are being assumed to be. That is, it acts non-modally within a context, although that context might itself be hypothetical. That, I think, is why it is true if the antecedent is false: because it is then assuming a contradiction, and it is assumed that anything follows from a contradiction. If it were operating modally, then the counterfactual antecedent would not necessarily be contradictory.

  4. Thanks for the clarification, though I am still not sure I see what the argument is supposed to be. So take Modus Ponens. It is valid if its form garuentees that IF the premises are true then the conclusion MUST be true. So take an instance of MP that has false premises, say,

    if the moon is made of green chees then I am a democrat
    the moon is made of green chees
    so I am a democrat

    The premises are false, does that mean that the argument is invalid? of course not because if the premises WERE true then the conclusion WOULD HAVE to be true…so to even understand why MP is a valid argument form you have to think counter-factually…

    As for material implication the suprising thing is supposed to be that it doesn’t seem to respect the common sense intuition that a molecular sentence’s truth value is not determined by the truth value of its atomic parts. So, take conjunction if a is false and b is false then so is a&b. but if a is false and b is false then a–>b is true…this fact is taken by many to indicate that ‘–>’ cannot mean the same as ‘if…then’. I have suggested that this is mistaken and that the actual suprise comes from realizing that the sentence is made true because of some possible world and that this is in fact what the (one of the) meaning(s) of ‘if…then’…Now, what is your alternative suggestion? If Santa is Fat then he is jolly is true when the antecedent is false because there is a contradiction? What is the contradition?

    I just don’t understand what you mean when you say “If it were operating modally, then the counterfactual antecedent would not necessarily be contradictory”

  5. I’m completely lost here, actually. But you said: “Now suppose that Santa in fact turns out to be neither fat nor jolly, does common sense expect (1) to be false? I don’t think so.” For some people, common sense expects (1) to be neither true nor false, since it expects (1) to be making a T or F assertion only when antecedent is true, and then as the consequent is T or F. So for some people (e.g. me) that IS why the material conditional feels counter-intuitive. But upon reflection I’m now sure that you’re right about other people’s intuitions.

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