A Short Reflection on Conditionals

Material implication is a very odd thing. At the very least, it reveals to me the awkward fact that what we mean when we make a conditional statement is not quite what is stated when the logician represents it with something like “p q”. It seems that something is lost in translation.

So what is the big deal? According to the logician, a conditional statement is true when both the antecedent and consequent are true or whenever the antecedent is false. This leads to there being seemingly contrary statements both being true at the same time. For instance:

A: If Mars were more massive than Earth, Mars would have a stronger gravitational pull than Earth.

This seems to be a rather uncontroversial claim, though counter-factual. So we say it is true. But what about this:

B: If Mars were more massive than Earth, Mars would not have a stronger gravitational pull than Earth.

As in A, B has a false antecedent. Accordingly, we might conclude that B is true. But how could that be? It seems that A is faithful to a certain physical understanding of the laws of the universe whereas B is just pure fantasy. B seems to presuppose not only a more massive Mars, but also an entire universe with a different set of physical laws — one where mass and gravity exhibit an inverse relationship from the one we observe every day in this universe. But that’s the paradox of material implication for you. A and B are both true since Mars is not more massive than Earth!

So there does seem to be a sense in which A is more true than B. We might even suggest that B is false, or at least extremely less likely to be true given the set of cosmological constants and physical laws we currently observe. So, how should we understand such conditionals? One way around this problem is to talk about such conditionals in terms of probabilities. A becomes something like:

C: Pr( If Mars were more massive than Earth, Mars would have a stronger gravitational pull than Earth).

or more simply,

C: Pr(M|G)

This might be helpful because we can then contrast C, with B’s probabilistic equivalent:

D: Pr(M|~G)

We can then assess the probabilities of the two statements so as to determine the likelihood that one is more probably true than the other. Of course this might give us certainty. We could imagine that in some possible world where Mars is more massive than Earth, it is also the case that greater mass diminishes the gravitational field. Such a world would be somewhat odd though. For it would not be clear how planets might naturally form. But, let us suppose that planet formation occurs by some other force, say electromagnetism. Thus we cannot conclude, with any degree of certainty that D is in some way false, just extremely implausible.

But now a more difficult matter. How can we assess conditionals where the antecedent is not logically possible. That is, there is no possible world in which the antecedent obtains. Consider, for instance:

E: If a married bachelor were to run a complete marathon, he would run at least 26.2 miles.

Is E true, false, probably true, probably false? How would we assess it? If we just depend upon material implication, then it seems the statement is true, for it is false that any married bachelors have ever run a marathon. That seems like a silly interpretation of the statement though. Here I think we must turn to modalities like possibly “” and necessarily ““. E seems to suggest some sort of relationship between running a complete marathon and the distance in miles that one would have run were one to run it. At the same time, married bachelors cannot exist. So, how do we assess this odd statement? How would we know, for instance, that in possible worlds where married bachelors exist, complete marathons are not 12.1 miles, or that such universes have dimensions where miles are intelligible? But that kind of question doesn’t really seem to help since there are no possible worlds were married bachelors exist. We cannot appeal to probabilities at all. We are left then considering whether E means:

F: ◊ (Mb⊃ R)


G: □ (Mb⊃ R)

Now it seems that both F and G are false, since it is neither possible, nor necessary that a married bachelor should occupy any world where running a complete marathon would imply that one has run 26.2 miles. Even though such is the case in this world, could we say that were married bachelors to exist and run a marathon, they would have run 26.2 miles? One would be forced to suppose that married bachelors possibly cohere with such a world. But how would we know that they do?

I would suggest that this has some radical philosophical implications, namely, that if a conditional is going to rise to the possibility of being true, it must contain terms that are logically coherent. Why is that radical? I will have more to say about this in posts to come. In the meantime, I would love to know if anyone disagrees with my assessment. In other words, could a probabilistic, possible, or necessary conditional be true if the antecedent and/or the consequent are logically impossible?

Posted on July 7, 2011, in Logic and tagged , , . Bookmark the permalink. 7 Comments.

  1. Time to get on board with Dialethism my friend!

  2. It's funny, I've thought about examples A and B before but I'm not sure how to tackle A or B, but for E, isn't that example similar to Russell's example about how the present king of France is bald? He works through the descriptive logic and the proposition doesn't hold because there is no king of France. Couldn't we apply that to the antecedent of E? However, I think it may show that the antecedent isn't true nor false, but just nonsensical.

  3. @rik hine… I don't think I can get on board with dialethism. All conditional statements would be true, and trivially true at that! But, if I were a dialethist, I wouldn't have a problem with the triviality of conditionals. It is also true that I would have a problem with the triviality of conditionals.

    Thanks for the reminder of dialethism. I have checked out a few books by Graham Priest on the topic to see if I can make heads or tails (or both) of it.

  4. @shaunmiller…

    I agree with you that E is nonsensical, but I think A, B, and the current king of France are not nonsensical.

    A is merely a fantastical counterfactual. B is not only fantastical, but physically impossible. The current king of France strikes me as more like A. Consider the following counterfactual:

    If there were a current king of France, he would be bald.

    This could mean that the most likely heir to the throne alive today happens to be bald.

    I think E is nonsense, and as I said to Rik, such nonsense is trivially true. I think it is important to distinguish conditionals which are trivially true from those which are not. This becomes especially important when a conditional is used in an argument that is intended to not be trivially true (IMHO).

  5. I think “nonsense” is a technical term for logicians (what else?). Nonsensical statements are neither true nor false, but simply nonsense. Consider this statement from Noam Chomkey:

    Colorless green ideas sleep furiously.

    It has syntax, but semantically it's nonsense. At least Wittgenstein and Russell would go toward that route.

    On the other hand Quine argued that for a sentence to be false, it would have to show that it was not true. Thus, he would consider the sentence above as false instead of nonsense.

  6. If colorless green ideas sleep furiously then….?

    If that is in the antecedent, anything could be in the consequent. But then, could you have a deduction with any force with that sort of non-sense. So, it seems to me that good modus tollens and modus tollens style deductions need not only true premises, but substantively true premises (if the conclusion is going to be substantively true that is).



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