Thursday, May 26, 2011

Tabulating →

Alexander Pruss had a post a while back giving an argument that indicative conditionals (if p then q) are material conditionals (Either q or not-p, as this would be understood in ordinary propositional logic). Longtime readers will know that I think indicative conditionals are almost never material conditionals and that the standard practice of representing them in logic as if they were will usually leave something important out (many indicative conditionals can indeed be represented for many logical purposes as material conditionals but at least most of those will leave something out that would be logically important under different conditions and many indicative conditionals are not material conditionals at all); indeed, claiming that they are is one of the ways you can sometimes really get me riled up. Not that Pruss's post riled me up, but this is all by the way of explaining why I've been intending to say something about this esoteric post for three weeks now.

Pruss's argument is:

(1) For any possible world w: (p at w) → (q at w) if and only if (p→q at w).
(2) For any predicates F and G, from "Every F is a G" (where "x is an F" is more euphonious way of saying that x satisfies F) together with the assumption that c exists, it logically follows that if c is an F, then c is a G.
(3) If (1) is true and → is non-hyperintensional, then indicatives are material.
(4) If (2) is true and → is non-hyperintensional, then indicatives are material.
(5) If (2) is true, then one has to assign the same truth value as the material conditional does to a number of paradoxical-sounding examples of indicative conditional sentences that are relevantly just like the standard alleged counterexamples to the thesis that all indicatives are material.

p → q would mean "If p, then q." Never mind the talk about non-hyperintensionality. Pruss's interest is primarily (5). What caught my eye was (1), which Pruss characterizes as plausible. I don't think it is plausible -- or to be more exact, I think it is plausible only under certain kinds of assumptions, assumptions that often are not true.

Consider the following way of thinking about possible worlds. Let every possible world be represented by a table, on which sentences are printed that represent what's true in that world. So a possible world in which "Dogs exist" is true would be represented as:

Dogs exist.

Now, we want to allow lots of possible worlds, and therefore many tables. But from one possible world one can infer a great many things about any of the possible worlds we are allowing (sometimes including itself). We can represent these as rules governing the tables themselves. There are many kinds of rules you can have; some of them will simply say that there's an allowed table somewhere that has a particular sentence on it, others might say that particular sentences are on every table we allow, others might say that if you have a particular sentence on one table there will be a particular sentence on another table.

So when we look at the left-hand side of Pruss's formula, (p at w) → (q at w), what this tells you is that whenever any allowed table (w) has the sentence "p" written on it the sentence "q" is should also be written on it. So if p is "Dogs exist" and q is "Dogs are friendly" then any table with p looks like this:

Dogs exist.
Dogs are friendly.

The left-hand side of the formula [(p→q at w)], however, would give us a w that looks like this:

If dogs exist, dogs are friendly.

Now, if the full formula [(p at w) → (q at w) if and only if (p→q at w)] were really true, these tables should look the same. In particular, they should both look like this:

Dogs exist.
Dogs are friendly.
If dogs exist, dogs are friendly.


Yet they don't. Why? Because Pruss's formula is only true if you make assumptions we haven't made yet.

In effect what this shows us in miniature is that Pruss's formula is only plausible if we assume that modality (the rules governing what sentences you can write on various tables) doesn't play any significant role in the meaning of indicative conditionals. Since I'm a constructivist about what philosophers call 'possible worlds', I think possible worlds in a sense just are tables modeling possiblity and the like: and since I'm a Humpty-Dumptyist about logical models, I can do whatever I please with tables, and therefore with possible worlds, and I dare both you and the tables to try to stop me. The essence of logic, like that of mathematics, is absolute freedom within the confines of ultimate consistency. So this is why I don't find Pruss's (1) very plausible at all.

Of course, one might not accept such views, but this is not essential to my main point, which is that Pruss's (1) is plausibly true only if certain assumptions are made about the modal landscape, and the assumption that whenever we find indicative conditionals, those assumptions are in fact assumed. I am not at all convinced that this is always true, and even one exception would make (1), which has no qualifications, false. (1) may have a more restricted true version; indeed, I think there surely must be a restricted form that is true of some modal set-ups, and maybe even all the more common ones. In any case, my point is just that any assumptions that might make (1) plausible are stronger assumptions than one might think.

2 comments:

  1. Dave M6:55 PM

    I didn't read all this, but I agree with you that it is completely crazy to say that indicative conditionals are material.  They mean different things!  My favorite example of this, or the one I always use anyway, is: "If this is Tuesday, we must be in Belgium."  Only sheer perversity can make you want to call that true simply because today is Thursday.  (Also, you have a typo in your def. of material conditional, which you have backwards.)

    ReplyDelete
  2. branemrys8:01 PM

    Thanks -- I'll fix.

    ReplyDelete

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