Fatalism

This is well-trod ground, but I was thinking about this old puzzle this afternoon and I wanted to work through it myself.

There’s an argument that Aristotle discusses, I think, to the effect that our present actions make no difference to future events. It goes like this. Let B be the proposition “There will be a sea battle tomorrow”, and let A be the proposition “The captain starts the attack.”

1. Either B or not B. (Premise)
2. Suppose B.
3. In that case, whether or not A, B.
4. So A makes no difference as to whether B.
5. Now drop the assumption that B, and suppose instead not-B.
6. In that case, whether or not A, not-B.
7. So again A makes no difference as to whether B.
8. So in any case, A makes no difference as to whether B.

That is, the captain’s decision makes no difference as to whether there will be a sea battle. But fatalism like this is crazy, isn’t it?

This argument, or something like it, has led some people to deny excluded middle for at least some sentences about the future. They say that there is no fact of the matter whether there will be a sea battle tomorrow, and only when tomorrow comes will the proposition become either true or false. Otherwise, they reason, how could we make free decisions that affect the future?

This response is unnecessary. We should stop and ask, what do we mean by the expression “Whether or not P, Q”? Here’s a reasonable thing to mean by it:

• (If P then Q, and if not-P then Q) or (If P then not-Q, and if not-P then not-Q).

In other words, either P and not-P equally well imply Q, or else P and not-P equally well imply not-Q. Intuitively, in no case does Q’s truth value depend on P’s.

But now we need to be careful about what we mean by “if”. In classical logic we take “If P then Q” to be logically equivalent to “Q or not-P.” (This meaning of “if” is called “the material conditional”.) On that understanding of “whether or not” and “if”, (3) logically follows from (2), and (6) logically follows from (5).

But in ordinary English usually what we mean by “If P then Q” is something stronger than just “Q or not P”. For instance, both of the inferences “It isn’t raining; so if it’s raining then it’s Tuesday” and “It’s raining; so if it’s Tuesday then it’s raining” sound weird at best, false at worst. Something closer to what we usually mean by “If P then Q” is “Necessarily, not P or Q” (or “In every relevant case, not P or Q”)—this may be too strong, but we’ll work with it. If we read “if” this way, then the analysis I gave for “Whether or not P, Q” is equivalent to “Necessarily, Q” or (“In every relevant case, Q”). And that sounds about right.

If we understand “whether or not” in the natural way, then the inference from “B” to “Whether or not A, B” is no good. It’s just like reasoning “B, therefore necessarily B.” On the other hand, if we insist on understanding “whether or not” in terms of the “if” of classical logic, then we shouldn’t allow the inference from “Whether or not A, B” to “A makes no difference as to whether B”. It sounds okay, but that’s just because we’re using the words “whether or not” in a funny artificial way. On neither of the two ways of understanding “whether or not” does the argument go through. We don’t have to deny that there are objective facts about the future in order to avoid fatalism.

A hasty nominalist argument

Here's a quick and dirty argument against the view that properties are things.

1. "If Fido is a dog then Fido has the property of being a dog" is a logical truth. (Premise)
   
2. Logical truths are permutation invariant: that is, they remain true when individuals are arbitrarily exchanged. (Premise)
   
3. Suppose "the property of being a dog" refers to an individual D, and "the property of being a cat" refers to an individual C. (For reductio)
   
4. Consider a model M in which C and D are exchanged. "If Fido is a dog then Fido has the property of being a dog" is true in M if and only if Fido has C—€”that is, if and only if Fido is a cat. So the Fido sentence is false in M.
5. But this contradicts (1) and (2). So (3) is false.
   

You can easily make parallel arguments against numbers, propositions, and any other domain where you think there are parallel logical truths. (E.g., "If there are eight planets then the number of planets is eight." "If snow is white then the proposition that snow is white is true.") The proposition case is very similar to an argument David Lewis makes in chapter 3 of Plurality—€”though he worries about necessary truth, rather than logical truth.

I don't find the argument especially convincing, but I think it's interesting anyway. And I think that's all I'll say about it just now.