Introducción

My proposal is to use only logically possible worlds in giving our semantics for counterfactuals. There are many ways to find that what a world represents is impossible. If using logic is enough to find that whatwrepresents is impossible, thenwis not part of the space of worlds I want to use in my semantics.

These worlds must be fully compositional with respect to the language of classical propositional logic. Since these worlds are logically possible, they must be maximal and closed under the relation of logical consequence. When the truth value of all propositional atoms is fixed, when all of them are either true or false, the truth value of all syntactically more complex formulas follows logi- cally. Since an instance of a failure of compositionality with regards to a logical language is logically impossible, these impossible worlds are compositional.

A consequence of this is that (Closure) is now valid again. If it is the case that ψ1∧...∧ψn → χ is a logical truth, it will be true at every world in the

model. If it was not, then logic would be enough to determine that that world was impossible. This is enough to make the proof of (Closure) given above go through.

More generally, the logic of the non-modal propositional fragment of the vacuist’s logic is just the same. Possible worlds and the impossible worlds I am using cannot be distinguished by propositional logic alone. If they could, then the impossible worlds would be logically impossible. This means that any valid- ity of the vacuist’s logic that involves no modal operator is kept intact. If you forget the modal operators, this counterfactual logicjust is the counterfactual logic of the vacuist.

Modal operators, when interpreted as concerning metaphysical modality, can distinguish between possible and impossible worlds. The truth conditions for the modal operators, if they are to have the intended interpretation, must be restricted to the former. This means that some validities involving modal operators will be lost, most notably vacuism itself.

But if we reinterpret the modal operators as representinglogical modality, everything can stay the same as in the vacuist semantics. Every validity in that logic which involves reference to metaphysical modality will now involve reference only to logical modality.

Metaphysical modality, then, is dissociated from the semantics of counter- factuals. I think that this is not a big loss for the logic of counterfactuals. The logic sketched above is quite robust. Metaphysical modality is not an ordinary notion, but a technical term in philosophy. If having a semantics of modality that respects speakers’ intuitions better while providing a strong counterfac- tual logic requires that some metaphysical impossibilities be represented, then so much the worse for metaphysical modality. I think that speakers’ intuitions of validity and their deductive behavior are not significantly affected by the transition of using metaphysical modality to using logical modality.

It might be that this is a problem for our theory ofmetaphysical modality. Maybe the association between counterfactuals and modal operators was more important to the modal operators than for the counterfactuals. We will consider whether that is so in the next chapter, but let us bracket that problem for now. This semantics is non-vacuist. Antecedents expressed by “If Bill Gates were my father” or by “If I were an iPhone” are now true at some worlds, albeit impossible ones. If a proposition is a metaphysical impossibility that is not a logical impossibility it will be true at some worlds and the respective coun- terfactuals will not be vacuous. There does not seem to be anything logically inconsistent about either mathematical platonism or nominalism. One is true at some worlds and the other is true at some others. This can be iterated for all metaphysical debates where the opposing positions are logically consistent. This is a significantly large class of propositions.

When we move closer to logical impossibility, things are not so simple. Con- sider analytical truths, where analiticity is conceived as what is sometimes called

Frege analyticity.10 A sentence is Frege analytic if it is a logical truth or it can be turned into a logical truth by the substitution of synonymous expressions. Examples are “Bachelors are unmarried” or “Vixens are female foxes”. What about counterfactuals with the negations of such sentences as antecedents?

Since they are not logically impossible, they will be true at some world. But in that world the substitution of synonymous expressions cannot be valid, otherwise it would represent a logical falsehood. If you think that synonymy facts are part of a larger class of semantic facts that can be true or false at a world, it would follow that these worlds are worlds where the actual semantic facts change. If the counterfactual is uttered in a context where the semantic facts are to be held fixed, the antecedents will not be true at any world. I

will treat this issue together with the main problem for my semantics in the following sections.

Countermathematicals are hard to analyse. This is due to the fact that saying something concrete about them involves taking some stand or other on the philosophy of mathematics and the semantics of mathematical discourse. If logicism is true, then mathematically impossible propositions would not be true at any world. If logicism is false, then countermathematicals will not be vacuous. Even in this case, it is possible that mathematical propositions collapse into the case of analytic propositions when the synonymy facts are substituted by mathematical definitions. I will largely ignore countermathematicals in what follows and hope that what I say in the other cases is sufficient to see how the story about them would go assuming a certain philosophy of mathematics.

I hope to have given some idea of how my semantics is non-vacuist while still providing for a strong counterfactual logic. I now turn to the main challenge it faces.