ENSACADO O EMPACADO

In the previous section, I showed how (v) validates (K),W’s consistency validates (D), and the connectedness, ininclusivity, fortification, and inexlusivity conditions validate the (T), (S4), (B), and (S5) axioms, respectively. When we interpretAas the property of being alpha,W as the property of being a world, andP as the parthood relation from Chapter 1, there is good though not conclusive reason to think that S5 correctly models metaphysical modality. For as long as we assume that the world behaves classically (as I outlined in section 3), there is reason to believe thatP obeys all these restrictions. Below, I explain these restrictions in the context of an applied semantics for metaphysical modality.

(K) and (D) are true if Principle (v) andW’s consistency hold. I’ve simply assumed that (v) is true and thatW is consistent, and I suppose someone could reject one or the other. But these are reasonable assumptions, nonetheless. Principle (v) seems to follow from our intuitive understanding of the material conditional. For if [φ⊃ψ] and [φ] are parts ofW, surely [ψ] is. Also, our world exemplifies the property of being a world, and I cannot fathom how an exemplified property could have inconsistent propositional properties as parts.

(T) is true ifP is connected, i.e., if the propositional properties which are part ofW are also part ofA. We have reason to think that this condition holds. First, being a world in general (W) is part of beingthisworld (A). Second, the transitivity of property parthood is a theorem of the property mereology. Therefore, if [φ] is part ofW, andW is part ofA, then [φ] is part ofA. So on the intended interpretation, we now have good reason to accept (K), (D), and (T).

Let me mention a potentially curious result. As part ofA,WinheritsA’s consistency: if there are no properties inAthat preclude one another, thena fortiorithere are no properties inW that preclude one another. Given (A),A’s consistency guarantees that there are no true contradictions.

Given (N),W’s consistency guarantees that there are no true contradictions necessarily. In a way, whatever secures the truth of the law of non-contradiction also seems to secure its necessary truth.

I introduced the next three restrictions in the previous section. I will explain the first of these restrictions by describing a special feature ofA. Take some propositional property [p1] which is

part ofA. Since [p1] is part ofA, presumably some true proposition p2 says that [p1] is part of A. So from (A), we infer thatp2’s corresponding propositional property [p2] is also part of A.

Therefore, if [p1] is part ofA, so is the property of being such that [p1] is part ofA. The privilege

of having properties concerning which properties are parts of a property is generally reserved for meta-properties via Inclusivity. But in virtue of being a property which accounts for all truths, even truths about itself,Ais ininclusive and has properties which concern which properties are parts of itself.

The propositional parts ofW tell us “what it takes” for any world to exist at all. These parts ofW concern the preconditions for the existence of a world in general, and I doubt whether the preconditions for worldhood could have been different. What could possibly alter the preconditions for the existence of a world in general? If the answer is “nothing,” the preconditions for anything’s being a world at all include thatW has the very parts it has. If [φ] is part ofW, [φ] is part ofW

necessarily. That is, if [φ] is part ofW then [φ]’s being part ofWis itself part ofW. LikeA,W is ininclusive. (S4) is true ifW is ininclusive.

On the current picture,W is part ofAand both are consistent. Now supposeφis true, that [φ] is part ofA. Then it seems reasonable to conclude that the preconditions for worldhood could not have included thatφis false. That is, it is a precondition that there is no such precondition thatφbe false. So not only is [¬φ] not part ofW (due toW’s being part ofAandA’s consistency). Also, the property of being such that [¬φ] is not part ofW is itself part ofW. So if [φ] is part ofA, it is part ofW that [¬φ] is not part ofW. This restriction fortifies the truth as necessarily possible and so secures the truth of (B).

I will introduce the final restriction by describing another special feature ofA. There are truths about which properties are not parts ofA. Given (A), the propositional properties corresponding to these truths are also parts ofA. Suppose that [p1] is not part ofA. Then, it is true that [p1] is not part

ofA. By (A), then, being such that [p1] is not part ofAis itself part of beingA. Some ofA’s parts

is a denizen of alpha, so there are truths about which properties are parts ofAand which are not.

A’s having or not having a part is part of how our world, alpha, is. In virtue of being the property of being alpha,Ais self-referential as both an ininclusive and inexclusive property.

W is also inexclusive, though for different reasons. Earlier, I gave a reason to think that the preconditions for the existence of a world in general are preconditions necessarily. Perhaps there is also reason to think that non-preconditions for the existence of a world are non-preconditions necessarily, i.e., that [φ] must not be part ofW if it is not part ofW. IfW is the kind of abstract object that could not have been different, then if [¬φ] is not part ofW, thenW’s not having [¬φ] as a part is itself part ofW. If there are necessary truths about which propertiesW does not have as parts, thenW has parts that concern which properties it does not have as parts. So, likeA,W is inexclusive. (S5) is true ifWis inexclusive.

On the modal intensionalist’s interpretation of the formalism, one can make a decent case that the S5 system is appropriate for modeling metaphysical modality. On that interpretation, there is good reason to believe thatP obeys (v), the consistency ofW, connectedness, fortification, and the ininclusivity and inexclusivity ofW. IfP is restricted in all these ways, (K), (T), (D), (S4), (B), and (S5) are all true.