An intensionalist approach to any logic requires that intensional inclusion relations among properties determine the assignment of truth-values. But an intensionalist approach to the modal propositional calculi cannot appeal to properties associated with subjects or predicates because there are no subjects and predicates in the propositional calculus. So intensional inclusion relations among properties may determine the assignment of truth-values to propositions only if the properties correspond to the unanalyzed propositions themselves.
There’s a function from propositions to the relevant properties. Start with the usual stock of proposition lettersp0, p1, p2, ... , which represent propositions not further analyzed. Modal
propositional logic contains non-atomic propositions, too, and those non-atomic propositions require connectives. So we’ll throw in the connectives written¬,∧,∨, and⊃, which represent negation, conjunction, disjunction, and the material conditional, respectively. Now let ‘[...]’ be a 1-1 function from propositions to their corresponding propositional properties. So for every propositionφ, there is a property [φ], orbeing such thatφ. Given this function, [φ] is identical to [ψ] just in caseφis identical toψ.11
There are at least two ways in which propositional properties differ from non-propositional properties. When something exemplifies a non-propositional property, the non-propositional property is part of being that thing. For example, when Fred is tall,being tallis part ofbeing Fred. But at least in simpler cases, when something exemplifies a propositional property, it is part of being that thing that some property is part of another. When alpha exemplifies the propositional property of being such that Fred is tall, for instance, it is part ofbeing alphathatbeing tallis part ofbeing Fred.
Secondly, propositional properties have a special connection to actuality and truth because of their additional structure. On the intensionalist approach, a proposition φis true just in case its corresponding propertybeing such thatφis part ofbeing alpha.12 So on the intensionalist approach,
there is an equivalence between truth and actuality.13 A propositional property’s being part ofbeing alphaand its corresponding proposition’s being true are one and the same. WhereAis the property being alphaand ‘<’ reads “is part of”:
(A) φis true=df.[φ]<A.
Here’s how this treatment “looks”:
Since the S5 system contains classical logic, and since I aim to offer an applied semantics for S5, I will assume the world behaves classically in ways that secure the theorems of classical logic. So I assume thatA, the property of being alpha, iscomplete: for every pair of propositionsφand¬φ,
11
Zalta (1993, 279, n. 7)
12
Most think that truth is “extensional” in the sense that when modal operators are not an issue, one may substitute any true proposition for any othersalva veritate. This is compatible with the present intensionalist treatment of truth.
13
Figure 3.2: An intensionalist approach to truth.
at least one of either [φ] or [¬φ] is part ofA. I also assume thatAisconsistent: for every pair of propositionsφand¬φ, at most one of either [φ] or [¬φ] is part ofA. Therefore, if [φ] is a part ofA, [¬φ] isn’t, andvice versa.A’s completeness secures the law of excluded middle and guarantees that there are no truth-value gaps. Its consistency secures the law of non-contradiction and guarantees that there are no true contradictions. One might rejectA’s completeness or consistency because of vagueness or liar paradoxes, but I ignore these complications because the first test for an alternative approach to modal logic should be whether and how it handles the classical systems.
A number of auxiliary assumptions seem to follow from our intuitive understanding of ‘...and...’ statements, ‘...or...’ statements, ‘if... then...’ statements and the like. For example:
(i) if [φ⊃ψ] and [φ] are parts ofA, so is [ψ],
(ii) [φ] is part ofAiff [¬¬φ] is,
(iii) [φ∧ψ] is part ofAiff [φ] and [ψ] are, and
(iv) [φ∨ψ] is part ofAiff either [φ] or [ψ] is.
These secure theorems and validate various inferences of propositional logic. For example, if [φ
∧ψ] is part ofA, then the conjunctionφ∧ψis true, by (A). But [φ∧ψ] is part ofAif and only if both [φ] and [ψ] are, by (iii). So if [φ∧ψ] is part ofA, then so are [φ] and [ψ]. If [φ] and [ψ] are both parts ofA, then bothφandψare true, by (A). Hence, (iii) validates the inferences from a conjunction to each conjunct.
Now we proceed from truth to necessary truth. Modal extensionalism treats necessary truth as truth in every possible world:
Figure 3.3: Necessary truth according to modal extensionalism.
But modal intensionalism says thatφis necessarily true if and only if its corresponding propositional propertybeing such thatφis part ofbeing a world. More formally, whereW is the propertybeing a world:
(N) φis true=df.[φ]<W.
In the S5 system, we may depict (N) as follows:
Figure 3.4: Necessary truth according to modal intensionalism.
Like the propositional parts ofA, I assume (reasonably, I think) that the propositional parts of
Wbehave in ways that secure our intuitive understanding of ‘...and...’ statements, ‘...or...’ statements, ‘if... then...’ statements and so on. This suggests a number of principles that govern how propositional
properties operate inW, including:
(v) if [φ⊃ψ] and [φ] are parts ofW, so is [ψ],
(vii) [φ∧ψ] is part ofW iff [φ] and [ψ] are, and
(viii) if [φ] is part ofW, so is [φ∨ψ].14
Joined with (N), these principles validate a number of intuitive modal inferences. Consider (vii), for instance. It can be easily shown that (vii) and (N) validate the inferences from(φ∧ψ) to bothφ
andψand from both of these back again to(φ∧ψ).
These sorts of principles, along with (N), imply that the theorems of propositional logic are necessarily true. For example, given (vii), if [φ∧ψ] is part ofW, [φ] is, too. Then, based on our intuitive understanding of ‘if... then...’, [(φ∧ψ)⊃φ] is also part ofW. We then infer from (N) that (φ∧ψ)⊃φis necessarily true. The property [(φ∧ψ)⊃φ] corresponds to a theorem of proposi- tional logic. Intuitively, all other theorems of propositional logic have corresponding propositional properties that are also parts ofW, so they are also necessarily true. These considerations justify an important inference rule in the weakest normal modal system K (a system that has all the theorems of propositional logic as theorems):
Necessitation Rule. Ifφis a theorem of K, then so isφ.
Next we define possibility in terms of necessity. ¬φis necessary just in caseφis impossible. So
φis not impossible—i.e., possible—just in case¬φis not necessary. On the intensionalist approach, then, a propositionφis possible when [¬φ]isn’tpart ofW:
(P) ♦φis true=df.[¬φ]≮W
Hence, what is possibly true corresponds to whatW’s parts do not preclude. Traditionally understood, the necessity and possibility operators are interdefinable: φis equivalent to¬♦¬φ(and¬¬φ
is equivalent to♦φ). Modal intensionalism justifies this equivalence. First, suppose that¬♦¬φis true. By (P),♦¬φis true when [¬¬φ] is not part ofW. So¬♦¬φis true when [¬¬φ] is part ofW. Given that [¬¬φ] is part ofW, [φ] is part ofW, too, by principle (vi). As a result,φis true, by (N). Therefore, if¬¬φis true, then so isφ. Now suppose thatφis true. By (N), [φ] is part ofW. Again, by principle (vi), since [φ] is part ofW, [¬¬φ] is also part ofW. And, as before,¬♦¬φis true when [¬¬φ] is part ofW. Therefore, ifφis true,¬♦¬φis true, which completes the proof.
14An analogue of (iv), the stronger biconditional principal that [φ∨ψ] is part ofWiff either [φ] or [ψ] is, presumably