Symbolizing Modal Propositions
Symbolization in modal logic is similar to symbolizing sentences in propositional logic, but with two new symbols: □ (box) and ◇ (diamond). Recall that propositional logic had only three rules of syntax:
- Any capital letter by itself is a WFF.
- Any WFF can be prefixed with “~”.
- Any two WFFs can be put together with “•, “∨”, “⊃;”, or “≡” between them, enclosing the result in parentheses.
The rules of syntax for modal logic are exactly the same, with one additional rule:
- Any WFF can be prefixed with “□” or “◇”.
Notice that rule 4 is similar to the rule for the tilde (rule 2). This means that the box and diamond symbols behave syntactically like the “~” in propositional logic: anywhere you could place a “~”, you can also place a “□” or “◇”.
In modal logic, the box symbol represents necessity. When a proposition is prefixed with “□”, this means that the proposition is necessarily true— i.e., true in all possible worlds. For example, “□P”—read as “necessarily P”—means that P is true in all possible worlds. Similarly, “□(P • Q)”—read as “necessarily, both P and Q”—means that the conjunction “(P • Q)” is true in all possible worlds.
The diamond symbol represents possibility. When a proposition is prefixed with “◇”, this means that the proposition is possibly true, i.e., true in at least one possible world. For example, “◇P”—read as “possibly P”—means that there is at least one possible way for P to be true; that is, P is true in at least one possible world. Similarly, “◇~(P ∨ Q)” means that the negation “~(P ∨ Q)” is true in at least one possible world, i.e., that there is at least one possible way for neither P nor Q to be true.
When a proposition is not prefixed with “□” or “◇”, this means that the proposition is true in the actual world—i.e., it is actually true; it is a true proposition in reality. For example, the proposition “(P ∨ ◇Q)” says that either P is actually true or Q is possibly true.
|expressions in English
It’s true that P.
P is actually true.
P is true in the actual world.
It’s possible that P.
P can be true.
P may be true.
P might be true.
P could be true.
P is true in some possible world.
It’s necessary that P.
P must be true.
P has to be true.
P is true in all possible worlds.
The box and diamond symbols can be used to represent any type of modality, including logical, conceptual, epistemic, physical, and metaphysical modality. For example, ~◇P symbolizes the sentence It’s not the case that P is possible; or, in other words, P is impossible. The meaning of that sentence, however, depends on the type of modality involved, which is usually determined by the context in which the sentence is expressed. Here are some of the propositions that can be represented using that same symbolization:
- When representing logical modality, ~◇P means that P is a logical contradiction.
- When representing conceptual modality, it means that P is inconceivable.
- When representing epistemic modality, it means that P is incompatible with your knowledge: it can’t be true, given what you know.
- When representing physical modality, it means P is forbidden by the laws of physics.
- When representing metaphysical modality, it means P is metaphysically impossible—that is, it’s impossible for some deeper reason, typically a reason involving facts about identity or essence. (See the discussion of metaphysical modality at the beginning of this chapter.)
As mentioned previously, we must exercise caution when dealing with arguments or propositions involving more than one type of modality. To avoid committing the fallacy of equivocation, usually it is best to symbolize only one type of modality at a time.
□~P symbolizes the sentence Necessarily, P is false. This is another way of saying that P is impossible, and is logically equivalent to ~◇P. Several other modal equivalences will be introduced on the next page.
◇(P • Q) symbolizes the sentence It’s possible that P and Q are both true. When the sentence refers to logical necessity, for example, this means that P and Q are consistent.)
□(P ⊃ Q) symbolizes the sentence It’s necessary that if P then Q. When referring to logical necessity, this means that P entails Q.
(◇P • ◇~P) symbolizes the sentence Possibly P, and possibly not P. When referring to logical necessity, this means that P is a contingency.
(P • □~P) symbolizes the sentence P is true but could have been false. In other words, P is a contingent truth.