Semantically, a scheme can be thought of as the conjunction of all its instances – since there are generally infinitely many such instances, this cannot be carried out syntactically! We say that a world/model satisfies a scheme if it satisfies all its instances. Note that an instance being satisfied in a Kripke model does not imply that the whole scheme is satisfied. For example, we may have a Kripke model in which all worlds satisfy ¬p q, but at least one world does not satisfy ¬q p; the scheme ¬φ ψ is not satisfied.

Equivalences between modal formulas

Definition 5.7 1. We say that a set of formulas Γ of basic modal logic semantically entails a formula ψ of basic modal logic if, in any world x of any model M = (W, R, L), we have x ψ whenever x φ for all φ Γ. In that case, we say that Γ ψ holds.

2.We say that φ and ψ are semantically equivalent if φ ψ and ψ φ hold. We denote this by φ ≡ ψ.

Note that φ ≡ ψ holds i any world in any model which satisfies one of them also satisfies the other. The definition of semantic equivalence is based on semantic entailment in the same way as the corresponding one for formulas of propositional logic. However, the underlying notion of semantic entailment for modal logic is quite di erent, as we will see shortly.

Any equivalence in propositional logic is also an equivalence in modal logic. Indeed, if we take any equivalence in propositional logic and substitute the atoms uniformly for any modal logic formula, the result is also an equivalence in modal logic. For example, take the equivalent formulas p → ¬q and ¬(p q) and now perform the substitution

p → p (q → p)

q → r → (q p).

The result of this substitution is the pair of formulas

p (q → p) → ¬(r → (q p))

(5.2)

¬(( p (q → p)) (r → (q p)))

which are equivalent as formulas of basic modal logic.

We have already noticed that is a universal quantifier on accessible worlds and is the corresponding existential quantifier. In view of these facts, it is not surprising to find that de Morgan rules apply for and :

¬ φ ≡ ¬φ and ¬ φ ≡ ¬φ.

Moreover, distributes over and distributes over :

(φ ψ) ≡ φ ψ and (φ ψ) ≡ φ ψ.

These equivalences correspond closely to the quantifier equivalences discussed in Section 2.3.2. It is also not surprising to find that does not distribute over and does not distribute over , i.e. we do not have equivalences between (φ ψ) and φ ψ, or between (φ ψ) and φ ψ. For example, in the fourth item of Example 5.6 we had x5 (p q) and x5 p q.

Note that is equivalent to , but not to , as we saw earlier. Similarly, ≡ but they are not equivalent to .

Another equivalence is ≡ p → p. For suppose x – i.e. x has an accessible world, say y – and suppose x p; then y p, so x p. Conversely, suppose x p → p; we must show it satisfies . Let us distinguish between the cases x p and x p; in the former, we get x p from x p → p and so x must have an accessible world; and in the latter, x must again have an accessible world in order to avoid satisfyingp. Either way, x has an accessible world, i.e. satisfies . Naturally, this argument works for any formula φ, not just an atom p.

Valid formulas

Definition 5.8 A formula φ of basic modal logic is said to be valid if it is true in every world of every model, i.e. i φ holds.

Any propositional tautology is a valid formula and so is any substitution instance of it. A substitution instance of a formula is the result of uniformly substituting the atoms of the formula by other formulas as done in (5.2).

For example, since p ¬p is a tautology, performing the substitution p →p (q → p) gives us a valid formula ( p (q → p)) ¬( p (q → p)).

As we may expect from equivalences above, these formulas are valid:

¬ φ ↔ ¬φ

(φ ψ) ↔ φ ψ.

To prove that the first of these is valid, we reason as follows. Suppose x is a world in a model M = (W, R, L). We want to show x ¬ φ ↔ ¬φ, i.e. that x ¬ φ i x ¬φ. Well, using Definition 5.4,

Figure 5.5. Another Kripke model.

x ¬ φ

i it isn’t the case that x φ

i it isn’t the case that, for all y such that R(x, y), y φ i there is some y such that R(x, y) and not y φ

i there is some y such that R(x, y) and y ¬φ i x ¬φ.

Proofs that the other two are valid are similarly routine and left as exercises. Another important formula which can be seen to be valid is the following:

(φ → ψ) φ → ψ.

It is sometimes written in the equivalent, but slightly less intuitive, form(φ → ψ) → ( φ → ψ). This formula scheme is called K in most books about modal logic, honouring the logician S. Kripke who, as we mentioned earlier, invented the so-called ‘possible worlds semantics’ of Definition 5.4.

To see that K is valid, again suppose we have some world x in some model M = (W, R, L). We have to show that x (φ → ψ) φ → ψ. Again referring to Definition 5.4, we assume that x (φ → ψ) φ and try to prove that x ψ:

x (φ → ψ) φ

i x (φ → ψ) and x φ

i for all y with R(x, y), we have y φ → ψ and y φ implies that, for all y with R(x, y), we have y ψ

i x ψ.

There aren’t any other interesting valid formulas in basic modal logic. Later, we will see additional valid formulas in extended modal logics of interest.