# ELI5: What is modal logic?

March 23, 2015

Someone recently asked me what sort of background knowledge is required to study modal logic. I thought I’d share my reply here, in case it might be of use to people.

My first taste of modal logic was Richard Montague’s intensional logic (IL), as applied to natural language. IL is a complex combination of higher-order typed logic (with typed λ-terms) and two types of modal logic (see the end of this post). Needless to say, I struggled long and hard to disentangle the pieces and understand them individually. In fact, for a while, I thought that modal logic was IL. It was only later that I discovered that, actually, even basic propositional logic can be modalized.

I strongly believe now that the study of modal logic (be it in a logic course or in a natural language semantics course) should begin with modal propositional logic, not modal predicate (quantificational) logic. Modality and quantification can much more easily be combined once they are each individually well understood.

If you’re new to modal logic and have, up to now, felt a little intimidated or confused by it, then hopefully the preceding paragraphs have eased your mind. If not, then hopefully the rest of this post will.

## Explain it like I’m 5: What is modal logic?

Modal logic is an extension of other, more basic types of logic, primarily classical propositional logic and predicate logic. You can learn modal logic once you’ve learned propositional logic. You do not need to know predicate logic to learn modal logic, although in practice many modal logics are extensions of predicate logic, which itself is an extension of propositional logic.

Here are some types of logic, in order (more or less) of increasing complexity.

• Propositional logic is the simplest logic. It contains propositional variables (p, q, r, …) and logical connectives (¬, ∧, ∨, →, …). You can create formulas like p, p ∧ q, (p ∨ ¬q) → r, etc. The truth value of a complex formula ϕ (one containing logical connectives) is a function of the truth values of the immediate subformulas of ϕ, based on the truth tables for the logical connectives.

• Non-quantificational predicate logic extends propositional logic by creating atomic formulas out of n-place predicate symbols (P, R, S, …), i.e. symbols that take n individual symbols (a, b, c, …) to create a formula, such as Pa, Rab, Sabc, etc. The individual symbols are taken from a domain (or universe) of individuals, D (or U). This logic still contains the logical connectives, so you can create complex formulas like Pa ∧ (Rab → ¬Sabc). As in propositional logic, the truth value of a complex formula ϕ is a function of the truth values of the immediate subformulas of ϕ, based on the truth tables for the logical connectives.

• Quantificational (first-order) predicate logic extends non-quantificational predicate logic by adding individual variables (x, y, z, …) and quantifier symbobls (∃, ∀) that operate over individual variables. You can create formulas like ∀xPx, ∃x∃yRxy, Pa ∧ ∀x(Sxbc → Px), etc. The truth value of a formula ϕ with a quantifier depends on whether the formula in the immediate scope of the quantifier is true of some (in the case of ∃) or all (in the case of ∀) individuals in the domain (roughly speaking). For example, ∃xPx is true iff P is holds of some individual in the domain, and ∀xPx is true iff P holds of every individual in the domain (roughly speaking). (Second- and other higher-order predicate logics are extensions of first-order predicate logic, which allow quantification not only over individuals, but also over sets (of sets (…(of sets)…)) of individuals.)

• Modal propositional logic extends propositional logic by adding intensional operators (e.g. ◇, □) and a domain of possible worlds and by making truth value assignments world-dependent, meaning that a formula ϕ only has a truth value relative to (or “in”) a possible world. This means that it no longer makes sense to say that ϕ is true or that ϕ is false; rather, we say that ϕ is true in (relative to) w1, false in w2, and so on. (A formula generally has different truth values in different possible worlds.) You can make formulas like p, p ∧ q, □p, □p → ◇q, etc. The truth value of a formula ϕ with an intensional operator depends on whether the formula in the immediate scope of the operator is true in some (in the case of ◇) or all (in the case of □) worlds accessible from the world relative to which ϕ is evaluated. For example, ◇p is true in w iff p is true in some world v which is accessible from w, and □p is true in w iff p is true in every world v which is accessible from w. Which worlds are accessible from which other worlds is determined by an accessibility relation. Accessibility relations play a huge role in modal logic.

• Modal predicate logic extends modal propositional logic in the same way that non-modal predicate logic extends non-modal propositional logic. It thus contains both intensional operators and quantifier symbols, so you can create complicated formulas like ◇(Rab → ∃x∀y(¬□Px ∧ □Sxy)).

Different intensional operators and different accessibility relations give rise to different modal logics, some of which have been used to model various real-world phenomena. Some examples:

• possibility and necessity (alethic logic)
• permission and obligation (deontic logic)
• knowledge (epistemic logic)
• belief (doxastic logic)
• time (temporal logic)

In the case of time, possible “worlds” model points in time and are ordered by an “earlier than” relation, e.g. “t1 < t2” means t1 is earlier in time than t2. Thus, the domain of possible worlds/times/states can have quite a rich structure; that is, it need not be a simple set, but can in addition have structure (relations, etc.) defined on it.

Montague’s intensional logic actually includes both worlds and times, so that a formula ϕ is said to be true (or false) in a particular world, at a particular time. Thus, ϕ can be true in w at time t1, but false in w at time t2; and ϕ can be true in w1 at time t, but false in w2 at time t; and so forth.