# Classical possible worlds semantics

It looks like to me that possible worlds semantics are closely associated with propositional modal logic (or interior/closure algebras).

Is there any literature where possible worlds semantics is associated with classical propositional logic (or ordinary Boolean algebras)?

Put differently, can we drop modalities when thinking about possible worlds semantics?

• Possible world semantics is the semantics/model side for a modal logic, the syntactical/theory side. – Mozibur Ullah May 20 '14 at 22:53

## 2 Answers

Possible-worlds models are relational structures (an underlying set equipped with a bunch of relations). E.g., a possible-worlds model M may consist of the following components: W, →, @, where W is a set of possible-worlds, → is an accessibility relation on W2, and @ is a distinguished world of W. A standard application of these models is the explication of modalities, but they can also be thought of as models of non-modal classical propositional calculus if thought of in the following way. Starting with the language:

Definition 1. (Language) Given a propositional letter p, the language of propositional logic is defined by the following grammar:   φ   :=   p   |   φ′   |   ¬φ   |   (φ ∧ φ),

we can continue our logical pursuits in one of two ways: (1) we can equip this language with a proof system (a set of axioms and rules of inference) and start deriving truths of propositional logic, and/or (2) we can equip this language with a semantics (an interpretation of its formulas in some recognized structure) and start reasoning semantically. The question under consideration is about the semantics for (Definition 1), so we will ignore proof systems of classical propositional logic and consider its semantics:

Definition 2. (Semantics) Models of classical propositional logic are truth-assignments.

Truth-assignments are functions that take propositional letters of the language of propositional logic to truth-values (which in the classical case means {0, 1}). A formula φ of the language of propositional logic is said to be true with respect to a truth-assignment v (symbolically: v ⊧ φ) just in case φ becomes true whenever the propositional letters occurring in φ are assigned truth-values according to v. For example, under assignment v = {p → 1, q → 0}, formula (p → q) becomes false.

Now, the question is whether this semantics has anything to do with possible-worlds, and the answer is that it does. The key is to observe that functions (including the truth-assignments) are also relations, and therefore truth-assignment functions are also relations, namely, relations that associate propositional letters to (unique) truth-values. The set of all truth-assignments is a relational structure, and we use it all the time when talking about tautologies and contradictions in classical propositional logic:

Definition 3. (Tautologies & Contradictions) Formula φ is a tautology ( ⊧ φ ) iff every truth-assignment makes φ true. Similarly, φ is a contradiction iff ¬φ is a tautology.

Truth-assignments can be thought of as possible worlds: all we have to do is extract from them those propositional letters that they map to 1 and we have a set of propositional letters that can be said to be true at that 'world'. For example, formula (p ∨ q) has 22 possible truth-assignment, 4 possible worlds: 00, 01, 10, and a 11 world. The first can be described as the world where neither p nor q hold; the second as the world where p holds but q doesn't, and so on. The formula will be neither a tautology, nor a contradiction (so a contingency), because there are worlds (namely: 01, 10, and 11) where it holds, and there is a world (namely: 00), where it doesn't.

That's the basic idea. For a standard treatment of truth-assignments look at:

Enderton, H. (1972) A Mathematical Introduction to Logic, 2nd Edition, § 1.2.

Yes you can! Possible world semantics for propositional modal logic can be modelled by Boolean algebras with operators. See B. Jonsson and A. Tarski: Boolean Algebras with Operators. American Journal of Mathematics 73 (1951) and R. Bull and K. Segerberg: Basic Modal Logic. In: Handbook of Philosophical Logic. Vol. 3.