Semántica clásica de los mundos posibles

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.

Ahora, la pregunta es si esta semántica tiene algo que ver con los mundos posibles, y la respuesta es que sí. La clave es observar que las funciones (incluidas las asignaciones de verdad) también son relaciones y, por lo tanto, las funciones de asignación de verdad también son relaciones, es decir, relaciones que asocian letras proposicionales a valores de verdad (únicos). El conjunto de todas las asignaciones de verdad es una estructura relacional, y la usamos todo el tiempo cuando hablamos de tautologías y contradicciones en la lógica proposicional clásica:

Definición 3. ( Tautologías y contradicciones ) La fórmula φ es una tautología ( ⊧ φ ) si y sólo si toda asignación de verdad hace que φ sea verdadero. De manera similar, φ es una contradicción si y si ¬φ es una tautología.

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) Una introducción matemática a la lógica , 2ª edición, § 1.2.

¡Sí tu puedes! La semántica del mundo posible para la lógica modal proposicional se puede modelar mediante álgebras booleanas con operadores. Consulte B. Jonsson y A. Tarski: Boolean Algebras with Operators. American Journal of Mathematics 73 (1951) y R. Bull y K. Segerberg: Basic Modal Logic. En: Manual de Lógica Filosófica . vol. 3.