Algebraic semantics for a modal logic close to S1

Abstract

The modal systems S1-S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1. We extend S1 by a Substitution Principle SP which generalizes a reference rule of S1. In system S1+SP, the relation of strict equivalence ϕ ≡ ψ satisfies the identity axioms of R. Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity). This enables us to develop a framework of algebraic semantics which captures S1+SP as well as the Lewis systems S3-S5. So from the viewpoint of algebraic semantics, S1+SP turns out to be an interesting modal logic. We show that S1+SP is strictly contained between S1 and S3 and differs from S2. It is the weakest modal logic containing S1 such that strict equivalence is axiomatized by propositional identity.connective ≡ such that formulas of the form (ϕ ≡ ψ) → (ϕ ↔ ψ) are theorems but the so-called Fregean Axiom (ϕ ↔ ψ) → (ϕ ≡ ψ) is not a theorem. ϕ ≡ ψ can be read as "ϕ and ψ have the same meaning" or "ϕ and ψ denote the same proposition". 1 If one forces the Fregean Axiom to be valid, then the underlying non-Fregean logic specializes to classical logic where models contain only two propositions: the True and the False; that is, a proposition is reduced to its truth-value. 2 Suszko, in his non-Fregean approach to modality [20,2], introduces necessity by the equational axiom scheme ϕ ≡ (ϕ ≡ ⊤), where the tautological formula ⊤ denotes the necessary proposition (situation). He defines two theories W T and W H whose models are certain topological Boolean algebras, also called interior algebras in the literature. Applying a classical result due to McKinsey and Tarski, Suszko then is able to conclude that the theories W T and W H correspond to the modal logics S4 and S5, respectively. Similar axiomatic approaches, where the modal operator is introduced by means of a given identity connective, were studied by Cresswell [4,5], Martens [17] and other authors; see also the historical note at the end of [20].In contrast to Suszko's approach, we shall work in this paper with the pure language of modal logic (i.e., the language of propositional logic augmented with a modal operator ). Instead of defining modality by means of an identity connective, we go the other way arround and define an "identity connective" by strict equivalence: ϕ ≡ ψ := (ϕ → ψ) ∧ (ψ → ϕ). We require that the so defined connective satisfies what we call here the axioms of propositional identity. These are the axioms that result from the adaptation of the identity axioms of basic non-Fregean logic ((e)-(h) of Definition 1.1. in [2]) to the language of modal logic. It turns out that it is enough to consider Lewis modal system S1 extended by an inference rule SR which is slightly stronger than the S1-rule of Substitution of Proved Strict Equivalents (see, e.g., [10]). Rule SR is equivalent with the Substitution Principle SP of non-Fregean logic and ensures that ≡ has the desired...

Epistemic extensions of combined classical and intuitionistic propositional logic

“…One immediately recognizes that all Lewis systems S1-S5 satisfy the axioms (Id1) and (Id2) of propositional identity if ϕ ≡ ψ is defined as strict equivalence (ϕ → ψ) ∧ (ψ → ϕ). 2 In [5,6] we proved that S3 is the weakest Lewis modal logic where strict equivalence satisfies all axioms of propositional identity (Id1)-(Id3). Moreover, in [5] we showed that logic S1+SP, i.e.…”

“…System S1+ SP then results from S1 by adding all instances of SP as theorems. We saw in [5] that S1+SP S1+ SP ⊆ S3. In particular, all instances of SP are derivable in S3, and S3 is the weakest among Lewis' modal logics with that property.…”

“…Note that the defining condition of EL5-models implies f (m) ≤ f (f (m)) and f ¬ (f (m)) ≤ f (f ¬ (f (m))), for all propositions m. In particular, every EL5-model is an EL4-model. The following is not hard to prove (see, e.g., [5]):…”

“…We expect the reader to be familiar with some basic lattice-theoretical notions such as (prime, ultra-) filters on lattices. We adopt the notation from [5,7] and write a bounded lattice as an algebraic structure…”

“…The approach presented in this paper relies on the assumptions of R. Suszko's non-Fregean logic (see, e.g., [2,10]) and on results of our recent research [7,5,6]. It is inspired by [4] and by ideas coming from Intuitionistic Epistemic Logic [1].…”