In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9-42, 1973)-nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ-versions (κ ≥ ω), introduce a hierarchy of κ-prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (κ-) compactness of the consequence relation together with the existence of a (κ-)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated. Mathematics Subject Classification (2000). Primary 03B22, 03G10; Secondary 03B20.
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...
We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. ∈K (Epsilon-K) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom Kiϕ → ϕ and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between ∈K and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic "paradoxes" can be solved in ∈K . Every specific ∈K -logic is defined as a certain extension of some underlying classical abstract logic.
A famous result, conjectured by Gödel in 1932 and proved by McKinsey and Tarski in 1948, says that ϕ is a theorem of intuitionistic propositional logic IPC iff its Gödel-translation ϕ ′ is a theorem of modal logic S4. In this paper, we extend an intuitionistic version of modal logic S1+SP, introduced in our previous paper [14], to a classical modal logic L and prove the following: a propositional formula ϕ is a theorem of IPC iff ϕ is a theorem of L (actually, we show: Φ ⊢ IP C ϕ iff Φ ⊢ L ϕ, for propositional Φ, ϕ). Thus, the map ϕ → ϕ is an embedding of IPC into L, i.e. L contains a copy of IPC. Moreover, L is a conservative extension of classical propositional logic CPC. In this sense, L is an amalgam of CPC and IPC. We show that L is sound and complete w.r.t. a class of special Heyting algebras with a (non-normal) modal operator.
Logics with quantifiers that range over a model-theoretic universe of propositions are interesting for several applications. For example, in the context of epistemic logic the knowledge axioms can be expressed by the single sentences ∀x.(K i x → x), and in a truth-theoretical context an analogue to Tarski's T-scheme can be expressed by the single axiom ∀x.(x : true ↔ x). In this article, we consider a first-order non-Fregean logic, originally developed by Sträter, which has a total truth predicate and is able to model propositional self-reference. We extend this logic by a connective '<' for propositional reference and study semantic aspects. ϕ <ψ expresses that the proposition denoted by formula ψ says something about (refers to) the proposition denoted by ϕ. This connective is related to a syntactical reference relation on formulas and to a semantical reference relation on the propositional universe of a given model. Our goal is to construct a canonical model, i.e. a model that establishes an order-isomorphism from the set of sentences (modulo alpha-congruence) to the universe of propositions, where syntactical and semantical reference are the respective orderings. The construction is not trivial because of the impredicativity of quantifiers: the bound variable in ∃x.ϕ ranges over all propositions, in particular over the proposition denoted by ∃x.ϕ itself. Our construction combines ideas coming from Sträter's dissertation with the algebraic concept of a canonical domain, which is introduced and studied in this article.
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