Kripke semantics

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Modal logic The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives (in this article and ), and the modal operator ("necessarily"). The modal operator ("possibly") is (classically) the dual of and may be defined in terms of necessity like so: ("possibly A" is defined as equivalent to "not necessarily not A"). A Kripke frame or modal frame is a pair , where W is a (possibly empty) set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibility relation. A Kripke model is a triple , where is a Kripke frame, and is a relation between nodes of W and modal formulas, such that for all w ∈ W and modal formulas A and B: if and only if , if and only if or , if and only if for all such that . We read as “w satisfies A”, “A is satisfied in w”, or “w forces A”. The relation is called the satisfaction relation, evaluation, or forcing relation. The satisfaction relation is uniquely determined by its value on propositional variables. A formula A is valid in: a model , if for all w ∈ W, a frame , if it is valid in for all possible choices of , a class C of frames or models, if it is valid in every member of C. We define Thm(C) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod(X) be the class of all frames which validate every formula from X.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (42)

Related courses (4)

Related lectures (11)

The Semantics of COALA in CO-OPN/2

This report presents the formal semantics of COALA, a design language based on the concept of Coordinated Atomic Actions (CA actions). COALA has been developped with the intent of providing a concrete

1999

Intermediate logic

In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic). A superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: 1. all axioms of intuitionistic logic belong to L; 2.

Completeness (logic)

In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.

Provability logic

Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel–Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4).

MATH-381: Mathematical logic

Branche des mathématiques en lien avec le fondement des mathématiques et l'informatique théorique. Le cours est centré sur la logique du 1er ordre et l'articulation entre syntaxe et sémantique.

MATH-483: Gödel and recursivity

Gödel incompleteness theorems and mathematical foundations of computer science

CS-550: Formal verification

We introduce formal verification as an approach for developing highly reliable systems. Formal verification finds proofs that computer systems work under all relevant scenarios. We will learn how to u

Music Semantics: Problems and Prospects

Explores music semantics, discussing inferences triggered by different musical elements and proposing a framework for understanding music's meaning.

Relational Semantics of Loops CS-550: Formal verification

Explores the relational semantics of loops in programs and the mathematical interpretation of loop structures.

Automating First-Order Logic Proofs Using Resolution CS-550: Formal verification

Covers first-order logic syntax, semantics, Skolemization, resolution, and normal form transformations.