model theory

The use of a model to test for the consistency of an axiomatized system is older than modern logic. Descartes's algebraic interpretation of Euclidean geometry provides a way of showing that if the theory of real numbers is consistent, so is the geometry. Similar mappings had been used by mathematicians in the 19th century, for example to show that if Euclidean geometry is consistent, so are various non-Euclidean geometries. Model theory is the general study of this kind of procedure: the study of interpretations of formal systems. Proof theory studies relations of deducibility between formulae of a system. But once the notion of an interpretation is in place we can ask whether a formal system meets certain conditions. In particular, can it lead us from sentences that are true under some interpretation to ones that are false under the same interpretation? And if a sentence is true under all interpretations, is it also a theorem of the system? We can define a notion of validity (a formula is valid if it is true in all interpretations) and semantic consequence (a formula B is a semantic consequence of a set of formulae, written A1…An |= B, if it is true in all interpretations in which they are true). Then the central questions for a calculus will be whether all and only its theorems are valid, and whether A1…An |= B if and only if A1…An ⊦ B. These are the questions of the soundness and completeness of a formal system. For the propositional calculus this turns into the question of whether the proof theory delivers as theorems all and only tautologies . There are many axiomatizations of the propositional calculus that are consistent and complete. Gödel proved in 1929 that the first-order predicate calculus is complete: any formula that is true under every interpretation is a theorem of the calculus.

Philosophy dictionary. . 2011.

Look at other dictionaries:

  • Model theory — This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. In mathematics, model theory is the study of (classes of) mathematical structures (e.g. groups, fields,… …   Wikipedia

  • Finite model theory — is a subfield of model theory that focuses on properties of logical languages, such as first order logic, over finite structures, such as finite groups, graphs, databases, and most abstract machines. It focuses in particular on connections… …   Wikipedia

  • Type (model theory) — In model theory and related areas of mathematics, a type is a set of first order formulas in a language L with free variables x1, x2,…, xn which are true of a sequence of elements of an L structure . Loosely speaking, types describe possible… …   Wikipedia

  • Institutional model theory — generalizes a large portion of first order model theory to an arbitrary logical system. The notion of logical system here is formalized as an institution. Institutions constitute a model oriented meta theory on logical systems similar to how the… …   Wikipedia

  • Inner model theory — In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V , or sometimes of a generic extension of …   Wikipedia

  • Actor model theory — In theoretical computer science, Actor model theory concerns theoretical issues for the Actor model.Actors are the primitives that form the basis of the Actor model of concurrent digital computation. In response to a message that it receives, an… …   Wikipedia

  • NIP (model theory) — In model theory, a branch of mathematical logic, a complete theory T is said to satisfy NIP (or not the independence property ) if none of its formulae satisfy the independence property, that is if none of its formulae can pick out any given… …   Wikipedia

  • Computable model theory — is a branch of model theory which deals with questions of computability as they apply to model theoretical structures. It was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and …   Wikipedia

  • Conceptual Model Theory — The Conceptual Model Theory of Human Understanding is a historically distinct theory of knowledge that is the first foundational epistemological theory to be validated by commercial artificial intelligence use (see… …   Wikipedia

  • Interpretation (model theory) — In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N . For example every reduct or definitional expansion of a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.