Figure 67: Dictionary definitions of "model" in logic

 

 

English definitions of „model“ in logic

 

 

A set of entities that satisfies all the formulae of a given formal or axiomatic system.

One of the respective citations is from William S. Hatcher: Logical Foundations of Mathematics (1982, i, 32):

By a model for F we mean any interpretation of F in which the proper axioms of F are all true.

 

 

A model is formally defined in context of some language L, following Tarski's concept of truth. The model consists of two things:

1. A universe set U which contains all the objects of interest (the "domain of discourse"), and
2. a mapping from L to U (called the evaluation mapping or interpretation function) which has as its domain all constant, predicate and function symbols in the language.

 

 

an interpretation of a formal system under which the theorems derivable in that system are mapped onto truths

a theory in which a given sentence is true

 

 

an interpretation of a theory arrived at by assigning referents in such a way as to make the theory true.

 

 

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?

 

 

Logic as a discipline » Features and problems of logic » Logical semantics

 

For the purpose of clarifying logical truth and hence the concept of logic itself, a tool that has turned out to be more important than the idea of logical form is logical semantics, sometimes also known as model theory. By this is meant a study of the relationships of linguistic expressions to those structures in which they may be interpreted and of which they can then convey information.

The crucial idea in this theory is that of truth (absolutely or with respect to an interpretation). It was first analyzed in logical semantics around 1930 by the Polish-American logician Alfred Tarski.

 

In its different variants, logical semantics is the central area in the philosophy of logic. It enables the logician to characterize the notion of logical truth irrespective of the supply of nonlogical constants that happen to be available to be substituted for variables, although this supply had to be used in the characterization that turned on the idea of logical form. It also enables him to identify logically true sentences with those that are true in every interpretation (in “every possible world”).