Interpretation Glossary
Definition
An interpretation is an assignment of values to the non-logical symbols (names, predicates and function symbols). Names are assigned individual members of the domain; predicates are assigned relations (sets of n-tuples from the domain); function symbols are assigned functions, which can be seen as a special case of relations. In terms of this, the truth values of formulae are determined.
Comments
In propositional logic, interpretation in the above sense amounts simply to assigning truth values to the atons. For full first-order logic, the more elaborate definition is needed.
Entailment: In terms of interpretation it is easy to define logical entailment semantically: X entails A iff A is true for every interpretation for which all formulae in X are true.
Model: An interpretation I is a model of a formula iff that formula is true for I. It is a model of a set of formulae iff all formulae in the set are true for I.
Examples
-
Interpretation for which (p
OR q) IMP NOTr is
false:
I(p) = 0
I(q) = 1
I(r) = 1 -
Interpretation produced as a "model"
by Logic for
Fun:
D | Adelaide Brisbane Canberra Darwin --+------------------------------------- F | TRUE FALSE TRUE FALSE D | Adelaide Brisbane Canberra Darwin --+--------------------------------------- f | Canberra Darwin Adelaide Adelaide
Written out in the "official" format:Domain D = {Adelaide, Brisbane, Canberra, Darwin}
I(F) = {Adelaide, Canberra}
I(f) = {(Adelaide, Canberra), (Brisbane, Darwin), (Canberra, Adelaide), (Darwin, Adelaide)}