The study of reasoning. Originally, the philosophical aim was to find deductive rules which prove statements from given premises and axioms. In formalizing rules for natural deduction, logical languages have been developed, designed to provide translations of statements. In these translations only certain logical aspects are made explicit, depending on the language chosen. Propositional logic and predicate logic are the languages of standard logic. A logical language is defined by a syntax and a semantics. The syntax defines the formulas of the language on the basis of a vocabulary of logical constants and other basic expressions, it specifies a set of axioms (expressed in conformity with the syntax of that language) and a set of explicit rules of inference for deriving further formulas from the axioms. The semantics provides a set of truth conditions defining when a formula is true given an interpretation of its basic expressions. The syntax and semantics of standard logic can be extended in different ways. The syntax can be extended with more logical operators (tense operators and modal operators) or with variables over predicates (second-order logic). That way we can incorporate tense and modal interpretations. A more radical departure from standard logic is type logic, in which all expressions are assigned to a particular set-theoretical category. Next to two-valued logics (in which propositions are exclusively true or false) three-valued logics (true, false and indefinite) and other many-valued logics have been defined. The characteristic of these languages is that they do not obey the principle of the excluded middle, which says that a formula must be either true or false.
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.