Propositional logic is the logical system which takes sentences and their combinations as primitives. The logical constants of the language are negation and the connectives &, v, ->, and <->. Propositional letters (also atomic propositions) are combined with these connectives into more complex propositional formulas according to the syntax of propositional logic. The semantics interprets the meaning of the logical constants in terms of truth-values. Propositional logic characterizes a particular class of valid arguments, like the one in (i).
(i) If the sun is shining, then John is happy The sun is shining Therefore, John is happy
When we translate the natural language statements in (i) into propositional logic (as in (ii)) we get the schema in (iii).
(ii) p: the sun is shining q: John is happy (iii) p -> q p ------ q
Translation into propositional logic makes it clear that the argument in (i) is valid because of certain logical constants. The validity of the schema in (iii) can be demonstrated with a formal syntactic deduction or by means of a truth-table.
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.