logic summary

logic, Study of inference and argument. Inferences are rule-governed steps from one or more propositions, known as premises, to another proposition, called the conclusion. A deductive inference is one that is intended to be valid, where a valid inference is one in which the conclusion must be true if the premises are true (see deduction; validity). All other inferences are called inductive (see induction). In a narrow sense, logic is the study of deductive inferences. In a still narrower sense, it is the study of inferences that depend on concepts that are expressed by the “logical constants,” including: (1) propositional connectives such as “not,” (symbolized as ¬), “and” (symbolized as ∧), “or” (symbolized as ∨), and “if-then” (symbolized as ⊃), (2) the existential and universal quantifiers, “(∃x)” and “(∀x),” often rendered in English as “There is an x such that …” and “For any (all) x, …,” respectively, (3) the concept of identity (expressed by “=”), and (4) some notion of predication. The study of the logical constants in (1) alone is known as the propositional calculus; the study of (1) through (4) is called first-order predicate calculus with identity. The logical form of a proposition is the entity obtained by replacing all nonlogical concepts in the proposition by variables. The study of the relations between such uninterpreted formulas is called formal logic. See also deontic logic; modal logic.