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# classical logic

Source:
A Dictionary of Logic

# classical logic

Any deductive system equivalent to the propositional or first-order logics described in, e.g., the Begriffsschrift of philosopher Gottlob Frege (1848–1925) or the Principia Mathematica of philosophers Alfred North Whitehead (1861–1947) and Bertrand Russell (1872–1970). Classical logic is most commonly described with a negation $¬$, conjunction $∧$, disjunction $∨$, and material implication $→$ (and existential and universal quantifiers $∃x$ and $∀x$ in the first-order case), often with one or more defined in terms of the others. As classical logic is functionally complete with respect to many sets of connectives, it can be described with many different combinations of connectives as primitive. Likewise, as the quantifiers may be expressed by means of the indefinite description operator $ε$, the epsilon calculus, too, is frequently considered a form of classical logic.

The boundaries of when a deductive system may be described as ‘classical’ are largely a matter of convention and are difficult to pin down. For example, two frequent means of expanding the classical base of first-order logic are the addition of novel quantifiers (e.g., the Keisler quantifier or second-order quantifiers) and the addition of modal operators (e.g., unary necessity operators). Although in many cases the enriched systems are conservative over classical logic, i.e., do not yield new theorems in the unenriched language, the addition of new quantifiers is generally regarded as ‘classical’ while modal logic is frequently thought of as ‘non-classical’.