Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well. They are characterised by a number of properties:
- Law of the excluded middle and Double negative elimination;
- Law of noncontradiction, and the principle of explosion;
- Monotonicity of entailment and Idempotency of entailment;
- Commutativity of conjunction;
- De Morgan duality: every logical operator is dual to another;
While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.
The intended semantics of classical logic is bivalent. With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.
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... a continuum of different intermediate logics ... Specific intermediate logics are often constructed by adding one or more axioms to intuitionistic logic, or by a semantical description ... Examples of intermediate logics include intuitionistic logic (IPC, Int, IL, H) classical logic (CPC, Cl, CL) IPC + p ∨ ¬p = IPC + ¬¬p → p = IPC + ((p ...
... In some logical calculi (notably, in classical logic) certain essentially different compound statements are logically equivalent ... Less trivial example of a redundancy is a classical equivalence between ¬P ∨ Q and P → Q ... Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one ...
... Computability logic is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal theory of truth integrates and extends classical ... Many-valued logic, including fuzzy logic, which rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1 ... Intuitionistic logic rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws Linear logic rejects idempotency of entailment as well Modal logic extends classical logic ...
... negation in classical logic ... For classical and intuitionistic logic, the "=" symbol means that corresponding implications "…→…" and "…←…" for logical compounds can be both proved as theorems, and the "≤" symbol ... Some of many-valued logics may have incompatible definitions of equivalence and order (entailment) ...
... truth with false and conjunction with disjunction In classical logic, with its intended semantics, the truth values are true (1 or T) and false (0 or ...
Famous quotes containing the words logic and/or classical:
“...some sort of false logic has crept into our schools, for the people whom I have seen doing housework or cooking know nothing of botany or chemistry, and the people who know botany and chemistry do not cook or sweep. The conclusion seems to be, if one knows chemistry she must not cook or do housework.”
—Ellen Henrietta Swallow Richards (18421911)
“Classical art, in a word, stands for form; romantic art for content. The romantic artist expects people to ask, What has he got to say? The classical artist expects them to ask, How does he say it?”
—R.G. (Robin George)