logics-by-purpose
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Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of an abstract rule, a rule that is not about any particular thing or property. In many definitions of logic, logical inference and inference with purely formal content are the same.
A formal system is an organization of terms used for the analysis of deduction.
CS has seen a multitude of logics suited for different purposes.
Modal logic is used for reasoning about concepts like possibility or necessity, extending the classical logic with non-truth-functional (modal) operators.
Probabilistic logic has the truth values of formulas as probabilities.
Temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that whenever a request is made, access to a resource is eventually granted, but it is never granted to two requestors simultaneously.
Moreover, these logics come in different flavors, usually admitting propositional, first-order, higher-order and intuitionistic presentations, as well as combinations of these and many ad hoc variants.
Non-classical logics are formal systems that significantly differ from standard logical (propositional and predicate) systems by way of extensions, deviations and variations, with the aim of constructing different models of logical consequence and logical truth.
Computability logic is a semantically constructed formal theory of computability (as opposed to classical logic, which is a formal theory of truth), integrating and extending classical, linear and intuitionistic logics.
Many-valued logics rejects bivalence, allowing for truth values other than just binary true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic, which deals with approximate concepts.
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that don't recognize the law of excluded middle, which is fundamental concept in classical logic.
Paraconsistent logics reject the principle of explosion.
Linear logic rejects principle of explosion, idempotency of entailment (contraction), monotonicity of entailment (weakening).
Relevance logic rejects monotonicity of entailment (weakening).
Non-monotonic logic rejects monotonicity of entailment (weakening). Nonmonotonic logic takes the stand that an established piece of knowledge may have to be retracted if additional facts are later known.
Non-reflexive (Schrödinger) logic rejects or restricts the law of identity.