> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/logic-types/_logics-by-purpose.md).

# logics-by-purpose

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.

## Logics by purpose

**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.

## Logics by presentation

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.

## Logics by properties

**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*.