A system of logic, also known as a logical calculus, or simply a logic, is a method by which to express and evaluate information in a logical manner.

Formal Language and Rules of Inference

Logical systems consist of a formal language of symbolic logic. This language defines:

A set of symbols to refer to formulae, including propositions and operators.

Grammar, that is rules of well-formation, on how formulae must be expressed.

The formal language of a system consists of, on one hand, the syntax of the language, and on the other, a method for expressing semantics within the system. The semantics of a system may be as simple as assigning truth-value to propositions and formulae, or more complicated, using predicate symbols to define non-logical relationships between formulae.

Systems also consist of rules of inference, which determine how expressions in the language may be used to draw new, previously unstated conclusions.

Common Systems of Logic

Classical Logics, the most common form of logical expression, including:

Aristotelian logic

Propositional logic

First-order logic

Second-order logic

Higher-order logics

Contextual Logics, which deal with non-truth-functional operators, and include:

Modal Logic, which deals with modal operators neccessarily and possibly.

Epistemic Logic, which reasons about knowledge

Doxastic Logic, which reasons about belief

Deontic Logic, which reasons about ethical obligation and permissibility

Temporal Logic, which reasons about propositions over time

Free Logic, which rejects the assumption that the domain is non-empty, that something exists

Fuzzy Logic, which rejects the law of the excluded middle

Intuitionistic Logic, which redefines truth values based on proof

Paraconsistent Logic, which allows contradictions without entailment of any other formulae

Relevance Logic, which requires a stronger link of relevance between premises and conclusion