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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.

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.

**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