# Systems of Logic

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:

• 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