Symbolic Logic

Symbolic logic is divided into propositional calculus, predicate calculus and modal logics.

Metalogic deals with the study of the properties of different logical systems. A logic system is generally defined as a language for expressing

well-formed formulas (wffs), and a set of transformation rules for deriving one formula from another. A subset of the set of all formulas is defined to be the set of axioms for that systems (this set may be finite or infinite, so long as it is recursively enumerable.) The set of theorems of the system

is the set of all axioms and all wffs that can be derived from the axioms by one or more applications of the transformation rules.

Once the system of logic can be defined, metalogical theorems can be derived concerning the properties of the system; and statements can also be derived

or tested within the system.

There are three main types of logical systems: propositional systems, predicate systems and modal systems.

Propositional calculus deals with the logic of individual sentences. There are a number of different systems of propositional calculus:

• classical -- the normal traditional system

• many-valued -- permits sentences to be more than just true or false, but also have intermediate truth values

• paraconsistent -- permits inconsistent sentences. Does not have ex contradictione quodlibet (from a contradiction anything follows)

• infinitary -- permits sentences to be infinitely long

• intuitionistic -- system of logic used in mathematical intuitionism

• relevant -- has only relevant implication

• substructural -- systems of logic weaker than classical logic

Predicate calculus deals with the logic of predication and quantification. Systems include:

• lower-order --

• higher-order -- permits quantification and predication of predicates

Modal logic -- also deontic logic, temporal logic, doxastic logic:

• various systems: K, M, S4, S5, B