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# Symbolic logic

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Symbolic logic is an attempt to codify and formalize the processes of reasoning, or logic. Logical statements are expressed as symbolic strings, in a precise, compact and unambiguous notation, similar to notations used in mathematics. Axioms, statements which are accepted without proof, are identified, and the valid rules of argumentation (transformation rules) are defined.

There are many different systems of symbolic logic. A system of symbolic logic has a number of components: the set of valid sentences, called well-formed formulas (or wffs); transformation rules for deriving new formulas from one or more initial formulas; the set of axioms, which is a subset of the set of wffs. The sets of wffs and axioms can be finite or infinite, so long as they are [recursively enumerable]?; i.e. so long as there exists a procedure for determining whether any given sentence is a wff or axiom, which could be carried out in a finite number of steps by a device such as a Turing machine.

The theorems of the system are the axioms and all the wffs that can be derived from the axioms by a finite number of applications of the transformation rules.

Once we have defined a system of logic, we can then seek to prove statements in the system. A proof of a statement in the system is a sequence of axioms and applications of transformation rules, which shows how we can derive the statement. If a valid proof exists of a statement in a system, it follows that the statement is a theorem of the system.

Metalogic is the study of different systems of symbolic logic, and their properties. We can not only prove statements in a logical system; we can also prove statements about logical systems.

There are three main types of logical systems which are used: propositional calculus, predicate calculus, and modal logic.

Propositional calculus deals with the logic of individual sentences. The primary system here is the classical system, which is the system most commonly used. Various alternatives to the classical system have been studied, including:

- 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 predicate calculus weaker than the classical

Predicate calculus extends propositional calculus to deal with predication and quantification. Again we have a classical system, and various deviations from this system. Most of the different systems of propositional calculus can be extended in this manner to produce systems of predicate calculus. There are two main varieties of systems of predicate calculus: first-order predicate calculus, which allows quantification over objects and is the standard logic used in mathematics (see set theory); and higher-order predicate calculus, which permits in addition quantification over predicates and predication of predicates.

The term modal logic has two meanings. In a restricted sense, it means the logic of necessity and possibility. But more broadly, it also includes other logics with a similar structure: deontic logics (which deal with ideas of permission and obligation), temporal logics (which deal with the relationships of past, present and future between events), and doxastic logic (which deals with belief). What all these systems have in common is that they involve the application of unary operators (called modal operators) to statements of either propositional or predicate calculus. It is in this broader sense that the term is used here.

The various systems of modal logic include K, M, S4, S5 and B. The systems differ in having different axioms.

See also Goedels Incompleteness Theorem

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Last edited August 15, 2001 10:26 am (diff)
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