Page 93 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Unit 6: Formalized Symbolic Logics

Notes

This diagram shows the syntactic entities which may be constructed from formal languages. The
symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas.
A formal language is identical to the set of its well-formed formulas. The set of well-formed
formulas may be broadly divided into theorems and non-theorems.

6.1.2 Propositional Logic

A proposition is a sentence expressing something true or false. A proposition is identified
ontologically as an idea, concept or abstraction whose token instances are patterns of symbols,
marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also
truth bearers. A formal theory is a set of sentences in a formal language. A symbol is an idea,
abstraction or concept, tokens of which may be marks or a configuration of marks which form
a particular pattern. Symbols of a formal language need not be symbols of anything. For instance,
there are logical constants which do not refer to any idea, but rather serve as a form of punctuation
in the language (e.g. parentheses). A symbol or string of symbols may comprise a well-formed
formula if the formulation is consistent with the formation rules of the language. Symbols of a
formal language must be capable of being specified without any reference to any interpretation
of them.

Formal Language

A formal language is a syntactic entity which consists of a set of finite strings of symbols which
are its words (usually called its well-formed formulas). Which strings of symbols are words is
determined by fiat by the creator of the language, usually by specifying a set of formation rules.
Such a language can be defined without reference to any meanings of any of its expressions; it
can exist before any interpretation is assigned to it – that is, before it has any meaning. Formation
rules are a precise description of which strings of symbols are the well-formed formulas of a
formal language. It is synonymous with the set of strings over the alphabet of the formal
language which constitute well formed formulas. However, it does not describe their semantics
(i.e. what they mean).

Formal Systems

A formal system (also called a logical calculus, or a logical system) consists of a formal language
together with a deductive apparatus (also called a deductive system). The deductive apparatus
may consist of a set of transformation rules (also called inference rules) or a set of axioms, or
have both. A formal system is used to derive one expression from one or more other expressions.

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