Symbols A symbol is an
idea,
abstraction or
concept,
tokens of which may be marks or a metalanguage 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 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 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).
Propositions 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
truthbearers.
Formal theories A
formal theory is a
set of
sentences in a
formal language.
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. Formal systems, like other syntactic entities may be defined without any
interpretation given to it (as being, for instance, a system of arithmetic).
Syntactic consequence within a formal system A formula A is a
syntactic consequence within some formal system \mathcal{FS} of a set Г of formulas if there is a
derivation in
formal system \mathcal{FS} of A from the set Г. :\Gamma \vdash_{\mathrm FS} A Syntactic consequence does not depend on any
interpretation of the formal system.
Syntactic completeness of a formal system A formal system \mathcal{S} is
syntactically complete (also
deductively complete,
maximally complete,
negation complete or simply
complete) iff for each formula A of the language of the system either A or ¬A is a theorem of \mathcal{S}. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an
inconsistency. Truth-functional
propositional logic and first-order
predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not
tautologies).
Gödel's incompleteness theorem shows that no
recursive system that is sufficiently powerful, such as the
Peano axioms, can be both consistent and complete.
Interpretations An
interpretation of a formal system is the assignment of meanings to the symbols, and
truth values to the sentences of a formal system. The study of interpretations is called
formal semantics.
Giving an interpretation is synonymous with
constructing a model. An interpretation is expressed in a
metalanguage, which may itself be a formal language, and as such itself is a syntactic entity. ==See also==