that 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 can be thought of as identical to the set of its well-formed formulas, which may be broadly divided into
theorems and non-theorems. A formal system has the following components, as a minimum: •
Formal language, which is a set of
well-formed formulas, which are strings of
symbols from an
alphabet, formed by a
formal grammar (consisting of
production rules or
formation rules). • Deductive system, deductive apparatus, or
proof system, which has
rules of inference that take
axioms and infers
theorems, both of which are part of the formal language. • In some cases an
inductive system, used to derive a proof by first establishing a simple case, then generalizing it. A formal system is said to be
recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are
decidable sets or
semidecidable sets, respectively.
Formal language A
formal language is a language that uses a set of strings whose symbols are taken from a specific alphabet, and operations used to form sentences from them. Like languages in
linguistics, formal languages generally have two aspects: • the
syntax is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) • the
semantics are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question) Usually only the
syntax of a formal language is considered via the notion of a
formal grammar. The two main categories of formal grammar are that of
generative grammars, which are sets of rules for how strings in a language can be written, and that of
analytic grammars (or reductive grammar), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.
Deductive system A
deductive system, also called a
deductive apparatus, consists of the
axioms (or
axiom schemata) and
rules of inference that can be used to
derive theorems of the system. In order to sustain its deductive integrity, a
deductive apparatus must be definable without reference to any
intended interpretation of the language. The aim is to ensure that each line of a
derivation is merely a
logical consequence of the lines that precede it. There should be no element of any
interpretation of the language that gets involved with the deductive nature of the system. The
logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger
theory or field (e.g.
Euclidean geometry) consistent with the usage in modern mathematics such as
model theory. An example of a deductive system would be the rules of inference and
axioms regarding equality used in
first order logic. The two main types of deductive systems are proof systems and formal semantics.
Proof system Formal proofs are sequences of
well-formed formulas (or WFF for short) that might either be an
axiom or be the product of applying an inference rule on previous WFFs in the proof sequence. Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a
decision procedure for deciding whether a given WFF is a theorem or not. The point of view that generating formal proofs is all there is to mathematics is often called
formalism.
David Hilbert founded
metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a
metalanguage. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the
object language, that is, the object of the discussion in question. The notion of
theorem just defined should not be confused with
theorems about the formal system, which, in order to avoid confusion, are usually called
metatheorems.
Formal semantics of logical system A
logical system is a deductive system (most commonly
first order logic) together with additional
non-logical axioms. According to
model theory, a logical system may be given
interpretations which describe whether a given
structure - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a
model of the logical system. A logical system is: •
Sound, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system. •
Semantically complete, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms. An example of a logical system is
Peano arithmetic. The standard model of arithmetic sets the
domain of discourse to be the
nonnegative integers and gives the symbols their usual meaning. There are also
non-standard models of arithmetic. ==History==