A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A formal system is essentially an "axiomatic system
In 1921, David Hilbert
proposed to use such system as the foundation for the knowledge in mathematics
. A formal system may represent a well-defined system of abstract thought
The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation
, for example, Paul Dirac
's bra–ket notation
Each formal system uses primitive symbols
(which collectively form an alphabet
) to finitely construct a formal language
from a set of axiom
s through inferential rules of formation
The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.
More formally, this can be expressed as the following:
# A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
# A grammar
consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed
if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
# A set of axioms, or axiom schema
ta, consisting of well-formed formulas.
# A set of inference rules
. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.
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 set
s or semidecidable sets
Inference and 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
A formal language
is a language that is defined by a formal system. Like languages in linguistics
, formal languages generally have two aspects:
* the syntax
of a language is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) studied in formal language theory
* the semantics
of a language are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
In computer science
usually only the syntax of a formal language is considered via the notion of a formal grammar
. A formal grammar is a precise description of the syntax of a formal language: a set
. The two main categories of formal grammar are that of generative grammar
s, which are sets of rules for how strings in a language can be generated, and that of analytic grammar
s (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. In short, an analytic grammar describes how to ''recognize'' when strings are members in the set, whereas a generative grammar describes how to ''write'' only those strings in the set.
, a formal language is usually not described by a formal grammar but by (a) natural language, such as English. Logical systems are defined by both a deductive system and natural language. Deductive systems in turn are only defined by natural language (see below).
A ''deductive system'', also called a ''deductive apparatus'' or a ''logic'', consists of the axiom
s (or axiom schema
ta) and rules of inference
that can be used to derive theorem
s of the system.
[Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971]
Such deductive systems preserve deductive
qualities in the formula
s that are expressed in the system. Usually the quality we are concerned with is truth
as opposed to falsehood. However, other modalities
, such as justification
may be preserved instead.
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 syntactic 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.
An example of deductive system is first order predicate logic
A ''logical system'' or ''language'' (not be confused with the kind of "formal language" discussed above which is described by a formal grammar), is a deductive system (see section above; most commonly first order predicate logic
) together with additional (non-logical) axioms and a semantics
. According to model-theoretic interpretation
, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. 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. Conversely, a logic system is 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
Early logic systems includes Indian logic of Pāṇini
, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long
(c. 325–250 BCE) . In more recent times, contributors include George Boole
, Augustus De Morgan
, and Gottlob Frege
. Mathematical logic
was developed in 19th century Europe
instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems
The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.
Examples of formal systems include:
* Lambda calculus
* Predicate calculus
* Propositional calculus
The following systems are variations of formal systems.
Formal proofs are sequences of well-formed formula
s (or wff for short). For a wff to qualify as part of a proof, it might either be an axiom
or be the product of applying an inference rule on previous wffs in the proof sequence. The last wff in the sequence is recognized as a theorem
The point of view that generating formal proofs is all there is to mathematics is often called ''formalism
''. David Hilbert
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.
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 notion of ''theorem'' just defined should not be confused with ''theorems about the formal system'', which, in order to avoid confusion, are usually called metatheorem
* Formal method
* Formal science
* Rewriting system
* Substitution instance
* Theory (mathematical logic)
* Raymond M. Smullyan
, 1961. ''Theory of Formal Systems: Annals of Mathematics Studies'', Princeton University Press (April 1, 1961) 156 pages
* Stephen Cole Kleene
, 1967. ''Mathematical Logic'' Reprinted by Dover, 2002.
* Douglas Hofstadter
, 1979. ''Gödel, Escher, Bach: An Eternal Golden Braid
'' . 777 pages.
* Encyclopædia BritannicaFormal system
Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.
* Peter Suber
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Category:4th century BC in India