deductive system

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A formal system is an abstract structure 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 "
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". In 1921,
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proposed to use such a system as the foundation for the knowledge in
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
. A formal system may represent a well-defined . The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of
notation In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include p ...
, for example,
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.

# Background

Each formal system uses primitive
symbols A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meanin ...
(which collectively form an
alphabet An alphabet is a standardized set of basic written symbols A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic ...
) to finitely construct a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
from a set of
axiom An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek () 'that which is thought worthy or ...

s through inferential
rules of formation In formal language theory In mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formed ...
. 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 In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the me ...
consisting of rules to form formulas from simpler formulas. A formula is said to be
well-formed Well-formed or wellformed indicate syntactic correctness and may refer to: * Well-formedness, quality of linguistic elements that conform to grammar rules * Well-formed formula, a string that is generated by a formal grammar in logic * 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 schemaIn mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical prop ...
ta, consisting of well-formed formulas. # A set of
inference rules A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax In linguistics, syntax () is the set of rules, principles, and processes that govern the structu ...
. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.

## Recursive

A formal system is said to be
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...
(i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or recursively enumerable set, semidecidable sets, respectively.

## Inference and entailment

The 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.

### Formal language

A
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...
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 and linguistics 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 (mathematics), set of String (computer science), strings. 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 generated, 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. 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. In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, 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).

### Deductive system

A ''deductive system'', also called a ''deductive apparatus'' or a ''logic'', consists of the
axiom An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek () 'that which is thought worthy or ...

s (or
axiom schemaIn mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical prop ...
ta) and rules of inference that can be used to formal proof, derive theorems 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 reasoning, deductive qualities in the formula (mathematical logic), formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modal logic, modalities, such as Theory of justification, justification or belief 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 Mathematical proof, derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any Interpretation (logic), interpretation of the language that gets involved with the deductive nature of the system. An example of deductive system is First-order logic, first order predicate logic.

### Logical system

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 logic, first order predicate logic) together with additional (non-logical) axioms and a Semantics of logic, semantics. According to Model theory, model-theoretic interpretation (logic), 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 Soundness, 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 Completeness (logic), 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_axioms#First-order_theory_of_arithmetic, Peano arithmetic.

# History

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.

## Formalism

### Hilbert's program

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gr ...
instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems.

## QED manifesto

The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

# Examples

Examples of formal systems include: * Lambda calculus * Predicate calculus * Propositional calculus

# Variants

The following systems are variations of formal systems.

## Proof system

Formal proofs are sequences of well-formed formulas (or wff for short). For a wff to qualify as part of a proof, it might either be an
axiom An axiom, postulate, or assumption is a Statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek () 'that which is thought worthy or ...

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#Theorems in logic, theorem. The point of view that generating formal proofs is all there is to mathematics is often called ''Philosophy of mathematics#Formalism, formalism''.
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gr ...
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. 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 decidability (logic), 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 metatheorems.

* Formal method * Formal science * Rewriting system * Substitution instance * Theory (mathematical logic)