Editing Formal system (section)

===Deductive system===
{{Cleanup|date=October 2023|reason=This section needs better organization and more citations.|section}}
{{Main|Inference|Logical consequence|Deductive reasoning}}
A ''deductive system'', also called a ''deductive apparatus'',<ref name=":1">{{proofwiki reference|id=Definition:Deductive_Apparatus |title=Deductive Apparatus |access-date=30 November 2024}}</ref> consists of the [[Axiom#Role_in_mathematical_logic|axiom]]s (or [[axiom schema]]ta) and [[rules of inference]] that can be used to [[formal proof|derive]] [[theorem]]s of the system.{{sfn|Hunter|1996|p=7}}

Such deductive systems preserve [[deductive reasoning|deductive]] qualities in the [[formula (mathematical logic)|formula]]s 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 [[logical 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.

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]].{{clarify|reason=This section doesn't really do a group job stating what an entailment actually is.|date=September 2017}}

An example of a deductive system would be the rules of inference and [[First-order_logic#Equality_and_its_axioms|axioms regarding equality]] used in [[First-order logic|first order logic]].

The two main types of deductive systems are proof systems and formal semantics.<ref name=":1" /><ref>{{cite book|title=Formal Semantics and Logic|url=https://www.princeton.edu/~fraassen/books/pdfs/Formal%20Semantics%20and%20Logic.pdf |last=van Fraassen |first=Bas C. |author-link=Bas van Fraassen |year=2016 |orig-date=1971 |publisher=Nousoul Digital Publishers|page=12|quote=Metalogic can in turn be roughly divided into two parts: proof theory and formal semantics... The division is not exact; many questions have been dealt with from both points of view, and some proof-theoretic methods and results are indispensable in semantics.}}</ref>

==== Proof system ====
{{Main|Proof system|Formal proof}}
Formal proofs are sequences of [[well-formed formula]]s (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. The last WFF in the sequence is recognized as a [[Theorem#Theorems in logic|theorem]].

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 point of view that generating formal proofs is all there is to mathematics is often called ''[[Formalism (philosophy of mathematics)|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 [[metatheorem]]s.

==== Formal semantics of logical system ====
{{main|Semantics of logic|Interpretation (logic)|Model theory}}

A ''logical system'' is a deductive system (most commonly [[First-order logic|first order logic]]) together with additional [[non-logical axioms]]. According to [[model theory]], a logical system may be given [[interpretation (logic)|interpretation]]s which describe whether a given [[Structure (mathematical logic)|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 (model theory)|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.
*[[Completeness (logic)#Semantic completeness|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 integer]]s and gives the symbols their usual meaning.<ref>{{cite book |last1=Kaye |first1=Richard |title=Models of Peano arithmetic |date=1991 |publisher=Clarendon Press |location=Oxford |isbn=9780198532132 |page=10 |chapter=1. The Standard Model}}</ref> There are also [[non-standard model of arithmetic|non-standard models of arithmetic]].