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{{short description|Study of the properties of logical systems}}
'''Metalogic''' is the [[metatheory]] of [[logic]]. Whereas ''logic'' studies how [[formal system|logical system]]s can be used to construct [[Validity (logic)|valid]] and [[soundness|sound]] [[argument]]s, metalogic studies the properties of [[Logical system|logical systems]].<ref>Harry Gensler, [https://books.google.com/books?id=jpteBwAAQBAJ Introduction to Logic], Routledge, 2001, p. 336.</ref> Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the [[formal language|languages]] and systems that are used to express truths.<ref name="metalogic">{{Hunter 1996}}</ref>

The basic objects of metalogical study are formal languages, formal systems, and their [[Interpretation (logic)|interpretations]]. The study of interpretation of formal systems is the branch of [[mathematical logic]]  that is known as [[model theory]], and the study of [[deductive system]]s is the branch that is known as [[proof theory]].

== Overview ==

=== Formal language ===
{{Main|Formal language}}

A ''formal language'' is an organized set of [[symbol (formal)|symbols]], the symbols of which precisely define it by shape and place. Such a language therefore can be defined without [[reference]] to the [[meaning (linguistics)|meanings]] of its expressions; it can exist before any [[Interpretation (logic)|interpretation]] is assigned to it—that is, before it has any meaning. [[First-order logic]] is expressed in some formal language. A [[formal grammar]] determines which symbols and sets of symbols are [[well-formed formula|formulas]] in a formal language.

A formal language can be formally defined as a set ''A'' of strings (finite sequences) on a fixed alphabet α. Some authors, including [[Rudolf Carnap]], define the language as the ordered pair <α, ''A''>.<ref name="itslaia">[[Rudolf Carnap]] (1958) ''[https://books.google.com/books?id=hAvVAgAAQBAJ Introduction to Symbolic Logic and its Applications]'', p.&nbsp;102.</ref> Carnap also requires that each element of α must occur in at least one string in ''A''.

=== Formation rules ===
{{Main|Formation rule}}

''Formation rules'' (also called ''formal grammar'') are a precise description of the [[well-formed formula]]s of a formal language. They are synonymous with the [[set (mathematics)|set]] of [[String (computer science)|strings]] over the [[alphabet]] of the formal language that constitute well formed formulas. However, it does not describe their [[semantics]] (i.e. what they mean).

=== Formal systems ===
{{Main|Formal system}}

A ''formal system'' (also called a ''logical calculus'', or a ''logical system'') consists of a formal language together with a [[Deductive system|deductive apparatus]] (also called a ''deductive system''). The deductive apparatus may consist of a set of [[Rule of inference|transformation rule]]s (also called ''inference rules'') or a set of [[axiom]]s, or have both. A formal system is used to [[Proof theory|derive]] one expression from one or more other expressions.

A ''formal system'' can be formally defined as an ordered triple <α,<math>\mathcal{I}</math>,<math>\mathcal{D}</math>d>, where <math>\mathcal{D}</math>d is the relation of direct derivability. This relation is understood in a comprehensive [[Sense and reference|sense]] such that the primitive sentences of the formal system are taken as directly [[formal proof|derivable]] from the [[empty set]] of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are so chosen that every first place member of <math>\mathcal{D}</math>d is a member of <math>\mathcal{I}</math> and every second place member is a finite subset of <math>\mathcal{I}</math>.

A ''formal system'' can also be defined with only the relation <math>\mathcal{D}</math>d.   Thereby can be omitted <math>\mathcal{I}</math>  and α in the definitions of ''interpreted formal language'', and ''interpreted formal system''. However, this method can be more difficult to understand and use.<ref name = "itslaia"/>

=== Formal proofs ===
{{Main|Formal proof}}

A ''formal proof'' is a sequence of well-formed formulas of a formal language, the last of which is a [[theorem]] of a formal system. The theorem is a [[Logical consequence|syntactic consequence]] of all the well formed formulae that precede it in the proof system. For a well formed formula to qualify as part of a proof, it must result from applying a rule of the deductive apparatus of some formal system to the previous well formed formulae in the proof sequence.

=== Interpretations ===
{{Main|Interpretation (logic)|Formal semantics (logic)}}

An ''interpretation'' of a formal system is the assignment of meanings to the symbols and [[Truth value|truth-value]]s to the sentences of the formal system. The study of interpretations is called [[Formal semantics (logic)|Formal semantics]]. ''Giving an interpretation'' is synonymous with ''constructing a [[Structure (mathematical logic)|model]]''.

== Important distinctions ==

=== Metalanguage–object language ===

{{Main|Metalanguage}}

In metalogic, formal languages are sometimes called ''object languages''. The language used to make statements about an object language is called a ''metalanguage''. This distinction is a key difference between logic and metalogic. While logic deals with ''proofs in a formal system'', expressed in some formal language, metalogic deals with ''proofs about a formal system'' which are expressed in a metalanguage about some object language.

=== Syntax–semantics ===
{{Main|Syntax (logic)|Formal semantics (logic)}}

In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof-theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'.

=== Use–mention ===
{{Main|Use–mention distinction}}

In metalogic, the words ''use'' and ''mention'', in both their noun and verb forms, take on a technical sense in order to identify an important distinction.<ref name="metalogic"/> The ''use–mention distinction'' (sometimes referred to as the ''words-as-words distinction'') is the distinction between ''using'' a word (or phrase) and ''mentioning'' it. Usually it is indicated that an expression is being mentioned rather than used by enclosing it in quotation marks, printing it in italics, or setting the expression by itself on a line. The enclosing in quotes of an expression gives us the [[name]] of an expression, for example:

{{poemquote|"Metalogic" is the name of this article.
This article is about metalogic.}}

=== Type–token ===
{{Main|Type–token distinction}}

The ''type-token distinction'' is a distinction in metalogic, that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the [[Type–token distinction|type]] of thing known as "The bicycle." Whereas, the bicycle in your garage is in a particular place at a particular time, that is not true of "the bicycle" as used in the sentence: "''The bicycle'' has become more popular recently." This distinction is used to clarify the meaning of [[symbol (formal)|symbols]] of [[formal language]]s.

== History ==
Metalogical questions have been asked since the time of [[Aristotle]].<ref>{{Citation |last=Smith |first=Robin |title=Aristotle's Logic |date=2022 |url=https://plato.stanford.edu/archives/win2022/entries/aristotle-logic/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=2023-08-28 |edition=Winter 2022 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref> However, it was only with the rise of formal languages in the late 19th and early 20th century that investigations into the foundations of logic began to flourish. In 1904, [[David Hilbert]] observed that in investigating the [[foundations of mathematics]] that logical notions are presupposed, and therefore a simultaneous account of metalogical and [[metamathematics|metamathematical]] principles was required. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by [[mathematical logic]] in academia. A possible alternate, less mathematical model may be found in the writings of [[Charles Sanders Peirce]] and other [[semiotics|semioticians]].

== Results ==
{{Prose|date=May 2024|1=section}}
Results in metalogic consist of such things as [[formal proof]]s demonstrating the [[consistency]], [[completeness (logic)|completeness]], and [[Decidability (logic)|decidability]] of particular [[formal system]]s.

Major results in metalogic include:

* Proof of the [[uncountability]] of the [[power set]] of the [[natural number]]s ([[Cantor's theorem]] 1891)
* [[Löwenheim–Skolem theorem]] ([[Leopold Löwenheim]] 1915 and [[Thoralf Skolem]] 1919)
* Proof of the consistency of truth-functional [[Propositional calculus|propositional logic]] ([[Emil Leon Post|Emil Post]] 1920)
* Proof of the semantic completeness of truth-functional propositional logic ([[Paul Bernays]] 1918),<ref name="reflections">Hao Wang, [https://books.google.com/books?id=wLLePwhDOMYC Reflections on Kurt Gödel]</ref> (Emil Post 1920)<ref name="metalogic"/>
* Proof of the syntactic completeness of truth-functional propositional logic (Emil Post 1920)<ref name="metalogic"/>
* Proof of the decidability of truth-functional propositional logic (Emil Post 1920)<ref name="metalogic"/>
* Proof of the consistency of first-order [[Monadic predicate calculus|monadic predicate logic]] ([[Leopold Löwenheim]] 1915)
* Proof of the semantic completeness of first-order monadic predicate logic (Leopold Löwenheim 1915)
* Proof of the decidability of first-order monadic predicate logic (Leopold Löwenheim 1915)
* Proof of the consistency of first-order predicate logic ([[David Hilbert]] and [[Wilhelm Ackermann]] 1928)
* Proof of the semantic completeness of first-order [[predicate logic]] ([[Gödel's completeness theorem]] 1930)
* Proof of the [[cut-elimination theorem]] for the [[sequent calculus]] ([[Gerhard Gentzen|Gentzen]]'s ''Hauptsatz'' 1934)
* Proof of the undecidability of first-order predicate logic ([[Entscheidungsproblem|Church's theorem]] 1936)
* [[Gödel's incompleteness theorems#First incompleteness theorem|Gödel's first incompleteness theorem]] 1931
* [[Gödel's incompleteness theorems#Second incompleteness theorem|Gödel's second incompleteness theorem]] 1931
* [[Tarski's undefinability theorem]] (Gödel and Tarski in the 1930s)

== See also ==
{{Portal|Philosophy}}
* [[Metalogic programming]]
* [[Metamathematics]]

==References==
<references/>

==External links==
*{{Commonscat-inline}}
*{{SpringerEOM
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 |author-last1=Dragalin
 |author-first1=A.G.
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{{Metalogic}}

[[Category:Metalogic| ]]
[[Category:Formal logic]]
[[Category:Metaphilosophy]]