Metalogic
Encyclopedia
Metalogic is the study of the metatheory
of logic
. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns itself with the theory of "sentences used to express truths of logic."
The basic objects of study in metalogic are formal language
s, formal system
s, and their interpretations
. The study of interpretation of formal systems is the branch of mathematical logic
known as model theory
, while the study of deductive apparatus is the branch known as proof theory
.
. 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 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.
of an expression, for example:
of formal language
s.
the essential feature of which is that it can
be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference
to any meanings
of any of its expressions; it can exist before any 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 formulas in a formal language.
A formal language can be defined formally as a set A of strings (finite sequences) on a fixed alphabet α. Some authors, including Carnap, define the language as the ordered pair <α, A>. Carnap also requires that each element of α must occur in at least one string in A.
s of a formal language. It is synonymous with the set of strings
over the alphabet
of the formal language which constitute well formed formulas. However, it does not describe their semantics
(i.e. what they mean).
s, or have both. A formal system is used to derive
one expression from one or more other expressions.
A formal system can be formally defined as an ordered triple <α,
,
d>, where 
d is the relation of direct derivability. This relation is understood in a comprehensive sense
such that the primitive sentences of the formal system are taken as directly 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 laid down in such a way that every first place member of
d is a member of 
and every second place member is a finite subset of 
.
It is also possible to define a formal system using only the relation
d. In this way we can omit 
, and α in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and work with.
of a formal system. The theorem is a syntactic consequence of all the well formed formulae preceding it in the proof. For a well formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous well formed formulae in the proof sequence.
.
s demonstrating the consistency
, completeness
, and decidability
of particular formal system
s.
Major results in metalogic include:
Metatheory
A metatheory or meta-theory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called metatheorems....
of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns itself with the theory of "sentences used to express truths of logic."
The basic objects of study in metalogic are formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s, formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s, and their interpretations
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
. The study of interpretation of formal systems is the branch of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
known as model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, while the study of deductive apparatus is the branch known as proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
.
History
Metalogical questions have been asked since the time of AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
. 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
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
observed that in investigating the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
that logical notions are presupposed, and therefore a simultaneous account of metalogical and metamathematical
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
principles was required. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
in academia.
Metalanguage–Object language
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
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
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. 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 nameName
A name is a word or term used for identification. Names can identify a class or category of things, or a single thing, either uniquely, or within a given context. A personal name identifies a specific unique and identifiable individual person, and may or may not include a middle name...
of an expression, for example:
- 'Metalogic' is the name of this article.
- This article is about metalogic.
Type–token
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 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 symbolsSymbol (formal)
For other uses see Symbol In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, tokens of which may be marks or a configuration of marks which form a particular pattern...
of formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s.
Formal language
A formal language is an organized set of symbolsSymbol (formal)
For other uses see Symbol In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, tokens of which may be marks or a configuration of marks which form a particular pattern...
the essential feature of which is that it can
be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference
Reference
Reference is derived from Middle English referren, from Middle French rèférer, from Latin referre, "to carry back", formed from the prefix re- and ferre, "to bear"...
to any meanings
Meaning (linguistics)
In linguistics, meaning is what is expressed by the writer or speaker, and what is conveyed to the reader or listener, provided that they talk about the same thing . In other words if the object and the name of the object and the concepts in their head are the same...
of any of its expressions; it can exist before any interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some 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 formulas in a formal language.
A formal language can be defined formally as a set A of strings (finite sequences) on a fixed alphabet α. Some authors, including Carnap, define the language as the ordered pair <α, A>. Carnap also requires that each element of α must occur in at least one string in A.
Formation rules
Formation rules (also called formal grammar) are a precise description of the well-formed formulaWell-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
s of a formal language. It is synonymous with the set of strings
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
over the alphabet
Alphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...
of the formal language which constitute well formed formulas. However, it does not describe their semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
(i.e. what they mean).
Formal systems
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axiomAxiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s, or have both. A formal system is used to derive
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
one expression from one or more other expressions.
A formal system can be formally defined as an ordered triple <α,
,d>, where d is the relation of direct derivability. This relation is understood in a comprehensive senseSense and reference
Sinn and bedeutung are usually translated, respectively, as sense and reference. Two different aspects of some terms' meanings, a term's reference is the object that the term refers to, while the term's sense is the way that the term refers to that object.Sinn and bedeutung were introduced by...
such that the primitive sentences of the formal system are taken as directly derivable
Formal proof
A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...
from the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are laid down in such a way that every first place member of
d is a member of and every second place member is a finite subset of .It is also possible to define a formal system using only the relation
d. In this way we can omit , and α in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and work with.Formal proofs
A formal proof is a sequence of well-formed formulas of a formal language, the last one of which is a theoremTheorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
of a formal system. The theorem is a syntactic consequence of all the well formed formulae preceding it in the proof. For a well formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous well formed formulae in the proof sequence.
Interpretations
An interpretation of a formal system is the assignment of meanings, to the symbols, and truth-values to the sentences of the formal system. The study of interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a modelStructure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
.
Results in metalogic
Results in metalogic consist of such things as formal proofFormal proof
A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...
s demonstrating the consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
, completeness
Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
, and decidability
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...
of particular formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s.
Major results in metalogic include:
- Proof of the uncountability of the set of all subsets of the set of natural numbers (Cantor's theoremCantor's theoremIn elementary set theory, Cantor's theorem states that, for any set A, the set of all subsets of A has a strictly greater cardinality than A itself...
1891) - Löwenheim-Skolem theorem (Leopold LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
1915 and Thoralf SkolemThoralf SkolemThoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
1919) - Proof of the consistency of truth-functional propositional logicPropositional calculusIn mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
(Emil Post 1920) - Proof of the semantic completeness of truth-functional propositional logic (Paul BernaysPaul BernaysPaul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...
1918), (Emil Post 1920) - Proof of the syntactic completeness of truth-functional propositional logic (Emil Post 1920)
- Proof of the decidability of truth-functional propositional logic (Emil Post 1920)
- Proof of the consistency of first order monadic predicate logic (Leopold LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
1915) - Proof of the semantic completeness of first order monadic predicate logic (Leopold LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
1915) - Proof of the decidability of first order monadic predicate logic (Leopold LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
1915) - Proof of the consistency of first order predicate logic (David HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
and Wilhelm AckermannWilhelm AckermannWilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
1928) - Proof of the semantic completeness of first order predicate logicPredicate logicIn mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
(Gödel's completeness theoremGödel's completeness theoremGödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
1930) - Proof of the undecidability of first order predicate logic (Church's theorem 1936)
- Gödel's first incompleteness theorem 1931
- Gödel's second incompleteness theorem 1931
- Tarski's undefinability theorem (Gödel and Tarski in the 1930s)