Metalogic is themetatheory oflogic. Whereaslogic studies howlogical systems can be used to constructvalid andsoundarguments, metalogic studies the properties oflogical systems.[1] Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derivedabout thelanguages and systems that are used to express truths.[2]
The basic objects of metalogical study are formal languages, formal systems, and theirinterpretations. The study of interpretation of formal systems is the branch ofmathematical logic that is known asmodel theory, and the study ofdeductive systems is the branch that is known asproof theory.
Aformal language is an organized set ofsymbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined withoutreference to themeanings of its expressions; it can exist before anyinterpretation is assigned to it—that is, before it has any meaning.First-order logic is expressed in some formal language. Aformal grammar determines which symbols and sets of symbols areformulas in a formal language.
A formal language can be formally defined as a setA of strings (finite sequences) on a fixed alphabet α. Some authors, includingRudolf Carnap, define the language as the ordered pair <α,A>.[3] Carnap also requires that each element of α must occur in at least one string inA.
Formation rules (also calledformal grammar) are a precise description of thewell-formed formulas of a formal language. They are synonymous with theset ofstrings over thealphabet of the formal language that constitute well formed formulas. However, it does not describe theirsemantics (i.e. what they mean).
Aformal system (also called alogical calculus, or alogical system) consists of a formal language together with adeductive apparatus (also called adeductive system). The deductive apparatus may consist of a set oftransformation rules (also calledinference rules) or a set ofaxioms, or have both. A formal system is used toderive one expression from one or more other expressions.
Aformal system can be formally defined as an ordered triple <α,,d>, whered is the relation of direct derivability. This relation is understood in a comprehensivesense such that the primitive sentences of the formal system are taken as directlyderivable from theempty 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 ofd is a member of and every second place member is a finite subset of.
Aformal system can also be defined with only the relationd. Thereby can be omitted and α in the definitions ofinterpreted formal language, andinterpreted formal system. However, this method can be more difficult to understand and use.[3]
Aformal proof is a sequence of well-formed formulas of a formal language, the last of which is atheorem of a formal system. The theorem is asyntactic 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.
Aninterpretation of a formal system is the assignment of meanings to the symbols andtruth-values to the sentences of the formal system. The study of interpretations is calledFormal semantics.Giving an interpretation is synonymous withconstructing amodel.
In metalogic, formal languages are sometimes calledobject languages. The language used to make statements about an object language is called ametalanguage. This distinction is a key difference between logic and metalogic. While logic deals withproofs in a formal system, expressed in some formal language, metalogic deals withproofs about a formal system which are expressed in a metalanguage about some object language.
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'.
In metalogic, the wordsuse andmention, in both their noun and verb forms, take on a technical sense in order to identify an important distinction.[2] Theuse–mention distinction (sometimes referred to as thewords-as-words distinction) is the distinction betweenusing a word (or phrase) andmentioning 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 thename of an expression, for example:
"Metalogic" is the name of this article.
This article is about metalogic.
Thetype-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 thetype 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 ofsymbols offormal languages.
Metalogical questions have been asked since the time ofAristotle.[4] 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 thefoundations of mathematics that logical notions are presupposed, and therefore a simultaneous account of metalogical andmetamathematical principles was required. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed bymathematical logic in academia. A possible alternate, less mathematical model may be found in the writings ofCharles Sanders Peirce and othersemioticians.
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Results in metalogic primally involve demonstrating properties of particularformal systems. Key properties includeconsistency,completeness, anddecidability. Other results are important to metalogic more for providing a tool utilized frequently in proving subsequent results. For example, the proof theuncountability of thepower set of thenatural numbers (Cantor's theorem 1891) which introduced the technique ofdiagonalization.
Major completeness or incompleteness results include:
Major consistency results include:
Major decidability results include:
Other important meta logic results include:
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