Metalogic

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their 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 systems is the branch that is known as proof theory. A formal language is an organized set of symbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings 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 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>. Carnap also requires that each element of α must occur in at least one string in A. Formation rules (also called formal grammar) are a precise description of the well-formed formulas of a formal language. They are synonymous with the set of 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). 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 axioms, 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, I {displaystyle {mathcal {I}}} , D {displaystyle {mathcal {D}}} d>, where D {displaystyle {mathcal {D}}} 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 so chosen that every first place member of D {displaystyle {mathcal {D}}} d is a member of I {displaystyle {mathcal {I}}} and every second place member is a finite subset of I {displaystyle {mathcal {I}}} . A formal system can also be defined with only the relation D {displaystyle {mathcal {D}}} d. Thereby can be omitted I {displaystyle {mathcal {I}}} and α in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and use. 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 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.