Metalogic
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.
(1) 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."
(2)The basic objects of study in metalogic are
formal languages,
formal systems, 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.
History
Metalogical questions have been asked since the time of
Aristotle. 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.
Important distinctions in metalogic
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
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
symbols of
formal languages.
Overview
Formal language
A
formal language is an organized
set of
symbols 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
formal 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>.
(3) Carnap also requires that each element of α must occur in at least one string in
A.
Formal grammar
A
formal grammar (also called
formation rules) is a precise description of a the
well-formed formulas 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).
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
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,
Scri(tI
,
Scri(tD
d>, where
Scri(tD
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
Scri(tD
d is a member of
Scri(tI
and every second place member is a finite subset of
Scri(tI
.It is also possible to define a
formal system using only the relation
Scri(tD
d. In this way we can omit
Scri(tI
, 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 sequences of well-formed formulas of a formal language, the last one of which is a
theorem 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.
Formal interpretations
A
formal 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 formal interpretations is called
Formal semantics.
Giving an interpretation is synonymous with ''constructing a
model.
Results in metalogic
Results in metalogic consist of such things as
formal proofs demonstrating the
consistency,
completeness, and
decidability of particular
formal systems.Major results in metalogic include:
See also
References
-
[ Harry J. Gensler, Introduction to Logic, Routledge, 2001, p. 253.]
-
[Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971]
-
[Rudolf Carnap (1958) Introduction to Symbolic Logic and its Applications, p. 102.]
-
[Hao Wang, Reflections on Kurt Gödel]
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