Formal system

A formal system is an

axioms by a set of inference rules.[1]

In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.[2]

The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of

notation, for example, Paul Dirac's bra–ket notation
.

Concepts

A formal system has the following:[3][4][5]

A formal system is said to be

semidecidable sets
, respectively.

Formal language

A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:

• the syntax is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
• the semantics are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)

Usually only the

analytic grammars (or reductive grammar[6][7]
), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.

Deductive system

A deductive system, also called a deductive apparatus,

rules of inference that can be used to derive theorems of the system.[9]

Such deductive systems preserve

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any

intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a logical consequence of the lines that precede it. There should be no element of any interpretation
of the language that gets involved with the deductive nature of the system.

The logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.[clarification needed]

An example of a deductive system would be the rules of inference and axioms regarding equality used in first order logic.

The two main types of deductive systems are proof systems and formal semantics.[8]

Proof system

Formal proofs are sequences of well-formed formulas (or WFF for short) that might either be an axiom or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a theorem.

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a decision procedure for deciding whether a given WFF is a theorem or not.

The point of view that generating formal proofs is all there is to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

Formal semantics of logical system

A logical system is a deductive system (most commonly

model
of the logical system.

A logical system is:

• Sound, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
• Semantically complete, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.

An example of a logical system is

nonnegative integers and gives the symbols their usual meaning.[10] There are also non-standard models of arithmetic
.

History

Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In more recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.

foundational crisis of mathematics, that was eventually tempered by Gödel's incompleteness theorems.[2] The QED manifesto
represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.