Formal proof

In

follows from the preceding sentences in the sequence, according to the rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable.^[1] If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but there may be no method by which we can reliably find proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.^[2]^[3]

The theorem is a syntactic consequence of all the well-formed formulas 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 formulas in the proof sequence.

Formal proofs often are constructed with the help of computers in

semi-decidable

, depending upon the formal system in use.

Background

Formal language

A formal language is a

meanings of any of its expressions; it can exist before any interpretation

is assigned to it – that is, before it has any meaning. Formal proofs are expressed in some formal languages.

Formal grammar

A formal grammar (also called formation rules) is a precise description of 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.

Interpretations

An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called

formal semantics. Giving an interpretation is synonymous with constructing a model

.

References

^ Kassios, Yannis (February 20, 2009). "Formal Proof" (PDF). cs.utoronto.ca. Retrieved 2019-12-12.
^ The Cambridge Dictionary of Philosophy, deduction
^ Barwise, Jon; Etchemendy, John Etchemendy (1999). Language, Proof and Logic (1st ed.). Seven Bridges Press and CSLI.
^ Harrison, John (December 2008). "Formal Proof—Theory and Practice" (PDF). ams.org. Retrieved 2019-12-12.

External links

"A Special Issue on Formal Proof". Notices of the American Mathematical Society. December 2008.
2πix.com: Logic Part of a series of articles covering mathematics and logic.
Archive of Formal Proofs
Mizar Home Page
Pr∞fWiki, Definition:Proof System/Formal Proof

[1] Kassios, Yannis (February 20, 2009). "Formal Proof" (PDF). cs.utoronto.ca. Retrieved 2019-12-12.

[2] The Cambridge Dictionary of Philosophy, deduction

[3] Barwise, Jon; Etchemendy, John Etchemendy (1999). Language, Proof and Logic (1st ed.). Seven Bridges Press and CSLI.

[4] Harrison, John (December 2008). "Formal Proof—Theory and Practice" (PDF). ams.org. Retrieved 2019-12-12.

[1]

[2]

[3]

v t e ‌Logical truth ⊤
Functional:	Truth value Truth function ⊨ Tautology
Formal:	Theory Formal proof Theorem
Negation	⊥ False Contradiction Inconsistency