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 interactive theorem proving (e.g., through the use of proof checker and automated theorem prover).[4] Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable, depending upon the formal system in use.
A formal language is a set of finite sequences of symbols. Such a language can be defined without reference to any 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.
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).
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.
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.
^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.
Part of a series of articles covering mathematics and logic.
Archive of Formal Proofs
Mizar Home Page
August 23, 2023
formal, proof, logic, mathematics, formal, proof, derivation, finite, sequence, sentences, called, well, formed, formulas, case, formal, language, each, which, axiom, assumption, follows, from, preceding, sentences, sequence, rule, inference, differs, from, na. In logic and mathematics a formal proof or derivation is a finite sequence of sentences called well formed formulas in the case of a formal language each of which is an axiom an assumption or follows from the preceding sentences in the sequence by a 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 not in general effective therefore there may be no method by which we can always find a 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 interactive theorem proving e g through the use of proof checker and automated theorem prover 4 Significantly these proofs can be checked automatically also by computer Checking formal proofs is usually simple while the problem of finding proofs automated theorem proving is usually computationally intractable and or only semi decidable depending upon the formal system in use Contents 1 Background 1 1 Formal language 1 2 Formal grammar 1 3 Formal systems 1 4 Interpretations 2 See also 3 References 4 External linksBackground EditFormal language Edit Main article Formal language A formal language is a set of finite sequences of symbols Such a language can be defined without reference to any 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 Edit Main articles Formal grammar and Formation rule 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 Edit Main article Formal system 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 Edit Main articles Formal semantics logic and Interpretation logic 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 See also EditAxiomatic system Formal verification Mathematical proof Proof assistant Proof calculus Proof theory Proof truth De Bruijn factorReferences Edit 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 Edit A Special Issue on Formal Proof Notices of the American Mathematical Society December 2008 2pix com Logic Part of a series of articles covering mathematics and logic Archive of Formal Proofs Mizar Home Page Retrieved from https en wikipedia org w index php title Formal proof amp oldid 1113095164, wikipedia, wiki, book, books, library,
