# Formal Proof

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 or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a 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 concept of natural deduction is a generalization of the concept of proof.

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 formulae in the proof sequence.

Formal proofs often are constructed with the help of computers in interactive theorem proving. 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.

### Other articles related to "formal proof, formal, proof":

Formal Proof - Background - 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 ...
Prices Of Production - Facts and Logic - Lack of A Formal Proof
... But this manoeuvre of itself cannot contain any formal proof of a necessary quantitative relationship between values and prices, nor a formal proof that capitals of the same size ... In that case, there is again no formal proof of any necessary relationship between values and prices, and Marx's manuscript really seems an endless, pointless ...
De Morgan's Laws - Formal Proof
... First we prove ¬(p ∨ q) ⇔ (¬p) ∧ (¬q) ... p q p ∨ q ¬(p ∨ q) ¬p ¬q (¬p) ∧ (¬q) 0 ... Since the values in the 4th and last columns are the same for all rows (which cover all possible truth value assignments to the variables), we can conclude that the two expressions are logically equivalent ...
Bew - Background
... Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a formal proof from a finite set of ... This task, known as automatic proof verification, is closely related to automated theorem proving ... The difference is that instead of constructing a new proof, the proof verifier simply checks that a provided formal proof (or, in instructions that can be followed to create a formal proof) is ...

### Famous quotes containing the words proof and/or formal:

There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.
Herman Melville (1819–1891)

It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.
Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)