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, formal proof, proof":

... 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 axioms is valid ... 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 correct ...

**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 but different compositions (and ... 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 ...

**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 ...

**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 ...

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

“There is no better *proof* of a man’s being truly good than his desiring to be constantly under the observation of good men.”

—François, Duc De La Rochefoucauld (1613–1680)

“The manifestation of poetry in external life is *formal* perfection. True sentiment grows within, and art must represent internal phenomena externally.”

—Franz Grillparzer (1791–1872)