Formal Proof
A Formal Proof 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.
- Context:
- It can be produced by a Theorem Proving Task.
- …
- Example(s):
- a Mathematical Proof.
- …
- Counter-Example(s):
- an Informal Proof.
- See: Proof, Proof Theory, Axiomatic System.
References
2013
- http://en.wikipedia.org/wiki/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.[1]
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.
- 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.[1]
- ↑ The Cambridge Dictionary of Philosophy, deduction
- http://www.proofwiki.org/wiki/Definition:Proof
- A proof is another name for a valid argument, but in this context the assumption is made that the premises are all true.
That is, a valid argument that has one or more false premises is not a proof.
Suppose $P$ is a proposition whose truth or falsehood is to be determined.
Constructing a valid argument upon a set of premises, all of which have previously been established as being true, is called proving $P$.
- A proof is another name for a valid argument, but in this context the assumption is made that the premises are all true.