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# The difference between proofs, rules of inference and deductive reasoning watch

1. Hi everyone, regarding the foundations of maths, there are a few concepts that I'm having trouble distinguishing between. I'd be really grateful to hear what others think are the difference between:

Rules of inference
http://en.wikipedia.org/wiki/Rule_of_inference

Deductive reasoning
http://en.wikipedia.org/wiki/Deductive_reasoning

Proofs
http://en.wikipedia.org/wiki/Mathematical_proof

It seems to me that they are all pretty similar, except for the fact that a proof is something you arrive at after deductive reasoning and rules of inference. I'd be really grateful to hear what others think on the matter.

2. Yes, deductive reasoning is the application of the rule of inference to things we already know (or define, or assume, or whatever) to be true. We tend to call something a proof if it's a deduction of a stated claim from accepted axioms or previous theorems.
3. I don't know a great deal about logic but it seems like this highlights the difference:

(Original post by Deductive Reasoning)
An argument is valid when it is impossible for both its premises to be true and its conclusion to be false. An argument can be valid even though the premises are false.
(Original post by Rule of Inference)
A rule of inference needn't preserve any semantic property such as truth or validity. In fact, there is nothing requiring that a logic characterized purely syntactically have a semantics. A rule may preserve e.g. the property of being the conjunction of the subformula of the longest formula in the premise set. However in many systems, rules of inference are used to generate theorems from each other (i.e. to prove theorems).

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