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I'm curious about what math (mathematical logic, metamath) says about the way we reason. This is going to be a vague question because I have not yet explored mathematical logic myself. My question: can we somehow develop, or construct, all of the valid rules of inference ways we reason about proofs (by contradiction, implication, and so forth) in some mathematical way? Can it be done from some simple principles?

Like, if you had some sort of model, and these ways of reasoning, as objects in that model, that could be derived. What would be the model?

user73738
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  • Do you mean philosophically, mathematically or both? – Lays Apr 22 '13 at 04:33
  • Well, preferably in a mathematical way. Like, if you had some sort of model, and these ways of reasoning, as objects in that model, could be derived. What would be the model? – user73738 Apr 22 '13 at 04:41
  • Here does this link help or maybe this and possibly this? – Lays Apr 22 '13 at 04:47
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    This question is quite vague. What do you mean by "reason about proofs"? – Qiaochu Yuan Apr 22 '13 at 04:48
  • I meant the rules of inference. The way we infer new truths from old ones. Can we derive the rules of inference? I guess "reason about proofs" didn't really make sense. – user73738 Apr 22 '13 at 04:51
  • Take a look at http://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem I suspect this is what you are interested on. It is worth saying that the completeness theorem fails in second-order logic. – boumol Apr 22 '13 at 06:25
  • @boumol what does Godel incompleteness theorem essentially say? – Lays Apr 22 '13 at 07:20
  • @Lays: Have you read the section "Statement of the theorem" in the wikipedia link? – boumol Apr 22 '13 at 07:25
  • @boumol yes, but my math skills aren't up to par to understand that part. – Lays Apr 22 '13 at 07:28
  • @Lays: Godel's theorem says what it says, so if you cannot follow wikipedia link I am afraid you need to improve your background. I suggest you take a deep look at http://plato.stanford.edu/entries/logic-classical/ to digest what completeness is. This entry is addressed to philosophers, so I suspect it will be easier to understand. – boumol Apr 22 '13 at 07:54
  • Formal logic can indeed be reduced to relatively small number of "self-evident" rules. I doubt that these rules can be derived from any other formal system, however. As a starting point, may I humbly suggest may tutorial/software available at http://www.dcproof.com. For a screen print of the logic menu, click on Features there. – Dan Christensen Apr 22 '13 at 15:33
  • @user73738: The thread I marked this as duplicate of may seem unrelated on the surface, but is actually a sketch of an answer to the core of your question, which is how we build things up from scratch and what philosophical assumptions we are making in the process. Also, Dan's website is useless for actual mathematical work, and moreover has some bogus articles. – user21820 Jun 07 '22 at 17:35

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Regarding what mathematical logic & metamathematics say about the way we reason, I think that you can see ML as a mathematical model of reasoning in Mathematics : Proof Theory, Model Theory, Computability Theory, all are about ways the mathematician (an idealize one) makes proofs and computing. All these activities are performed with language: so ML has to start with some idealized models of language (formal systems for First Order Calculus, and so on); those model are very much simplified, but they are very useful.

Mauro ALLEGRANZA
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