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Can a useful, formal system of logic be constructed with no axioms or assumptions, or perhaps using only Occam's razor?

This may seem a silly and fun question, but what is the most complex proof that can be constructed in such a way?

Suppose nothing exists. Therefore supposition exists; a contradiction. Therefore something exists.

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    Natural Deduction has only rules and no axioms. – Mauro ALLEGRANZA Mar 28 '17 at 18:38
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    Occam's razor is an heuristic principle; do you think to use it as axiom ? – Mauro ALLEGRANZA Mar 28 '17 at 18:40
  • @MauroALLEGRANZA axiom: Occam's razor decides every decision. – samerivertwice Mar 28 '17 at 19:31
  • In your proof you have used a Proof by contradiction or Indirect Proof, i.e. a logical law or rule of inference. – Mauro ALLEGRANZA Mar 28 '17 at 19:35
  • Considering that Occam's razor is a selection principle, and it needs something to select from, no. And existence of a supposition does not follow from "suppose nothing exists" without a whole lot of other suppositions about assertions, figures of speech, etc. – Conifold Mar 28 '17 at 19:36
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This depends on the sense in which you mean the word 'useful' -- mathematicians try to make a study of the logical objects which are 'useful' or interesting in some sense, and pretty much any set-theoretic encoding (or type-theoretic encoding) of this information will require assumptions, which are identical to axioms from the standpoint of philosophy of set theory.

Perhaps this is not true by necessity, though. There is currently a development underway that is referred to as the 'reverse mathematics of second-order set theory', which seeks to understand what assumptions are necessary to prove certain well-known theorems in mathematics. This is a reversal of classical mathematics, in the sense that we are moving from a place of

'we believe this theorem is true in most reasonable contexts -- what is the bare minimum that we need to assume to prove it so',

as opposed to

'here is what we are assuming is true, what are the consequences'.

A development with minimal axiomatic use for maximum proof capability would probably fit somewhere into this hierarchy of assumption strength. I don't know how one would proceed with no assumptions, however -- even your suggestion of Occam's razor is ultimately still just an assumption.

  • I'm in favour of this development because all "facts" are in fact only facts based only upon the evidence used to deduce them. This is akin to saying, what is the evidence I must accept as true (axioms), in order to make such a thing (conclusion) true. As regards Occam's razor, sometimes we have apparently conflicting or competing facts. Ultimately I think Occam's razor is the fundamental underlying tool by which we select the facts we accept as true. If you look at your axiom selection... why did you not take as an axiom, "the flying spaghetti monster made it so". Occam's razor is the answer. – samerivertwice Jul 27 '17 at 9:23
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I like this question, because for me it relates to the argument or question about the nature of logic itself - the problem, for me, arises with the assumption that we need to construct a system of logic, in the first instance. This suggests we already have an idea of what kind of a system we would like to construct, and perhaps a system of logic that we "construct" would necessarily destroy its own epistemic status, as a consequence of our having "constructed" it, and this I believe is precisely what Wittgenstein was trying to show in his Tractatus. I think we are able to develop a fully coherent knowledge and understanding of what an internally consistent position on the nature of logic would entail - and it should also be possible for us to speak meaningfully, via propositions, about our position on the nature of logic, but these propositions would not necessarily constitute axioms or assumptions built into the system itself. Rather, they would be propositions regarding a set of ontological positions we hold, which then relate to our understanding of the nature of logic.

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