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Math Help - are there logics based on consistency "up to n proof steps"?

  1. #1
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    are there logics based on consistency "up to n proof steps"?

    That is, in logic there are two basic ways of accepting a theory:
    (1) it is consistent
    (2) it is at least equi-consistent with some convenient theory, usually arithmetic

    However, the human brain only worries about consistency as far as one has time to think (having evolved a logic that was more concerned with escaping predators than working out the next quantum gravity theory); a theory is accepted provisionally until something better comes along, and contradictions which may or may not appear with more thinking are not a cause of worry until it happens. So, if one wishes a logic to imitate human thinking, it would seem useful to have a formal logic which had some concept of "it's consistent up to a certain level of implication". However, the details might get sticky (just as finite models are often less convenient than infinite ones). So, does such a logic exist? If so, what is it called, and can anyone give a good link to explain the nitty-gritty of the logic? Thanks.
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  2. #2
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    Re: are there logics based on consistency "up to n proof steps"?

    Why would one want a 'logic to imitate human thinking'? The whole point of logic is to improve human thinking. You seem to be saying 'because most people aren't very intellectual, our logic should not be intellectual'. That's like saying 'because most can't run very fast, we should design automobiles that won't run very fast'.
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    Re: are there logics based on consistency "up to n proof steps"?

    Logic has long passed from being considered "how people should think" to a general study of algebraic systems based on axiomatic foundations. (OK, this definition could have been better phrased, but you get the idea.) Thus, just as Geometry no longer fits Euclid's concept of studying lines etc as they are in the "real world" (whatever that is) but has passed to studying opposing geometries, so too has logic room for the study of many different systems. One kind of system (among many) which has many applications is a logic of how humans think. (You may remember the title of Boole's famous book was "The Laws of Human Thought".) For example, an intelligent computer which interacts with humans must be able to handle human foibles until such time as humans stop being humans and turn into computers. (It could do the field of psychology no end of good if psychologists would listen, but that is a side point.) Of course I am not advocating turning the whole field of logic over to this direction of study -- far from it; physicists would tear me to pieces for such heresy-- but it is a corner of the field that deserves attention. (And it does, as you can see by a brief survey of the Journal of Symbolic Logic nowadays.)
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    Re: are there logics based on consistency "up to n proof steps"?

    Thanks from Deveno and topsquark
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  5. #5
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    Re: are there logics based on consistency "up to n proof steps"?

    Hi.

    Thatīs a very interesting question indeed. Its one of my reasearch topics.

    Hopefully, what you ask for is well known in proof theory.
    See Rosserīs theorem in Handbook of Mathematical Logic (Studies in Logic and the Foundations of Mathematics), edited by J. Barwise


    With my best regards.
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    Re: are there logics based on consistency "up to n proof steps"?

    Just bumping this over the spam
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