Hal Finney wrote:
>I'm not sure what you're getting at here, but let me make one point.
>It's not clear that you can say that our universe is based on a particular
>form of mathematical logic. Does our universe in any sense rely on or
>embody the principle that we can't have P and not-P? I don't see it.
>Our universe is based on atoms and quarks and fields. When we say
>that P and not-P can't both be true, the logic is in our heads, not in
>the universe.
Well, logic deals with propositions, so if you treat statements about
physics as propositions then in that sense it seems like our universe does
obey standard first-order logic. For example, you can't shuffle the
equations of Newtonian physics to simultaneously show that F=ma and F!=ma.
Likewise, for specific predictions that come out of these laws, you'll never
have a situation where you can use a single theory to get two contradictory
predictions (there can be cases where different successful theories produce
contradictory predictions, like maybe the conflict between general
relativity and quantum mechanics, but in those cases physicists always
assume that one or both of the theories is incorrect, not that the laws of
physics fail to obey the laws of logic).
More generally, it's my understanding that all mathematics, even alternate
forms of logic, can be formulated in terms of the inference rules of
first-order logic + additional axioms. For example, you could probably
discuss multivalued logics in terms of propositions about the truth-value of
various statements, and even if the truth-values were allowed to take on
more than two values, the propositions *about* each statement's truth-value
would still be either true or false. If this is correct, then the notion of
"alternate forms of logic" is misleading in a sense, since all mathematical
structures that we can conceive of do obey the rules of first-order logic if
you formulate them in this way.
Jesse Mazer
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Received on Wed Apr 23 2003 - 20:39:43 PDT