Is logic "universe-dependent"? Does logic changes depending in the world we exist, or is logic universal for all existing beings regardless of where they exist, and how can we prove or refute this? Is there a way to refute something like this?
If we take the example of arithmetic logic:
Imagine a universe in which reasoning about numbers was impossible because arithmetic was invalid i.e., objects could not be counted, measurements could not be made, etc.
In that universe, the distance between the sun (if it existed) and the earth (if it existed) could be one millimeter, one light year, red and three-quarters, or yes (your choice). The number of electrons in a helium atom could be zero, one billion, hot, or tasty- whatever you prefer.
It is difficult to imagine how life as we have come to know it could possibly evolve in such a place. This suggests that if such a place did in fact exist, we wouldn't be there to experience it.
Define universe. If we could do that, we'd have a definite answer.
Classical/Greek logic, is only one of three schools of logic - along with Nyaya/Indian, and the suppressed Chinese Mohist tradition. Many core assumptions of classical logic are deeply suspect and open to challenge, and I'd describe the transition to negation-as-failure as basically taking up digital logic as foundational.
I'd say the fundamentals of digital logic, relate to geometrical relations in space, & they emerge from very basic relationships. Nand is functionally complete, & can constitute all Boolean expressions. I'd suggest that is fundamental, in the Turing-machine sense.
Edited to add: Predicate calculus is a formalisation of rules of inference, without second-order & more self-reference. It is essentially 'just a language', and like any language it depends on the modes of life it is relevant to and tacit assumptions of speakers, it is only that it attempts to model itself on the consistent application of rules found in mathematics and geometry - a programme ended in it's attempts at total convergence, by Godel's theorems.
A language of formalised inference of creatures of a much smaller scale, the rules of inference would be those of quantum mechanics, so including superpositions rather than the excluded middle, or binary outputs. Creatures without hard boundaries would use a different intuitive basis for their number system, and equivalent of subitism. Etc. But such languages would still have Turing-equivalence, amounting to them being inter-translatable through the medium of digital logic. Edit ends
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