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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?

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    Considering that we have no access to other "universes" (i.e. possible worlds) there is nothing to prove or refute, it is a matter of convention. And the usual convention in modal logic is to keep the logic fixed across all possible worlds. This is expressed by the necessitation rule, any logical theorem is necessarily true, i.e. true in all possible worlds. But there is nothing stopping one from considering a set of worlds with varying logics, if there is a point to it.
    – Conifold
    May 23, 2021 at 22:06
  • Logic is intended to be an a priori science, meaning that it is what you can know to be true without observing anything about the universe. Assuming that logicians are successful in picking out only the a priori, then logic must be universe independent. Anything that is universe dependent requires you to observe the universe to see which possible universe you are in. May 24, 2021 at 5:28

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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.

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    If objects cannot be counted, there are no objects. Objects are individuals that we can see, think, speak of as "individuated", i.e. separate, permanent, and so on. Thus we can "name" them and numbers are simply "abstract names". Thus, if there are objects, we can - in principle - count them. May 24, 2021 at 8:29
  • It's like saying the view that truth is relative, means truth is meaningless & everything is up for grabs. Logic is relative.
    – CriglCragl
    May 24, 2021 at 14:07
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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

You might be interested in Tegmark & Wu's AI Physicist.

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  • There is also modern symbolic logic which is none of the above and which is what almost everyone in mathematics, philosophy or computer science means when they say "logic". Digital logic isn't even logic, strictly speaking; it's just an abstraction of a certain class of digital circuits. May 24, 2021 at 5:24
  • @DavidGudeman: Happier?
    – CriglCragl
    May 24, 2021 at 21:27
  • @CrigCagl, you are speculating, not answering, and what you say is so idiosyncratic I doubt you have any background in logic beyond Wikipedia. Logic is not "deeply suspect and open to challenge"; It does have its limitations which are well known. These limitations don't make anything "deeply suspect" (and digital logic has the same limitations). Predicate calculus is not just a language; it contains a formal system of inference. Your comments about number systems are also odd and unsupported, and you seem to think digital logic is the propositional calculus. May 25, 2021 at 16:23
  • Good job I didn't say that then - I said some of the assumptions are, and specifically had in mind en.wikipedia.org/wiki/Law_of_excluded_middle#Criticisms Hence Russell's paradox etc.
    – CriglCragl
    May 25, 2021 at 17:07
  • yes, I followed the link before. The issues surrounding the law of the excluded middle is one of the things I meant. It isn't a problem with logic as such; instead it limits you to studying sentences which can only be either true or false. There are lots of declarative sentences which do not fit in this category such as the Liar and sentences with fuzzy meanings. There are also modifiers such as "believes" that cannot be handled by the predicate calculus, but as long as you are aware of these limitations, there is nothing problematic about the logic itself. May 25, 2021 at 23:07

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