While classical mechanics has a logic that is based on a Boolean algebra of subsets of the state space, Quantum logic is based on the subspaces of a complex Hilbert space.


Many examples in quantum logic are counter intuitive as de Morgan's law does not hold.

I would venture that it is more intuitive for us to conceptualise a problem space using set theory than it is using a vector space representation.

Can others think of domains of knowledge that are better conceptualised in a vector space and that cause philosophical confusion because we mistakenly represent them by set relations?

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    This may not quite be what you are looking for, but the first part of ‘Logic, Logic, and Logic’ by George Boolos discusses some of the philosophical implications of set theory, esp. in connection with second-order logic and plural quantification. – MarkOxford Aug 3 '17 at 11:32
  • Thanks @MarkOxford I will check that out. It will be handy to flesh out my background knowledge on the subject. – ChrisGuest Aug 3 '17 at 12:09
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    @ChrisGuest Especially relevant from that volume would be "To be is to be the value of a variable (or some values of some variables)" where he discusses plural vs set-theoretic interpretations of second-order logic and some implications for semantics and ontology. Most of the general issues I can think of would be of the sort that Boolos discusses there. – Dennis Aug 6 '17 at 19:53

Every domain of discourse that uses vague or fuzzy adjectives (or classified nouns, which are reified adjectives). Which is every domain of discourse that uses adjectives. Which is every domain of discourse.

Discreteness is not an aspect of macroscopic reality, things made of particles do not have clear boundaries. And individuation is not an aspect of quantum reality, particles are not completely separable -- they overlap, and they entangle. But these concepts are central to our notion of 'thing'. As Quine points out in his discussions around vagueness, similarity and natural kinds, set theory models something we only wish could be true. But we cannot remove the basic idea of identifying and classifying individuals from our logic.

From a narrative psychology point of view, we are so biased by our personal notion of individual beings that we project individuation onto nature. But it is not an aspect of nature. It is not even an aspect of human beings. It is only really an aspect of people in stories. But we cannot shake it as an organizing principle because our picture of the world is made up of stories, not observations.

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Any introductory book on QM will show that states in QM are not points in a Hilbert space, but lines; this the actual mathematical structure to be considered is projective Hilbert space.

This might then suggest that the project of Quantum Logic is misconceived, but that would itself misconceive what both Neumann & Birkhoff werre attempting when they first presented this proposal, that is to open up the space of concepts by which we can think about quantum mechanics.

The main philosophical problem with set theory is its ontology, where it only considers what is as foundational, with process being a derived concept; it's main rival, category theory and it's variants, take process and flow as conceptually foundational.

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One very prominent problem would be the world-as-a whole. In metaphysics we collide head-on with Russell's paradox and have to solve it for progress. Paul Davies discusses this issue at length in his book Mind of God. Many problems in metaphysics can be traced to set-theoretic assumptions that embody dualism and have to be overcome. Davies discusses this in terms of the 'Mindscape', the set of all possible ideas. Clearly this is an idea, and yet it is not in the set. A study of this problem can be very rewarding. I recommend Mind of God.

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Do you really think that set theory is good for logic? Here are some thoughts which should raise doubts:

Every entry of the Cantor-list (applied to prove uncountability of the set of real numbers) differs from the antidiagonal. From this it is concluded that all entries differ from the antidiagonal and therefore the antidiagonal is not in the list.

Every digit of the decimal expansion of a real number is insufficient to determine this real number. From this it is not concluded that all digits of an entry or of the antidiagonal are insufficient to determine a real number or the antidiagonal.

Every rational number can be indexed by a natural number. From this it is concluded that all rational numbers can be indexed by natural numbers and the set is countable.

Every natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that all natural numbers leave the overwhelming majority of rationals without index.

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    But these are all the same objection---you don't like operations on anything but finite terms. I don't think that's actually a point of contention for most people. – Canyon Aug 5 '17 at 15:11

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