I tried to read the paper by Hillary Putnam, but there were too many difficult equations — Dusty of Sky
I tried to read the paper by Hillary Putnam, but there were too many difficult equations, so I'm hoping that someone here can make his case in more ordinary language. — Dusty of Sky
Putnam gives an example of the double-slit experiment on pp180-181. On his view, the photon goes through (slit A1 or slit A2) and hits region R, yet it is not the case that the photon (goes through slit A1 and hits region R) or (goes through slit A2 and hits region R). — Andrew M
I can't conceive how an observation which violates principle of distributivity could be possible. What would that even look like? Putnam is an accomplished and respected philosopher, so I'm sure there's something I'm missing. I'll try to read the paper again at some point and hope I have better luck with it. But if you think you understand what he's saying, please try to explain. — Dusty of Sky
Addendum: after glancing at the equations on pp180-181, I see that he's using probability. I don't have a good grasp of probabilistic logic, but my understanding is that it adds a number of layers of complexity and uncertainty to classical logic. I may be wrong, but I think that the principles of probabilistic logic are quite contentious among logicians. Perhaps quantum mechanics only violates ordinary principles of probabilistic logic, not classical logic. — Dusty of Sky
Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic.
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For Putnam, the elements of L(H) represent categorical properties that an object possesses, or does not, independently of whether or not we look. Inasmuch as this picture of physical properties is confirmed by the empirical success of quantum mechanics, we must, on this view, accept that the way in which physical properties actually hang together is not Boolean. Since logic is, for Putnam, very much the study of how physical properties actually hang together, he concludes that classical logic is simply mistaken: the distributive law is not universally valid. — Quantum Logic and Probability Theory - SEP
Perhaps quantum mechanics only violates ordinary principles of probabilistic logic, not classical logic. — Dusty of Sky
That’s because the photon doesn’t go through A1 OR A2, it goes through A1 AND A2. — Pfhorrest
Putnam is saying that the photon going through (slit A1 or slit A2) and hitting region R describes an interference experiment. That is, you don't know which slit the photon went through but, on conventional realist assumptions, it went through one slit or the other. However the photon need not hit region R if you do measure which slit it went through. Now we know this already since this is just what QM predicts. But Putnam's claim is that those two experimental observations are the left-hand-side and right-hand-side of the principle of distributivity, and so violate it. — Andrew M
However I will maintain that there is a sense in which we choose, amongst possible logical systems, that which best fits what it is we are trying to explain.
SO what might very loosely be described as the empirical component of logic is no more than choosing a logic that fits our purpose, when our purpose is describing how things are — Banno
I don't think this necessarily contradicts the principle of distributivity. It seems that measuring which slit the photon goes through affects the conditions of the experiment. So if you don't measure, then both ((A1 or A2) and R) and ((A1 and R) or (A2 and R)) are true. If you do measure, then both are at least potentially false because not-R can be true. Am I still missing something? — Dusty of Sky
1 / | \ A1 A2 R \ | / 0
(A1 or A2) and R = 1 and R = R
(A1 and R) or (A2 and R) = 0 or 0 = 0
Also, here's a derivation of one side of the principle of distributivity. The principle follows from more basic logical principles, so if you reject distributivity, you must also reject at least one of the other principles used in this derivation. — Dusty of Sky
Even if we treat it as false in quantum mechanics, I don't think we must interpret this as invalidating the principle's universality. — Dusty of Sky
Which goes to show that "laws of thought" - including the principle of distributivity - don't have to be as rigid and universal as people often assume. We can adopt different logics for different uses. — SophistiCat
I think I understand, but please tell me if I am missing something. So when the photon hits R, as long we hold to a realist interpretation, we must assume that it passes through either A1 or A2. So R is true, and (A1 or A2) is true. The former is verified by observation and the latter by realist assumptions. Therefore, their conjunction is true. — Dusty of Sky
But neither (A1 and R) nor (A2 and R) can be verified as true, since we don't observe the photon to pass through either A1 or A2. So the statement "(A1 and R) or (A2 and R)" is evaluated as false, because neither disjuncts can be verified. (Or can they in fact be verified as false? When we measure A1 and A2 individually, do we never or only sometimes detect a photon passing through them?) — Dusty of Sky
I don't think that this proves that the principle of distributivity fails. It may be useful to not apply distributivity when dealing with quantum phenomena, but that doesn't mean that the principle is false. It is inconceivable for the principle to be actually false. If R is true and A1 or A2 is true, then either R and A1 is true or R and A2 is true. That's a simple tautology. Just because we can discover more in quantum mechanics by not applying a principle does not necessarily mean that the principle is false. And if we have reason to believe that the principle is necessarily and universally true, as I think we do in the case of distributivity, then its usefulness in quantum mechanics should make no difference. Even if we treat it as false in quantum mechanics, I don't think we must interpret this as invalidating the principle's universality. — Dusty of Sky
We must now ask: what is the nature of the world if the proposed interpretation of quantum mechanics is the correct one? The answer is both radical and simple. Logic is as empirical as geometry. It makes as much sense to speak of 'physical logic' as of 'physical geometry'. We live in a world with a non-classical logic. Certain statements - just the ones we encounter in daily life - do obey classical logic, but this is so because the corresponding subspaces of H(S) form a very special lattice under the inclusion relation: a so-called 'Boolean lattice'. Quantum mechanics itself explains the approximate validity of classical logic 'in the large', just as non-Euclidean geometry explains the approximate validity of Euclidean geometry 'in the small'. — Putnam: The logic of quantum mechanics, p184
Perhaps it is true that either (A1 and R) or (A2 and R), but since we can verify neither disjunct, we treat it as false, not because it is false in reality because our measurements fail to demonstrate it. (Or, if our measurements in fact demonstrate the contrary, that the photon passed through neither, then we would have to interpret the act of measurement as affecting the photon). — Dusty of Sky
But otherwise, yes, it comes down to the measurement problem. — Andrew M
If one of the disjuncts were true in reality, it would be a hidden variable. But that would require a non-local interpretation, per Bell's Theorem. — Andrew M
OK, so it's interesting to consider Putnam's argument here. He notes that you could says exactly the same thing about Euclidean geometry. It might be considered necessarily and universally true, but it nonetheless fails to describe the world we live in. — Andrew M
It seems arbitrary to me that we should make the realist assumption that (A1 or A2) is true, even though this assumptions can't be empirically verified, but not also assume that the principle of distributivity holds just because we can't empirically verify either (A1 and R) or (A2 and R). — Dusty of Sky
My claim is that a logic in which the principle of distributivity is false does violate the laws of thought such that any claim made in such a logic, regardless of its usefulness, amounts to nonsense if we actually try to conceive of its meaning. — Dusty of Sky
God on the sixth day [the last day] of Creation created all the living creatures and, “in his own image,” man both “male and female.” — Britannica
Logic without distributivity is not as problematic as you think. You may find this recent article interesting: Non-distributive logics: from semantics to meaning. — SophistiCat
But this is if you look at quantum logic as making an absolute metaphysical statement about quantum mechanics, rather than simply treating the logic instrumentally, or as usefully capturing some aspect of the phenomenon without pretending to the ultimate truth. — SophistiCat
Did God create the universe for us or were we created for the universe? If the former then logic isn't empirical and if the latter it is. — TheMadFool
If conjunction and disjunction (∨ and ∧) are interpreted differently than in classical logic, then it does not seem so surprising that the principle of distributivity might fail. But this does not entail that the principle does not hold universally. The principle does hold universally (it seems to me) so long as we interpret the conjunction and and disjunction symbols (and whatever other symbols might also be relevant) to mean what they mean in classical logic. If we change their meanings, then it makes (classically) logical sense that we'd get a different set of theorems. — Dusty of Sky
But I admit that much of what I read in the introduction went over my head. — Dusty of Sky
We come to understand logical truths, like the principle of non-contradiction, by abstracting from various experiences — Dusty of Sky
contradictions — Dusty of Sky
Ex falso quodlibet is Latin for “from falsehood, anything”. It is also called the principle of explosion. In logic it refers to the principle that when a contradiction can be derived in a system, then any proposition follows. In type theory it is the elimination rule of the empty type. — Google
I think I see how this could be problematic. Suppose you did a double slit experiment with two entangled particles separated by a significant distance. Then turning on the detector for one of them would communicate an effect to the motion of the other which would travel faster than the speed of light. Is that what you have mind? — Dusty of Sky
I don't think Euclidian geometry is necessarily universally true in the same way that classical logic is. A principle like "there is exactly one straight line passing through any two points" is always true with regard to our perception of space. We can't imagine a non-Euclidian realm in which the principle does not hold. But just because non-Euclidian space is unimaginable does not mean it's inconceivable. Non-Euclidian space violates the principles of sensory perception but not the principles of rational thought. My claim is that a logic in which the principle of distributivity is false does violate the laws of thought such that any claim made in such a logic, regardless of its usefulness, amounts to nonsense if we actually try to conceive of its meaning. It is impossible to think the proposition "((A1 or A2) and R) and not ((A1 and R) or (A2 and R))". You can write it out in symbols and claim that it is true, but you don't actually have a concept of what you are affirming any more than you have a concept of a married bachelor. — Dusty of Sky
So is Putnam's argument that we ought to sacrifice the universality of classical logic in order to preserve realism and locality? — Dusty of Sky
psi = 1/sqrt(2)(A1 + A2)
A2 | | + psi | + | + +------------ A1
So based on this state space geometry, quantum logic is the general case and classical logic is the special case (where states are definite and have unique complements). — Andrew M
With this general sense of logic, it is indeed possible to have a logic in which conjunction and disjunction mean something different than what they mean in classical logic, but play broadly similar roles. — SophistiCat
Honestly, sometimes I feel all this is just a game. Anyway, the point is there are logics that can handle contradictions pretty well or so some tell me. — TheMadFool
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