## Is Logic Empirical?

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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.

But it seems to me that logic cannot be empirical. If a thought defies a logical rule, then it is meaningless. The sentence "the triangle has four angles" is not a meaningful sentence because the predicate negates the subject. If the triangle has four angles, then it does not have three angles, so it is not a triangle, in which case our proposition does not refer to a triangle, in which case it has no subject and refers to nothing. No empirical observations could confirm a contradiction because contradictions are meaningless. They cannot even be conceptualized, much less observed. How could you observe something of which you cannot form a concept?

Now apparently quantum mechanics gives us reason to alter basic logical concepts. I am happy to admit that we must be careful about how we apply logical inferences to quantum phenomena, but I am doubtful that quantum phenomena defy foundational logical laws such as non-contradiction, identity and excluded middle. And if an interpretation of quantum mechanics does lead to violations of logic e.g. the cat is both alive and not-alive, then I think the problem lies in the interpretation of empirical observation. The conflict is not between observation and logic but between a false or unrefined interpretation and logic.
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Yeah I get real tired of people thinking quantum mechanics violates classical logic. Being in a superposition of two states is not the same as being classically in both contrary states at the same time.
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I tried to read the paper by Hillary Putnam, but there were too many difficult equations

As with most analytical philosophy, it's just a waste of life. Don't be hard on yourself with Putman, analytical philosers have an arrogant style, they just can't help themselves, it makes them feel superior.

I think you have asked an excellent question here.
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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.

Here's a link to Putnam's paper, republished as "The logic of quantum mechanics".

Putnam's argument is that the principle of distributivity fails for quantum mechanics. That is, he claims that there are instances where (A and (B or C)) is true, yet ((A and B) or (A and C)) is false.

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).

If his view is correct, then that is an example where classical logic fails for empirical reasons.
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That’s because the photon doesn’t go through A1 OR A2, it goes through A1 AND A2.
• 53
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).

I don't see how that's possible. "The photon goes through A1 and hits R, or the photon goes through A2 and hits R" is just a less efficient way of saying "the photon goes through A1 or A2 and hits R". Just like "you will eat eggs for breakfast and chicken for lunch, or you will eat pancakes for breakfast and chicken for lunch" is a less efficient way of saying "you will eat eggs or pancakes for breakfast and chicken for lunch". 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.

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.
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That’s because the photon doesn’t go through A1 OR A2, it goes through A1 AND A2.

That would be an interpretation. As SMBC puts it, 'Sweetie, superposition doesn't mean "and", but it also doesn't mean "or"'.

It's a complex linear combination of going through both slits.
• 1.1k
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.

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.

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.

Putnam is rejecting classical logic - see statement (10) on p190, where he states the principle of distributivity and says that it fails in quantum logic. Also:

Mathematically, quantum mechanics can be regarded as a non-classical probability calculus resting upon a non-classical propositional logic.
...
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.

Perhaps quantum mechanics only violates ordinary principles of probabilistic logic, not classical logic.

Most quantum physicists accept and use classical logic. But QM can be understood as a generalization of probability theory that, in addition to positive numbers, also allows negative and complex numbers (i.e., probability amplitudes).
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That’s because the photon doesn’t go through A1 OR A2, it goes through A1 AND A2.

Or perhaps in the sense that the photon goes through neither A1 nor A2 because there is never any particle photon at any instant. A wave version follows all paths but can only be realized as a photon particle hit in one of the patterns seen on the detector?
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You are correct and SMBC made my overarching point better than me.
• 53
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.

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?

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.
1. Show (A and (B or C)) implies ((A and B) or (A and C))
2. A and (B or C) - assumed conditional derivation
3. Show (A and B) or (A and C)
4. not-((A and B) or (A and C)) - assumed indirect derivation
5. not-(A and B) and not-(A and C) - 4 De Morgan's Law
6. not-A or not-B - 5 simplification and De Morgan's Law
7. not-not-A - 2 simplification and double negation
8. not-B - 6 7 disjunctive syllogism
9. B or C - 2 right side simplification
10. C - 8 9 disjunctive syllogism
11. not-not-C - 10 double negation
12. not-A or not-C - 5 simplification and De Morgan's Law
12. not-A - 11 12 disjunctive syllogism
13. 7 and 12 contradict one another, so the indirect derivation is complete
14. Line 3 is proven, so the conditional derivation is complete
• 9.4k
The change of meaning issue, from the cited article; interesting.

The contention is that the distributive rule is dysfunctional in quantum mechanical descriptions. Putnam rejects the notion that disregarding the distributive rule would amount to an arbitrary change of meaning. I think he is right here; it seems that classical logic cannot be re-interpreted in such a way that it can encompass quantum mechanics.

It's a pretty good argument against conventionalism, understood as the notion that logical rules are a tradition or habit.

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.
• 53
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

Although we may have to modify the way we apply logic depending on what purpose we are employing it for, it seems to me that the most basic principles of logic, as well as the principles which can be derived from them, such as distributivity, must always remain the same. Isn't a proposition or inference which violates a basic principle (e.g. identity, non-contradiction, excluded middle) nonsensical and impossible for us to conceptualize. What purpose could a logic which deals in incomprehensible nonsense possibly serve?
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Though it raises the issue of what seem like contradictory statements, in a specific case, cannot be dismissed via logic. IOW prior to QM one might have thought that one could deduce that two statements cannot both be true, when in fact they both can be true. So the application of logic to reality is of course affected by assumptions that can be faulty. 'At any given moment in time and knowledge can what seems to violate logic may not do that. And applications of logic may seem necessarily correct but not be. I think there could be more humility about this.
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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?

If you don't measure, you don't know whether those propositions are true. So it would be an interpretation.

Alternative interpretations are (A1 and A2 and R) and ((neither A1 nor A2) and R) as suggested by @Pfhorrest and @magritte earlier.

Instead of trying different combinations of classical conjunctions and disjunctions, Putnam instead reinterprets them based on a non-distributive lattice. Here's an example of how it works:

   1
/ | \
A1 A2 R
\ | /
0


The rule for disjunction is that while there is no common node, go up the lattice. The rule for conjunction is that while there is no common node, go down the lattice.

So for the photon going through (slit A1 or slit A2) and hitting region R, we have:

  (A1 or A2) and R
= 1 and R
= R


Whereas for the photon (going through slit 1 and hitting region R) or (going through slit 2 and hitting region R), we have:

  (A1 and R) or (A2 and R)
= 0 or 0
= 0


For a nice explanation of this, see Alex Wilce's talk, A Gentle Introduction to Quantum Logic. At 34:20, Wilce connects this to von Neumann and Birkhoff's quantum logic - if subspaces are ordered by set inclusion, we have a non-distributive lattice.

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.

See 37:00 where Wilce mentions non-unique complements which affects negation. Also see Disjunction in quantum logic.
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Sure, what is said should be consistent.
• 53
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. 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?)

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. 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).
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Even if we treat it as false in quantum mechanics, I don't think we must interpret this as invalidating the principle's universality.

You don't have to use quantum logic in quantum mechanics either; classical logic works perfectly well there - it's just one way of thinking about QM that some find useful or entertaining. 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.
• 53
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.

There might be certain contexts (although I can't think of any) in which it might be useful to assign certain contradictions as true. But I still think there is a law of though which makes contradictions inconceivable. It seems to me that a violation of the principle of distributivity is likewise inconceivable. If I turned on my blinker and turned either left or right, then does it not necessarily follow that I turned on my blinker and turned left or turned on my blinker and turned right? Can you conceive of the former as true and latter as false? Would that not violate the laws of thought?
• 1.1k
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.

Yes.

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?)

They can be verified as false. In this case, we observe the photon pass through either A1 or A2 (since we place detectors there), but we never observe it hit R (i.e., we observe it hit some other region). Which is to say, no interference occurs in this case.

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.

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. Whereas if you drop the parallel postulate you get non-Euclidean geometry which does describe the world we live in. That relegates Euclidean geometry to a special case of non-Euclidean geometry that just so happens to approximate what we observe in everyday experience.

Similarly, if you drop distributivity you get quantum logic. Classical logic is a special case within that more general logic that approximates what we observe in everyday experience. That is, the Boolean lattice that characterizes classical logic emerges as a special case within a more general non-distributive lattice. So, for example, if you fired bullets at a suitably robust double-slit apparatus, they would clump behind each slit whether or not you measured which slit they went through. Interference is rarely observed at a macroscopic level, so classical logic closely approximates what we normally observe there. As Putnam puts it:

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'.

In a sense, we can ask what the world would look like if distributivity was generally false, but was approximated in everyday macroscopic experience. Well, it would look just like our world.

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).

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.

But otherwise, yes, it comes down to the measurement problem.
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But otherwise, yes, it comes down to the measurement problem.

I just refreshed my memory on the double slit experiment, and it seems practically certain that setting up a device to determine which slit the particle goes through will affect where it lands. So if you place a detector on A1 or A2, then it is certain that the particle will not hit R. If you turn off the detector, then the particle will hit R, but you won't know whether it went through A1 or A2. The particle behaves like a wave if you don't track its motion and like a particle if you do. So since the detector clearly affects the conditions of the experiment, then I think it makes sense to assert that the principle of distributivity holds when the detector is off. The statement "(A1 and R) or (A2 and R)" is true, we just don't know which disjunct is true and which disjunct is false. When we turn the detector on, both disjuncts are false because tracking the particle changes the way it moves. 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). In fact, I think rejecting the realist assumption would be preferable. I can conceive of a particle teleporting straight onto R without passing through either slit. This would only violates the nomological law of continuous motion. But I can't conceive of the principle of distributivity as being false because it is a logical tautology. Like I said before, distributivity follows necessarily from more primitive logical axioms, so the principle's universality can't be denied without also undermining the universality of some of the most basic laws of thought.

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.

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? So is Putnam's argument that we ought to sacrifice the universality of classical logic in order to preserve realism and locality? If so, I don't think that's advisable. I don't think there's anything wrong with using quantum logic as tool, but I think we should still maintain that the principles of classical logic are always true. Maybe in the future, there will be something to explain how localism and realism are compatible with classical logic. Maybe future experiments will deliver definitive evidence against localism and or realism. I know that some very strange paradoxes would seem to follow if localism were false, but this seems like nothing in comparison with the confusion that would follow if we relativize logic, which is the foundation of all rational thought.

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.

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. The only reason that we are even capable of working out the principles of non-Euclidian geometries and non-classical logics is that we are capable of using classical logic. The principles of classical logic underlie everything rational we think and do in science, philosophy and daily life.

Is there some specific axiom of classical logic that you think we can afford to relativize? Distributivity is a theorem, not an axiom, so to reject the theorem would require rejecting an axiom. I know some people point to the law of excluded middle (which allows for double negation) as possibly dubious, but I don't think so.
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I think you make a good point here:

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).

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.

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.

Logic without distributivity is not as problematic as you think. You may find this recent article interesting: Non-distributive logics: from semantics to meaning.
• 7.3k
From an evolutionary perspective, brains, logic with it, were later additions to our repertoire of abilities. The brain adapted/adapts to the enviroment and not the other way round.

From a creationist perspective:

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

Here there's some uncertainty. Did God create us, our brains/minds, to be compatible with the universe or did God create the universe to be compatible with our brains/minds? 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.
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Logic without distributivity is not as problematic as you think. You may find this recent article interesting: Non-distributive logics: from semantics to meaning.

I read the paper's introduction, and two pieces jumped out at me:
1) "For instance, a natural question is whether relational semantics of (some) non-distributive logics can provide an intuitive explanation of why, or under which circumstances, the failure of distributivity is a reasonable and desirable feature; i.e. whether a given relational semantics supports one or more intuitive interpretations under which the failure of distributivity is an essential part of what ‘correct reasoning patterns’ are in certain specific contexts. Perhaps even more interestingly, whether relational semantics can be used to unambiguously identify those contexts. Such an intuitive explanation also requires a different interpretation of the connectives ∨ and ∧ which coherently fits with the interpretation of the other logical connectives, and which coherently extends to the meaning of axioms in various signatures."

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.

2)
"the graph-based semantics supports a view of LE-logics as hyper-constructivist logics, i.e. logics in which the principle of excluded middle fails at the meta-linguistic level (in the sense that, at states in graph-based models, formulas can be satisﬁed, refuted or neither)"

So it seems like the law of excluded middle might also be rejected in these logics. There's nothing strictly illogical about operating within a system where true and false aren't the only possible values a statement can have. It might sometimes be useful make 'unknown' a third option, and maybe other logics can incorporate probability such that a statement can have any value between 0 and 1. But I still think it is true that, in reality, every meaningful proposition must be either true or false. The proposition either corresponds to reality (e.g. the Eiffel Tower is in France) or it does not correspond to reality (e.g. the Eiffel Tower is not in France). There is no third option.

But I admit that much of what I read in the introduction went over my head. So perhaps I'm misinterpreting the quotes I pulled out.

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.

I agree that if we look at logic as an instrument, then it does not matter whether its axioms and theorems correspond to objective truths about reality. But I think that we should look at the principles of classical logic as being objectively true.

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.

To be clear, I think that, phenomenologically, logic is at least somewhat empirical. We come to understand logical truths, like the principle of non-contradiction, by abstracting from various experiences, and all experience involves sensory phenomena, whether it's directly received from the senses or reproduced in the imagination. And we communicate logic using words and symbols, which we process via our senses. But we should distinguish the cause of our coming to understand logical principles from the cause of the certainty of logical principles. Once we understand logical principles, it becomes apparent that their certainty does not depend on the specific experiences whereby we came to understand them. I may have realized that non-contradiction is true by considering that a triangle cannot be a square, but I could have just as easily realized it by considering that a bachelor cannot be married or by considering any other contradiction. If the principle were empirical, then I'd have to admit the possibility that non-contradiction may only apply to my specific set of experiences. But once I admit this possibility, the entire structure of thought is undermined. If contradictions are possible, then perhaps experiential evidence against contradictions (e.g. I've never observed a square triangle) might really be evidence for contradictions. I don't have empirical evidence to tell me how I ought to interpret empirical evidence, so if contradictions are possible, perhaps I should interpret my experiences to signify the opposite of what they seem to signify. We arrive at our knowledge of logic through a combination of sensory experience and intuition. But once we grasp logic adequately, it becomes a necessary and indispensable foundation for our understanding of the world. So when I say that logic is not empirical, what I mean is that we should not seek to adjust the fundamental principles of logic in order to accommodate empirical evidence. We need logic to make sense of empirical evidence, so, as long as we are committed to the search for truth, we must regard logical axioms and theorems as a priori and self-evident.

The only reason for you to think that our capacity for logic is a product of evolution or intelligent design is that these explanations are (or seem) reasonable. But if the implication of these explanations is that logical principles are mere tools for interpreting the world or contingent structures of thought that might have been created otherwise, then the foundation of reasonable thought is undermined. In that case, you would have no rational justification for believing in either evolution or intelligent design. I think the best explanation for why we use logic is that logical axioms correspond to basic features of all existence: things are what they are, they are not what they are not, and for any given property, a thing either has or does not have that property.
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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.

If you identify logic with classical logic, or something with a close family resemblance, then yes. But formal logic in general is less specific than classical logic, even though it still has to do with reasoning, with inference. Which is to say that the patterns of reasoning that are available to us go beyond those that are covered by classical logic. 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. Perhaps the example on pp. 12-13 will help to illustrate the point, though admittedly, taken in isolation it may not look very convincing.

Yeah, I am out of my depth here as well. Perhaps one of our resident mathematicians will come along and enlighten us :)
• 7.3k
We come to understand logical truths, like the principle of non-contradiction, by abstracting from various experiences

:ok:

First thing to notice here is we're worried about logic and reality not in agreement. The one thing that gives us sleepless nights is the law of non-contradiction, the possibility of it being violated, and for good reason I suppose. If contradictions are ever allowed, then the specter of ex falso quodlibet threatens to cause havoc:

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

However, there are systems of logic that allow contradictions, and such systems are designed in such a way as to forestall ex falso quodlibet. 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.

However, what I can't wrap my head around is how contradictions can be thought of as possible. Suppose there's this blank space: (.......). I now write in this blank space a proposition, say about god: (God exists). Then I contradict myself and say god doesn't exist: (God exists). In effect, I'm back to square one, the blank space: (.......). After all, (God exists) = (.......). There's no net effect on the blank space i.e. a contradiction is not a proposition at all ( :chin: ). How then can it be true? Unless of course, in the case of my example, God doesn't exist is not God exists. As you can see, we need to look at "not" or negation differently.
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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?

Yes, although the problem can be demonstrated with just the standard double-slit setup. Just place a single detector on one of the slits, say A1. That is sufficient to make the interference pattern disappear, even for just those photons that go through A2 that one might think should not be physically disturbed from their path.

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.

We can observe the surface of a sphere which is a non-Euclidean surface. Consider a geodesic such as the Earth's equator. A geodesic in non-Euclidean geometry generalizes the notion of a straight line from Euclidean geometry to instead be the shortest path between two points on a surface. In the special case where a surface is flat, the geodesic is a straight line.

Similarly, consider a flipped coin where the result is heads or tails but we don't know which. The coin is definitely either heads or tails, which is a classical disjunction. Now consider a quantum coin that is in a linear superposition of heads and tails. Quantum logic generalizes classical logic to include superpositions. In the special case where the state of the coin has collapsed to a definite state, quantum logic just reduces to classical logic.

So is Putnam's argument that we ought to sacrifice the universality of classical logic in order to preserve realism and locality?

I'm not sure if Putnam was aware of Bell's theorem at that time. But his position can be seen as rejecting realism in Bell's sense, i.e., as rejecting counterfactual-definiteness. To give a tangible/visualizable sense of what quantum logic is, I'll outline a geometric proof-of-concept in a separate post.
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Here is a geometrical proof-of-concept for quantum logic and how classical logic emerges as a special case in normal experience.

The quantum superposition state (psi) in the double-slit experiment (where A1 and A2 represent the photon going through each respective slit) is

psi = 1/sqrt(2)(A1 + A2)


where the probability of the photon being measured at either slit is the square of 1/sqrt(2), i.e. 1/2.

This can be geometrically represented by the following state space where the potentially definite states A1 and A2 are orthogonal axes of unit length, and psi is a 45° diagonal line of unit length (imagine the plus signs form a solid line to the origin).[*] Psi is essentially a North-East arrow which has a North component (A2) and an East component (A1).

A2
|
|        +  psi
|     +
|  +
+------------ A1


We can see here that the superposition state psi is not the same as either of the axis states A1 or A2. So, absent a measurement at the slits, the question of which slit the photon goes through has no definite answer (i.e., it's counterfactually indefinite). On a measurement at the slit, psi is projected (collapses) onto one of those axes, at which point the question of which slit the photon goes through becomes definite.

Now we can ask what is the smallest subspace that contains both axis lines. It is the 2D plane. The 2D plane is the span of the union of the two axis lines. However the plane contains any line that lies on it, and we can see that psi is another line on the plane.

So, in quantum logic, that span of the union is what is meant by disjunction. Intersection is what is meant by conjunction (in this case, the lines intersect only at the origin). The complement of a line is what is meant by negation (in this case, each line has multiple complements - every other line on the plane - which makes the logic non-distributive.)

Now, absent a measurement at the slits, if we ask whether the photon goes through slit A1 or A2, we can see that psi is contained in the span of the union of A1 and A2, so the answer is yes. That is, the disjunction itself is definite even though the question of which disjunct is true is indefinite.

There is a special case where psi may already be on one of the axes when it is measured. This is the case if a second detector is placed at the slits. If it is measured by the first detector as having gone through slit A1, then it will be measured by the second detector as having gone through slit A1 as well. The fact that no interference pattern is observed on the back screen just is a consequence of that. It's a consistent history, so to speak.

Which brings us to classical observations. Why, when an experiment is performed firing bullets instead of photons, and we don't measure which slits the bullets go through, is no interference pattern observed?

The reason, in effect, is that the measurement has already been done by the environment (termed decoherence). And the rest of what is observed follows consistently from the initial measurement. Thus what we are doing when we ordinarily observe something is to measure a value where psi has already been projected onto one of the axes. The worst case is that we don't know which axis it was initially projected onto, which is just a case of ordinary probability (e.g., whether a hidden coin is heads-up or tails-up). So, even absent our measurement, the disjuncts of which slit the bullet went through are definite. And that special case, which is always and everywhere observed except in quantum interference experiments, is the empirical and intuitive basis for classical disjunction.

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).

See here for a helpful definition of quantum logic and the axioms.

--

[*] As a fun exercise for those who don't have degrees in this sort of thing, see if you can figure out where the 1/sqrt(2) comes from in the geometry.
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