## Is Logic Empirical?

• 49
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.
• 3.1k
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.
• 596
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.
• 1.1k
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.
• 3.1k
That’s because the photon doesn’t go through A1 OR A2, it goes through A1 AND A2.
• 49
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.
• 1.1k
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).
• 20
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?
• 3.1k
You are correct and SMBC made my overarching point better than me.
• 49
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
• 9.2k
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.
• 49
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?
• 1.6k
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.
• 1.1k
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.
• 9.2k
Sure, what is said should be consistent.
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