One of my favorite Sci-Fi short stories is “Mimsy Were the Borogoves” , whose title is taken from Lewis Caroll’s nonsensical poem “Jabberwocky” . (Spoiler alert, skip to the 4th paragraph if you want to read the story for yourself) The story starts with a distant future post-human scientist who sends some super-futuristic educational toys back in time. Some of the toys reappear in 1940s America, and the rest reappear in late 19th century England. In the 1940s, two young siblings pick up the toys and start playing with them. Soon after, they start exhibiting strange behaviors and superhuman powers, so much that their parents are scared and take them to a psychologist. The psychologist concludes that the toys must be alien and are somehow teaching the children a logic and perception of space and time radically different from ours. The kids’ new behavior seems incomprehensible to adults, because they are too old to learn this original way of seeing things (presumably they have already internalized classical logic and euclidean geometry and can no longer think otherwise).
In 19th century England, the second set of toys is found by Alice Liddell, who also starts thinking in strange and new ways. Based on what she learned from the toys, she recites some strange poetry to family friend Lewis Caroll, who later includes the verses in his fantasy works.
Back in the 1940s, the two aforementioned siblings discover “Through the Looking Glass” and read the poem Jabberwocky, at which point they figure out the missing part of some outlandish space-time equation and use it to teleport themselves to a parallel dimension while their horrified father watches.
It is of course just a sci-fi story, but I find the idea beautifully compelling: What if indeed the laws of logic were not absolute, but more of a custom we acquired as children, like our native language, and that we couldn’t unlearn later as adults? Which brings me to the topic of this blog post: Can there be other logics beside the classical logic we are all familiar with? Other logics that have different, perhaps counter intuitive rules?
Aristotle was the first to formalize the rules of logic, establishing the laws of non-contradiciton and the excluded middle, and providing rules for deductive reasoning. In the 19th century, George Bool formalized Aristotle’s laws into what became known as Boolean Logic, or Boolean Algebra – which is probably the logic most of you are familiar with. Gottlob Frege took this a step further and developed predicate logic, which in turn got extended into First-Order logic (No, that is not a Star Wars pun!!).
Together, all of these fall under the general heading of classical logic, in that they cover what most people intuitively mean by the term ‘logic’, and all are really just extensions and formalizations of Aristotle’s original syllogistic logic. When someone mentions that they applied reason to solve a problem, or that they have proved a theory or an abstract argument, they implicitly mean that they used the rules of classical logic to do so.
But this almost begs the question: Where do rules of classical logic come from and how do we know if they’re always the same? It seems that if anything was transcendent, immutable and independent of the facts of the world, it had to be the laws of logic – they were the only domain where reason alone was enough to provide us with knowledge of the truth. Even hardened Empiricists like Hume, Wittgenstein, or the logical positivists accorded logic a special status, though they would never place it in any ontologically separate realm à la Plato.
But are we sure that classical logic all there is and is it really that transcendent and certain? After all, Buddhists had talked about a logic that differed radically from Aristotle’s for centuries, could they be right?
By the 1930s, logic dominated Western thought: Logical Atomists and Logical Positivists sought to reduce all of language and philosophy to the analysis of logical propositions, and Russell and Whitehead thought that they had succeeded in building all of mathematics from the ground up using the basic axioms of logic and set theory. But right at its apogee, the edifice started to crumble: from the theoretical side, Kurt Gödel drove a knife through the heart of Russell and other logicists dreams with his infamous incompleteness theorems, while from the experimental side, relativity and quantum mechanics were messing with all of our preconceptions, include our very notions of time and reality.
With all of this going on, it was only a matter of time before someone would start questioning the axioms of logic and wondering whether even our laws of reason might be put in doubt.
Birkhoff and Von Neuman proposed in 1932 that the paradoxes of Quantum Mechanics can be explained if we abandoned classical logic and used some form of Quantum logic instead. Such a Quantum logic would change (or get rid of all together) some of the rules of classical logic, and would be a perfect case of logical axioms arrived at by observation.
W.V.O Quine, in his 1951 paper “Two Dogmas of Empiricism” questioned the analytic-synthetic distinction, and showed that even analytic propositions were dependent on empirical evidence. Since the rules of logic were analytic propositions par excellence, they too, were ultimately dependent on empirical data, and were not absolute laws.
Later in 1968, Hilary Putnam picked up where Birkhoff and Von Neuman left, in his paper “Is Logic Empirical?”, later republished as “The Logic of Quantum Mechanics.”. In it he argued that, just as empirical physical results – namely relativity – forced us to abandon Euclidean geometry, so it is possible that the results of quantum mechanics will force us to abandon classical logic.
Did this mean we should all ditch Aristotle and Frege, and start learning how to think like mystical half-dead cats?
Quantum logic never lived up to the initial hype. Although Quantum logic is still an active field of study up to the present day, it is does not get much attention from most philosophers and had been abandoned completely by physicists. Those few who do study the topic view it mainly as a tool for studying the mathematical foundations of Quantum phenomena, or an interesting order theoretical oddity, not as a fundamental revision to the current classical rules of logic.
The main problem that is faced by Quantum logic (or any such radical revision of logic, empirically justified or otherwise), is that we tend to think and communicate in classical logic. It would be very difficult, or in Kantian fashion, outright impossible for us, to perceive and discuss the world in anything other than classical logic – it seems to be hardwired into our brains. Although the logical atomist program failed as a metaphysical theory, it did show us just how ingrained classical logic is into our linguistic and mental structure. As Wittgenstein said in the Tractatus, the limits of language are the limits of the world: No one can place themselves outside of logic, and then pick among different logics to reason and argue with, even if those alternative logics are justified.
Consider the distributive property of classical logic:
p and (x or y) = (p and x) or (p and y)
In Quantum Logic, the distributive property no longer holds, so that:
p and (x or y) ≠ (p and x) or (p and y)
This might seem esoteric to you. To show just how strange and inconceivable quantum logic is, we will turn the above formulas into a practical everyday example. Consider the following propositions:
- p: ‘The coffee is on the table’
- x: ‘The food is in the fridge’
- y: ‘The food is in the microwave’
In classical logic, the statements:
- a = p and (x or y): ‘The coffee is on the table and the food is in the fridge or in the microwave’
- b = (p and x) or (p and y): ‘The coffee is on table and the food is in the fridge, or the coffee is on the table and the food is in the microwave’
(a) and (b) have exactly the same meaning, and whenever (a) is true, so is (b). But in quantum logic one can be true without the other (because of the uncertainty principle, it is possible that a is true while b is false).
Now can you imagine trying to reason or argue with someone for whom propositions of the forms (a) and (b) didn’t have the same meaning? Can you imagine trying to read a paper written using such logic, or arguing a case in court with such logic?
The sheer inconceivability of such a non-classical logic in everyday speech is what critics of Birkhoff and Von Neuman’s proposal meant when they argued that no one could ever replace classical logic, because it was hardwired into our brains. In fact, Birkhoff, Von Neumann, and Putnam all abandoned Quantum logic later in their careers
Those non-classical logics which have been successful (fuzzy logic, modal logic, intuitionistic logic) are those that extend classical logic, as opposed to replacing it, or at least respect classical truth tables in the limiting case.
But then, the far fetched possibility hinted at in “Mimsy were The Borogoves” remains: What if this limitation, not being able to conceive of something like Quantum Logic, is something only adults suffer from. What if it is indeed more a question of habit than of nature, and children can be taught at a very young age to think along other logics besides classical logic and it’s extensions?
The possibilities would be endless,…it would be a beautiful Quantum Jabberwocky indeed, when the quarks could gyre and gimble, and y leptons would raths till the mbranes bryllyg with no grabe or no wabe. One can always dream.
This post is an expansion of an answer I posted to the question “Is logic subjective?” on a philosophy QA forum.