I heard several times that the results of quantum mechanics (double-slit experiment for instance ) challenge our logic. One example of that is the famous physicist Lawrence Krauss.

He keeps saying that after our discoveries of quantum mechanics, "logic" is becoming flawed. He wears a 2 + 2 = 5 T-Shirt to support his case.

Does anyone know exactly what are physicists referring to when they introduce the word "logic" to say it shouldn't be taken for granted ? Would "logic" be common-sense? Would "logic" be sensory experience? If yes, then I'd highly agree. While we can't experience aurally frequencies out of 20-20khz range we are certain that they do exist.

On the other hand, if "logic" refers to our deductive and inductive logic then I don't understand.
How could we say that the result of lots of year of deductive and inductive logic (from Thales to Newton to our modern science) points out that deductive and inductive logic is flawed.
Mathematics also rely heavily on deductive logic. Mathematics is also the language in which physics is formally expressed.
Denying or not taking for granted even our "deductive logic" would be not taking for granted Mathematics in which case would end Physics.

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    The 2+2=5 t-shirt also has the sub-caption "for extremely large values of 2", which indicates it's absurdist (joking) nature [I tend to interpret it as a commentary on how physicists are willing, in some cases, to abuse formal mathematics].
    – Dave
    Commented Jun 20, 2013 at 18:03
  • "the reason i didn't post in philosophy communities is that they certainly know a lot less ( if anything ) of quantum mechanics than most of you here." –nerdy, physics.SE
    – stoicfury
    Commented Jun 21, 2013 at 17:46
  • 3
    In general, posting the same question to multiple SE sites is discouraged; you should pick the best place for your question and go with it. If you are displeased with your answers on one site, you can request it be transferred to another, instead of making a whole new question. :)
    – stoicfury
    Commented Jun 21, 2013 at 17:49
  • @Dave: I always thought the "2+2=5" joke is about approximation error; e.g. the "true" calculation being done here is 2.3 + 2.3 = 4.6, but all of the parameters involved are only being reported to one significant figure of precision.
    – user6559
    Commented Jul 8, 2018 at 19:42

6 Answers 6


Von Neumann in the early days of Quantum Theory came up with Quantum Logic. He noticed that that the mathematical apparatus describing a system can be seen as a generalisation of ordinary logic. This of course only a formal similarity but he believes in taking this as a serious hint about how to think about Quantum Theory.

Formally he interprets Quantum Theory (solely in its mathematics) as a probability calculus based on a non-classical lattice. (Technically speaking it is measure theory generalised from sigma-algebras to ortho-algebras). This lattice is not distributive & is ortho-complemented, whereas the classical boolean lattice is distributive and complemented.

The lattice is non-classical because there are measurements we can make that cannot be done simultaneously. This is the content of Heisenbergs Uncertainty principle.

Now as far as Logicians are concerned this is only a fragment of a logic. There is no mechanism for inference, nor its truth-functionality. That is we cannot say whether the logic is complete or sound - which to Quine was a serious argument for taking FOL as logic.

Well before Krauss, it was Finkelstein & Putnam in the sixties that began to aggressively push the non-classical logic centre-stage, mainly it seems to get the mainstream take these kinds of logics seriously.

More recently, Isham & his colloborators are using Topos Theory to provide semantics & a proof theory to quantum logic in the form of the Bohr Topos. Notably the proof system is not classical either it is higher order typed intuitionistic logic. It is also complete & sound. One interesting result is that the kinematics of the Bohr Topos when viewed internally is classical & when viewed externally is quantum-mechanical.

This is a new way to take seriously Von Neumanns dictum that it is the quantum logic that is essential in QM. Its far too early to say how important the results are though.

Finally, the double-slit experiment doesn't challenge our logic but our physics as then conceived. That is then Physics had a simple deterministic interpretation - symbolically the 'clockwork universe'. In fact this experiment does have a classical interpretation via Bohmian Mechanics - but it is non-local. One should note that the Greek atomists did include indeterminism in their physics. As Newton did preface his Principia with an excerpt from Lucretious De Rerum Natura, it probably would not have come as a surprise to him to see its advent again.

And yes, 2+2=5 is a joke - and not to be taken seriously - other than to provoke a little thought amongst the mathematically or logically inclined.


This SEP article on Quantum Logic is a good place to start.

Although Krauss is right that Quantum Logic is non-classical (most presentations I've seen either reject the law of excluded middle or go paraconsistent), and so therefore not the standard logic we are used to, he is vastly overstating the claim if he says that "logic is flawed" (without heavy qualification) and if he thinks that anything about Quantum Mechanics shows that 2+2=5 is possibly true.

At best you can claim that classical logic is not the appropriate logic for reasoning about quantum phenomena--- this much seems right. If you held some sort of view that there could only be "one true logic" (this is a claim that, e.g, Quine made about the classical first-order predicate calculus; see this article by Gillian Russell if you're interested in the "one true logic" thesis and challenges to it) then maybe you could claim that Quantum Physics shows us that classical logic isn't the one true logic, and that instead it is some sort of paraconsistent logic (though how 2+2=5 follows from this is anyone's guess; "paraconsistent" doesn't mean "anything goes"). But better to simply give up the claim that there is "one true logic".

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    I recently wondered whether Quine actually caused any harm by his claim of "one true logic". Especially the requirement of completeness seems arbitrary. If he would have asked about first-order logic for finite models (as currently used in computer science), he would have also lost completeness. Does Quine wants to deduce from this that the notion of being "finite" is not well defined? Let's see what I learn from the article you linked. Commented Jun 19, 2013 at 23:51
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    @ThomasKlimpel I tend to think that Quine has caused some harm. So much discussion revolves around what should count as logic, with the question seeming much more pressing under the assumption that there could only be one thing called "logic"--- I feel like this is really wasted energy, a "pseudo-problem", if you will. I get the impression that this attitude is waning in contemporary philosophy. A sort of logical pluralism seems to be gaining popularity.
    – Dennis
    Commented Jun 20, 2013 at 2:33
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    I think Wittgenstein sorted all this out in his later works, and to some extent his earlier works, and the foregoing discussions are full of just the kind of confusion he resolved long ago. Logic not true or false of anything. There are no facts of logic, only rules of logic systems. Such rules are arbitrary, like rules of grammar. There are various systems of rules, formal and informal, and some of them are useful, others not. Talk of logical systems are true or false is confused. (Same is true of mathematics.)
    – adrianos
    Commented Jan 17, 2014 at 19:05

Before we can evaluate the logic of some statement or theory, we must first have a system of logic that serves as the standard of evaluation. Logic is a fundamental tool for validating statements relative to their content. Classical (Aristotelian) logic introduced the fundamental principles of any applied logic: 1. The law of identity. 2. The law of non-contradiction 3. The law of the excluded middle. These principles are axiomatic based on the natural universe, a universe that has identity. Thus the law of identity holds that everything that is, is what it is; it has a specific nature that defines its capabilities and limits. The law of non-contradiction holds that an assertion cannot be both true and false about the same existent in the same respect. The law of the excluded middle holds that any assertion about existents must be either true or false; there is no middle ground of truth.

Clearly logic, a tool of epistemology (which answers such questions as 'What is knowledge, and how is knowledge achieved') must rest on a system of metaphysics (which answers such questions as: 'What is being and what is the nature of being qua being. What do we know from the sheer fact that something 'is'?

Classical physics is based on a metaphysics that views the world as having a non-contradictory identity and on an epistemology that views knowledge as the process of identification and integration of the facts of existence, through the power of reason, using its tools of induction and deduction constrained by the laws of classical logic.

Based on my layman understanding of it, quantum logic appears to be an inherently ambiguous concept. It generally refers to the special nature of quantum 'facts'. These include 1. That no absolute mathematical facts about instances of existents are possible at the quantum mechanical level, since all facts about the measurable properties of existents are stochastic (defined entirely and exclusively in terms of probabilities, not actualities), 2. Probability, in the QM view permeates the universe thoroughly and metaphysically, implying that at the QM level (which quantum theory holds to be the foundational level of all existence) nothing has an ontological determinate value, all properties of existents being absolutely ontologically stochastic, 3. Human knowledge (including scientific knowledge) of concrete existents requires measurement (that is any form of epistemological 'looking') and that the very act of 'looking' causes the quantum phenomenon under observation to collapse its probabilistic wave into an determinate transient 'fact'.

This QM epistemological view is, of course, in direct contradiction to classical views of knowledge that hold that facts are neither caused by nor influenced by acts of consciousness (perceiving, thinking, wishing, etc.). So here is where the epistemology/ ontology of QM jars the rational mind: The assertion that looking or measuring alters the state of the physical world and causes facts to come into (as well as their contraries go out of) existence. This view is a species of the primacy of consciousness, which holds that consciousness, on its own without body, can cause reality to come into existence. This is the basis of theories of God as the Creator of the Universe. Modern science has, until QM theory, repudiated the primacy of consciousness doctrine. Now it is part of QM doctrine. No wonder theologians are now encouraged by QM!

Part of the problem in understanding QM logic is that QM theory does not seem to embrace the established meaning of the concept of probability. Probability is an epistemological not an ontological concept. It refers to limitations of knowing about a fact in a given context (such as a coin toss), where the initial conditions could result in any one of a number of possible (potential) outcomes (actualizations). Probability is a mathematical tool for quantifying the uncertainty of stochastically determined potential outcomes. Probability, in this view, is not an ontological statement about facts of existence in the objective world (independent of consciousness). The world itself and its existents are what they are at any moment. What is possible to entities as an outcome of their previous states is determined by their internal determinate nature. The view that the universe embodies natural laws means that the universe is determinate and that any fact about the universe that we care to study has a determinable nature. This part of what it means to say that facts (not our ability to know them in a given cognitive context) are determinate and not merely probable. This is what Einstein meant by his assertion: 'God does not play dice with the universe.'

For an excellent discussion of Einstein's views on Quantum Theory and its relationship to reality, see Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality by Manjit Kumar, W. W. Norton & Company, 2008. See especially Chapter 13 Quantum Reality for a full discussion of the ERP (Einstein, Rosen, Podolsky) critique of QM as incomplete in its lack of correspondence mapping between its concepts and the physical (real) world.

From my blog article: http://bioperipatetic.com/aristotle-and-the-philosophical-crisis-of-quantum-theory/

This problem of ‘becoming’, of process and temporal continuity of being is relevant to quantum mechanics and attempts to overcome the static reality view of the Copenhagen interpretation. The Copenhagen interpretation of quantum mechanics is not the only embraced interpretation. There are others, equally or even more bizarre. Some have reduced the degree of strangeness by introducing explicitly Neoplatonic ideas into their interpretation. Most prominent and widely embraced, and least paradoxical, is the ingenious theory of David Bohm, which is based on the Cusanian idea of the universe as the unfolding (explicatio) of a transcendent enfolded (implicatio) world whose nature is fundamentally mathematical. This is the fundamental vision underlying Bohm’s ‘implicate order’, illustrated by his famous thought experiment: the Glycerine Machine. What make David Bohm’s model of ‘the implicate order’ so significant is that it introduces the fundamental missing element of quantum mechanics: process and continuity, what Arran Gare would call the dimension of ‘becoming’. Here are David Bohm comments about the absence of the concepts of movement, process and continuity in quantum mechanics:

You see, the present quantum mechanics does not have any concept of movement or process or continuity in time. It really deals with one moment only, one observation, and the probability that one observation will be followed by another one. But there is obviously process in the physical world. Now I want to say that that process can be understood from the implicate order as this activity of re-projection and re-injection. So, the theory of the implicate order, carried this far, goes quite beyond present quantum mechanics. It actually deals with process, which quantum mechanics does not, except by reference to an observing apparatus which in turn has to be referred to something else. — from Morphic Fields and the Implicate Order: A dialogue with David Bohm, p. 7.


He wears a 2 + 2 = 5 T-Shirt to support his case.

It was a tasteless joke. Refer to comment#39, by Krauss, here. There is no point analyzing this any further. You will not get a defense, even from Krauss.

Krauss' point was only that common-sense logic, no matter how well-applied can lead you astray when dealing with quantum or relativistic domains. To start with, one can't just add velocities like 2+2 when it comes to relativistic ranges. (I did not specify the units for these numbers). Then, use all the notions you have from relativity and try to make sense of the measured speeds of breaking entanglement. I don't know what kind of logic will explain the double slit experiment either.

  • The joke makes much more sense than if it were something serious. Although Krauss is prone to overstating his case (c.f. the recent "Something from Nothing" controversy between him and David Albert), I couldn't imagine him making that leap. Thanks for providing more context, I was a bit baffled. I actually think the joke is kinda funny when you hear the whole thing "2+2=5 for large values of 2".
    – Dennis
    Commented Jun 19, 2013 at 22:28
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    The double-slit experiment does have a realist (the wave function is real) but non-local interpretation - Bohmian Mechanics. Interestingly enough the Born Rule is a deduction here rather than an additional postulate in the standard QM theory. Commented Jun 20, 2013 at 4:32

I think the 2+2=5 joke is based on quantum tunneling, which is when an event happens even when there is not enough energy for it to happen. For example, two atoms could fuse even if there is not enough energy present to fuse. Lawrence Krauss's T-shirt is suggesting that the same should happen is mathematics: 5 could be the sum of 2+2, even though there is not enough 'value' for it to happen. However, in quantum tunneling energy is not created, rather a barrier is passed over. The same idea does not apply in mathematics. It is a clever T-shirt, but nothing more.


Quantum mechanics does not challenge logic. Quantum mechanics in the Heisenberg picture describes the behaviour of physical objects using Hermitian operators called observables, see




When you observe the outcome of a measurement of some observable you see a value that corresponds to one of the projectors of that observable. That outcome sometimes can only be explained by taking into account the way some representative set of observables changes. That is, you have to take into account not only the other possible outcomes of the measurement you made but also changes in observables you didn't measure. The processes that account for the outcome of these experiments in some respects resemble multiple versions of the same system interacting with one another. People often seem to get confused about this and somehow think it requires changing logic or saying quantum mechanics is non-local or something similarly drastic. But quantum mechanics doesn't require any such change, it just requires applying logic consistently to describing quantum systems using sets of representative observables.

What you end up with when you do this is a theory in which physical reality consists not just of the universe we see around us but a larger structure called the multiverse which, under some circumstances and in some approximations, resembles multiple non-interacting versions of the universe as described by classical physics. In other circumstances, such as interference experiments or EPR experiments, you can't properly explain what is happening in terms of a single universe. See:


  • The challenge is to represent quantum mechanics directly in a deductive system, not via a complicated encoding over multiple levels (like encode Hilbert spaces in first-order logic + ZFC, and then encode Quantum mechanics in Hilbert spaces). One challenge then is that classically propositions P(x), P(x,y), "for all x P(x)" and "for all x P(x,y)" are fine, but for Quantum mechanics (and projective geometry) at least the last proposition might make no sense, because the cartesian product is "not well behaved" for Quantum mechanics (and projective geometry). Commented May 15, 2014 at 10:41

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