Many physicists use 'reality cannot have a true paradox, only apparent ones in our theory' as a very common maximum and guide to theory checking. As with all universal rules, Paul Feyerabend disagrees[1] that this is useful rule across science, and suggests that there are facts that are best described by inconsistent theories.

Unfortunately he only gives one such example[2]: optical illusions where a static object appears to be moving. He asserts that "[t]he only phenomenologically adequate description is 'it moves, in space, but it does not change place'".

In a footnote (ft 92, ch 16) Feyerabend tries to address the obvious criticism (raised to him by A.J Ayer and G.E.L. Owen) that the above statement deals with appearances by saying that putting "it appears to move" in front of the statement just pushes the question to the level of appearences but doesn't resolve it.

However, it seems to me that Feyerabend is completely missing the point, because he is assuming that we are saying (with glorious use of brackets added by me) that:

[A] "(it appears that it moves in space but does not change place)"

but any person actually making that statement actually means:

[B] "(it appears that it moves in space) but does not change place".

I would agree with Feyerabend that the statement [A] doesn't achieve anything except introduce an extra judgement statement, but statement [B] (which is what I think most people mean) separates two domains of 'appearance' and 'reality' and thus is not contradictory.


Feyerabend spends a lot of time talking about appearance and reality elsewhere, so I am worried that maybe I am missing some subtlety that makes statement [B] contradictory. Am I missing some subtlety in Feyerabend's critique of self-consistency?

Alternatively, if this particular argument is so weak then it seems like for such a bold claim he would support it more elsewhere. Are there other articles where Feyerabend (or subsequent authors) more closely defends his critique of self-consistency? Alternatively, are there more convincing examples of the failure of self-consistency?

Sources and further information:

[1] In his typically colourful language he writes (on pg. 245-6 of 2010 reprint of Against Method):

the idea that things are well defined and that we do not live in a paradoxical world leads to the standard that our knowledge must be self-consistent. Theories that contain contradictions cannot be part of science. This apparently quite fundamental standard which many philosophers accept as unhesitatingly as Catholics once accepted the dogma of the immaculate conception of the Virgin loses its authority the moment we find that there are facts whose only adequate description is inconsistent and that inconsistent theories may be fruitful and easy to handle while the attempt to make them conform to the demands of consistency creates useless and unwieldy monsters.

[2] His actual example is on pg. 202:

fixate on a moving pattern that has come to a standstill, and you will see it move in the opposite direction, but without changing position.

He would not approve of my use of the word 'optical-illusion' in the description, so I don't mean to smuggle in the naturalist interpretation, I am just giving the easiest way to remind you of the phenomena. I think the Rotating Snakes would be a fine and easier to see example of the sort he meant.

  • There are proposal about Inconsistent Mathematics , Paraconsistent Logic , G.Birkhoff and J.von Neumann's Quantum Logic. In principle, it seems that we may tolerate (some form of) contradiction in our description of the world. The real issue is : are those approaches fruitful ? Are them able to give us result comparable to those produced by "classical" logic and mathematics and physical science ? Mar 9 '14 at 9:57
  • @MauroALLEGRANZA I am vaguely familiar with para-consistent logics as paraphernalia in math, and they are certainly a requirement to start making sense of Feyerabend's point, but they are largely philosophical oddities than parts of science, hence less convincing. Quantum logic is different from classical logic, but not inconsistent in itself. Mar 9 '14 at 10:28
  • The issue is (for me) : may we admit theories that "tolerates" contradictions in our description of the world ? The concept of a "contradictory world" makes little sense for me. How we may "speak of" a world without a theory that gives us a description of it (where is the ding an sich ?) Mar 9 '14 at 10:31
  • @Kaznatcheev: If one accepts, quantum logic as a logic (in the formal sense) then its quite likely that there are inconsistent logics in Physics (in the formal sense); one clue is that dual-intuitionistic is paraconsistent. One may then have to ask why are we choosing to examine the consistent side of the logic, rather than its inconsistent dual. Mar 9 '14 at 13:37
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    When thinking, talking and writing of inconsistencies as part of the world, what "postmoderns" want is to divide the peoples, instead of allowing us to unite. It reminds me of the "doublethink" that was part of "newspeak" in Orwell's "1984" book. Have you ever seen their projects by this angle?
    – Rodrigo
    Apr 9 '14 at 12:09

Inconsistent theories do not mean that they are trivial, but they may be. For example in classical propositional logic once a contradiction is admitted the theory becomes trivial and essentially useless.

To adopt a non-classical logic may mean that one is simply playing games with the formal structure of the logic where one adopts or tries to adopt an interpretation of what the logic means - its semantics. For example classical logic uses Tarskian semantics, the same semantics that Wittgenstein adopted in his Tractatus which is that truth corresponds to existence and falsity to non-existence.

The BHK semantics, first put forward by Brouwer, is to assign truth the semantics of proof. It turns out then that the most natural logic is intuitionistic where the excluded middle is denied, and it is a multi-valued logic. This is not a logic that allows contradictions, but its dual does.

One might adopt such a paraconsistent logic to model epistemological truth where what is known may be in conflict with each other. So boundaries between domains of knowledge need policing. One might then argue that this is knowledge as represented to us, not truth in itself.

This leads us into the deeper waters that Feyerabend is pointing out. If one adopts the principle that the Jain philosophers expounded on anekantevada or more than one point of view as being absolute, in a similar sense that Wittgenstein said of logic being the limits of our world and not the world, then inconsistent logic is inevitable. Feyerabend took at least some of his cues from Hume and Kant who placed logic in the realm of human knowledge as opposed to being a property of the world itself.

This is still different from dialethism or true contradictions. Formally for example one solution of the liar's paradox is to adopt a three-valued logic which allows a statement to be both true and false. One still of course has to explain what this means. The Dao begins with a famous line:

The Tao that is the Tao is not the True Tao

One can take a dialethist position on this statement, or an epistemological position, in that truth that can be grasped by the human mind is always partial and 'true' truth always eludes us. The Catuskoti tetralemma shows also that the formal consequences of considering dialethism was considered formally - and one possible consequence of this, in at least Buddhist logic, is that an atom can be both a point and not a point. This gives a new picture of the continuum which as Aristotle argued against the Greek atomists must have cohesion. This is not far from the conception the intuitionistic continuum where a similar view is taken.

Hegel is probably the major philosopher in the West to have taken contradictions seriously. From Being and Non-Being comes Becoming is one of his early triads that sets his Geist in motion, or one might say is his Geist, for contradiction is "the root of all movement and vitality."

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    Although this is an insightful discussion of the general logical and philosophical points, it is really not what I am looking for. I am not looking for how we could potentially implement some non-consistent logic, but instead for actual cases in the history of what-most-people-consider science within a single theory (as opposed to the boundaries of two incompatible theories as we get with say quantum mechanics and folk physics). Mar 12 '14 at 12:32

The two-fold problem of observation was described in general terms in Plato's dialogs, the Theaetetus and the Cratylus.

The first problem is that appearances vary from person to person, and over time.

The second problem is that objects themselves physically change depending on aspect of presentation, such as parts and wholes, as well as over time. An apple will ripen then rot.

Science, especially physical experiments, will attempt to momentarily control these variables to obtain a snapshot observation, so data can be extracted through measurement. The consistency of data depends on repeating the same momentary control, both from a theoretical perspective, and also from an environmental and instrumental perspective.

Feyerabend's suggestion that there are facts that are best described by inconsistent theories, is one recognition that scientific aspects and perspectives are not absolute, just as personal observations are not absolute.

One great example is the Penrose Andromeda Paradox, http://en.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument . According to Penrose, who is one of the most prominent physicists of the last 50 years, a person walking toward the Andromeda galaxy will see an Andromedan supernova up to a week earlier than a person who is moving in the opposite direction. This is due to special-relativistic foreshortening of space, becoming apparent at that 2-million lightyear distance.

The observable universe varies in directional extent and color depending on the observer's velocity. There are more and bluer galaxies in the direction of our motion, and fewer and redder galaxies in the opposite direction.

This paradox is mind-bending because science, for the most part, does not distinguish the observed world from a commonsense or hypothetically real world. In physics, what you see is what you get.

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