The Inconsistency of Science

Karl Popper

Karl Popper

Can a good scientific theory be logically inconsistent with itself? If you think not you’d be in good company: Karl Popper, the famous philosopher of science, believed that consistency was the sine qua non of a good scientific theory. It seems obvious why. If a theory is internally inconsistent, after all, then anything follows from it. If anything follows from it then it wouldn’t matter what experimental results we get, the theory would not be falsified. To Popper an unfalsifiable theory cannot even be counted as science, never mind bad science.

The Square of Opposition

The Square of Opposition

It is Classical Logic that tells us that anything follows from a contradiction. This is the (in)famous ex contradictione quodlibet or logical explosion. If a theory contains the proposition A and the proposition not-A, then it doesn’t exclude any proposition at all. Imagine if someone told you that Marianne is wearing jeans, and moments later told you that it is not the case that Marianne is wearing jeans. Such a person has not, in his utterances, excluded any possibility has he? You have no idea at all what Marianne is wearing. Nothing at all she might be wearing falsifies the conjunction of these two propositions.

Niels Bohr

Niels Bohr

It would seem, though, that logical consistency as a constraint on a good scientific theory, is often flouted. There are, in the literature, many cases of scientific theories that have been inconsistent. And these are not any old theories either – they include Bohr’s Theory of the Atom, Classical Electrodynamics, Newtonian Cosmology, and the early calculus.

This cries out for explanation. But how should it be explained? It would seem that there are only two possible explanations:

  1. we are wrong to think these theories are internally inconsistent;
  1. we are wrong to insist classical logic is a constraint on scientific theorising.

During the weekend school on 9/10 January we had representatives of both explanations.

Peter Vickers

Peter Vickers

Peter Vickers, of Durham University and author of the book on which the weekend was based[1], believes that most of the canonical examples of inconsistency in science are not examples of inconsistency at all. In fact he can find only two inconsistencies, one is the contradiction between Bohr’s postulates and Paul Ehrenfest’s adiabatic principle  (see chapter 4, section 3.3. pages 58-71). The other can be seen only on a particular reading of the main theses of Newtonian Cosmology (see chapter 5).

Neither of these inconsistencies, furthermore, is a worrying inconsistency, says Vickers. Neither is a doxastic inconsistency – a case of scientists believing a contradiction, both are pragmatic inconsistencies, similar to using approximations, idealisations or abstractions. There is no threat, therefore, to Classical Logic from any scientific theory that is both internally inconsistent and a good scientific theory[2].

Dunja Seselja

Dunja Seselja

Our other speakers, though, Christian Straaser  and Dunja Seselja  of The Institute for Philosophy II at Ruhr University Bochum, Germany disagree. They believe that pragmatic inconsistencies are every bit as important to our reasoning as doxastic inconsistencies. The inconsistencies identified by Vickers, therefore, are a threat to Classical Logic: as internal contradictions are not always and everywhere treated by scientists as a threat to their theorizing, the logic implicitly used by scientists cannot be classical logic.

Both Strasser and Seselja believe that the inconsistencies found in science motivate the introduction of some Paraconsistent Logic; a logic that permits contradiction without its leading to logical explosion. Such logics – and there are many – can accommodate inconsistencies. According to those who embrace such logics much scientific reasoning, especially in the early stages of a theory, can only be described in terms of some paraconsistent logic.

So do you think that scientific theorising accommodates inconsistencies? Or do you think that any appearance of inconsistency must be either misleading, or taken as a black mark – a sign that the theory is false?

If you find the topic of this weekend school fascinating (as most participants at the weekend school did) you can, by joining the OUDCE Philosophical Society, listen to the podcasts of the lectures. Alternatively you can read the books and papers on the reading list:

Reading List: 

More important

Seselja, D., and C. Strasser (2014): ‘Concerning Peter Vickers’s Recent Treatment of Paraconsistencitis’, International Studies in the Philosophy of Science 28, Issue 3, pp.325-340.

Vickers, P. (2013a): Understanding Inconsistent Science. Oxford: OUP.

Recommended additional reading

Harman, G. (1986): Change in View, Cambridge, MA: MIT Press. **Especially Chapter 1**

Saatsi, J. and Vickers, P. (2011): ‘Miraculous Success? Inconsistency and Untruth in Kirchhoff’s Diffraction Theory’, British Journal for the Philosophy of Science 62, no.1, pp.29-46.

Vickers, P. (2013b): ‘A Confrontation of Convergent Realism’, Philosophy of Science 80(2), 189-211.

Weingartner, P. (1993): ‘Can there be Reasons for Putting Limitations on Classical Logic?’, in P. Humphreys (ed.) Patrick Suppes: Scientific Philosopher, Vol.3, Kluwer, pp.89-124.

[1] Peter Vickers, Understanding Inconsistent Science, Oxford University Press, 2013, 273pp., $65.00 (hbk), ISBN 9780199692026.

[2] Vickers is adamant that the concept of ‘theory’ carries no weight in his argument. In fact he relies on ‘theory eliminativism’ throughout his book, thinking of an inconsistency as holding between propositions, rather than between, or within theories. This, he argues, avoids all the confusions and vaguenesses of ‘theory’ talk.

About Marianne

Marianne is Director of Studies in Philosophy at Oxford University's Department for Continuing Education
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17 Responses to The Inconsistency of Science

  1. David Lilley says:

    Briefly. Bishop Berkley could legitimately write about vision but Russell shouldn’t have. Vision had passed from philosophy to natural science to a bona fide scientific discipline, esoteric and beyond the scope of philosophy. Similarly it is illegitimate for a philosopher to leave his discipline and comment on an extremely esoteric scientific discipline like quantum mechanics, quarks, black holes. In these subjects the philosopher is a layman and can make no contribution. You cannot be a Hume and a Dirac at the same time. You would have to be amazing to be just one of them. Similarly, Einstein and Hawkins should not comment on the existence or otherwise of an absolute being or the best diet.

    The only line on inconsistencies is that it indicates error and the need for a new tentative solution. And please let that come from the core discipline and not the layman.

    • Marianne says:

      Vision has indeed passed from philosophy to science. But perception hasn’t. Vision is necessary to (one form of) perception, but it isn’t sufficient (animals see things, it is not clear they perceive things (i.e. bring them under concepts).

      I agree that there are certain things that the philosopher has no business commenting on(anything empirical), but I disagree that this leaves him with no scow for talking about physics – especially the highly theoretical end. Descartes was a mathematician and a philosopher – but those days are largely gone. But there is still a philosophy of mathematics:

      Paraconsistent logicians do not agree that inconsistency = error.

  2. Rossetti says:

    Kind people,

    The whole of science is unfalsifiable in its basic approach. According to a crude man like Popper, if he is consistent, the whole of science is bad science. We can not consider the witterings of such a low-caliber brain, except polemically.

    Logic is not necessary to human life (even the concept of inconsistency, taken in any strict scientific sense, is not necessary, of course one notices it in a broad sense in ordinary speech), and the one used is derived, historically, however much modified, from Greece through a theory of the alien and culturally rare or even unique notion of essence (being, ousia) as the grounding of the concept of identity (e.g. there is no word for is in ancient Hebrew nor Arabic and the Romans found the Greek logic strange). The standard of consistency as a securer of truth is useless (i.e., it is either a convention or a non-necessary rarity of cultural outlook). It might never have been developed, and so its specific manner of seeking would have not come along, it is not necessary to a search for truth. Which means that the only standard in such cases can be what produces the results folks want. Consistency is simply a matter of expedience in a given working field; it’s merit to be judged ad hoc and in accordance with the results wished for by the practitioners.

    Science is not what it claims to be. It does not produce purely theoretical laws in any domain, such as can latter be truly applied (only in an ad hoc practical manner). It is, rightly thought, a kind of uber handicraft that can make no special claim to true knowledge. No more than a shoemaker.The imposing character of all so-called science is peculiar to the so intensive and so extensive practice, for it is the chief activity of great institutions such as our dear universities. And at the time of the highest populations ever known.

    I hope this Cassandraing will not fall on deaf, but rather on sensible, ears.

    • Marianne says:

      Oh dear, I am afraid that reading your first sentence I was already disliking intensely not WHAT you were saying (which I wasn’t really taking in), but the WAY you were saying it. On this website I should like people always to recognise the Principle of Charity. We do not call people names, or say they are stupid. Karl Popper is a hugely respected philosopher of science. In calling him a ‘crude man’ you do not argue against him, you just insult him.

      Of course logic is not necessary to life. There are plenty of living things that do not use logic. But arguably logic is necessary to a full human life. To the exert that human beings are rational animals they will not flourish unless they use logic.

      • Rossetti says:

        But which logic? I think in common sense we say, “If Kiyoaki says she doesn’t speak French, but the next day I hear here conversing fluently in French, I must judge that she is being contradictory”, as only in extreme speculative terms can there be said to be a chance that she might have learned in her sleep.

        I’m not sure what the conundrum is about. Is it for science that one needs logic. Or for life? Why a specific strict form of logic is necessary for life is not clear to me. And if it is for science that is just a matter of what gets the job done isn’t it?

        If someone uses brutish rhetoric to contentiously conceal their guff the only thing to do is observe that they are stonkingingly crude. As was the reason given by his would-be colleagues for Popper being rejected by the University of Chicago. If you use polite words, illogical, no substance, don’t you just treat the people you’re talking to as fools. Do you deny that there are idiots?

        Put more simply: Does the Principal of Charity extend to non-philosophers?

        There was something funny on this point in the FT recently:

      • Marianne says:

        Which logic is not determined. There are many different ideas about the logic that we use in everyday life (classical logic, intuitionistic logic, paraconsistent logic….). But that we use some (or several) logic(s) is not controversial. This is true in science, but also in everyday life. The Principle of Charity extends to everyone, and arguably Popper didn’t use it often enough. I followed your link, but couldn’t find what it was that you might be referring to….?

  3. J. Rubinstein says:

    In deference to your first respondent I should say that I am not a scientist but I am very interested in its progress. Surely the default position of science is doubt. I’ve lost track of the number of times I have heard scientists say how excited they are about holes and inconsistencies in their theories because these enable them to challenge and develop their understanding. I have heard a scientist describe their discipline as developing highly nuanced doubt, and in addition, that once doubt is removed, only engineering is left.

    When I look, by contrast, at the imaginations of conspiracy theorists, they are often logically consistent. They just happen to be, in my judgment, probably wrong.

    It is surely the ability to look at logical inconsistencies and subject even while logically consistent theories to scientists that distinguishes scientific theories from religious ones.

    • Marianne says:

      Karl Popper would agree with you that it is the possibility of falsification that demarcates scientific theories from non-scientific theories. Marianne

  4. rossetti says:

    kind people,
    Thanks for your engagement with these augments.

    Dawkins will not suffer from your own demonstrations of his feebleness. Popularity and idiocy are wholly compatible states. Why do we have to listen to people like Popper, people who talk guff? Sheer prejudice of the cannon.

    Which logic is not determined. There are many different ideas about the logic that we use in everyday life (classical logic, intuitionistic logic, paraconsistent logic….). But that we use some (or several) logic(s) is not controversial. This is true in science, but also in everyday life. The Principle of Charity extends to everyone, and arguably Popper didn’t use it often enough. I followed your link, but couldn’t find what it was that you might be referring to….?

    We don’t use intuitionist or paraconsistent logic in everyday life. Those are only used by mathematicians. I don’t understand what could be meant by that. And I’m not making a pedantic point, but a simple and literal one. Hegel, for instance, does not use intuitionist logic either, although everyone seems to guff on about him doing so, surely not in any practical sense (whatever else happens in extreme and possibly imaginary speculative theory). In life we just notice in one way or another when people are roughly inconsistent or consistent.

    A true logic, one determined strictly by truth, would not be of much help for life so far as I understand the issue. For the very simple reason that one must first produce premises which the logic can then operate on. Everything of any issue happens prior to the instrumentality of logical inference.

    I think the issue is simply that one does not consider something like ad hominem can only be taken as an extreme guideline. And only a simple idiot, someone who falls into wells and such things, can treat it as determining. Life demands that we ask about the source of the argument. Lucy Kellaway, the author of that article, is well aware that folks like Popper waste our time with illogical guff. Very serious thinkers at The University of Chicago have already had this time wasting experience. Why can’t we have some common sense and drop the guff from the cannon?

    • Marianne says:

      Hi Rosetti, my heart sinks when I see posts as long as this. I try to do this page alongside my full time job, and other things! Also you have left much of my post in yours which makes it harder to read.I would appreciate it if you would take this into consideration when you post again.

      Popper does not talk ‘guff’. He is a famous philosopher of science for a reason. His claims about falsificationism are extremely important. Logicians try to capture formally the way we reason in everyday life. You cannot say that we do not use either of these logics. There is reason to think we use both (not conclusive reason of course). Logic is not determined by truth. A correct logic will help us determine truth. The truth (to my mind anyway) is there before logic.

  5. rossetti says:

    “You cannot say that we do not use either of these logics.”

    1.Can you give one serious example?

    I really don’t understand that. Although I can, with charity, see quasi-examples, like regurgitating one’s cake, and then saying, I have my cake and eat it too. But they are frivolous. In life, so far as anyone of the slightest common sense can detect, “logic”, which is a strange Greek project, means only consistency, i.e., telling the truth. Speaking truly or falsely. Logos means speech, Popper hardly knew that. So bad was his grasp of the Greeks which got him refused for a job at the University of Chicago by his manifest betters, who deemed him an illogical and wholly a waste of time.

    I think all you know about Popper is that he is connected to the catchphrase falsification. But I am not so sure even that makes so much sense as people seem to take it to on faith.

    2.Is the theory that the earth spins “falsified” by the fact that it is, now, standing still under my feet?

    The principle of charity is ridiculous except if it is applied to serious thinkers. With the aim of confronting their thought. When applied to someone like Popper and Dawkins it is a frivolous time waste excusable only on the basis that their popularity is dangerous to serious thought and truth.

    • Marianne says:

      I am sorry Rosetti, I have approved your comment in case anyone else wants to reply, but I am not going to reply to a comment which so obviously violates the Principle of Charity. This is my website, and I make the rules.

  6. Chris Matthews says:

    Love, hate, epiphanies and the placebo.

    Human experience and individuals are by nature full of contradictions.

    We can only know or comprehend and ‘think’ within the limits of the human experience; unless you are willing to include faith, believe and ideas about God or Gods – and even then our ‘experience’ and understand (of same) would be through the lens of human perspective.

    Therefore is is not necessary for scientific theory or investigation to be constrained by the notion that any such activities and ideas mush be internally bond by an arbitrary set of dogmas (that may include logical consistency).

    In many ways, science and those who toil to investigate, hypothesize and expand our ‘human’ understanding; would be wise to include some measure of uncertainty; doubt and humility in their approach to scientific discovery.

    I think it is well known, that many of the most amazing discoveries (in the field of science) happened accidentally (unintentionally) whilst in pursuit of solving another problem and/or even a simple lab error like leaving a petri dish out overnight.

    However, to add a bit of a twist. I think that being humble, flexible and even intentional about the use of or inclusion of logically inconsistent ideas within same; can and should be rational.

    In other words, if in an attempt to explain or summarize the interim steps into the realm of quantum mechanics; it becomes necessary to create a theory (that is ostensibly) illogical or inconsistent with the contemporary ‘rules’ of physics and the maths used for same – it is a rational exercise to employ theories and hypothesis about ideas that are by nature irrational and illogical behaviour(s) to support the work until a more complete understanding and discoveries can be made to hopefully explain it (all).

    I believe it has taken almost 60-80 years for mathematicians and material science to agree and a ‘theory of everything’ to explain and marry the wide differences (observed) in the nano scale with larger physical multiverse – within the limits of our human perspective and comprehension.

    PS – thank you for introducing me to both the person and work of Karl Popper. I thoroughly enjoyed reading up on him.

    • Marianne says:

      Hello again Chris, another long post! You really would get a reply much more quickly if you could cut down your words! Who is suggesting that scientific investigation should be ruled by dogma? Or are you suggesting that logical consistency constitutes dogma? There are those who’d agree with you (as you probably realise, anyone who embraces paraconsistent logic).I think doubt and humility ought to the the sine qua non of science. Unfortunately scientists (like everyone else) are only human.

      The trouble with neglecting logic is that there are then no constraints on what you do, or how you interpret something. Contradiction may be a barrier, but it is a bearer because it places a constraint – it says ‘stop here’. That’s what you are objecting to, of course, but without constraints, anything goes, and that’s very scary – because where do you stop? How do you recognise truth except in contrast to the falsehoods signalled by contradiction? Having said that QM is not of course neatly fitted into the rules of classical logic – and there’s nothing to say that classical logic is the logic we should use.


  7. Łukasz says:

    Well, I recommend the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072. In this paper, a proof of the consistency of the Arithmetic System was published. This proof had been done within this Arithmetic System.
    Of course, this result can be considered as wrong, because when anyone is hearing about this result, thinks at once about Gödel’s Second Incompleteness Theorem.

    However, on the other hand, (this fact is often neglected), Gödel’s Second Incompleteness Theorem tells that consistency of Arithmetic System within this System is unprovable, UNLESS this System is INCONSISTENT.

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