The closest to this idea I could find was talk of ceteris paribus laws, but it was hard to tell how much this concerned physics, even just potentially/speculatively. But having decided to believe that most or all the axioms of set theory, say, "break down" the farther one goes, I had cause to wonder if the laws of physics have natural exceptions, too, or rather if one could conceive of such a law as had exceptions.

Something like a halting moment for a (hyper)computer, maybe? Note also that this idea is not identical to Lee Smolin's "changing laws of physics," at least if the change is more permanent on Smolin's picture (I don't know the details, though). And there seem like there might be some affinities between this fail-rate concept and the "new" riddle of induction.

  • We're, as usual, interested in the one thing that matters when it comes to failure/success. Mar 3, 2023 at 10:32
  • What do you mean by "fail rate" or exception? Like all of science is intrinsically wrong and has a margin of error. And if you find an exception you've reduced a law to a cp law and revitalized the search for a better formulation of that law.
    – haxor789
    Mar 3, 2023 at 10:47
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    Physicists distinguish between "fundamental" and "effective" theories, and the latter have a range of applicability beyond which their laws break down. Newton's laws, for example, break down at high velocities, the Standard Model at high energies, general relativity at small scales, etc. On empiricist epistemology, all laws are only effective, this is supported by the renormalization approach in QFT, see Cao-Schweber, pp. 66ff:"We thus obtain an endless tower of theories... in which none can ultimately be regarded as the fundamental".
    – Conifold
    Mar 3, 2023 at 12:46
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    Another option is the "dappled world" realism of Nancy Cartwright, where the laws are just statistically stable effects of "nomological machines", "a fixed (enough) arrangement of components, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of regular behavior that we represent in our scientific laws”. This view of all laws as ceteris paribus is shared by the "disunity mafia" in philosophy of science
    – Conifold
    Mar 3, 2023 at 13:09
  • @haxor789 Popper characterized laws of physics as universally quantified propositions, IIRC (David Gudeman covered this in an answer to a separate OP on this SE). But unrestricted quantification is problematic, and here, let us suppose the world is continuous, and so laws can potentially have uncountably many instances. But so is there some ordinal, covered by the cardinality of the Continuum, such that a hypercomputer, on that ordinal input, halts? And then if a law is equated with the action of a hypercomputer, we have an infinite quantification with at least finitely many exceptions. Mar 3, 2023 at 15:55

3 Answers 3


Of course, many derived laws are ceteris paribus in a trivial way. It concerns especially laws in sciences like chemistry or biology, but there are plenty of those in physics as well. Models they come out of make simplifying assumptions applicable in special contexts only, and are not expected to work universally. Dynamics of ideal fluids is one example. I will describe two more interesting conceptions that support ceteris paribus laws "all the way down", one empiricist and one realist.

The laws of specialized models/theories mentioned above are called "effective" in science, and opposed to "fundamental" laws that are meant to be universal and exceptionless. Some theories originally intended as fundamental turned out to have a limited range of applicability beyond which their laws break down. Newton's laws, for example, break down at high velocities, the Standard Model at high energies, general relativity at small scales, etc.

Pessimistic induction on this historical experience leads some empiricists to submit that all laws are only effective, to be supplanted by those of later theories. Renormalization approach in QFT naturally lends itself to such philosophy because the renormalization procedure presupposes a high energy cutoff, at which the theory presumably breaks down, see Cao-Schweber, The Conceptual Foundations and the Philosophical Aspects of Renormalization Theory, pp. 66ff:

"Since nobody knows what the renormalizable theory at the unattainable higher energies is, or even whether it exists at all, we have to probe the accessible low energy first and design representations that fit this energy range. We then extend our theory to higher energies only when it becomes relevant to our understanding of physics. We thus obtain an endless tower of theories, in which each theory is a particular response to a particular experimental situation and none can ultimately be regarded as the fundamental.

[...] The epistemological position supported by the recent developments in renormalization theory, especially by the EFT approach, is empiricist in nature. Physical theories are justified by empirical data from which the theories are abstracted. They are effective instruments for organiz ing the data by imposing local order and coherence, and they conceive and express local causal regularities... This position rejects uncompromisingly the idea successively ad vanced during the last fifteen years by grand unified theorists, supergravity theorists, and superstring theorists that the development of fundamental physics will end with the discovery of an ultimate, definitive, and conclusive mathematical formalism. Rather, the development is taken as a process of successive extrapolations that is assumed not to have an end, with every step of the extrapolation being justified by a collective reinterpretation of theory and observation before and after the extrapolation."

The realist option also rejects the Galilean (or, perhaps, Platonist) conception of nature as a "book written in the language of mathematics" expressing its exact laws. But it adopts instead Aristotelian ontology of objects having individual causal powers ("propensities", "capacities", "dispositions"), whose exercise and interaction produces pockets of stable regularities under the right circumstances, ceteris paribus. And pockets of disorder otherwise. This is the "dappled world", in Nancy Cartwright's metaphor, that leads to the idea of disunity of science, advanced e.g. by the Stanford disunity mafia (Dupré, Hacking, Suppes, Cartwright) in opposition to the traditional optimistic unification through reductionism.

The most developed ontological proposal in this direction are Cartwright's "nomological machines":

"It is a fixed (enough) arrangement of components, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of regular behavior that we represent in our scientific laws."

Timpson in Quantum Bayesianism, pp.25ff describes how such ontology would support statistical interpretations of quantum mechanics:

"But how to think of this unspeakable micro-level? It is here that we can begin to engage with some off-the-shelf ontology; for an immediate thought that might be suggested is that we should opt for a conception in which the basic systems have largely modal or dispositional characteristics, rather than occurrent, categorical ones. Thus the systems primarily have dispositions to give rise to various events when they interact with one another. Some of these events produced will be the outcomes of interaction of smaller systems with the larger composite systems (for example, measuring devices) with which we are familiar; and it is these events that the Bayesian agent will update upon.

The ontological picture being borrowed from here is of course that of Nancy Cartwright (Cartwright, 1999) who advocates an ontological picture for science in which objects primarily have dispositions or powers and it is only when these powers interact in highly contrived, or highly specialised, situations that they will give rise to the repeatable, regular behaviour that can be described by the kinds of general statements we traditionally think of as laws of nature, or as lawlike truths. Where things differ in our case is that we are imagining that at the fundamental level there are no situations, however specialised, in which we will obtain lawlike behaviour. Interactions of the powers of our micro systems always give rise to unruly results."

The "dappled world" picture faces challenges in the face of laws that do appear to be exact, see McArthur, Contra Cartwright for a critique.


As others have pointed out, there are many laws of physics that 'fail' outside their scope of applicability. In that sense the laws are approximately true in a limited set of circumstances.

There are statistical laws in physics that are good models for large ensembles, but 'fail' when applied to small samples.

There are patterns in physics in which you can spot a clear trend which 'fails' with exceptions. For example, the heavier elements are radioactive, but technetium is an exception.

There are many cp rules in physics. For example, a heavy disk is harder to flip than a light disk, but not necessarily if the light disk is spinning.

However, if what you mean is a law that holds almost all the time but with inexplicable exceptions- eg electrons all having the same charge, except for the odd one now and again being found momentarily to possess two thirds of the charge- then I am struggling to think of one. I suppose the reason might be that we only call a pattern a law if it generally holds.

  • The laws don't fail, what fails are models you build out of them, if applied to real world processes for which those models are insufficient..
    – Karl
    Mar 4, 2023 at 19:52
  • @karl You are talking at cross-purposes. You are using the word law in a different sense to my use of it. Mar 4, 2023 at 20:09

Most if not all laws of physics have well-defined ranges of operation where they furnish accurate predictions of experimental outcomes, but if applied outside those ranges, they do not- because new physical phenomena must be taken into account in those circumstances.

For example, any law that contains a term like 1/(distance)^2 is doomed to fail when (distance) becomes small enough, because then 1/(distance)^2 approaches infinity, and there are no infinite quantities in the realm of physics.

As another example, Newton's law of gravity is known to fail when the mass of the gravitating object becomes large enough that its escape velocity approaches the speed of light. Einstein's laws of general relativity then take over in that realm.

In addition, Newton's laws of dynamics are known to fail when the velocities of the moving masses that appear in those laws approach the speed of light. They also fail when those masses become very, very small- which is the realm of quantum dynamics.

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