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These are given not just as nice rules of thumb to follow or ways that one should think. Aristotle identified these as necessary conditions for thought. People sometimes try to produce counter examples to these Laws by pointing out how statements can become true or false depending on the conditions; e.g. "It is raining" might be true now, but was false yesterday" or "it is half way between raining and not-raining." But these attempts always involve changing the reference of the statement. Once we get the reference of the statement clear and explicit, it does not seem possible for a statement to make sense and violate these laws. At this point, the 2 thousand year debate over the nature of reference begins, and the shape of philosophy is drawn.

This was said in reference to the idea that the law of identity could have some kind of counterexample, but I was wondering if there could be a counterexample of the law of identity?

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    I have reservations about the idea of quantum physics as counter example to the law of identity. Sure one can often hear sentences like "the particle is both here and there" or "the particle went though both slits", but it appears to mostly be confusion raised by the fact that everyday language is not adapted to describe quantum reality (and how would it be, as all the reality we experience directly is macroscopic). But rigorously speaking, each particle or field has a well defined state vector and is rigorously defined as a mathematical object.
    – armand
    Jun 7 at 8:23
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    "But rigorously speaking, each particle or field has a well defined state vector and is rigorously defined as a mathematical object." --- not when multiple particles interact, they don't. Then we have only a state vector describing the whole. The motivation behind Schrödinger logic is that we should not speak of the identity of individual particles when they are indistinguishable by any measurement, even in principle.
    – Bumble
    Jun 7 at 14:51
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    @Hypnosifl Thanks. But even if our 'things' are bijections, it opens up a distinction between what we normally treat as 'objects' within our domain and these 'things'. I think Steven French is making a similar point in section 5 of the SEP article plato.stanford.edu/entries/qt-idind/#Self-Ind
    – Bumble
    Jun 7 at 18:25
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    @bumble I don't think it changes anything, because then this system of interacting particles is still a well defined set of numbers, always equal to itself and that can be used to conduct rigorous calculations. Again, the confusion arises from trying to instinctively grasp the idea of interacting particles, which escapes our capacity and depends on various interpretations of QM. But the mathematical object is unambiguous.
    – armand
    Jun 8 at 3:10
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No. There is no counter-example in the real world. Any purported counter-example is fallacious. And this is not a no-true-Scotsman fallacy, because "E = F" literally means that the object that "E" refers to is the same as the object that "F" refers to. So, obviously, "E = E" is true for any expression "E" that refers to some object. If you want to use another kind of 'equality', that is your choice, but completely irrelevant to the fact that the law of identity is valid for reality.

An analogy is that "Sayaman is an SE user" is obviously true, according to appropriate interpretation of the words involved. If you choose to take an alternative interpretation, that is your business, but it has zero impact on the truth of the statement according to the proper interpretation.

Similarly, the law of identity is a fact if you interpret the equality symbol correctly. That interpretation is a part of the law of identity itself. Note that any statement can be made false by using some stupid interpretation of it.

For example, "==" in many programming languages (e.g. C, Java, Python, ...) is not the same as classical equality; given a procedure f in the language, "f(0) == f(0)" may not evaluate to true even if f is purely functional (i.e. has no side-effects). I repeat, this does not yield a counter-example to the law of identity, but merely shows that "X == Y" is different from "X = Y". In fact, it is ill-defined to even ask whether "f(0) = f(0)" is true or not, until you define what object "f(0)" refers to. If f(0) does not halt, what do you define "f(0)" to mean?

In short, the entire question is based on the erroneous notion that you can talk about soundness of a logical law in the absence of well-defined semantics. Truth and falsehood are always with respect to some semantics, and if you do not precisely specify the semantics you want, then it is meaningless to ask for counter-examples to a law (i.e. examples where the law yields a falsehood).

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Judging from the perspective of Modal Logic and World Semantics, I recall a section in "Modal Logic as Metaphysics" by Timothy Williamson, that according to permanentism, always everything is always something and identicals are always identical in classical Modal Logic. I personally would argue that a thing is always identical to itself, but in temporal logic, it doesn't have to be identical to an earlier version of itself. For example a (concrete) chair is necessarily identical to itself, the same chair a year later is still identical to itself, but not necessarily identical to itself a year earlier.
Modal logically, it is necessary that the law of identity holds in every logically possible world. and can only be invalid in impossible worlds, but those would be so exotic that there is no sound method to logically argue about them either way.

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We would need to clarify what you mean by asking for a counterexample. There is no point in looking for an object that is not identical with itself unless you start with some criteria of what identity is.

In logic, identity (or equality) is a dyadic predicate. We could write it as Identical(x,y) but it is more handy to use some syntactic sugar and write it as x=y. We normally have to 'interpret' predicates, along with names and functions. Interpretation here is a technical term from model theory. It is a function that, for a given domain, assigns referents to names, functions to function-symbols, and relations to predicate-symbols.

Now we have a choice. We could treat identity like any other predicate and interpret it. This is called "first-order logic without identity". On this approach, it is down to the theory to provide the axioms that will determine whether x=y is true for some x and y. Such axioms typically have to do with reflexivity, symmetry, transitivity and substitutability. They are spelled out in a little more detail in this article.

But with the vast majority of theories, we end up using the same axioms for identity, so we can help ourselves to a shortcut and treat identity as a logical constant. This is called "first-order logic with identity". It transfers responsibility for defining the properties of the identity predicate from the theory to the logic. Under this convention, the law of identity is a logical truth.

But occasionally, just occasionally, we may come across a domain of discourse where our requirements concerning identity are different from usual. We may instead wish to interpret identity using some other equivalence relation. So we can stick with first-order logic without identity and interpret the identity predicate. Under some such interpretations we may find cases where distinct individuals a and b satisfy a=b. An interpretation where this happens is said to be non-normal, while an intepretation where it does not happen is said to be normal.

So, your question, Is there a counterexample? really amounts to asking, is there some domain of discourse in which it makes sense to use an interpretation of identity with non-normal models? The short answer is yes. The longer answer involves philosophical arguments about the nature of identity.

In quantified modal logic, it may be that an individual exists in more than one possible world, and we require an identity predicate that operates cross-world, and another that operates intra-world. When dealing with intensional contexts, we might require an identity predicate that is substitutable in such a context, and another that is not. As has been mentioned in the comments, in the case of the elementary particles of quantum mechanics, it is arguably impossible to speak of the identity of individual particles, since they are indistinguishable, even in principle. This is a motivation for Schrödinger logic. There is a longer discussion of this in this Stanford Encyclopedia article.

One last point. Rationalism is BS. You cannot make progress in logic, or any other discipline, by just saying, "Well this is just obvious to me, so it must be so." Our knowledge, including our knowledge of logic, is the product of centuries of blood, sweat and tears by lots of clever people. Nothing is obvious. Everything has to answer to critical enquiry. The history of philosophy is littered with things people used to think were obviously true, but which to us now would seem not just false but absurd.

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    I don't think you have a good definition of rationalism.
    – CriglCragl
    Jun 8 at 1:24
  • @CriglCragl Fair comment. I'm criticising the position that anything that people think is obvious and undubitable is therefore true and that anything that people cannot imagine being true is therefore false. Although this is not a definition of rationalism, it does tend to be the way rationalists think. When people think this way, all they are doing is expressing the limitations of their imagination.
    – Bumble
    Jun 8 at 14:08
  • 'The argument by lack of imagination', I call it
    – CriglCragl
    Jun 8 at 16:03
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I would say all the examples Buddhists use to establish 'anatta'. Recently discussed here: Is there any exception or rebuttal to the law of noncontradiction?

'I cannot step in the same river even once' you might say.

Reframe the situation, an previously identical things can become different, and previously different identical. Linking the logic picture to a physics/information theory picture: Why is a measured true value “TRUE”?

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Everything is a counterexample to the concept of identity. All equality is relative to some intended purpose, and no two things, no matter how closely related, are ever truly identical.

Heraclitus has a point. You never step into the same river twice.

An object in the real world is never equal to itself in the sense of obeying Leibniz Law of identity, that two things are identical if and only if they have the same values for all properties. Any object you refer to in reality will have a different age, by the time one has moved on from making the first reference to making the second. Since age is a property of all real objects, nothing real ever fulfills this requirement.

Things are only equal up to some point of differentiation, and you have to make sure the context into which you intend to substitute does not use some property that cannot be prevented from changing.

The river is the same river only up to the purposes of basic geography. It is not the same river physically, the water has changed. Socrates from half a second ago is no longer made up of the same atoms as the Socrates before you now...

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    This misunderstands the laws of Aristotle. One thing is the concept of identity, which is something that exists without mutations along time in our subjective understanding, and another thing is the object such concept refers to. Either the object is a rock or a river, both change continuously and are never the same factually, but the concept remains in time. How does the subjective concept relates to the objective phenomenon is out of scope here.
    – RodolfoAP
    Jun 8 at 16:17
  • @RodolfoAP. It is unlikely that when Aristotle referred to individuals in syllogisms, actual human beings, he considered them to be something that existed without mutations. That seems like a Scholastic embellishment. Besides which, this question is not set within Aristotle, it is a real question of everyday relevance. Jun 8 at 16:20

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