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When we say that the law of identity doesn't exist, does it mean it's always A=A is always false or that A=A is not always true? It seems that saying the law of identity doesn't exist, doesn't mean we can't say anything, but we can say more things, is this true? The only other implication is that we can't prove something that relies on the proof of identity as being always true, but true conditionally on the fact that the law of identity is valid for an object X.

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  • The context you are using is bad. You are making the law of identity seem like a rule of inference. It is not like a modus ponens or disjunction syllogism and so on. The law of identity is a consistency concept. If one argues that A means one thing & later uses A in a different context that is a fallacy. It should be called out. So the law is important. If A has the same context then A is either always true or always false. We can compare propositions in this way.
    – Logikal
    Jun 5 at 5:47
  • In logic x=x is valid i.e. always true. In High-Order logic we can derive it from the Definition of Identity Jun 5 at 10:19
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Nobody says that the law of identity does not exist. To start, it exist at least as a concept. Then, it exists as an essential rule of logic.

If you don't understand its meaning, Google for "Aristotle's laws of logic". In syntesis:

  1. A=K
  2. A=1

From both rules, you can conclude, by logic, that K=1. Because the law of identity says that A has the same identity in 1 and 2.

Without the law of identity A is not the same A in both steps. So, K=9 is valid. But you cannot prove it, given that A has no identity.

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It is important to clarify what you are saying in your question. You write the law as "A=A" but this is misleading. It is conventional in simple introductions to logic to use capital roman letters to stand for propositions, lower case letters near the beginning of the alphabet as names of things, and lower case letters near the end of the alphabet for variables. So, "a=a" means the individual named 'a' is identical with itself. "For all x: x=x" means that every thing that is ranged over by the variable x is identical with itself.

If 'A' is used in the conventional way to mean a proposition, then "A=A" has nothing to do with the law of identity. Identity is a relation over individuals, not over propositions (at least in first order logic). Strictly speaking, "A=A" is not even a well-formed formula. I know some clever dick will point out examples of mathematicians who use "A=A" to express the biconditional "A↔A" but this is just sloppy usage and should be deprecated. Identity is a predicate: it is a two place relation between individuals.

So, the law of identity is the claim that every individual is identical with itself. It does not say anything about propositions. It is not the claim that if a proposition is true then it is true, or anything like that. It also has nothing to do with words having multiple meanings or with equivocation, despite what Wikipedia might say. It is a claim about things.

In logic, we distinguish between the logical constants, such as ¬∧∨→, and predicates. Predicates are normally said to require an interpretation in order to determine a truth value. If we ask: is it true that F(a)? We can only answer if we interpret the predicate F and the name a. If F is interpreted as physicist and a is interpreted as Richard Feynman then it is true. If F is interpreted as kangaroo and a is interpreted as Boris Johnson then it is false. Logical constants on the other hand do not require interpretation, because their meaning is fixed (for a given logic). We do not need to do any interpreting to know that A→A is true, or that ¬(A∧¬A) is true, since these are logical truths.

Identity is a predicate, so it might seem natural to treat it like any other predicate and require an interpretation. But it is such an important predicate, and plays a vital role in so many things we want to do in logic, that it is common (though not universal) to treat the identity predicate as a logical constant in its own right. Logicians speak of "first order logic with identity" or "first order logic without identity".

If we are using first order logic with identity, then a=a is a logical truth and does not require interpretation. If we are using first order logic without identity, then we need to interpret the identity predicate, which is to say, we need to know how it maps onto the individuals in our domain. Conventionally, and in the vast majority of uses of logic, we want to say that every individual is identical with itself. It sounds almost trivial, though actually it is not. In some strange applications of logic, such as with elementary particles, we may want an interpretation that does not require that every individual has a unique identity.

Now to answer your question. If we do choose to interpret the identity predicate in such a way that the law of identity does not hold, it means only that it is not a logical truth that everything is self-identical. It does not follow that everything is not self-identical. It does not even follow that anything is not self-identical. It might be contingently true for some domain that everything happens to be self-identical. It means only that being self-identical is not a logical truth.

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Start with the statement, “A is A”, which says All A is A. The denial is Some A is not A. The denial replaces the certainty of the first statement with the randomness of the second.

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According to wiki reference here:

In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A.

In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation. That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings without introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term.

So if law of identity doesn't exist, there'll be no Equivocation fallacy any more and we can say more equivocally confused things and we're still considered logically valid, but cannot be sound since our macro empirical world certainly doesn't look like this by any measure, even the same word clearly corresponds to different things under different contexts.

However, in our quantum world this law may not exist according to the same wiki reference above:

Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understand identity as meaning indistinguishability (agreement with respect to attributes).

So in this imagined quantum world with really indistinguishable objects as identical per Schrödinger logic, you know there're many objects but you cannot have a predetermined clear way to choose a specific object, not even based on their relative space position as if still can be easily meshed and have a choice function to pick one out from our macro experience.

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