## Law of Identity

• 763
Sure it does. While I did not quote the paper in which this is done, non-reflexive logics can and have been formulated within a metatheory that itself was non-reflexive. It's not that identity is entirely dismissed, it's restricted in scope.

"Quasi-set theory Q is a first-order ZFU-style set theory. Its underlying logic is just like first-order classical logic without identity (the system L mentioned before), with a significant difference: the semantics for that system L underlying Q is described in a non-reflexive metalanguage; just like classical logic has a semantics developed in classical set theory, this particular system of logic (i.e. L) has a semantics developed in a non-reflexive set theory (more on that topic soon). So, the system in question is not exactly classical logic, but it formally coincides with classical logic, although it is semantically different (since its semantics is provided in a non-reflexive metalanguage)." — Classical Logic or Non-Reflexive Logic? A case of Semantic Underdetermination
• 200
Sure it does.

So it does not necessarily. Or how is this solved?
• 763
It does hold necessarily. Basically, the logic is structured to have two types of terms: Terms to which identity holds and terms to which it does not. Similarly, identity is defined as applying to the appropriate kind of term. So identity holds necessarily (it is a tautology in such logics), but that doesn't mean it can be generalized to all types of terms in the logic (or, correspondingly, to all types of objects).
• 200
Terms to which identity holds and terms to which it does not.

But this is just the point: My Thesis was that the set of meaningful statements made in a discussion is just the set of the terms for which identity holds. Those do assert themselves.
• 763
The problem is this isn't obvious in e.g. quantum mechanics. We seem to have plausible examples of objects which are not self-identical (I don't think the "asserting themselves" is quite the right characterization). That's the motivation for the type of logic previously described.
• 200
This just is the difference between form and content.
Say we have a system where t denotes a binary truth-predicate: t(x, true), t(x, false)
This is contradictory. x is said to be true and false alike. We could use formalisms to deduce further things in this system if more clauses were given.
But the clauses themselves are always thought to be true if they appear in it.
If you start expanding: "p" means "p is true" you cannot express the truth-predicate using itself.
t(x, true) would mean t(t(x, true), true), which would mean t(t(t(x, true), true), true) and so on.
This is why I think you cannot formulate non-reflexive-logics using itself.
The truth-"predicate" is self-identical if "x" asserts "x".
• 763
This has nothing to do with the truth-predicate though. That some class of terms may not be self-identical does not mean that the terms aren't true when they appear. I don't quite understand what you're trying to say, maybe this is just a limitation on my part. If identity is restricted in a logic it simply means either identity is not an assumed law (such as in first-order classical logic without identity) or the terms are sorted into two categories, so that there is a failure of application when you try to assert something like "∀x(x = x)".

There's no contradiction because it's not asserting that ~(∀x(x = x)) or anything similar. Identity isn't false so there's no way to derive a contradiction here. Truth-predication isn't even directly involved, I think. We can still say say true things of non-identical objects, we just cannot (supposedly) truthfully say they are self-identical.
• 200
so that there is a failure of application when you try to assert something like "∀x(x = x)".
How so if "∀x(x = x)" does not mean "∀x(x = x)" but something else? You cannot make a statement which does not assert itself without... an extra-ordinary amount of freedom what can be written without any possiblilty of someone marking it as an error.
• 200
Truth-predication isn't even directly involved, I think.
Do you mean that sentence to be taken as truth?
• 763
By requiring that all terms carry a subscript to separate them into two categories. One where that's valid and the other where it isn't:

We allow the usual connectives, quantifiers, an identity relation symbol, and punctuation symbols. For one of the kinds of terms (variables and constants), let us say T p T2, ..., terms of the first kind, we allow that identity holds as usual. In the intended model they represent the individuals. For terms of the second kind, tp t2, ..., identity is not an allowed relation. In the intended interpretation those terms denote non-individuals, items with no identity conditions

Do you mean that sentence to be taken as truth?

I don't even know what you're trying to say now. That I believe what I'm saying is true does not entail that it's impossible to give a coherent formalism where objects are not self-identical. Identity doesn't seem directly related to truth-predication, that's what I was saying. So restricting identity doesn't somehow prevent one from predicating truth to the purportedly non-self-identical objects.
• 200
That I believe what I'm saying is true does not entail that it's impossible to give a coherent formalism where objects are not self-identical.
Expressing such a formalism in itself is:

For terms of the second kind, tp t2, ..., identity is not an allowed relation. In the intended interpretation *those terms* denote non-individuals, items with no identity conditions
It'd be unclear what "those" refers to if the "terms" would not be the "terms", don't you think?
• 763
The law of identity is not a law about reference, it says that everything is self-identical (the conclusion of investigating the formalism being we can even show that in the metatheory identity is not assumed). So long as the law is syntactically restricted to terms of a stipulated kind the other kinds of terms are thereby not subject to it.
• 656
"As we have said in the previous section, in non-reflexive logics we do not accept the negation of the reflexive law of identity. Also, we don’t have to accept that it must fail in at least some interpretations. Rather, we adopt its restriction in the form of its inapplicability. Here, ‘inapplicability’ is couched in terms of identity not making sense, not being a formula, for some kinds of terms." - Arenhart

Exactly. Per the bank account metaphor, identity is not applicable to the melted coins since they are not individual coins. So to ask whether the coins are self-identical would be a category mistake (not false). What we really have is molten metal which is good for five coins when withdrawals are made.
• 200
The law of identity is not a law about reference
Sure it is. It models that a mental object that was defined stays the same. A quantum particle, in contrast to it's definition, does not.
• 3.3k
Why do you think the Law of Identity is required in classical logic? I'm guessing here that, as one of the three laws of thought, it is a necessity for logic.

I'm not sure but the concept of identity in philosophy isn't very clear.

Anyway, I think the Law of Identity has to do with symbols and semantics both:

1. Symbolic: ''Box'' here at one time = ''Box'' there at another time

2. Semantic: ''Box'' means a container and this ''box'' = that ''box'' at another place in the conversation

Without this basic agreement conversation would be impossible right?
• 3.3k

Can you read the above post? Thanks:smile:
• 763
Reference and self-identical aren't the same thing.

Why do you think the Law of Identity is required in classical logic? I'm guessing here that, as one of the three laws of thought, it is a necessity for logic.

It's not literally required. Classical logic without identity is already a well studied formal system. But it's clearly very useful and an obvious starting point for a set of axioms.
Without this basic agreement conversation would be impossible right?

Sure we shouldn't equivocate. But if you go back to some of the papers Andrew M and I were quoting, there's no equivocation here. The idea isn't that you should violate identity by saying it's false or by changing the meaning of terms mid discussion. But that there may be some class of objects where applying identity doesn't make sense (like a category mistake).
• 200
or by changing the meaning of terms mid discussion
How would you recognize a change in meaning if the term wasn't identical to itself?
What is the difference between "a=2, b=3" and "a=2, a=3"? I for my part say that they are visually distinct.
• 767
You are confusing terms of language, or written symbols, with entities that are designated by them. You have essentially reproduced the confused argument of the OP.
• 200
Elaborate on that, please. I do not quite see where. If there is a determined something the law of identity applies.
• 5.7k
Anyway, I think the Law of Identity has to do with symbols and semantics both:

1. Symbolic: ''Box'' here at one time = ''Box'' there at another time

2. Semantic: ''Box'' means a container and this ''box'' = that ''box'' at another place in the conversation

Without this basic agreement conversation would be impossible right?

I don't think that this is correct, the law of identity is not concerned with the symbol, nor the semantics (meaning) of the symbol, it is concerned with the particular thing which is identified through the use of the symbol. So, if we are talking about "the chair", the law of identity is not concerned with that symbol, nor what it means to be a chair, it is concerned with that particular entity which we have identified as "the chair". The law of identity says that this particular thing has an identity, regardless of the symbol we use to refer to it ("the chair" in this case), and what is implied about that thing (what "chair" means to us), through the use of that particular symbol chosen to represent the thing.
• 200
it is concerned with the particular thing which is identified through the use of the symbol.
Correct. But the symbol "a" just establishes an abstract identity. This is why you can know that I am talking about the same symbol when I now write, "a" was introduced at an earlier time.
• 5.7k

The question is, whether an "abstract entity" qualifies as an entity to which the law of identity is applicable. The abstract entity is a class of things, a type, like horse, dog, cat, etc.. So if the "a" signifies an abstract entity, then one instance of "a" is the same as another, by being the same type, an "a", just like one horse is the same as another, by being the same type, a horse.

It is doubtful whether the law of identity applies in the identification of a type, as an abstract entity, but let's suppose there is such an entity, an abstract entity, which is signified by "a". Each time you use "a", you signify this abstract entity. It is not the case that each instance is "the same symbol", but each time it is a different instance of an "a", and therefore a different symbol, but each instance of the symbol, despite the differences, is recognized as symbolizing the same abstract entity.
• 200
When showing that an "x" does not exist this does not extinct the letter. But this would be the case if the expression would be equal to the expressed.
If you take words like "yellow" or "green", in quantum mechanics you get "yellow=green". I sometimes wear a green tie. Can you imagine this? Under those conditions? Writing "yellow=green" is no problem. Having an idea of what this really means is - because yellow is yellow, and green is green. Or was it brown? Or blue? Sorry, I really don't get this.

If I define "x" as "a sentence that does not exist.", what do we have then?
"x" - as a letter which refers to
"x" - as the idea of it being a variable which refers to
nothing (a sentence which does not exist)

We still can talk about x and sentences that do not exist: Such x'es do not require much typing.
• 5.7k
If I define "x" as "a sentence that does not exist.", what do we have then?
"x" - as a letter which refers to
"x" - as the idea of it being a variable which refers to
nothing (a sentence which does not exist)

We still can talk about x and sentences that do not exist: Such x'es do not require much typing.

I think the issue here is that you have identified something as "a sentence that does not exist". So "x" signifies this thing which you call by that name. This is just like when we said "a" stands for an abstract entity. We have identified a thing which is being called an "abstract entity", and "a" represents that thing. Likewise, you have identified something as "a sentence that does not exist", and you represent this thing with "x"..

There is no problem with identifying and talking about abstract entities, and non-existent things, so long as we adhere to the law of identity. The thing identified must be the thing identified, and not something else. It's when we allow that the thing which is identified is something else, other than what it is identified as, like it has another identity as a distinct different thing, that we run into problems. I believe this is what happens in QM, there is a problem with the continuity of existence of the identified thing, so the thing is given another identity to create the guise of continuity. But the continuity is false because there are two distinct identities for what is said to be one and the same thing.
• 3.3k
But that there may be some class of objects where applying identity doesn't make sense (like a category mistake).

That makes sense. You're talking about the concept of identity aren't you?

There is no ''discernible'' difference between two electrons, for example. So, identity, as in uniquness, is a problem for electrons. Am I getting your point?

Such an idea, however, is subjective rather than objective. Continuing with the electron example, let's take electron A and electron B. No scientific analysis can distinguish A from B. So, we conclude A = B or, in your case, we give up the notion of identity altogether.

Not to be nitpicky but there is a difference between A and B electrons. They're at different loci in space. Don't you think, therefore, that we can still retain the concept of identity for such situations?

So, if at all there's a problem with the concept of identity it lies with our inability to see the difference between electron A and electron B. It's subjective. But we know there IS a difference in location between A and B. That's objective.
• 5.7k
Such an idea, however, is subjective rather than objective. Continuing with the electron example, let's take electron A and electron B. No scientific analysis can distinguish A from B. So, we conclude A = B or, in your case, we give up the notion of identity altogether.

Not to be nitpicky but there is a difference between A and B electrons. They're at different loci in space. Don't you think, therefore, that we can still retain the concept of identity for such situations?

There is something called the Pauli exclusion principle which I think distinguishes electron A from electron B. I believe it is based in a combination of distinct properties intrinsic to the electron, and relative positioning as well. The electrons have different energy values and this is very important in chemistry, contributing to the concept of valence. An electron may lose energy, releasing a photon, but this must be more than just a change of location, it must be a physical change to the electron itself.
• 763
There is no ''discernible'' difference between two electrons, for example. So, identity, as in uniquness, is a problem for electrons. Am I getting your point?

By "indiscernible" it is meant they are ontologically indiscernible, not that we merely lack the means by which to tell them apart. So this:

So, if at all there's a problem with the concept of identity it lies with our inability to see the difference between electron A and electron B. It's subjective. But we know there IS a difference in location between A and B. That's objective.

Is not right. As an example of this, we have the Hong–Ou–Mandel effect. Similarly, standard quantum theory (to the limited extent that I can understand it, granted) seems to suggest that quanta cannot been distinguished or even labeled. Even those that disagree will usually say that their physical properties cannot be distinguished, but want to maintain some kind of individuation must be there.
• 700
In the sentence "a is a" "a" is used formally. That is to say that it refers to some (generalized) object beyond itself. In the sentence " a is not a because one a is on the left side of the copula and the other a is on the right side," "a" is used materially. Which is to say that "a" means the symbol "a" and not what "a" indicates. As material and formal predication are different, your argument is uses "a" equivocally, and so is fallacious.

As to Aristotelian vs modern logic, they are not logic in the same sense. Aristotelian (and, more broadly, intentional) logic is defined as the "science of correct thinking (about reality)." Modern logics are not concerned with thought processes per se, but with rules of symbolic manipulation. Since they deal with different subject matter, they are not directly comparable.

For example, in Aristotelian logic universal propositions have existential import. That is because propositions cannot be true unless they are based on our experience of reality. In modern logic, propositions need not be justified by real cases, and so universals need not have existential import.
• 267

Yes, I argue the same point, however I will have to look it up in my notes-forums so I can copy and paste the argument because it leads to a contradiction inherent within the law of excluded middle and several other issues, such as the fallacy of equivocation being inherent within the law.
I also provide a new argument as to what the law of identity should be..

But until then, yes you are correct.
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