## Law of Identity

• 4
I wanted to get some opinions from people who are more knowledgeable than I am in logic. Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical. I was actually going to use this in an argument but it sounds too cute so I thought I'd ask people who knew the subject better if this is a valid point, Is there some technical reason why it doesnt work and in general what your thoughts were. Has Aristotelian logic been subjected to the same critiques as Euclid's geometry. In other words is there a non Aristotelian logic to be derived by a critical examination of it's axioms?
• 2.3k
One 'is' is the is of predication, stating a property or relation of an entity. Like 'the morning star is a star', one 'is' is the is of identity, stating that an entity is equivalent to other entity - or perhaps more subtly that the symbol on the left of the copula and the symbol on the right are different names for the same thing. Like 'the morning star is the evening star'. A nice historical reference here is this.

The use of both in ordinary language is more complex, like the relationship between designation/naming and the entities named. As usual there's a good discussion of surrounding issues (in analytic philosophy) on SEP here for theories of naming and here for theories of identity.
• 762
Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical

Being on the RHS or the LHS is not a property of a, but a property of the sentence "a is a".
• 4
What about the second part of the question. Has there been an equivalent of noneuclidean geometry in Aristotelian logic? Is the law of identity kin to euclidean axioms? Thanks for the help though, this is a complex field and its easy to think you've got a handle on things when you dont.
• 1.6k
What about the second part of the question. Has there been an equivalent of noneuclidean geometry in Aristotelian logic? Is the law of identity kin to euclidean axioms? Thanks for the help though, this is a complex field and its easy to think you've got a handle on things when you dont.

As I mentioned in another thread recently, Peter Geach has been an advocate of the thesis of relative identity. According to this thesis, two objects A and B can both be, and fail to be, identical depending on what sortal concept they are made to fall under. For instance, as applied to the Christian doctrine of Trinity; the Father, the Son and the Holy Spirit can be deemed to be the same unique God but three different persons. To take a less contentious example (albeit still contentious) the original ship of Theseus might be the same functional artifact as the later ship that has been maintained thought replacing the old planks, although both of those ships aren't the same historical artifact. Under that interpretation, the ships A and B are the same functional artifacts but not the same historical artifact.

The thesis of relative identity still is very contentious. I much prefer Wiggins' thesis of the sortal dependence of identity, which, unlike Geach's thesis, remains consistent with Leibnitz' Law (of indiscernibility of identicals). Under that new thesis, while it's still true that what it is that determines whether the referents of A and B are identical is the individuation criteria associated with the sortal concept that they both fall under, objects that fall under different sortal concepts always are distinct objects. Hence, for instance, the original functional artifact and the original historical artifacts that we may both call ambiguously "the Ship of Theseus" are two different objects even though they may, at an early time in history, have occupied the same spatial location and have had the exact same material constitution. They have, though, separate later histories and aren't individuated according to the same criteria.
• 762
Has Aristotelian logic been subjected to the same critiques as Euclid's geometry. In other words is there a non Aristotelian logic to be derived by a critical examination of it's axioms?

Yes and no. "Yes" in the sense that, just as with geometry, we now know of more than one logic. "No" in the sense that we did not find other geometries by proving that some Euclidean axioms are wrong, and neither did we find other logics by proving that some axioms of the Aristotelian logic are wrong.

Nothing is wrong with Euclidean geometry, and nothing is wrong Aristotelian logic. It's just that at some point we decided that the concept of "logic" doesn't have to be limited to Aristotelian logic, and just as there is now a generalized concept of "geometry" that covers any number of geometries (including both familiar, practical geometries, and completely abstract, made-up ones), there is a generalized concept of "logic" that covers any number of logics. We have also found that the same logic can be axiomatized differently, i.e. two different axiomatic systems can have all the same implications.
• 762
I should also clarify that modern formal logic is not quite the same thing as traditional Aristotelian logic - not just because it can have different laws (axioms), but because it is a different thing conceptually. Although it is possible to reconceptualize traditional logic in the modern paradigm, it wouldn't be what people used to think of as "Aristotelian logic."
• 5.7k
Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical.

The Arostotelian formulation of the law of identity is that a thing is the same as itself. It may be that "a is a" is a representation of this. The thing to remember then is that "a" is a symbol which represents the thing which is the same as itself. So if we take the symbol "a" and ask if one "a" is the same thing as another "a", clearly they are not the same, by the law of identity, as they are distinct things. And when we say that one symbol, one instance of "a" is the same as another instance of "a", we are using "the same" in a way which does not correspond to the law of identity. Beware of equivocation.
• 22
I see.
Two pennies are alike, When one contrasts and compares two pennies, they might say that they are the same, opposed to being different.

But they are not the same penny.
• 3.3k
I'm not knowledgeable but what I think is...

The law of identity (A=A) is a logical necessity.

Imagine A is not A. We would then have the logical contradiction A & ~A, violating the law of non-contradiction.
• 5.3k
The law of identity (A=A) is a logical necessity.

The best way to think of it is as a definition of "=".
• 1.6k
The law of identity (A=A) is a logical necessity.

Imagine A is not A. We would then have the logical contradiction A & ~A, violating the law of non-contradiction.

You seem to be using "A" as the name of a proposition rather than the name of a particular. The "=" usually names the numerical identity relation, which obtains between a particular and itself. If you use "A" as the name of a proposition, and thereby "A=A" to express the claim that the proposition A is identical with itself, then, the negation of this claim isn't expressed as "A & ~A" but rather as "~(A=A)".
• 3.3k
The best way to think of it is as a definition of "=".
:up:

You seem to be using "A" as the name of a proposition rather than the name of a particular. The "=" usually names the numerical identity relation, which obtains between a particular and itself. If you use "A" as the name of a proposition, and thereby "A=A" to express the claim that the proposition A is identical with itself, then, the negation of this claim isn't expressed as "A & ~A" but rather as "~(A=A)".

I guess there are fine nuances I'm unable to see.

As far as I know logical equality comes in two flavors:

1. Identity (the particular you refer to): George Washington = First president of the USA

2. Logical equivalence (propositional meaning): If A=God exists then A :: A.

Both types have logical significance.

Does the Law of Identitiy (LoI) refer to both types of logical equality?

a=a type 1 seems necessary for any form of logical argumentation since if this were not true we would be making the fallacy of equivocation. Denying this would be what you refer to as ~(a=a)

A = A type 2 is also a necessity because then we would be saying ~(A :: A) which is a contradiction of the form A & ~A

A :: A is true IFF A<->A is a tautology. So if ~(A :: A) then ~(A<->A) which is a contradiction. But I wonder how one would express the contradiction so obtained? A &~A? You seem to disagree.

Perhaps A & ~A is a generic form of a contradiction and can be used here to express what I'm getting at.
• 1.6k
A :: A is true IFF A<->A is a tautology. So if ~(A :: A) then ~(A<->A) which is a contradiction. But I wonder how one would express the contradiction so obtained? A &~A? You seem to disagree.

I appreciate your separating the case of particulars from the case of propositions.

What I am unsure of is what it might mean to be denying that a proposition A is (numerically?) identical with itself. It is unclear to me that it is equivalent to denying that "A<->A" is a tautology. Maybe you are glossing "A=A" as equivalent to "A :: A", but I also am also unclear about the rationale for that. The relation of numerical identity just makes more sense to me as applied to particulars, or Fregean objects, and maybe also to Fregean functions (or properties). The issue of the individuation of propositions (either Fregean or Russellian propositions) is trickier.
• 5.7k
Two pennies are alike, When one contrasts and compares two pennies, they might say that they are the same, opposed to being different.

But they are not the same penny.

That's right, two pennies are "the same" in the sense of the same type of thing. But they are not the same in the sense of the law of identity which would mean that they would have to be one and the same penny.

The best way to think of it is as a definition of "=".

I think that this is incorrect. Equality (=) implies two distinct things with equal value. The law of identity identifies one thing as itself. "Equals" and "the same" do not have the same meaning.in the sense that the law of identity implies for "the same".
• 3.3k
I think the Law of Identity, as the name implies, refers to particulars as you said.

I was just pointing out that if we were to consider logical equivalence then we may understand the law of identity as just another way of stating the law of non-contradiction

1. ~(A<->A)
2. ~[(A->A) & (A->A)]....1 ME
3.~(A->A)....2 Taut
4.~(~A v A)...3 MI
5. ~~A & ~A...4 DeM
6. A &~A....5 DN

6 is a contradiction which means A<->A is true
• 763
The law of identity (A=A) is a logical necessity.

Imagine A is not A. We would then have the logical contradiction A & ~A, violating the law of non-contradiction.

"Logical necessity" is not some extra-systematic modality applying to everything, it's determined by the set of logical truths of a given logic. Dropping the Law of Identity does not entail that "A is not A". We already have a perfectly well-understood logic without identity. Helpfully, it is known as "first-order predicate logic without identity". Identity is simply not part of that formulation of classical logic, so it's not a logical truth there.

Dropping a logical law is not the same as assuming the negation of that law, that's silly. Logics without predicates are not making the assumption that predication is incoherent or something. Another way to accomplish this is to modify the Law of Identity by defining it to only apply to some class of objects and not others. Newton da Costa and others have done work on these non-reflexive logics. But yet again, no contradiction appears in these formalisms, they're consistent systems where identity is not generalizable to all objects because the intention of such systems is to give a logical representation of ontologically indistinguishable yet non-identical objects (for use in QM). See the SEP on this.
• 200
Regarding the Law of identity "a is a" is it wrong to argue that a is not a because one a is on the left side of the copula and the other a is on the right side, and having different properties they are clearly not identical.
Obviously the left a is not the right. This only means that, when talking about the law of identity, we do not mean the self-identity of the letters, when saying "a is a". If the law of identity was not given the left a would not be the left a. By exclusion this would mean the left a would be the right a and vice versa: a=a would be true. Which would be a contradiction to the law of identity not being given.
• 3.3k
What I ''feel'' is that the Law of Identity is required in logic. May be it's only classical logic that requires it because if you deny that water=water then how are we to have a meaningful discussion on water? Isn't consistency in the meaning of words and terms a requirement for sound argumentation?

If, in a discussion, the meaning of ''sex'' changes from gender to intercourse we would have a problem:

Name: John Smith
Age: 24 years old
Sex: Daily with my partner OR Male??!!
• 5.7k

That's not the law of identity, that's equivocation. The law of identity is used to identify specific things, it is distinct from a definition. Logic can proceed from definitions, without any specific object being identified as fulfilling the criteria of the definition. The law of identity dictates how we apply logic, to specific things in the world. Each object must be recognized as distinct from every other object, by the law of identity.

So the law of identity is really used independently of the logic, enforcing the idea that if this is the identified thing, which the logic is being applied to, then it and only it, is the thing which the logic is being applied to. A definition is a generalization, which allows exclusion to the law of identity. If object A has the same defining properties as object B, then the same logic, with regard to those properties, may be applied equally to the two objects. In this way we disregard the law of identity which states that the two objects are not the same, and we treat them as the same, by applying the same logic to them.
• 763
What I ''feel'' is that the Law of Identity is required in logic. May be it's only classical logic that requires it because if you deny that water=water then how are we to have a meaningful discussion on water? Isn't consistency in the meaning of words and terms a requirement for sound argumentation?

What does "require" mean here? I'm assuming we don't want to beg the question and say "We need identity because otherwise things aren't identical" or something like that. Identity appears in basically every logic (even non-classical systems), but that's not because it's impossible to modify it or do without it (it just seems like such an obvious thing to assume). Nor does it follow if you limit identity that "water=water" is false. Take this bit from Krause & da Costa:

We begin by recalling the infamous Problem of the Identical Particles. According to a widely held interpretation of non-relativistic quantum mechanics, there are many situations in which one cannot distinguish particles of the same kind; they seem to be absolutely indiscernible and that is not simply a reflection of epistemological deficiencies. That is, the problem, according to this interpretation, is seen as an ontological one, and the mentioned indiscernibility prompted some physicists and philosophers alike to claim that quantum particles had "lost their identity", in the precise sense that quantum entities would not be individuals: they would have no identity. Entities without identity such as quantum particles (under this hypothesis) were claimed to be non-individuals."

The logic they use to formalize this idea doesn't drop into incoherency because it's not saying identity is false, but that it's inapplicable within a certain domain. It's a non-issue for me if the above is correct or not, I merely want to say there's no technical impossibility of doing this sensibly.

If, in a discussion, the meaning of ''sex'' changes from gender to intercourse we would have a problem:

Name: John Smith
Age: 24 years old
Sex: Daily with my partner OR Male??!!

That's just an equivocation though, it doesn't really bear on claims about identity being limited in some cases.
• 517
So the law of identity is really used independently of the logic, enforcing the idea that if this is the identified thing, which the logic is being applied to, then it and only it, is the thing which the logic is being applied to.

That's an interesting way of looking at it, and it would explain why Aristotle actually didn't formulate the Law of Identity as such, didn't seem to think it that important, and didn't connect it through to the Law of Non-Contradiction (which really was Aristotle's thing). All that - the way we think of the Law of Identity today - seems to be a later development with some of the Schoolmen, Leibniz and Locke.

Aristotle said "Every thing is a something." And I think that's the core idea that's important - the idea that beings have identities, natures, specific ways they are and aren't. And then you get into the whole thing of actuality and potentiality and all the rest of it.
• 200
Identity appears in basically every logic (even non-classical systems), but that's not because it's impossible to modify it or do without it (it just seems like such an obvious thing to assume). Nor does it follow if you limit identity that "water=water" is false.
The point where I can not follow this is:
If you proof p(x) - does x have the property p() then? And was p(x) proven?
You do not seem to get around the fact that writing "p" asserts "p".
• 5.7k
That's an interesting way of looking at it, and it would explain why Aristotle actually didn't formulate the Law of Identity as such, didn't seem to think it that important, and didn't connect it through to the Law of Non-Contradiction (which really was Aristotle's thing). All that - the way we think of the Law of Identity today - seems to be a later development with some of the Schoolmen, Leibniz and Locke.

There is a natural progression from the law of identity to the law of non-contradiction. The goal is to know, or understand the object. First we identify the object, you might say we point to it, or assign a name to it. If we can do this, then we can say that it has an identify according to the law of identity. Having an identity validates the claim that it exists, as an object. Next, we describe the identified object, and we must do this according to the law of non-contradiction. We cannot assign contradictory properties to the identified object because this is repugnant to the intellect, making the object unintelligible. These principles are designed so as to make the object intelligible, they are what appeals to the intellect in its goal of knowing, or understanding the object.
• 517
There is a natural progression from the law of identity to the law of non-contradiction. The goal is to know, or understand the object. First we identify the object, you might say we point to it, or assign a name to it. If we can do this, then we can say that it has an identify according to the law of identity. Having an identity validates the claim that it exists, as an object. Next, we describe the identified object, and we must do this according to the law of non-contradiction. We cannot assign contradictory properties to the identified object because this is repugnant to the intellect, making the object unintelligible. These principles are designed so as to make the object intelligible, they are what appeals to the intellect in its goal of knowing, or understanding the object.

Yeah, I agree with that, so far as navigating everyday life goes; but zooming out a bit more, I see identification as secondary (or subsequent to) to discovery, or the knowledge-gathering process. One identifies what is already known, but to bring things into knowledge is a different process, a process of generate-and-test. That's a process of punting, guessing at, possible identities the thing could have (possible coherent bundles of features that are logically interlinked, etc.), and then testing the implications of that possible identity as the object bumps into the rest of the world (including one's experiments and interventions with it). If it doesn't behave as expected, then either we try on another possible identity, or adjust the one we had.

I think we went through this in a long (fun!) argument we had on the other board ages ago, where we disagreed about the priority of public language vs. private identification being foundational.
• 763
I somewhat confused. In the part you were quoting, I was talking about whether it's possible have a logic to represent the idea that some objects might not be such that Identity is applicable to them. Quoting Arendhart:

In a nutshell, non-reflexive systems of
logic are systems that violate the so-called ‘Reflexive Law of Identity’ in the form
∀x(x = x). In its ‘metaphysical reading’, the Reflexive Law of Identity is known as a
version of the ‘Principle of Identity’, roughly stating that everything is self-identical.
Versions of this law are restricted in systems of non-reflexive logic, and those systems
are said to incorporate in a rigorous fashion the idea of entities somehow losing their
identity.

As they and others go on to point out, this is a restriction on identity by means of separating the terms of language into those to which identity applies and those of which it does not. Whether identity applies to all objects or not doesn't seem to invalidate that if you proof of p(x) then x has that property predicated of it.
• 5.7k
Yeah, I agree with that, so far as navigating everyday life goes; but zooming out a bit more, I see identification as secondary (or subsequent to) to discovery, or the knowledge-gathering process. One identifies what is already known, but to bring things into knowledge is a different process, a process of generate-and-test. That's a process of punting, guessing at, possible identities the thing could have (possible coherent bundles of features that are logically interlinked, etc.), and then testing the implications of that possible identity as the object bumps into the rest of the world (including one's experiments and interventions with it). If it doesn't behave as expected, then either we try on another possible identity, or adjust the one we had.

I think you are using "identity" here in a way other than that prescribed by Aristotle's law of identity. When you say "guessing at, possible identities the things could have", you imply that identity is what we give to the object. But this is exactly what the law of identity seeks to avoid. Identity is not what we give to the object, it is not the description we make of it, it is what the object has inherent within itself, its own identity, as the thing which it is. That's why the law of identity states that a thing is the same as itself, it's identity is inherent within it, not what we assign to it. The descriptive terms which we assign to the thing are something completely different from the identity which the thing has within itself.
• 517
you imply that identity is what we give to the object. But this is exactly what the law of identity seeks to avoid.

The fact that you are trying to guess at an identity doesn't mean you can't in fact hit upon it. Of course the identity you're looking for is the one the thing actually objectively has, but since you don't have a hotline to God or backchannel to reality, you have to work on the principle of generate-and-test.
• 649
I was talking about whether it's possible have a logic to represent the idea that some objects might not be such that Identity is applicable to them. Quoting Arendhart:

"In a nutshell, non-reflexive systems of
logic are systems that violate the so-called ‘Reflexive Law of Identity’ in the form
∀x(x = x). In its ‘metaphysical reading’, the Reflexive Law of Identity is known as a
version of the ‘Principle of Identity’, roughly stating that everything is self-identical.
Versions of this law are restricted in systems of non-reflexive logic, and those systems
are said to incorporate in a rigorous fashion the idea of entities somehow losing their
identity."

As they and others go on to point out, this is a restriction on identity by means of separating the terms of language into those to which identity applies and those of which it does not.

That's a nice paper - here's a link to a pre-print for anyone interested.

Just to summarize the problem, Leibniz' Identity of Indiscernibles says that no two objects have exactly the same properties. However quantum mechanics says that two particles can have exactly the same properties. For example, see the Hong–Ou–Mandel effect.

A natural way to resolve this conflict is to say that Leibniz' principle is only applicable to substantial objects, that is, objects that emerge as the result of quantum interactions (or measurements). Substantial objects have identity and are always distinguishable from other substantial objects. Whereas outside an interaction, quantum particles are only accounted for in a formal sense and lack substance and identity.

A metaphor can illustrate this. Suppose that the nation's currency consists only of similarly-marked metal coins worth $1 each. Coins can be deposited at your bank where they are melted down in a furnace. At that time, the coins have no substantial existence (or identity). However, formally, if you deposit five coins, your account balance will be$5. Additionally, the molten metal materially backs your account balance. If you need coins, you can push a button and a new coin is immediately minted for your use.

So the coins in circulation have identity. But the melted coins in the bank have only an aggregate cardinality (you formally own five coins).
• 763
Yes I rather like the bank account metaphor. I'll do you one better (hope Sci-Hub links are allowed), here's the link to the full paper:

The article is really insightful elsewhere, as it makes this point that I thought was really profound (even if pro logicians might see it as obvious), because it's so often misunderstood by those making very strong claims about logic:

"This is also connected with a second point. What exactly is meant when we say that
we deny a tautology (or a logical law, or a logical necessity)? In denying that an axiom
of classical logic is valid in general, don’t we have to accept that this ‘axiom’ is false
in at least one interpretation of an alternative system in which the same formula may
be expressed? Consider, for instance, intuitionistic logic. In denying the validity of
some instances of the law of excluded middle, it is not the case that intuitionists accept
its negation in its place. However, they do accept that the law may be false sometimes
(mostly when we deal with infinite collections)."
[...]
"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. Recall from our discussion in
the previous section that this is the formal counterpart of the idea of something not
having TI. So, if this is correct, the link between metaphysics and logic that underlies
the non-reflexive formulation of the RV is reasonable, in fact, but it does not go in the
same lines as we think it is reasonable to reject some classical principles of logic in
any non-classical logic with the same vocabulary."
• 200
I somewhat confused. In the part you were quoting, I was talking about whether it's possible have a logic to represent the idea that some objects might not be such that Identity is applicable to them.
I know :)
But the question you answered to was how we could meaningful discuss things without a=a-identity. I guess we cannot as the formulation of sentences and statements does not follow the logic we find in quantum-mechanics. You could not formulate non-reflexive logic using non-reflexive logic (if I'm not wrong).
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