Law of Identity
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?
Comments (71)
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.
Being on the RHS or the LHS is not a property of a, but a property of the sentence "a is a".
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.
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.
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.
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.
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.
The best way to think of it is as a definition of "=".
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)".
:up:
Quoting Pierre-Normand
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.
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.
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.
Quoting Banno
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".
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
"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.
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.
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 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.
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:
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.
That's just an equivocation though, it doesn't really bear on claims about identity being limited in some cases.
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.
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".
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.
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.
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.
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.
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).
http://sci-hub.tw/https://link.springer.com/article/10.1007/s11229-015-0997-5
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:
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).
So it does not necessarily. Or how is this solved?
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.
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".
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.
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.
Do you mean that sentence to be taken as truth?
Quoting Heiko
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.
Expressing such a formalism in itself is:
It'd be unclear what "those" refers to if the "terms" would not be the "terms", don't you think?
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.
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.
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?
Can you read the above post? Thanks:smile:
Quoting TheMadFool
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.Quoting TheMadFool
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).
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.
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.
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.
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.
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.
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.
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.
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.
By "indiscernible" it is meant they are ontologically indiscernible, not that we merely lack the means by which to tell them apart. So this:
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.
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.
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.
and
2. E!x defined: (Some F)(Fx).
3. (All x)(x=x <-> E!x), is a theorem.
If either or both do not exist then x=y is provably false.
4. (All x)(x=x) is not valid.
example
(The present King if France)=(The present King if France), is false, even though
(All F)(F(The present King if France) <-> F(The present King if France)), is tautologous.
IS the sentence about the marks on either side of the "="?
More generally is it incorrect to point out that some unknown property may exist between "a" and "a" that makes them different? Would claiming no property exists because it cannot be proven otherwise not be committing an argument from ignorance fallacy?
When you write "a" and "a" as two distinct things, and ask about the difference between these two things, you have given us the premise that they are two distinct things. The need here would be to support, justify that premise, that they are distinct. We can see that they are distinct things because they occupy different places. So despite the fact that they look the same, the claim that they are distinct things is justified by that fact, that they occupy different places.
That would be confusing use with mention.
If you're not familiar with the use/mention distinction, here are a couple easy examples:
"Dogs" has four letters. Dogs have no letters.
The "mention" is marked off by quotation marks above. The "use" isn't. "Mention" concerns the expression as an expression. "Use" is what the expression is about. It's what the expression "points to," the referent of it.
Another example, courtesy of Wikipedia's page on the distinction (https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinction):
Use: Cheese is derived from milk.
Mention: 'Cheese' is derived from the Old English word ??se.
So when you talk about one A being on the left and the other on the right, you're talking about the mention.
But the principle of identity isn't about anything in the mention sense. It's about the use sense. In the use sense, there aren't two different As. We're simply required to write or say it that way a la a mention.
Not trying to do that. I can word it a different way like how do you prove an unknown property doesn't exist between this letter ==> "a" <== and this letter without commiting an argument from ignorance fallacy?
What are you saying, that a thing might be different from itself? So I don't get your point. You point to "A", and ask if there is a property of that thing which is also not a property of it?
Yeah at best we can say that it's so obvious that A is A that there's no conceivable reason why it wouldn't be. But I don't see why there couldn't exist some inconceivable reason why it's not.
It's a principle, "A is the same as itself". If there is no conceivable reason why this wouldn't be true then it's a solid principle. If you allow that there might be a "reason" why it is not true, or might not be true in some cases, then by the use of that word, "reason", you allow that it is conceivable. Then we might doubt that principle and seek the reason. But to say "inconceivable reason" is contradictory and doesn't give us any reason to doubt the principle.
The principle serves to help us understand things. And our understanding is only as solid as the principle. if things start going wrong with our understanding of things, evidence comes forward that our understanding might really be a misunderstanding, then we might start to question our principles, to determine where the problem is, why is there an appearance of mistake. If we start at the bottom, and the law of identity is pretty much the bottom, we can consider whether there is any reason to doubt this principle. But it doesn't make sense to look for a reason which is inconceivable. What kind of reason would that be, and how could we ever look for it?
I'm also applying the principle that something is not true because it cannot be proven false. It's an argument from ignorant fallacy to assume that. When I construct theories I allow for the possibility of being wrong for some reason I don't understand currently. Maybe reason is an illusion.
Consider that the principle which we call the law of identity, is not necessarily true, it's just a useful principle. So long as it serves us well, we'll use it. But if we start finding misunderstandings, and mistakes, like I described in the last post, then we might question this principle to make sure that it isn't leading us astray. So we don't really assume that it's true, just because it hasn't been proven false, there's two factors. We assume it true because it has served us well and it hasn't misled us. The latter, "it hasn't misled us" is similar to "it hasn't been proven false", and the former "it has served us well" is similar to being proven true.