Understanding the Law of Identity
The law of identity is expressed in various ways: A is A. Whatever is, is. A man is a man. Whatsoever is white is white.
I confess that the law seems tautological, trite, useless. Id be embarrassed to write a sentence like A is A What the point?
Id dismiss the Law of Identity as nonsense if it werent for the fact that Plato, Aristotle, Leibniz, Boole, Heidegger and other philosophical heavyweights seem to have taken it seriously. So, what can the law mean? What can be its significance?
(Definition of identity interpretation)
Maybe A man is a man is meant to tell us something about the word is? For instance, we could indicate the meaning of the equals sign by saying A man = a man. Then if we write A man = male human being we have an idea of what is meant. Similarly, we understand A man is a male human being.
(Universal/instantiation interpretation)
Does A man is a man mean that any instantiation of the universal man is a man, i.e., that the instantiation necessary shares qualities of the universal? Once we decide to call someone a man then necessarily that someone has certain chromosomes, genitals, and other properties of the universal man?
(Heideggers interpretation)
Wikipedia: https://en.wikipedia.org/wiki/Law_of_identity has this about Heidegger.
Martin Heidegger . . . links the law of identity "A=A" to the Parmenides' fragment (....for the same thing can be thought and can exist). Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being.
Any insights into the law of identity would be welcome.
I confess that the law seems tautological, trite, useless. Id be embarrassed to write a sentence like A is A What the point?
Id dismiss the Law of Identity as nonsense if it werent for the fact that Plato, Aristotle, Leibniz, Boole, Heidegger and other philosophical heavyweights seem to have taken it seriously. So, what can the law mean? What can be its significance?
(Definition of identity interpretation)
Maybe A man is a man is meant to tell us something about the word is? For instance, we could indicate the meaning of the equals sign by saying A man = a man. Then if we write A man = male human being we have an idea of what is meant. Similarly, we understand A man is a male human being.
(Universal/instantiation interpretation)
Does A man is a man mean that any instantiation of the universal man is a man, i.e., that the instantiation necessary shares qualities of the universal? Once we decide to call someone a man then necessarily that someone has certain chromosomes, genitals, and other properties of the universal man?
(Heideggers interpretation)
Wikipedia: https://en.wikipedia.org/wiki/Law_of_identity has this about Heidegger.
Martin Heidegger . . . links the law of identity "A=A" to the Parmenides' fragment (....for the same thing can be thought and can exist). Heidegger thus understands identity starting from the relationship of Thinking and Being, and from the belonging-together of Thinking and Being.
Any insights into the law of identity would be welcome.
Comments (56)
Actually it is quite problematic.
Some years ago I think I discovered a paradox I called Angelo Cannatas paradox, since I havent found it anywhere else. it is exactly about the law of identity.
Another problem arises if we consider the subjectivity involved in whatever we think of.
No. This is a tautology.
A non-tautological identity is A is B.
Can you better explain, please?
How does this assist us?
Identity also becomes tricky in metaphysics. For example, many forms of nominalism embrace trope theories, which is the idea that objects that share traits (e.g. triangularity) do so by sharing tropes. Tropes are not universals, that is abstract objects that exist outside of their instantiations.Tropes exist only as properties that objects have. For example, there are no triangles outside actual triangular objects in trope theory.
Within these trope theories, many theories posit that objects are identified by their tropes. Sum up all the things that can be said about an object and that is the identity of the object.
But this view has problems. Imagine two identical red balls, ball A and ball B. We'd like to say that these two balls are two different things, that we have two balls if we pick both up. We probably do not want to say "ball A and ball B are actually the same ball in two different places at once." However, if we claim that things are the sum of their traits, we end up in a pickle, because now the red balls DO seem to be identical and thus they are one ball in two places.
Saying they are two different balls because they are in two different locations is not that helpful either. Relative location is a derived trait, one that changes with context. If such derived traits are part of identity then you would be a different person when you're north of your house than you are when you're south of it.
Going back to the link above, this has some real world implications, because small objects like atoms are very much like our balls A and B. But, to make things extra tricky, they are not even definitely in one place or another at any given time, and when we measure the location of such a particle in a system it is impossible for us to know "which one" we have measured with certainty.
This brings up a larger philosophical question: if the things we are composed of seem to lack a unique identity, or rather they share an identity with many of their kind, all of whom exist in different locations at once, how do we get our unique identity?
Look at how the subject of identity is handled in formal logic, philosophical logic, mathematical logic, set theory, and mathematics. Those clear up a lot of questions (though there are still some philosophical questions that arise).
Water is H2O
George Washington is (fill in your description)
9 is 4 + 5
Hesperus is Phosphorus
I've studied the law of identity for quite some time. It is rather perplexing, and requires a good deal of thought to properly understand. It is stated as 'a thing is the same as itself'. Despite the appearance of meaninglessness, there are a number of facets to this law, which we can consider.
The first thing is that this law puts the identity of the thing within the thing itself. This is important to understand, because it means that a thing's true identity is not what we, as human beings say it is, it's true identity is what it is, itself. So if we say "that person is a man", "man" is not that thing's true identity. And even if we say "that person is Art48", "Art48" is not that thing's true identity, because the thing's true identity is the thing itself. The thing itself is the thing's identity
The next thing is that the law of identity allows that a thing might be continuously changing, yet maintain its status as the same thing. This is very difficult to conceive of, because normally when a thing changes, we would say that it ceases to be what it was, and it becomes something different. But since we've allowed that the thing's true identity is within the the thing itself, then what we say about the thing is irrelevant to the thing's identity. So despite the fact that we might say that the thing has changed from being this, to being that, within itself the thing has maintained its identity, as itself, and it may continue to be the same thing which it was.
The law of identity may be denied, as I believe Hegel did. But doing this renders the other laws of logic, noncontradiction, and excluded middle, as useless. This is because these laws put restrictions on what we can truthfully say about a thing, by determining what is impossible for us to truthfully say about a thing. However, if a thing has no identity within itself, then there is no such thing as a thing, and it would be meaningless to talk about what we can truthfully say about a thing.
Interesting.
Water is H20 is Identity - same thing, different names
George Washington is - insert the value statement of your choice - not about identity
9 = 2 + 5 - the law of identify allows us to say that 5 + 4 equals 9.
Hesperus is Phosphorus - is a property of a thing - not identity
The fact that one can refer to properties of a thing does not make the property identical to the thing.
Identity - A is A - A thing is what it is
Non-contradiction - A is not not A - A thing isn't what it isn't
Excluded middle - A or not A - No thing is neither or both.
These axioms seem to allow us to have maths and reason. The questions people seem to ask - are these structures in the human mind, or are they are like Platonism?
The 'law of identity' is the most truthful, possibly the only completely truthful, instance of 'is' or 'equals'. If you trace the western philosophical tradition back to Parmenides, as that quote you mention suggests, the idea is that only 'what is' truly is, and cannot be otherwise. The reason this perplexed that ancients, is that all of the objects of perception - the denizens of the empirical domain - constantly change, decay and turn into something else - so what are they really? Is there anything about them which is real or are they merely illusory?
Those kinds of questions are a long way removed from the attitude of modernity. I'm no expert in any of it, but am trying to re-trace the steps, so to speak, to understand the issues. It should be said that Parmenides and his contemporaries are representative of what is called the 'axial age' philosophies, which also include the Buddha and Lao Tzu. 'The term 'Axial Age,' coined by German philosopher Karl Jaspers (1883-1969), refers to the period between 900 and 300 BCE, when the intellectual, philosophical, and religious systems that came to shape subsequent human society and culture emerged.'
There's an interesting and little-known philosopher with the unusual name of Afrikan Spir, a German-speaking Russian (actually Ukrainian) neo-kantian. I notice the Wikipedia entry on him summarised his ontology thus:
I've actually found a very good PDF of Afrikan Spir's book wherein this is developed, although I haven't tackled it yet.
~A v A excluded middle.
A = B identity.
(I guess, in terms of propositions, whatever proposition implies itself.)
When we start talking about Hamlet or the Moon, then we've already presupposed identity.
Otherwise, what would we be talking about? Meaning would be lost; meaning presupposes identity.
Not that the world has to oblige, though.
Going the other way around, we could say that tautologies are true by definition.
Well, actually, we do.
Within some (logical) system, a proof of a proposition could be showing that the proposition is related to a tautology via bi-implications.
1. This pen is good.
Ergo,
2. This pen will go to heaven.
Is the law of identity about consistency in the meaning of symbols or does it also extend over metaphysical identity? In other words, does the law of identity state something about things themselves and not just the words used to refer to them?
Spot on. One of the best examples of how logic can make our everyday language clear.
It feels odd that, unlike the other two laws of thought, there's an awkwardness to constructing a truth table for the law of identity.
A = A probably means A [math]\leftrightarrow[/math] A but then now we're talking about propositions and logic is about propositions. So, the law should be the law of identity (of propositions).
Identity of indiscernibles: entities x and y are identical if every property possessed by x is also possessed by y and vice versa.
indiscernibility of identicals: if two entities are identical with each other then they have the same properties.
Precisely. In ordinary math we have an identity (valid for all or most values) and conditional equation (valid for a select set of values). 2(x+1)=2x+2 and 2(x+1)=3. Rough definitions.
Words matter! Mass-Energy/Acceleration-Gravity [s]identity[/s] equivalence!
:strong: :up:
It's a useful convention, allowing us to apply logic, make inferences, abstract and generalize etc etc... enabling us to built up knowledge.
It's important to keep in mind that the law of identity, and logic in general, is not about the world, but about language only.
We arbitrarily split classes of particular things off from the whole/the flux of existence by giving them labels, and decide that classes of things that are given the same labels are equal to themselves.... even though 'in reality' only particulars are equal to themselves, and only at the exact same time.
The fact that x is not exactly equal to x generally, doesn't matter all that much, because it still works for our intents and purposes. And we need this basic 'falsification', because without it we wouldn't be able to abstract from particulars to something more general... any kind of knowledge would be impossible.
Yes, they matter, and I chose my words carefully. The identity of indiscernibles is a metaphysical principle of strict identity, not a physics statement.
Example: Joe describes object A to you. He gives you a complete list of of A's properties, including specifically where it was located at a specific point in time.
Separately, Tom described to you an object he calls B, describing a complete set of B's properties including precisely where and when.
Now, you compare A properties to B properties, confident both sets of properties are complete. You notice there's no discernible difference. Identity of indiscernibles implies A is B. The labels A and B refer to the same object. They have the same identity.
This is a good post overall, but the bit about Hegel needs an addendum.
For Hegel, the real is the rational. He's all about trying to walk his readers off the ledge of radical skepticism. Part of the way forward from doubting everything is to accept that the world is rational, not illogical. Because if the world isn't rational, then nothing necessarily follows from anything else, and knowledge becomes impossible.
Hegel definitely wants to keep rationality at the center of his thought. In his early theological writings he sees Logos, unifying reason (identified with Christ in The Gospel of John), as the element in faith that can lift man up and give him freedom.
Hegel's problem with identity as it is commonly formulated comes from his attempt to rebuild logic. He wants to create a logic that is aware of the self as thinking subject, and of objects as existing for that subject. He wants to move past propositions such as, "the apple is red," that take the apple and its redness as existing outside of the perceiving mind. Identity has to be different because identity changes and grows more complete over time as our knowledge grows (as the dialectical progresses). And he doesn't want to look just at the apple as being a part of an individual subject's mind, since he is not a solipsist or subjective idealist, but how it is for all minds.
Thus, the apple's identity can unfold through the progression of history. The truth of the apple is the whole, and so covers the apple as well as the apple seed, the sprout from that seed, the apple sapling that comes from the sprout, the branch that comes off the sapling-turned-tree, the bud on that branch, and the apple that springs from the bud, complete with new seeds. A = A can't fit the truth. Is the seed the tree, the United States in 1795 the United States in 2025? Since the truth of a thing is the process of its becoming the law of identity as commonly used becomes defunct, but it isn't just tossed aside.
[quote=Master Yoda]Hit the bullseye, you have![/quote]
Space and time are properties. Missed that completely!
So, is for every predicate P if Px & also Py, x = y with especial care taken to include locus & tempus among the predicates.
What's your take on two cars of the same model? Would you still say identity of indiscernibles or would you switch to equivalence of indiscernibles?
Muchas gracias again for clearing up the matter for me! There's more to discuss but leave that for another day!
None of these looks the same. Arguably, for almost all (maybe all) the systems we experience in everyday life, they actually are not the same. No pie is the same all the way through. No system is set up so that the distribution of molecules and the energy of individual molecules is evenly distributed throughout. Getting one slice of pie that is the size of half a pie is different from getting four slices that add up to half the area of the pie.
Got me thinking about how much of what we can do in the sciences is the results of being able to abstract away differences. This thread didn't go anywhere but it brings me back to the idea of synonymity. If things only exist as they exist for other things (e.g., information theoretic approaches) than you have a shifting amount of synonymity between different identities. Water = H20 = identity for many reactions. However, push enough water in one direction and tiny differences in energy levels result in (as of today's physics) totally unpredictable and chaotic turbulence. You move from differences that don't make a difference, to differences that make a large, macroscopic difference.
When people say the three laws of thought are
A=A ... identity
A v ~A .... excluded middle
~(A & ~A) ... non-contradiction
they are using 'A' for two different things.
For identity, 'A' ranges over objects.
For excluded middle and non-contradiction, 'A' ranges over propositions.
Indeed, that is not a good presentation. 'A' should not be mixed up that way.
A better statement is:
For all individuals x, we have x=x.
For all propositions P, we have P v ~P and we have ~(P & ~P).
But the "three laws of thought" paradigm does not express the full scope of reasoning about identity or reasoning about propositions. There are other principles that are also needed for reasoning about identity and for propositional logic. The paradigm has been surpassed by those of symbolic logic that are more comprehensive.
His invention of the predicate calculus is great intellectual wisdom.
If it's not too much trouble, can you please refresh my memory on the 3 identity laws in logic? Danke!
There's one that I haven't forgotten:
1. x = y
2. Px
Ergo,
3. Py
I can't remember the name of the rule though. :sad:
There are 2 more, one's called symmetry. That's all I have on the identity laws. :smile:
x=x ... reflexivity
with
(x=y & Px) -> Py ... indiscernibility of identicals (aka substitutivity)
is a complete axiomatization of identity theory and they imply:
x=y -> y=x ... symmetry
and
(x=y & y=z) -> x=z ... transitivity
The converse of the indiscernibility of identicals is the identity of indiscernibles. Interestingly, if the language has infinitely many predicates, then the identity of indiscernibles is not expressible.
Another complete axiomatization (from Wang) is:
Ex(x=y & Px) <-> Py
That proves
x=x
and
(x=y & Px) -> Py
I made a bad typo. I just now corrected it.
This doesn't remove the problem I showed in the paradox.
No problemo, you corrected it before anyone saw it! :smile:
Did you read this? I'm sure you're in the know about what relativist is talking about.
I have a minor use-mention quibble with the penultimate sentence, but otherwise it seems to me that he or she gave a reasonable explanation of the identity of indiscernibles. Why do you ask?
Neither, strictly speaking, because there will be differences (e.g. the VIN number).
These identities lead to consideration of essentialism and natural kinds. "Electron" is a natural kind: all electrons share the same set of properties (except for spatiotemporal location). That set of properties is the essence of electron-ness. Any object possessing that exact set of properties, is necessarily an electron.
Quoting Relativist
Sure, but what if there are not differences, since you bring up electrons? Two electrons Bill and Ted enter from opposite directions a shared space and interact, and leave via different trajectories than their incoming one. Which exiting electron is Ted? Do particles have identity? They seem very much not to. A molecule perhaps does, but a molecule is nearly a classical thing. There's no evidence that they have spatiotemporal location until measured, so that doesn't distinguish them. The topic is about identity of particulars, not shared properties of a universal.
I'm not talking about an epistemological distinction. I'm not asking if it's possible to measure which one is Ted. I'm just asking if one of them is in fact Ted, however much Bill has the same properties.
Given relation R and elements a, b, and c, we may define many properties, but there are 3 of interest,
When a relation has these 3 properties it is called an equivalence relation. Examples are congruence, similarity, "has the same birthday as", and (of course) =.
But there are many common relations which violate one or more of these properties and, thus, are not equivalence relations. Consider
and so on
Electrons however can't be observed like billiard balls above. Nobody knows whether two electrons simply pass through each other without any interaction or behave exactly like billiard balls as described above. In other words, after two electrons, call 'em A and B, have been put on a collision course and they occupy the same spatial region (collision occurs in case of billiard balls), we can't identify which is A and which is B. However, this isn't exactly a counterexample to Leibniz's identity of indiscernibles rule as the electrons can be identified if we get our hands on all the information we need.
The Son/Father is the father/son of The Son/Father!
[quote=Real Gone Cat]"is the sibling of" : violates Reflexive only (assuming sibling means sharing the same mother and father)[/quote]
The Father/Son is the sibling of The Son/Father!
Sancta Trinitas Unus Deus :pray: :pray: :pray:
Then why the roundabout way of stating ~A = A is false? Is it hat we don't want to introduce the "not equal" connective?
Quoting Count Timothy von Icarus
And mathematically there are no actual triangular objects in the physical world, merely approximations.
Quoting Count Timothy von Icarus
Good point
Quoting Metaphysician Undercover
unless change is part of the thing's identity, as a whirlpool for instance, or the human body's continuous process of food intake and subsequent evacuation.
Quoting Count Timothy von Icarus
You might find E-Prime relevant to the above.
https://en.wikipedia.org/wiki/E-Prime#:~:text=E%2DPrime%20(short%20for%20English,conjugations%2C%20contractions%20and%20archaic%20forms
I think we are identified by our pulse, thus A is the man is wrong by your standard of A(being his presence). Having the presence of a man is equal to taking his head- you have his head, metaphorically.
Identity seems to be stringently the pulse as it identifies all of a bio.
These are activities, not things. Activities are attributed to things, as what a thing is doing, so the law of identity doesn't apply. This is partly why it is very difficult for us to gather a complete understanding of activities.
It is that we want to keep the axioms in the most basic form possible. ~A != A (or ~A = A is false) are derived statements that can be simplified to A = A.
Metaphysics (identity & change) of/in Logic.