That's what it is - a tool for working through those apparent contradictions — Banno

I have seen only stipulative resolutions to the murky aspects of identity, and frankly such resolutions seem to have no significant point and to be of little value.

But I guess it all comes down to personal tastes and interests.

1. $x = y$ [assume]
2. $\diamond ( \neg x = y)$ [assume]
3. $\diamond (× = y)$ [from 1]
4. $\diamond (x = y) \land \diamond (\neg x = y)$ [from 2 and 3]
5. $\diamond (x = y \land \neg x = y)$ [from 4]
6. $\neg \diamond (x = y \land \neg x = y)$ [contradictions impossible]
7. $(\diamond (x = y \land \neg x = y)) \land (\neg \diamond (x = y \land \neg x = y))$ [from 5 and 6]
8. $\neg \diamond (\neg x = y)$ [2 to 7 reductio ad absurdum]
9. $\Box \neg (\neg x = y)$ [from 8]
10 $\Box (x = y)$ [from 9]
11. $(x = y) \to \Box (x =y)$ [1 to 10 conditional proof]

QED

https://www.umsu.de/trees/#(x=y)~5~8(x=y)

1. $x = y$ [assume for conditional proof]
2. $\Diamond (x = y)$ [from 1]
3. $\neg \Diamond (x = y \land \neg x = y)$ [LNC]
4. $\Box \neg (x = y \land \neg x = y)$ [from 3]
5. $\Box (x = y \lor \neg x = y)$ [from 4]
6. $\Box (x = y) \lor \Box(\neg x = y)$ [from 5]
7. $\Box (x = y) \lor \neg \Diamond (x = y)$ [from 6]
8. $\Box (x = y)$ [from 2, 7]
9. $(x = y) \to \Box (x = y)$ [1 to 8 conditional proof]

QED

Not just valid, also sound?

(9) is a tautaology.

(9) is a tautaology — Banno

First, thanks for the awesome link. I'll need it. Does the site also explain the rules and latest notations of natural deduction?

Second, yep, it's a tautology. Is that a bad thing? It's a property of the system as opposed to being a property of the world?

It's a useful page, although I find trees a bit hard to follow, so I mostly use it to check validity.

Since it's a tautology it is necessarily true, ie, true in all possible worlds. "Property of the system" if you like; it's a grammatical structure that cannot be wrong.

It follows Kripke's argument in showing that any identity is a necessary identity.

How do you think it helps here? Kripke suggests that if x=y is known by observation, then ☐(x=y) is also known by observation, but @Mww continues to deny this, claiming despite the examples given, that x=y is never known by observation but is always a priori. I don't see that this will convince him otherwise. No helping some folk.

Edit: Try this one: https://www.umsu.de/trees/#(x=y)~5~8(x=y)||universality . S5 is easier to follow.. Simple accessibility relations. w and v being possible worlds.

@Banno Perhaps @Mww doesn't agree that once x = y is observationally confirmed, $\Box (x = y)$ (tautology). Does s/he mean that if, observationally x = y, it is possible that, observationally ~x = y. I believe that amounts to a contradiction.

Observationally x = y $\to \Diamond (x = y)$

If, as Mww claims $\neg ((x = y) \to \Box (x = y))$ then $\Diamond (x = y \land \neg x = y)$, but $(x = y \land \neg x = y)$ is a contradiction (impossible) i.e. $\neg \Diamond (x = y \land \neg x = y)$

Ergo, reductio ad absurdum, $(x = y) \to \Box (x = y)$.

The law of identity allows that a thing could continue to be the same thing, despite undergoing change. — Metaphysician Undercover

How can that be ?

In logic, the law of identity states that each thing is identical with itself. Gottfried Wilhelm Leibniz expressed it as "Everything is what it is". Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A".

Object A, from the law of identity, is identical with itself.

I am taking object A as something that exists in the world, not as the name "object A". Object A and "object A" are different things. Object A exists in the world and "object A" exists in the mind. "Object A" is the name for object A.

All the properties object A has are necessary for object A to be object A, and all the properties object A has are essential for object A to be object A.

If object A changes into Object B over time, even if it has lost only one molecule, then object B cannot be the same as object A.

All the properties object A has are necessary for object A to be object A, and all the properties object A has are essential for object A to be object A.

If object A changes into Object B over time, even if it has lost only one molecule, then object B cannot be the same as object A. — RussellA

Are you content with this account? Is the property of having writ that post essential to your being who you are? Might you have not written it, yet remained RussellA?

The heat moves from one body to the other, in a process that can be described with mathematical predictability. — Banno

I don't see what relevance this has to the topic, unless you are claiming that the motion of molecules causes heat. — Banno

I agree that Kripke has put forward his case that true identity statements are necessary before introducing the examples of names, heat and my pain. However, if the examples he uses of true necessary identity statements are not in fact true necessary identity statements, then this casts doubt on the case he has previously made.

For example, he is using the word "heat" in two different ways, and when he writes "heat is the motion of molecules", it is unclear in what sense he is using the word "heat". "Heat" can refer to the cause and "heat" can also refer to the effect.

Specifically, Kripke gives an example of a true necessary identity statement as "heat is the motion of molecules"

Heat is the transfer of energy between objects due to a temperature difference between them. Bodies don't contain heat.

Problem one - as regards heat transfer by convection, a body having internal energy moves from location A to location B. As bodies don't contain heat, how can heat be the motion of molecules ?

Problem two - as regards heat transfer by conduction and radiation, no molecules move from hotter object A to cooler object B. How can heat move from A to B, if heat is molecules in motion, and no molecules move from A to B ?

Sure, it's actually in this room. But it might possibly have been in the other. — Banno

On logic…..

“…. Notwithstanding, there lies so seductive a charm in the possession of a specious art like this (…) any attempt to employ it as an instrument (organon) in order to extend and enlarge the range of our knowledge must end in mere prating; any one being able to maintain or oppose, with some appearance of truth, any single assertion whatever. Such instruction is quite unbecoming the dignity of philosophy…”

It is possible for there to be a lectern in another room. It is possible for a lectern to be anywhere. According to the example in question, “this very object, in the room it is in fact in, even at this very time” cannot possibly be in any other room.

Try as I might, some folks I just can’t help. Horse/water kinda thing, I guess.

“this very object, in the room it is in fact in, even at this very time” cannot possibly be in any other room. — Mww

And yet the lectern might have been in the other room, you might not have written your reply, Kant might have gone into fishing rather than philosophy... all of these things might have happened. They didn't, but they may have.

Recent modal logic gives us a way to deal with such suppositions. What you have proposed, does not.

In this possible world, the lectern is in this room, not in the other room. In some other possible world, it is otherwise, without contradiction.

Logic has advanced somewhat since Kant.

How can that be ?

In logic, the law of identity states that each thing is identical with itself. Gottfried Wilhelm Leibniz expressed it as "Everything is what it is". Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A". — RussellA

Yes, each thing is identical with itself, everything is what it is. And, things are changing as time passes. Therefore being identical with itself, or being what it is, means changing as time passes. There is no inconsistency between the two.

If it were the case that when a thing changes it is no longer the thing that it is, then an object could not have any temporal extension. At every moment, as time passes, and the object changes, it would become a new object. So, in order that an object can have temporal extension, and maintain tis identity as the thing which it is, while time passes and it also changes, Aristotle proposed the law of identity. This is the reason for his "hylomorphism", a material thing has two aspects, form and matter. The form is changing (actual) and the matter stays the same (potential).

If object A changes into Object B over time, even if it has lost only one molecule, then object B cannot be the same as object A. — RussellA

This is where we have to be careful not to be fooled by sophistry. When object A requires a new description (because it's properties change due to the passing of time), this does not mean that it has become a different object. That's the very reason for the law of identity, to allow us to say that a thing maintains its identity as the same thing, which it is, despite changing as time passes.

Without this law, all sorts of logical sophistry ensues. Instead of being within the object itself, as is the case with the law of identity (as the same as itself), an object's identity is what we say about a thing. And as Banno is demonstrating, we can say very strange things about objects, and show how these strange things are consistent with our axioms of identity. Of course, when the fundamental axiom is that the object's identity is what we say about it, rather than what it is in itself, we can make its identity whatever we want, and there is no truth to the matter.

Logic has advanced... — Banno

"Declined" is probably a better word to use here. Look at the confused mess of quantum mechanics as an example.

Are you content with this account? Is the property of having writ that post essential to your being who you are? Might you not have written it, yet have remained RussellA? — Banno

Yes. I'm not saying that I have Dissociative Identity Disorder, but there are two distinct RussellA's.

There is the RussellA that exists in the world as fundamental particles and forces, and whose identity is constantly changing. At each moment in time, RussellA's properties are both necessary and essential in order for RussellA to be identical with itself. Even if RussellA loses only one fundamental particle, then RussellA's identity would have changed

There is also the "RussellA" that exists on this Forum and whose identity stays the same over time. A "RussellA" that exists in society, exists in the mind and exists in language. "RussellA" has what Kripke calls rigid designators, ensuring that "RussellA's" identity remains the same even in all possible worlds. "RussellA" has necessary and essential properties. What these are no one knows for certain, but I believe that I have them, even if I don't know what they are. I might have had to go shopping, not had time to write this post, yet remained "RussellA". However, RussellA would inevitably have changed, gained weight, lost some more hair, etc.

If I had not written this post, I would still be "RussellA" but I wouldn't be RussellA.

all of these things might have happened. They didn't, but they may have. — Banno

Nobody cares about what might have happened, when they are only affected by what does.

Recent modal logic gives us a way to deal with such suppositions. What you have proposed, does not. — Banno

Correct. Not much need to deal with might-have-beens. Psychologists excepted, but (sorry, Isaac) no proper philosopher cares about them anyway.

Logic has advanced somewhat since Kant. — Banno

Logic has changed. Whether it has advanced, is questionable. All the basic conceptions of modern modal logic are already contained in Kantian metaphysics, and have been classified as such since Aristotle.

Admit it, Good Sir: you’re grasping at straws. All the cool stuff has already been done, and you missed the boat.

When object A requires a new description (because it's properties change due to the passing of time), this does not mean that it has become a different object. That's the very reason for the law of identity, to allow us to say that a thing maintains its identity as the same thing, which it is, despite changing as time passes. — Metaphysician Undercover

A person can maintain their identity as the same thing yet at the same time have different properties.

But how can an object maintain its identity as the same thing yet at the same time have different properties ?

A person can maintain their identity as the same thing yet at the same time have different properties.

But how can an object maintain its identity as the same thing yet at the same time have different properties ? — RussellA

Permit me to take a stab at that.

Properties
1. Essential i.e. critical to identity e.g. the 3 sides + the 3 angles of a triangle.
2. Incidental i.e. not critical to identity e.g. the color of the triangle above.

Permit me to take a stab at that. Properties 1. Essential i.e. critical to identity e.g. the 3 sides + the 3 angles of a triangle. 2. Incidental i.e. not critical to identity e.g. the color of the triangle above. — Agent Smith

There are objects in the world and there are "objects" in the mind

Assume in the world is something that has a set of properties: being in a flat plane, three straight sides, having three angles, 10cm large, colour green, rotated at 45 degrees, located in Paris, made of paper, makes no sound, has no smell, etc

What determines that some of these properties are essential to the identity of this something and some properties are incidental ? A mind-independent world cannot make this determination, it can only be made by a person. However, in the event that one person judges that colour is an essential property whilst another person judges that colour is an incidental property, how is it determined who is correct ?

If it is a person who is judging which properties are essential and which incidental, then they are not referring to something that exists in the world as a set of properties, but they are referring to something that exists in the mind as a concept.

I agree that concepts in the mind maintain their identity even though different instantiations of the concept in the world may be associated with different properties.

Logic has changed. Whether it has advanced, is questionable. All the basic conceptions of modern modal logic are already contained in Kantian metaphysics, and have been classified as such since Aristotle. — Mww

Hrmm... I think I'd say logic has changed considerably since Kant, and I'd say that it's for the better too. While Kant has the modalities as categories I'd say that's a problem with his logic -- Camus even makes a joke about that in The Myth of Sisyphus, so I always presumed it was understood that the modal categories are kind of funny in that they don't really spell out either a relation between objects (causality) or properties of objects (quality and quantity), but rather pick out judgments of a certain kind.

Modal logic is more specific than Kant's.

Furthermore, the categories are part of a transcendental logic, right? So we can easily see Kripke as contributing to logic, as such, to use Kant's distinctions. This is a pure logic rather than a transcendental logic. At least, this is how I'd put things. (The difference between logic as such and transcendental logic is... not easy to spell out. If this doesn't click, then this is probably as clear as I can be without more work.)

Those are good questions. I guess it would depend on, inter alia, the studier's goals and/or, colloquially speaking, brains and/or circumstances.

I'll try to illustrate with examples.

1. Goals: If my goal is finding the area of a triangle then I'd need to focus on the angles and sides (how many there are and their values).

2. Brains: Properties are abstractions and to home in on those properties that are essential takes brains. Everybody saw objects fall to the earth, the moon traverse the night sky, but it took a Newton to realize that these two were the same thing.

3. Circumstances: A biologist may be interested in a tiger's penis :rofl: but if you were to encounter one in the forest, your eyes would immediately zoom in on its claws and fangs.

It looks like I consider essential is contextual - it depends on the situation. Do I mean that there are no essential properties, a sine qua non for classification/categorization/identification? At this point, I call upon the all-too-familiar phrase ceteris paribus - the world is chaotic, but every once in a while there's a lull in the storm and when we place (an) object(s) in that quiet zone, its/their true essence comes into view.

I think I'd say logic has changed considerably since Kant, and I'd say that it's for the better too. — Moliere

You wouldn’t be alone.

Modal logic is more specific than Kant's. — Moliere

Maybe; dunno. Specific in what way? As I said in another thread… one division containing two books containing five chapters containing eight sections containing 179 pages…..and an appendix. All as only one of two divisions in an rather thorough exposition of a very specific human logical functionality.

Can’t help but think the moderns have that exposition, as the ground of their own presuppositions. Likely, since Kripke actually begins this article with a reference to it.

But this isn’t the place for Kant himself, so…..

Maybe; dunno. Specific in what way? — Mww

Just in the way that multiple people can work with it, understand it, communicate about it, and even -- sometimes -- use it. Speaking about Kant we don't really use his categories as much as argue whether or not they are necessary for all the other stuff we do. It's a confusing logic, even if it is ultimately correct. With Kant's categories he is so certain that we know what he's talking about that he says we already know what he's saying.

Yet, here we are -- reading a transcript of a talk about different interpretations of modalities.

Not that one couldn't work this into Kant's project, necessarily... that's why I posited the as-such/transcendental distinction between different notions of logic. Especially because @Banno was emphasizing how this is just a way of talking, rather than a metaphysics. I think transcendental logic gets close to metaphysics in the wider sense of the philosophical tradition, while demarcating what is and isn't metaphysics by Kant's philosophy.

Ok; all good….

With Kant's categories he is so certain that we know what he's talking about that he says we already know what he's saying. — Moliere

….except for that. He was pretty certain we commoners hadn’t a clue what he was talking about, even though he says every one of us is doing what his theory suggests. But I see what you mean: after his explanation, we can say…oh hell yeah, that’s right!!!!

@Banno, do you notice any issues having to do with the fact that logic in the context of the issue being discussed in this thread is pure symbol manipulation, of course, as per some agreed up, rational rules.

As you mentioned, $(x = y) \to \Box (x = y)$ is a tautology in the sense its truth is independent of reality - it's, some would say, just the way the system works.

I pulled out the old Pluhar to look and see what I was talking about -- and I think you're right. I'll just post the quote that was in my head upon finding it to offer clarification on what you quoted:

In this treatise I deliberately refrain from offering definitions of these categories, even though I may possess them. I shall hereafter dissect these concepts only to a degree adequate for the doctrine of method that I here produce. Whereas definitions of the categories could rightly be demanded of me in a system of pure reason, here they would only make us lose sight of the main point of the inquiry. For they would give rise to doubts and charges that we may readily relegate to another activity without in any way detracting from our essential aim. Still, from what little I have mentioned about this, we can see distinctly that a complete lexicon with all the requisite explications not only is possible but could easily be brought about. The compartments are now at hand. They only need to be filled in; and a systematic [transcendental] topic, such as the present one, will make it difficult to miss the place where each concept properly belongs, and at the same time will make it easy to notice any place that is still empty. — CPR Pluhar translation, A83/B109

Here's a blatant appeal to authority:

These arguments overturned the conventional view, inherited from Immanuel Kant (1720–1804), that identified all a priori propositions as necessary and all a posteriori propositions as contingent. — Britannica

Your first RussellA is rather ephemeral, dissipating in an instant. An individual hardly worthy of the title.

I think the physics here is misguided. We might consider other examples. For instance, that water is H₂O was found to be the case by experiment. Yet in any possible world, if what is called "water" is not H₂O, then in that world "water" refers to something else.

We can plug this in to (x=y)→◻(x=y), with "water" substituted for x and "H₂O" substituted for "y".

Hence, and as it is more commonly phrased, water is necessarily H₂O. Moreover, that water is H₂O is known by investigating how the world is, not by examining the terms involved.

...just the way the system works. — Agent Smith

Well, yes, although I would phrase this more specifically, as that logic sets out how we can talk about stuff while maintaining consistency. So if one grants the use of the predicate "=" to be that the thing on the left is the very same as the thing on the right, one must grant (x=y)→◻(x=y).

...pure symbol manipulation — Agent Smith

Again, yes, although one should avoid the temptation to deny the consequences for how we talk. So the discovery that water = H₂O leads us to conclude that necessarily, water = H₂O, or leave consistency behind.