## Law Of Identity And Mathematics Of Change

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The law of identity is one of the most basic laws in mathematics. The law of identity states that a thing is itself: A=A. While this is true absolutely of things that don't change, the living things (and many non-living things) are constantly changing; and, as impacting on the living things - as well as many non-living things - that change, there needs to be a supplement to this law.

I am Ilya Shambat, and I have always been Ilya Shambat. However I am different now in many respects than I was when I was a toddler, and different also in many respects than I was five years back. A=A in some ways but not in others. A more complete understanding therefore is this:

Something is itself in addition to the changes that it has undergone over time.

Mathematically, this can be seen by taking the A=A equation and replacing it with A1=A0+D. A0 is the initial state; A1 is the later state; and D is the change that has taken place between the initial and the later state.

D - the total change - is a multiple of the time that the change has occurred and the rate of change: D=t*r, where t is time and r is rate of change. The rate of change does not have to be constant, and it does not have to be positive. Change occurs, in all sorts of directions, all the time. And just as, in physics, "work" can be positive and negative, there is also positive change and negative change.

The faster the rate of change, or the greater the time that the change has happened, the greater the change. A big change that takes a short period of time, or small changes accumulated over greater time, both result in a large change.

When there is no time - when t is 0 - then D is also 0, and A is again equal to A. Same is the case when the rate of change is 0. The exception to that rule is if t=0 and r is infinite, or if r=0 and t is infinite; in which case t*r, and thus D, can be anything at all. If either term is infinite and the other term is non-zero, then infinite change is realized.

Change takes time; it also takes speed of change. For any non-infinite time, zero rate of change will produce zero change; and for any non-infinite rate of change, zero time will produce zero change as well.

As Newtonian physics is a subset of larger physics when taken over small speeds and distances, so the law of identity is a subset of change mathematics where either the time or the rate of change is zero and the other term is not infinity. A=A when no change has happened; A1=A0+D when change has.

What A1=A0+D means in reality is that something is itself as it was at the original state, plus or minus the changes that have taken place since that time. This should come as no surprise; but many people do not realize the D factor - the factor of change that takes place in all living beings and in many non-living ones. Based on this miscalcuation people tend to treat others the way they'd known them years previously and not realize the change that they may have undergone during that time. This kind of attitude prevents growth and improvement in people and pigeon-holes them in places that are no longer appropriate. Thus, people may treat contemporary Germans as if they were Nazis, or treat contemporary Jews as if they were Caiaphas, when vast changes have happened in Germans since Second World War and in Jews since 1st century AD. A man may treat his wife based on how she may have been 20 years prior and not realize that she no longer has the same attitudes as she did back then. A person may treat an ex-classmate, 30 years down the road, as though he were still what he was when he was 7. And further on down the line.

The rational response to this misuse of the law of identity is: Living things change. Over time, and with any rate of change, A<>A. A1=A0+D.

The failure to compute change results in all sorts of destructive outcomes. Things are treated as if they were what they'd been in the past without realizing that a lot in them has altered. The attitude of failing to acknowledge change prevents positive change in people from occurring. But it also keeps people from being able to exercise creative intelligence and implement positive changes or keep up with the changes that take place in the world.

The other part of this equation is that A is still A for as long as A exists as itself. Whatever changes I undergo as a person, I am still identifiable as myself. When I am no longer identifiable as myself, I cease to exist. In this case, A1=0; D=(-A0).

Change takes non-zero time, and a non-zero rate of change, for all non-infinite situations. The faster the change, and the greater the time over which it takes place, the greater the change that transpires; the greater the difference between the object at the initial state and the object at a later state.

With the law of identity remaining in place, it is possible to look at another, less obvious, feature: And that is as follows. As change becomes embedded into the fabric of things, so is the time that the change has taken to transpire. Time, through this mechanism, becomes part of things as they are and is encoded in the reality of things. One obvious example is the year rings that we see in the trees; but the same dynamic can be found in all sorts of less obvious situations. An 80-year-old person carries the mark of time, which a toddler does not.

The law of identity is therefore a subset of reality; something that happens when either the time or the speed of change is zero, and the other term is not infinite. In a larger picture, things both change and remain the same. This is something of course that many people understand intuitively; but it takes reasoning and mathematics to understand it at a rational level.
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The law of identity is one of the most basic laws in mathematics. The law of identity states that a thing is itself: A=A. While this is true absolutely of things that don't change, the living things (and many non-living things) are constantly changing; and, as impacting on the living things - as well as many non-living things - that change, there needs to be a supplement to this law.

The law of identity states that a thing is the same as itself, or identical to itself. This does not deny the possibility of change, because despite the fact that the thing is changing it still remains the thing that it is, i.e.the same as itself. What makes a thing a thing, and what makes a thing the thing which it is, "itself", are completely different questions which are not answered by the law of identity. The law of identity simply states that a thing is the thing that it is. And this is regardless of change.
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The mathematics of biology involves a lot of differential equations, which are equations that show exactly how something changes over time. Perhaps you'd find them of interest.

https://en.wikipedia.org/wiki/Differential_equation

A related idea is that we can use differential equations to describe two interrelated changing systems. For example, when there are more predators than prey, the predators eat all the prey; then there aren't enough prey and the predators starve, reducing their population ... which allows the prey to survive and reproduce more, so that there are more of them to eat, and then the predators grow in numbers again. This eternal cycle of predator and prey populations is modeled mathematically by a couple of differential equations.

https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations

The law of identity, by the way, is not a law of mathematics. It's more primitive than that, it's a law of logic. Mathematics inherits the law of identity from logic; math doesn't posit or explicitly assume it.

The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology. The individual components of our model at any instant don't change; and then we can introduce a time variable to account for change from one moment to the next.

In fact basic calculus is the model here.

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change ...

https://en.wikipedia.org/wiki/Calculus

Another point of interest is dynamical systems, which Wiki describes as

... a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

https://en.wikipedia.org/wiki/Dynamical_system

I've always felt that "dynamic systems" is more euphonious as a matter of English usage; but dynamical systems it is.

Change takes non-zero time, and a non-zero rate of change, for all non-infinite situations. The faster the change, and the greater the time over which it takes place, the greater the change that transpires; the greater the difference between the object at the initial state and the object at a later state.

Ah. Perhaps you already understand everything I wrote and you're making a more subtle point, which I'll take a stab at interpreting. You are right. In mathematical modeling, we model time by the real numbers, and then we say that "at time t" the world is in such and so state, no more and no less, unchanging as if frozen in ice. And at some later moment it's in a different state, so Newton showed us how to calculate the limit of this process as the time interval gets small, to thereby assign something we call the "instantaneous rate of change."

If perhaps you are pointing out that this is somewhat of a bogus or artificial abstraction, I quite agree. After all nobody knows whether time itself is accurately modeled by the standard model of the mathematical real numbers. That's a philosophical assumption made by science. It bumps into quantum theory. There are good reasons to doubt the mental model of static states as a function of time, and the standard real numbers as the official model of time. That viewpoint has been pragmatically successful for a few hundred years, but as to its ultimate truth, that's unknown.
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The law of identity should stated as: $\forall x (x=x)$. However, when you write:

because despite the fact that the thing is changing it still remains the thing that it is, i.e.the same as itself.

I believe you are importing metaphysical claims into the law of identity. The law itself is completely neutral with respect to whether or not an object is the same (or different) after undergoing certain change.

But you are right to point out that the OP also seems to add content to the law of identity. That is, it seems the OP wishes to say that the law of identity only applies to things that do not change. But this is also incorrect, as things can change and still remain the same object.

For example, let's consider the following description $D(x)$:

$x$ is a lawyer in a start-up firm in Brooklyn.

Now let's suppose that John ($j$) is a lawyer in a start-up firm in Brooklyn. It would follow that $D(j)$ is true for some time $[t,t+n]$ (i.e., from the moment that John started his job at the start-up firm, to the moment he either left that job to work elsewhere, or the moment the firm was established and no longer a start-up).

But consider that this was not always true about John, and that, indeed, there is an interval of time for which $\neg D(j)$ is true. Despite this, $j=j$, and there are certainly no doubts about this. In short, change can be captured by what set of sentences are true (and false) about a given object at different times.

We could take a radical metaphysical position and insist that objects can only be self-identical for any given time slice $t$. But here too, the law of identity would apply at any given time slice. The law is completely neutral here.

I think this is right, in any case. If someone has any references to philosophers who argue that the law of identity is not neutral with respect to these metaphysical questions, please send me the reference! Thanks!
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I believe you are importing metaphysical claims into the law of identity. The law itself is completely neutral with respect to whether or not an object is the same (or different) after undergoing certain change.

I think the law of identity is itself a metaphysical claim. So it's not a matter of me importing metaphysical claims into the law of identity, it already is a metaphysical claim.

We could take a radical metaphysical position and insist that objects can only be self-identical for any given time slice t tt. But here too, the law of identity would apply at any given time slice. The law is completely neutral here.

Are you familiar with the two distinct forms of identity, sometimes called qualitative identity and numerical identity? Qualitative identity allows that two distinct things, with the very same description, are "the same". Two cars off the same production line may be called "the same". In this case, identity is a function of the thing's description, "what" the thing is. Two human beings are "the same", by virtue of being within the same category, human. I call this logical identity, or formal identity.

Numerical identity, on the other hand, distinguishes one distinct thing from all other things. So the two cars from the same production line are not really "the same" car according to numerical identity. But numerical identity is based in the material existence of the thing, it is not based in a description of "what" the thing is, nor is it based in any particular logical formula whatsoever. I would say that it's based in an observed temporal continuity of existence. This is why the same car can get scratches and dents, new parts and new paint job, and still continue being the same car. This type of identity, which I call material identity, is based in the ontological assumption, "that" the thing is (an existing thing), it is not based in "what" the thing is. Are you familiar with "The Ship of Theseus"? This ancient riddle conflates the two distinct forms of identity (which were not well distinguished at the time), to pose an interesting question.
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If perhaps you are pointing out that this is somewhat of a bogus or artificial abstraction, I quite agree. After all nobody knows whether time itself is accurately modeled by the standard model of the mathematical real numbers. That's a philosophical assumption made by science. It bumps into quantum theory. There are good reasons to doubt the mental model of static states as a function of time, and the standard real numbers as the official model of time. That viewpoint has been pragmatically successful for a few hundred years, but as to its ultimate truth, that's unknown.

You might like this calculus identity:

$\frac{d f}{d g}=\frac{{d}f}{{d}t}\frac{{d}g^{-1}}{{d}t}=\frac{df}{dt}\frac{dt}{dg}$

One can imagine measuring the time it takes a kettle to boil by the heartbeat of the person watching it, the clock measuring both factors out. In that regard time's an instrumental variable for any bijective continuously differentiable function of it.
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I think the law of identity is itself a metaphysical claim. So it's not a matter of me importing metaphysical claims into the law of identity, it already is a metaphysical claim.

The law of identity is a law of logic; it is not an ontological principle. Perhaps you mean Leibniz's law of indiscernibles?

There is a notable and important difference between them.

The law of identity

$(\forall x)(x=x)$

The identity of indiscernibles:

$(\forall x)(\forall y)[(\forall F)(Fx\leftrightarrow Fy)\rightarrow x=y]$

The law of identity is most certainly a principle of logic, not of metaphysics.
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The law of identity is a law of logic; it is not an ontological principle. Perhaps you mean Leibniz's law of indiscernibles?

No, I mean Aristotle's law of identity, which is an ontological principle. It states that a thing is the same as itself. It is ontological because it assumes the existence of the thing. Without the existence of the thing the principle makes no sense. So if any logicians make use of this principle, they are making use of an ontological principle.

The law of identity is most certainly a principle of logic, not of metaphysics.

It may be the case that logicians make use of the principle, but to classify the principle itself, we need to see what validates it, and that is an ontological assumption about the existence of a thing. So it is a metaphysical principle. For example, there are many "scientific principles", and this means that the principles are verified by scientific methods. But when some scientists speculate about metaphysics, and employ metaphysical principles, we cannot call these principles scientific principles just because scientists are using them. Likewise, when logicians employ the law of identity, they are employing a metaphysical principle not a logical principle. It is ontology which states that a thing cannot be other than itself, not logic. What sort of logic do you think one could use to determine that a thing could not be other than itself?
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It isn't empirically clear when two things are identical or different, even when comparing two 'identical' photographs.

Therefore, to my mind A=A should be rewritten A <--> A' to denote a rule of inter-substitution between two entities that are treated as being the same.
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I'm surprised that nobody has considered 'identity of A' as 'continued contextual functionality as A'.
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The law of identity, by the way, is not a law of mathematics. It's more primitive than that, it's a law of logic. Mathematics inherits the law of identity from logic; math doesn't posit or explicitly assume it.

The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology.
Yep. Or basically what we talk is about a bijection. Or set theory.

The law of identity is therefore a subset of reality
No.

Something being basically logic, on a "lower level" as Fishfry said to modeling reality isn't a subset in this way. It would be like saying that arithmetic is a subset differential calculus or probability theory. Or that math is a part of physics… because everything, like our minds, are made of particles.
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It doesn't exclude change, it only excludes equivocation. Whatever you're referring to with "A" needs to be the same in all instances of "A." Otherwise it's the fallacy of equivocation.
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That's basically what I was trying to tell Kornelius. It's an ontological principle because it produces the logical necessity that there is such a thing as what is being referred to with "A", or else the principle is just nonsense. If there was not a particular thing which is referred to with "A" you could refer to anything as A. So the law of identity necessitates the existence of the thing identified.
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No, I mean Aristotle's law of identity, which is an ontological principle. It states that a thing is the same as itself. It is ontological because it assumes the existence of the thing. Without the existence of the thing the principle makes no sense. So if any logicians make use of this principle, they are making use of an ontological principle.

But we know now, because of mathematical advances in logic, that this principle does not assume the existence of anything. The statement $(\forall x) (x=x)$ is made true by any model that assumes no objects: it would be vacuously satisfied, and therefore true.

This makes sense, in any case, since it is a logically true proposition, i.e., it is true in every model, including all models in which no objects exist.

It is simply incorrect to say that the statement that every object is identical with itself implies (or presumes) that an object exists. It does not.

It may be the case that logicians make use of the principle, but to classify the principle itself, we need to see what validates it, and that is an ontological assumption about the existence of a thing.

I am sorry to be blunt, but this is simply incorrect. As I said: every model validates it, no matter whether no objects, some objects or infinitely many (countable or uncountable) objects exist.
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But we know now, because of mathematical advances in logic, that this principle does not assume the existence of anything. The statement (∀x)(x=x) (∀x)(x=x)(\forall x) (x=x) is made true by any model that assumes no objects: it would be vacuously satisfied, and therefore true.

I'm not familiar with your use of symbols, but there is an object assumed, or else there is nothing identified. The object need not be a physical object, are you familiar with mathematical objects? If your statement identifies a mathematical object, then this is an ontological statement, it gives reality to that mathematical object, as an identified object. Perhaps your symbol is the object itself, I don't know what your symbol symbolizes. And a model with no objects makes no sense to me, because the model is itself an object.

It is simply incorrect to say that the statement that every object is identical with itself implies (or presumes) that an object exists. It does not.

That's true, the law of identity itself, does not give existence to any objects. But when the law of identity is used, when an object is identified, then the object necessarily exists, as the object which it is. Otherwise the law of identity is violated. You cannot claim that a specified object is identical to itself, and also say that there is no such object, without launching yourself into nonsense.

I am sorry to be blunt, but this is simply incorrect. As I said: every model validates it, no matter whether no objects, some objects or infinitely many (countable or uncountable) objects exist.

You can say that, but your claim is wrong. Try to demonstrate it, why don't you? Show me a model with no objects which validates the law of identity.
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Once again, I am surprised. This time that some posters on this forum do not understand the difference between a formal coherent model, like classical logic, and potential problems in its application to what we call 'the world'. Kornelius os correct as far as logic based on 'set theory' irrespective of whether an 'object' or ' member of a set' can be said to 'exist in the world'. Indeed 'existence' is a whole other ball game transcendent of the one we usually call 'formal logic'

The law of identity is therefore a subset of reality; something that happens when either the time or the speed of change is zero, and the other term is not infinite. In a larger picture, things both change and remain the same. This is something of course that many people understand intuitively; but it takes reasoning and mathematics to understand it at a rational level.

The problem with this quote taken from the OP is that phrases like 'a subset of reality' are already kowtowing to the 'logic' they are seeking to transcend. The only way out of this would seem to be to resort to neologisms (as for example in Heidegger), or to compare and contrast different 'logics' (as in 'fuzzy sets')
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The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology.
— fishfry

Yep. Or basically what we talk is about a bijection. Or set theory.
ssu

Identity is deeper than bijection. There's a bijection but not identity between {1,2,3} and {a, b, c,}.
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The 'mathematical modelling' you suggest may already operate under names like 'nested systems theory' or 'state transition theory'. But the 'intuitive rationality' involved tends to take us away from a naive view of independent 'objects' towards constructivism and the role of language in promoting ideas of 'persistence'.
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One can imagine measuring the time it takes a kettle to boil by the heartbeat of the person watching it,

Yes Galileo used his heartbeat as a timer.

the clock measuring both factors out. In that regard time's an instrumental variable for any bijective continuously differentiable function of it.

Ah but no. The continuity of the real numbers are the mathematical model of time. But we don't know for sure if time itself is continuous. That was my point. I don't necessarily take differential equations for reality. It's the map/territory thing.
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Ah but no. The continuity of the real numbers are the mathematical model of time. But we don't know for sure if time itself is continuous. That was my point. I don't necessarily take differential equations for reality. It's the map/territory thing.

Eh that's fair. The point I was trying to make was that calculus has the tools to 'internalise' indexical time to any process which (sufficiently smoothly and bijectively) scales with it. In that regard, the evolution of one system with respect to another always gives a derived sense of time.

So, since it's arbitrary for the math, you can think of time relationally; as the pairing of systems creating an index; rather than as the index by which systems evolve.

Edit: or if you want it put (overstated) metaphysically, instead of conceiving as becoming as being changing over time, you can consider time as being's rates of becoming.

Edit2: in a broader mathematical context, time as an (ultimately redundant) output from the coupling of differential operators rather than the medium in which their coupling is expressed.
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Since we've both referenced Lotka-Volterra in previous posts, I'm thinking of the following procedure:

\begin{align} \frac{dy}{dt}= \delta xy - \gamma y\\ \frac{dx}{dt} = \alpha x - \beta xy\end{align}

implies

\begin{align}\frac{dy}{dx} = \frac{\delta xy - \gamma y}{\alpha x - \beta xy} \end{align}

time cancels out without a loss of information.
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I am Ilya Shambat, and I have always been Ilya Shambat.

Or, alternatively, there is no real 'IIya Shambat'. It may not be satisfying, but it's certainly more parsimonious!
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So, since it's arbitrary for the math, you can think of time relationally; as the pairing of systems creating an index; rather than as the index by which systems evolve.

I think you have it a little backwards. We should think of time in relation to physical "clocks," such as heartbeats, diurnal cycles, pendulums or electromagnetic oscillations - because how else can we think of it? That this can be expressed in the form of the chain rule when modeling processes using differentiable functions is just a consequence. The backwards reasoning from a mathematical model to reality is inherently perilous, because mathematics can model all sorts of unphysical and counterfactual things.

Edit: or if you want it put (overstated) metaphysically, instead of conceiving as becoming as being changing over time, you can consider time as being's rates of becoming.

Yes, except that when you ask what "rate" is, time creeps back in. I don't think you can completely eliminate time from consideration, reduce it to something else. You can put it in relation to something else, such as a clock (heartbeats, etc.), but that relationship is not reductive: it goes both ways. Clocks are just as dependent on time as time is on clocks.
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He's just saying that if you use the variable to refer to something, then that thing exists as something, whether it's just an idea or description or whatever it is.
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Kornelius os correct as far as logic based on 'set theory' irrespective of whether an 'object' or ' member of a set' can be said to 'exist in the world'. Indeed 'existence' is a whole other ball game transcendent of the one we usually call 'formal logic'

Yes, but my point is that the law of identity transcends formal logic as well as the notion of 'existence", and that is why it is an ontological principle rather than a principle of logic. It is evident that the law of identity transcends logic by the fact that there are two incompatible forms of identity, what is referred to as qualitative and numerical identity. That these two are incompatible, and cannot be synthesized into one, is demonstrated by the riddle of The Ship of Theseus.

Whichever of the two forms of identity that you choose to employ in your logical endeavours, will determine the outcome of your logic, like a premise. Sure you can take "what a thing is", without that thing having existence (like a symbol which represents nothing), and proceed to apply logic to this "what a thing is", but then you necessarily use qualitative identity. However, the law of identity clearly deals with numerical identity, the thing itself. So all you do in this case is separate "what the thing is", from the thing itself, and circumvent the law of identity. Therefore this logic which you refer to, does not actually employ the law of identity, it avoids the force of the law by hiding behind the illusion that qualitative identity is identity in the sense of the law of identity. But it is not, so it violates the law of identity by choosing qualitative identity as a principle, instead of numerical identity, which is required by the law.
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(Edit problems)
I can't see that the law of identity makes any ontological claim at all other than that 'objects' might have static fixed identity rather than dynamic continued functionality. But that is the essence of the OP and the basis of the pseudo-problem of the Ship of Theseus. If that is what you are driving at then I agree.
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I can't see that the law of identity makes any ontological claim at all other than that 'objects' might have static fixed identity rather than dynamic continued functionality. But that is the essence of the OP and the basis of the pseudo-problem of the Ship of Theseus. If that is what you are driving at then I agree.

No, it's not really what I'm driving at at all. To think that the law of identity states that an object has a static fixed identity is a misunderstanding of the law. What the law does is place the identity of the object within the object itself, rather than within a description or a name. Therefore the identity of the object is just as dynamic as the object itself is, because an object's identity is the same thing as the object. "An object is the same as itself". But what the law does, which requires a metaphysical assumption, is to state that there is something there with a temporal continuity, an object. This is required in order that it may actually have "an" identity, rather than a multiplicity. And, despite all the changes which are occurring, there is something which is remaining the same, which has an identity as "the object". This is why it is an ontological principle.

The Ship of Theseus is a pseudo-problem because it starts with the ontologically based premise that there is an identified object called the ship of Theseus, and that this thing has some sort of temporal extension. Once you recognize that there is no necessity by which such an assumption is produced, the problem disappears because the name could be applied arbitrarily.
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Understood. However I still see the issue as one of 'applicability' rather than one of 'metaphysical assumption'. Nietzsche's dismissal of the distinction between 'description' and 'reality' seems to be be relevant to our case.
And in terms of 'applicability' we might remember Niels Bohr's adage: ''No, no...you are not thinking...you are just being logical ! ''
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I agree, the distinction between description and reality is relevant here. The law of identity attempts to get right to reality, independent of what we say about it, by placing the identity of the thing right within the thing itself. There's a critical point which needs to be understood though, and that is that anything we say about the thing is always going to be something said about it, and not part of its real identity, what's within it. So the law of identity itself is always going to suffer that problem of being something said about reality (descriptive), though its intent is to say something true, real. That is why it is an "assumption". The formulators of the law have looked for the most fundamental, the most widely applicable principle in relation to "reality". So, recognizing that anything we say about reality will necessarily be descriptive, the law of identity is an attempt to say the most important thing about reality which can be said, and that is to emphasize this separation, and put the real identity of the thing within the thing itself, rather than within what we say about it. The assumption is that this separation is true, real. That's what the law of identity gives us, is an indication of the separation between the true identity of the thing, which is within the thing itself, and the identity which we assign to the thing. The thing itself is an object, but in grammar the object is represented as the subject, and this is that separation, predication is of the subject. And that separation must be maintained.

To be able to properly apply the law of identity requires that one understand the law. Leibniz's "identity of indiscernibles" is an application. Simply put, it tells us that if there are two distinct things, then they are not identical (i.e. not the same thing), and conversely, if there is no difference between what appears like two distinct things, then it is actually one thing (the same thing).
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I'm not familiar with your use of symbols, but there is an object assumed, or else there is nothing identified. The object need not be a physical object, are you familiar with mathematical objects? If your statement identifies a mathematical object, then this is an ontological statement, it gives reality to that mathematical object, as an identified object. Perhaps your symbol is the object itself, I don't know what your symbol symbolizes. And a model with no objects makes no sense to me, because the model is itself an object.

Hi MU,

I apologize: I should not have assumed you were familiar with this; that is completely on me. I am employing standard first-order logic notation. The statement $(\forall x)(x=x)$ says "for all x, x is identical to x."

Yes, I do know about abstract objects but, no, the statement would be true in a model where there are no objects at all, whether abstract, physical, etc.

The symbol is also not an object.

You cannot claim that a specified object is identical to itself, and also say that there is no such object, without launching yourself into nonsense.

This is incorrect. I will show this in my reply to this:

You can say that, but your claim is wrong. Try to demonstrate it, why don't you? Show me a model with no objects which validates the law of identity.

This is easy to show, and it is something that would be taught in an introductory course in formal logic in every philosophy and mathematics department. I will try my best to elucidate the concepts as best I can since you mentioned that you are not familiar with first-order logic and model theory. I strongly recommend studying these topics; it is an absolute must for philosophy!

Let $\mathcal{L}$ be the standard first-order language in which $(\forall x)(x=x)$ is expressed.

(*This is just to say that we are talking about a sentence in first-order predicate logic, using the usual syntax of a first-order language, and only allowing quantification over objects (and not over predicates). None of this is actually important)

Let $\mathcal{M}$ be a structure whose domain is $\varnothing$ as well as the usual interpretation for the symbols in $\mathcal{L}$. There is no need to specify the interpretations for our purposes.

(*A structure in logic is a set equipped with functions that assign a semantic value (or interpretation) to the non-logical symbols in the language. So, for example, the symbol "=" in our language will get assigned the usual interpretation (equality), etc. This is the only assignment that is relevant here in any case, since the symbol $\forall$ is a logical constant, so does not get re-interpreted)

Now, a sentence $P$ is valid in $\mathcal{M}$ if (and only if) the structure/model $\mathcal{M}$ entails $P$. We write: $\vDash_{\mathcal{M}} P$. This would mean that $P$ is (logically) valid in $\mathcal{M}$.

Indeed, if for any model $\mathcal{M}^*, \vDash_{M^*} P$, then $P$ is a LOGICAL TRUTH. This is just to say that it is true in every model.

Now it follows, vacuously, that $\vDash_{M} (\forall x)(x=x)$ since there are no objects in $\varnothing$. If you do not see that it is vacuously satisfied, consider this:

$(\forall x)(x=x)$ is logically equivalent to $\neg (\exists x)\neg(x=x)$. That is, it is logically equivalent to the proposition that: it is not the case that there exists an object that is not identical to itself. It is obvious, then, that $\vDash_{\mathcal{M}} \neg (\exists x)\neg(x=x)$. Thus, $\vDash_{M} (\forall x)(x=x)$.

In short, the universe of discourse in the structure $\mathcal{M}$ we just considered is empty, i.e., there exist no objects. And this structure satisfies the law of identity.

Indeed, the law of identity is true in EVERY model $\mathcal{M}^*$ with any domain of objects (empty or not).

I understand that this might seem overly technical if you haven't been exposed to logic, but I hope I made it as accessible as I could in such a post. Please let me know if there is any step that isn't clear!

Cheers.
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