Law Of Identity And Mathematics Of Change
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.
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.
Comments (57)
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.
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.
https://en.wikipedia.org/wiki/Calculus
Another point of interest is dynamical systems, which Wiki describes as
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.
Quoting Ilya B Shambat
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.
The law of identity should stated as: [math] \forall x (x=x)[/math]. However, when you write:
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 [math]D(x)[/math]:
[math] x [/math] is a lawyer in a start-up firm in Brooklyn.
Now let's suppose that John ([math]j[/math]) is a lawyer in a start-up firm in Brooklyn. It would follow that [math]D(j)[/math] is true for some time [math][t,t+n][/math] (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 [math]\neg D(j)[/math] is true. Despite this, [math]j=j[/math], 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 [math]t[/math]. 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!
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.
Quoting Kornelius
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.
You might like this calculus identity:
[math]\frac{d f}{d g}=\frac{{d}f}{{d}t}\frac{{d}g^{-1}}{{d}t}=\frac{df}{dt}\frac{dt}{dg}[/math]
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.
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
[math] (\forall x)(x=x)[/math]
The identity of indiscernibles:
[math] (\forall x)(\forall y)[(\forall F)(Fx\leftrightarrow Fy)\rightarrow x=y][/math]
The law of identity is most certainly a principle of logic, not of metaphysics.
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.
Quoting Kornelius
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?
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.
Yep. Or basically what we talk is about a bijection. Or set theory.
Quoting Ilya B Shambat
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.
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.
But we know now, because of mathematical advances in logic, that this principle does not assume the existence of anything. The statement [math](\forall x) (x=x)[/math] 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.
Quoting Metaphysician Undercover
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.
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.
Quoting Kornelius
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.
Quoting Kornelius
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.
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')
Identity is deeper than bijection. There's a bijection but not identity between {1,2,3} and {a, b, c,}.
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'.
Yes Galileo used his heartbeat as a timer.
Quoting fdrake
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.
Since we've both referenced Lotka-Volterra in previous posts, I'm thinking of the following procedure:
[math]\begin{align} \frac{dy}{dt}= \delta xy - \gamma y\\ \frac{dx}{dt} = \alpha x - \beta xy\end{align}[/math]
implies
[math] \begin{align}\frac{dy}{dx} = \frac{\delta xy - \gamma y}{\alpha x - \beta xy} \end{align}[/math]
time cancels out without a loss of information.
Or, alternatively, there is no real 'IIya Shambat'. It may not be satisfying, but it's certainly more parsimonious!
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.
Quoting fdrake
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.
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.
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.
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.
And in terms of 'applicability' we might remember Niels Bohr's adage: ''No, no...you are not thinking...you are just being logical ! ''
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).
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 [math] (\forall x)(x=x)[/math] 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.
Quoting Metaphysician Undercover
This is incorrect. I will show this in my reply to this:
Quoting Metaphysician Undercover
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 [math] \mathcal{L}[/math] be the standard first-order language in which [math] (\forall x)(x=x)[/math] 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 [math]\mathcal{M}[/math] be a structure whose domain is [math]\varnothing[/math] as well as the usual interpretation for the symbols in [math]\mathcal{L}[/math]. 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 [math]\forall[/math] is a logical constant, so does not get re-interpreted)
Now, a sentence [math]P[/math] is valid in [math]\mathcal{M}[/math] if (and only if) the structure/model [math]\mathcal{M}[/math] entails [math]P[/math]. We write: [math]\vDash_{\mathcal{M}} P[/math]. This would mean that [math]P[/math] is (logically) valid in [math]\mathcal{M}[/math].
Indeed, if for any model [math]\mathcal{M}^*, \vDash_{M^*} P[/math], then [math]P[/math] is a LOGICAL TRUTH. This is just to say that it is true in every model.
Now it follows, vacuously, that [math]\vDash_{M} (\forall x)(x=x)[/math] since there are no objects in [math]\varnothing[/math]. If you do not see that it is vacuously satisfied, consider this:
[math](\forall x)(x=x)[/math] is logically equivalent to [math]\neg (\exists x)\neg(x=x)[/math]. 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 [math]\vDash_{\mathcal{M}} \neg (\exists x)\neg(x=x)[/math]. Thus, [math]\vDash_{M} (\forall x)(x=x)[/math].
In short, the universe of discourse in the structure [math]\mathcal{M}[/math] 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 [math]\mathcal{M}^*[/math] 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.
But the variable need not refer in certain models, since certain models may be empty. But it is true nonetheless since it is a quantified statement. Consider my previous post. But to be clear, a variable does not refer to a particular object. It is open for a semantic assignment (i.e. open to be assigned an object, though not in quantified statements, since the variable does not occur free). It has been a while for me, but I think all this is correct...
Now as it concerns "referring to ideas, descriptions, etc.", I am not too sure I follow and I am not sure how this would impact the discussion and the law of identity (which applies to objects, not descriptions, etc.).
You know, the bolded bit amused me, because avoiding the reification of a time beyond or above the unfolding of processes was precisely what I wanted to do. The idea that a clock is simultaneously a measurement of and a definer of time is a bit weird (@Banno @Luke @Fooloso4 @StreetlightX for Wittgenstein thread stuff :) ). I think it's better to think of periodic phenomena as operationalisations of a time concept which is larger than them; ways to index events to regularly repeating patterns.
Another thing the chain rule there reveals, though obviously in some poetic sense, is that there are multiple 'times' and their rates of unfolding differ. Everest slowly increasing in height is effectively a zero rate from the buzz of city life, but from the perspective of stellar accretion Everest's process of increasing height is like driving past the speed limit.
Quoting SophistiCat
Thought experiment here - suppose that the universe is a process of unfolding itself, how can there be a time separate from the rates of its constitutive processes? What I'm trying to get at is that we should think of time as internal to the unfolding of related processes, rather than as an indifferent substrate unfolding occurs over. Think of time as equivalent to the plurality of linked rates, rather than a physical process operative over all of them. Just like 'the kidney' is not an organ of the body, but kidneys are.
Yes, exactly, clocks (periodic processes) don't define time in the way definitions usually work, i.e. by completely reducing one concept to one or more other concepts; instead they operationalize time.
Quoting fdrake
I agree with you here: it wouldn't do to think of time as just mathematical time of scientific models, or as an abstract metaphysical entity that exists independently of the world of physical processes. Just as there is no movement without there being moving things, there is no time without there being processes, unfoldings, etc.
And yet... how can there be processes, what could unfolding possibly mean, what are we to make of rates - without referring to the concept of time? I still insist that, although all these physical concepts in the first part of the sentence - let's refer to them as clocks for brevity - serve to operationalize time, they do not define time away; they are not more primary in our understanding than time itself is. And while we cannot understand time without referring to clocks, neither can we understand clocks without referring to time.
There's a lot going on in the question.
(1) There's an epistemological issue - which Kantian/phenomenological considerations fit into - how are clocks (operationalisations of time) interpretively pre-structured by the categories of the understanding or by experiential temporality.
(2) There's a cultural issue - what are the origins of the unified concept of time, what kind of understandings do people have to learn to grok time?
(3) Then there's an ontological (well, also ontic) issue about unfolding/becoming being dependent upon time for it to unfold.
The interesting issue here is (3), but we need to talk about what not to do given (1) and (2).
I would like to posit that insofar as (1) experiential temporality, or the transcendental structure of time are related to the issue, we shouldn't index ontical unfolding - natural time/temporality - as a development of experiential temporality. Only our understanding of ontic time is facilitated by experiential temporality. Experiential temporality allows the issue to be raised in the first place, but is otherwise irrelevant to providing a good exegesis of the interdependence of time and unfolding. The first eye opening was one event in the natural flows that subtend our existence.
In regards to (2), just like we can't say that a mathematical entity must have a corresponding entity in a nature for a theory of nature which has that mathematical entity in it for that theory to be correct (example: infinite plane waves, sum over 'all histories' approach in quantum mechanics), we should not be so sure that cultural artefacts and norms of interpretation vouchsafe the necessary existence of a referent of words. Nature informs our vocabulary through our understanding, but light's frequencies are nevertheless not arranged in a colour wheel.
I'm quite suspicious, therefore, that something like time would have a unique ontic correlate - for there to be a pattern of nature which is time - just because the unfolding of processes requires a time concept to think. To me, it appears that something like the type-token distinction is at work here; the word time is a sortal we learn that synthesises the operationalisations that we are first exposed to, the mathematical abstractions of periodic processes, numbers inscribed on clockfaces, the rhythm of our hearts and so on. Our understanding is densely populated with things and strategies of thought that are not in concordance with the unfolding of nature, and do not help us to reveal its structure.
From this I think we should resist saying that the progression of the physical entity of a clock depends upon a concept we have derived from the clock; as if the clock would not tick without the operationalisation of time that it embodies in our understanding. Or if it would not tick without experiential temporality stretching along with it.
This speaks to learning what time is by learning the role it plays in (our interpretations of) life, rather than the role it plays in nature itself. I think it suggests we should reject the ontic relevance of time as a unified concept, just like we can reject the idea of mathematical entities necessarily having an existence in nature (if someone kicks over a rock and discovers [math]\aleph_{\omega}[/math] I would be incredibly surprised). As the chain rule thing shows, it doesn't matter whether we have [math]t[/math] or (smooth, bijective) [math]f(t)[/math] in our physical theories, as it just requires scaling the laws (imagine if seconds were instead 2 seconds, divide the time term in a law by a half or multiply by 2 depending on the context, sorted).
I think it's important to think ontic time immanently, and processes being 'clocks' for each other might provide a vantage point from which to do this.
This is the problem then. That is not the law of identity. The law of identity does not allow that there is more than one X. When you say "for all X...", you have already allowed the possibility of more than one X, thus breaking the law.
Quoting Kornelius
What is not clear is how you get from the law of identity, as commonly stated, to your formulation of it. And I'm sorry to be the one to inform you of this, but your example fails because it utilizes a formulation of the law of identity which is already itself in violation of the conventional law of identity.
?
I am not sure where you are getting this and why you think it is true. Could you clarify? In no suitable formulation of the law of identity would it be valid only in a model with exactly one and only one object. How would you even formulate this? I take it something like this:
[math] (\exists x) (x=x) \wedge \neg(\exists y)\neg(x=y)[/math]
But this is no law of logic, and certainly not a law of identity. It is fairly simple to provide a model for which the statement is false. Therefore, it is not a law of logic. Logical laws are true in every model, not just some models.
Quoting Metaphysician Undercover
I am employing the very familiar and standard notion from logic.
-Wiki
If your position requires you to reject a basic principle of formal logic, I would reconsider it carefully, especially since you admitted that you are not familiar with logic. We have an obligation not to be confident in our pronouncements if we are not entirely sure about all that goes into asserting them. Moreover, we should be open to reconsidering our position. So far in our discussion, everything is pointing to the conclusion that you (and not I) are confused about the law of identity. The formulation I have given for the law of identity is not mine, it is the one learned by everyone in the first course on logic in philosophy or mathematics.
I believe I have provided relatively clear explanations for what are elementary concepts in logic (i.e. the law of identity). If anything is unclear, please let me know.
Yes, and thank you for a comprehensive response.
Quoting fdrake
Oh but I don't think that we derive the concept of time from the clock. From the moment of the first eye opening we already have some intuitive understanding of time. Observing clocks helps us to further contextualize, structure, and quantify that understanding, and more careful observation and reflection leads to more sophisticated understanding of the structure and measure of time in terms of mathematical models and measuring devices.
So when you ask yourself, "What is time?" you can point to periodic processes or to theoretical models, but then if you ask, "What validates those explanations?" you still have to go back to the phenomenology (including, of course, the phenomenology of clocks), because what else would we go back to? That doesn't mean, of course, that we have to hang on to every prejudice and intuition, but our explanations have to be true to something, or else they just hang free, like abstract mathematical entities.
What does a clock show? What does it mean to say that this iteration is prior to that? If we reject mathematical models as inadequate for exhaustively answering empirical questions, I am afraid that an answer can only be provided by gesturing, tautologically, towards some sort of unfolding. Tautologically because, of course, our notion of unfolding is already informed by the notion of periodic processes.
Anyway the law of identity is simply a constraint put on logical entities e.g. proper nouns and propositions so that valid inferences can be made. For instance, if the proposition P = It is raining, then as we construct an argument around P the gist of P must remain unchanging throughout. Don't you think?
The law of identity states that a thing is the same as itself.
Quoting Kornelius
That is our point of disagreement. My claim is that the law of identity is not a law of logic, it's a metaphysical assumption. You think it's a law of logic. Because of this disagreement, I do no think we will ever find an expression of the law of identity which we both agree with.
This is from your wiki quote:
So we seem to agree at this point. My question to you is how do you proceed from the proposition "each thing...", to your formulation "for all x...."? Notice that the former refers to particular, individual things, and the latter refers to a group of things. Your formulation appears to have a veiled inductive conclusion, inherent within. You must apply inductive logic to "each thing is identical to itself, to derive "all things are identical to themselves". Therefore your formulation is one which has been polluted by inductive logic.
I don't want to reject mathematical models, far from being a mere philosophical point; if I thought that I would have to change job! Specifically, I think mathematical models really do allow us to find things out about nature. What I was trying to highlight was that the use of time in mathematical models doesn't really tell us much about it, as any smooth bijective function of time could be used to parametrise them.
All that really says is that the time parameter in mathematical models is often rather arbitrary, and when thinking about what ontological commitments to form based on mathematical models, we should be very careful with attributing existence to something which may be chosen so freely.
In terms of the Lotka Volterra example earlier, the relevant dynamic the equations seek is the reciprocal dependence of predator numbers on prey numbers. Predator numbers and prey numbers are something it makes sense to have a commitment about, and the rate of change of one with respect to the other is the target of the model. It's what the equations try to capture.
Time in the model, in that regard, is a useful independent parameter that you can evaluate both populations at. I'm not saying we should do away with it.
Quoting SophistiCat
What I have in mind is a few procedures for giving an account of the unified concept of time.
(A) One takes the plurality of rates, synthesises that through some phenomenological considerations, and outputs a concept of time which is necessary in our understanding.
(B) One takes the time variable, synthesises that through some phenomenological considerations, and outputs a concept of time which is necessary in our understanding.
(C) One takes the plurality of rates, synthesises that through our capacities of understanding more generally, and outputs a time concept which is tied speculatively to time in nature.
(D) One takes a time variable, synthesises that through our capacities of understanding more generally, and outputs a concept of time which is tied speculatively to time in nature.
You can see that (A,B) and (C,D) are grouped structurally, I don't really care which approach is taken within (A,B) or (C,D), they denote the development of a phenomenological understanding of time indexed to humans and a use of whatever that time concept is to understand time in nature.
What I'm trying to point out here is that we should not take answers from the (A,B) group of questions as answers to the (C,D) group of questions. Even if one has, like in Kant, linked the unity of the time concept to the sensory manifold and the transcendental unity of apperception, one still has the independent branch of questions about time in nature; like what Riemann and Einstein and even Bergson aimed at; that cannot be given answers in this way. (C,D) questions are possible to address, and are of philosophical merit. They just require a different workflow to address than the 'link to a priori structure of experience' machine, as there is time in nature irrelevant of experiential temporality.
The problems posed by (C,D) do influence how we should think of experiential temporality - perhaps it is not 'primary' in all senses, humans evolved in the presence of a time which is not our own, and in that regard the 'merely ontic' notion of time targeted in (C,D) is primary. But here what really matters is that they're different question groups with different methodologies to attack. (C,D) weaponise experiential temporality to 'carve nature at its joints'.
My love of the chain rule example is that it suggests one way to exploit the arbitrarity of the time variable to 'internalise' it to other concepts; of differentials of unfolding. While time and unfolding are probably interdependent, time is often seen as unitary whereas unfolding is a plurality of links which we know have affective power in nature. It invites an immanent thought of time, whereas the times thought in (A,B) and the hypostatised 'indifferent substrate' of time are both marred by their transcendental character.
Edit-imprecise summary: time is something empirically real, not just something transcendentally ideal. The empirically real component requires different methodology to attack than the usual Kantian/phenomenological interpretive machines, and is still of philosophical interest.
All things are identical to themselves. Which is exactly the formulation I discussed and exactly the principle that implies nothing with respect to the number of existant objects.
Quoting Metaphysician Undercover
The law of identity is a law of logic. What was stated: an object is identical to itself, IS a law of logic. Period. This is why I think you may not be expressing what you really want to express clearly. I think you may have a different identity claim in mind that may, in fact, be a metaphysical claim. The one you expressed, however, is not.
Quoting Metaphysician Undercover
They are equivalent. "For all x" and "each x" is logically equivalent. "Each" is a universal quantifier expression. I could easily have said that [math](\forall x)(x=x)[/math] says: each x is self-identical OR
for each x, x is identical to x.
Quoting Metaphysician Undercover
Not at all. You are misconstruing the statement "each thing" for "each thing I observe now". That is NOT the law of identity. That is, "each thing observed up to this point has been identical to itself" is NOT at all what "each object is identical to itself" states.
For this reason, no induction is required AT ALL to get the law of identity. As stated, the law of identity is an axiom of logic. It is a logical principle through and through because, as I showed, it is true in every model.
That being said, if you want to discuss the empirical claim regarding the objects we've observed, then by all means. However, we do not come to the conclusion that an object is identical to itself via observation anyway.
Otherwise, I am not sure what you are getting at. I think, perhaps, you are trying to say that claims about persistent identity are metaphysical. Here I am in complete agreement with you. Identity over time is metaphysics.
Do you not recognize the difference between "a thing", and "all things"?
The law of identity states that a thing is the same as itself. What justifies you formulation, that all things are the same as themselves, other than induction? The law of identity is not an inductive principle.
In logic, it is important to note that arbitrary reference is equivalent to referring to all objects in a domain of discourse.
This is why it is crucial to be precise when employing the indefinite article. When one uses an indefinite article like "a", one must precisely articular whether one is using one of two quantificational statements: [math]\exists[/math] (for some, there exists a, etc.) or [math]\forall[/math] (for all, every, etc.)
The reason this is important is that some may want to argue that the law of identity does not employ the unversal quantification. If I say "a thing is identical to itself", do I mean to say:
1) [math] (\exists x) (x=x)[/math]
OR
2) [math](\forall x)(x=x)[/math]
MU argues that the identity principle is just the statement (1), and he further states that someone who says it is (2) is confused:
Quoting Metaphysician Undercover
It is very important to note that everyone agrees that (2) is the formulation for the law of identity. It is easy to show why. So here we go:
Proposition: (1) is not logically valid, where (1) refers to the proposition [math] (\exists x) (x=x)[/math]
PROOF: By definition, a proposition is logically valid if and only if for any model [math]\mathcal{M}[/math], (1) is satisfied (is made true) in that model, i.e., [math]\vDash_{M} (\exists x) (x=x)[/math].
Consider a model [math]\mathcal{M}[/math] with domain [math]\varnothing[/math]. Since the domain is empty, we have that [math]\vDash_{M} \neg(\exists x) (x=x)[/math] from which it follows that [math]\nvDash_{M} (\exists x) (x=x)[/math]. Thus, (1) is not valid in [math]\mathcal{M}[/math], and thus not a logical truth. [math]\square[/math]
INFORMAL ARGUMENT: Since the proof may use some technical devices we may not be familiar with, here is the essential idea. In order to show that (1) is not a logical truth, we just have to show that we can imagine a possible situation in which (1) is false. Indeed, in a universe where no objects exist, (1) is false. Therefore, (1) cannot be a logical truth.
What is interesting to notice as well, is that (2) does not imply (1). If this were the case, then any model in which (2) is true, (1) is also true. However, in the model I just showed you (see proof above) (2) is true (see one of my previous posts in this discussion if you want the technical proof) but (1) is false. Therefore, (2) does not imply (1).
The reason (2) does not imply (1) is because (1) implies the existence of at least one objects. (2) does not imply the existence of an object. Here are some logical equivalences that might help you see this:
[math]\begin{align*}& (\exists x)(x=x) \Leftrightarrow \neg(\forall x)\neg(x=x)\\& (\forall x)(x=x) \Leftrightarrow \neg(\exists x)\neg(x=x)\end{align*}[/math]
The Law of identity is held as a law that is logically true. Indeed, (2) is logically true, i.e., it is true in all models.
The proper way to state the law of identity is:
All objects are self-identical (whether many, one or no objects exist)
Or, alternatively, we can say:
There exists no object that is not identical to itself (whether many, one or no objects exist).
Both of these statements are logically equivalent and, most importantly, do not imply that an object exists.
That's the point, the law of identity is a metaphysical assumption, so of course it's not logically valid.
Quoting Kornelius
No, the law of identity is not held as a law which is logically valid. It is held to be true, and perhaps even sound, depending on how you define "sound", but it is not held to be logically valid. The three fundamental laws, identity, non-contradiction, and excluded middle, are all held to be true, but not one of them, on its own, is logically valid.
Recommended reading:
The Logic Book (good for practice exercises)
A Mathematical Introduction to Logic
Set Theory, Logic and their Limitations
Suppose you went back in time and encountered your 2-year old self. Two distinct individuals standing side be side, with clear physical differences cannot be considered the identical person. You do not even share the same set of memories, you only share a 2-year subset (and your memories of that shared subset are fuzzier). Your DNA isn't even identical - our DNA gradually changes a little over time.
Even without time travel, to maintain identity over time, there must be something that endures. What is that?
Well, there is this position, to which I am somewhat sympathetic, that the abstract (mathematical) entities that we find to be indispensable in explaining (modeling) the world thereby exist. Of course, as you point out, time may not even be all that indispensable, or even if some time was necessary, there is no one definite form of it that we are forced to adopt. But then the latter problem is basically what Einstein's relativity tackles, where time is quite substantive, even if it is very much a reference- and coordinate-dependent entity.
Quoting fdrake
You don't even need a smooth function in order to convey this idea: really, what it comes down to is variable substitution: expressing one quantity in terms of another. This works even for ragged and discontinuous relationships. However, to return to my reservations about this thought as a justification for what is, I think, a physical and/or metaphysical thesis, the same abstract manipulation can be applied in ways that are less physically meaningful and certainly don't warrant a parallel conclusion. For example, in the famous predator-prey example, instead of looking at populations of wolves and hares, we could look at the population of wolves and the amount of manure excreted by hares, which of course is closely related to the population of hares. Does this mean that we can therefor dispense with hares in this system? Well, we could for the sake of modeling the population of wolves (or the amount of shit, if that is what we are interested in), but surely our ability to do so doesn't indicate that hares lack substance!
(By the way, for me the Lotka-Volterra problem was one of the more memorable experiences from learning mathematics. It becomes even more dynamically interesting in 3D, if you add another variable into the system, such as grass.)
Quoting fdrake
Thanks for this, I know I haven't addressed much of what you've said - but that's because I would like to think more about it.
I don't think relativity tackles the problem, really. To be sure, it makes time immanent, which is a good step. It makes space, motion, time and mass have reciprocal relationships and intricate interdependence. But it makes it immanent by fleshing out couplings between time and space and motion and mass in an abstract 4 dimensional vector space of which time is an independent direction of variation. You can still do the same trick with a smooth bijection to get another 'time' and make, say, the time direction a function of the oscillation between hyperfine states of a hydrogen atom (as we do to operationalise it now), or through any other physical process of unfolding.
I would like to have my cake and eat it too, and say that time is relational in a deeper sense, but that it still makes sense to think of it as an independent direction of variation in the relativity sense. Less Wrong has an interesting thought experiment on the matter:
This 'global rate of motion' being seen as pregnant in the above understand of general relativity is just what I would like to problematise. I think this is consistent with general relativity, as to think of the universe as having a 'global time coordinate' or 'global rate of change' forgets that time is one of the directions of variation of the universe; it's already baked in.
When we imagine the universe unfolding over time, we fix our frame of reference to the mind's eye independent of it all, and this is good as we have freedom of choice to define how we measure one process with another - and what processes we use for such measurement - but it hides that such an independent direction of variation must still be pregnant in the processes which make up the universe rather than exterior to them all.
That we could externalise time in a manner 'exterior to them all' is more about our imagination than about the ontic status of time as transcendent/immanent with respect to the universe's processes, time is already something interior; so it must have something to do with the plurality of processes which unfold.
One clue that time is relational capacity of systems would be that in the absence of a suitable relationship of coupling or correlation, no unfolding would be observable, and to my knowledge this is just what we see:
So the relational character of time is something that comes out of general relativity conceptually and quantum experiments demonstrably. I would like to say something like this is poetically suggested by basic calculus too:
[math]\frac{df}{df}=1=\frac{dg}{dg}[/math]
the evolution of the function is indiscernible when you measure that evolution through its own unfolding.
You actually have to be very careful with how you transform variables to preserve their meaning. You could surject the real line onto {0,1} and lose so much that the new scale is no longer a clock, it's an indicator of a discrete property. In order to preserve trends, for example, the variable transformation should be sufficiently smooth for the problem tackled and definitely monotonic. The smoothness varies, if one requires to estimate the second derivative of a function from a curve you should only transform using functions which have at least a differentiable second derivative.
The take home message here is that the ability to use any smooth bijection of time equivalently to time is actually rather odd in these terms; most variable substitutions which preserve the interpretable relationship between the variables and the model definitely don't have this property.
Isn't it obvious? Not one of the three fundamental laws of logic is a valid logical conclusion. For example, suppose there are rules which must be followed in order to produce a valid logical conclusion. It is impossible that the rules themselves are valid logical conclusions, because they are necessarily prior to any logical conclusion.
The problem is that the formulation of the law of identity, which Kornelius used in the proof that the law of identity is a valid logical conclusion, is not a true representation of the law of identity. The law states that a thing is the same as itself. Konelius' formulation stated "for all things". So in stating that all things have something in common, they are the same in this sense, Kornelius has already violated the law of identity which states that "sameness" can only refer to the relationship between a thing and itself.
Quoting tim wood
That you might be able to prove anything without such a law does not prove that the law is a valid logical conclusion. It only points to the usefulness of the law as a tool for understanding.
Quoting tim wood
The point being argued was the nature of the law of identity. I said it is an ontological principle, and Kornelius argued that it is a logical principle. My point was that despite the fact that the principle may be adapted and used by logic, it is grounded by, and justified by ontology. Therefore it is an ontological principle, not a logical principle, in the same way that ontological principles which are used in science, are not scientific principles despite the fact that they are used by science.
If Kornelius changes the meaning of "same" from how it is properly expressed in the law of identity, to prove that the law of identity is logically valid, then this proof is based on an equivocation and therefore invalid due to that fallacy.
Quoting tim wood
No acceptable proof has been demonstrated yet. As I explained, the proof provided is based on an adapted version of the law of identity. And, as I've argued this adapted version actually violates the law of identity as expressed in its proper form.
Quoting tim wood
An ontological principle is a statement, or proposition which claims something about the nature of being. The point I was making is that it is an assumption, rather than something proven by logic.
I believe that to understand why an ontological principle is a fundamental assumption rather than an inductive conclusion requires an analysis of the difference between subject and predicate. Once the subject is distinguished from the predicate as that which is described in the act of predication, then we can proceed toward understanding the distinction between the subject and the object (this might be described in Kantian terms of phenomenon/noumenon). An inductive conclusion is based on predication, and therefore makes a statement concerning a commonality in predication. The sameness which leads to the inductive conclusion is found in the predicate. So the sameness which is referred to with inductive conclusions is a sameness which is produced by predication.
Now we must validate the sameness of the subject, and this is the fact that we call distinct objects by the same name, because they are the same type of object. But this type of sameness can only be validated by predication as well, they are the same type of object, because the same thing can be predicated about them. This leaves us in a vicious circle whereby the object itself is inaccessible, and nothing can be validly said about the object, as Kant described with the concept of noumena.
So to say anything about the object itself, is to simply make an assumption about it. The first assumption that we make is that it is an object, a being, a thing, and therefore it has an identity as such. This is the law of identity, it's based in the assumption that there are real existing things, and that they have within themselves, their own identity, independent of the identity which we give them, which is as a subject. Once we have given the real object real existence, through this assumption, in this Aristotelian manner, we can proceed toward understanding what this real existence consists of, what validates this assumption. This is first and foremost, temporal extension, which the concept of "matter" accounts for.
No I don't think that this is the case, because a proposition is a type of statement, and one can hold an assumption without stating it. But I don't think this distinction is relevant anyway.
Quoting tim wood
Yes, this is the difficult thing. We do make claims about such general things, universals. What does it mean to be a human being, to be an animal, to be alive, etc.. Notice that I phrased it as "what does it mean", There are many such examples, what does "colour" mean, what does "number" mean. When we make a statement which claims something about these ideas, we are generally trying to clarify the meaning of the term. Do you agree that this type of expression, clarifying the meaning of terms, is distinct from predication? These claims which we have, hold, or make, about the meaning of the terms, are what I call assumptions.
So if someone makes a claim about "being" this is an expression of what that person believes is the meaning of the term. Maybe it could be called defining the term. If you look, you'll notice that such definitions are generally assumptions. For example, let me take something very simple, like numbers, and start with the numeral "one". That word refers to a unity, an individual. Next we have "two". Two refers to one individual together with another, making an artificial unity of "two". Notice that I distinguished the unity which is referred to by "one", from the unity which is referred to by "two", by calling the latter "artificial" (whether or not this term is adequate is not the point). It is necessary to do this because the use of "unity" which refers to one is distinct from the use of "unity" which refers to two. These are two completely distinct types of unity. "Two" implies that the unity referred to as two, is already intrinsically divided into two, whereas "one" implies divisibility (of infinite possibility), with no such division having been made already. So the unity referred to by "two" is a false unity because it is of necessity already divided. In the use of "two", we must recognize a sort of contradiction, a unity, one thing referred to with "two", which already has a defined division into two equal parts, so it is not really a unity. Whereas "one" represents a unity without any such division. Therefore the "unity" of one is distinct from the "unity" of two, three, four, etc.., and we cannot say that "two" refers to a unit in the same way that we say "one" refers to a unit without equivocation. These are some of my "assumptions" concerning numbers.
Quoting tim wood
This is a good question as well, and I'll tell you what I assume is the answer to it. The problem is that we do not have access to see, touch, or in any way sense the vast majority of things in existence. Therefore we do not have the capacity to make proper inductive conclusions concerning "all things". (Incidentally this is the biggest problem with what I consider the best arguments for God, formulations of the cosmological argument. They start from principles which appear to be inductive principles, but are really not drawn from sound induction, and so are dismissed by atheists as faulty assumptions). This is why ontological principles are better characterized as assumptions rather than inductive conclusions. If we start allowing that these are proper inductive conclusions, it sets a bad example.
Instead, ontological assumptions are drawn from examining all sorts of evidence, and drawing conclusions from who knows what sort of logic, mixed in with different intentions and pragmatic concerns. So it's better to call them assumptions than inductive conclusions.
Quoting tim wood
I don't agree with this. It may be the case that predication is required for deductive reasoning, but there are other forms of reasoning as well. Induction for example, though it often involves predication, does not require it. But, as mentioned it is difficult to draw a line between good induction and faulty induction. We can apply induction, for example, to different activities, deciding whether certain activities are successful for achieving desired ends. The process of trial and error allows us to focus in on the successful activity, and when it is found that a certain activity consistently produces the desired result, we might produce an inductive conclusion concerning cause and effect. The process of determining the correct activity is not a matter of predication, though it is a matter of reasoning.
Quoting tim wood
You must have misunderstood what I said. The "sameness" recognized through predication is a false sameness. It is the "sameness" which is found within inductive reasoning (which is really similarity), and is not the "sameness" expressed by the law of identity. That's the problem, Kornelius switched the "sameness" of the law of identity (often called numerical identity), for the "sameness" of inductive reasoning (often called qualitative identity, which is really a similarity), so that the formulation of the law of identity expressed by Kornelius was based in an equivocation of the word "same".
Quoting tim wood
I don't see the basis for this claim, I think it's drawn from a misunderstanding of what I said.
Quoting tim wood
No, it's the object we assume into existence. The subject has real presence to us, within our minds, but the object is what is assumed. That's why there is such a thing as radical skepticism concerning the sensible world.
Quoting tim wood
We're discussing the law of identity, and this was expressed by Aristotle, and the proper expression of it is maintained as the Aristotelian expression even today. So if we are to understand "the law of identity" we need to understand the Aristotelian principles behind that law. But if your intent is to replace that law with something else, then we ought not call it "the law of identity", because of the risk of creating ambiguity and equivocation.
Quoting tim wood
Let me explain the difference. We can define "thing" as "that which is the same as itself", or we can look at different individual things and make the inductive conclusion that all things are the same as themselves. The latter, as explained above, is a faulty inductive conclusion because it is very likely that the vast majority of individual things are hidden from our senses. So, the law of identity, which defines what a "thing" is, is not supported by inductive logic, it's more of a stipulation. Therefore it is not a logical principle, i.e. it is not a logical conclusion. I will not deny that it is supported by some sort of reasons, and some sort of "necessity", but it is more of a necessity in the sense of "needed for" the purpose of understanding, and not in the sense of a necessary conclusion, which requires some sort of understanding as a prerequisite for logical process.
Quoting tim wood
Again, I do not understand the relevance of this.
Quoting tim wood
How would this be possible? To discuss the law of identity in terms of logic would be to reformulate it into logical terms, which would destroy its essence, as Kornelius did.