The implication is then that X does not equate to any value but its own. X=X. — Mac

Except this is totally wrong. An apple is an apple. It is also a fruit. That something is itself says nothing. It is semantically empty. As are all tautologies. Tautologies cannot be exclusionary because they are not even inclusive: they say nothing.

The most you could say is that an apple being an apple precludes it from being a not-apple (¬X), but this too says nothing at all. It is an extentional and not intensional distinction.

The only way around this is Leibniz's way, which was to pull the entire world into the 'identity' of any one thing, such that relations of inclusion and exclusion are relations of compossibility and incompossibility of entire worlds. Leibniz needed God to balance the whole equation, so that seems like a bit of a lost cause too.

However, when we formally define equality, we do have to explicitly postulate self-identity - it doesn't fall out of other postulates. — SophistiCat

Totally. Insofar as one is setting out a grammar for logic, this makes total sense. But this speaks to a certain use of language - it certainly doesn't amount to some kind of biological or even metaphysical principle.

A note from the Wikipedia entry on the Russian philosopher, said to be an influence on Tolstoy and Nietszche, by the name of Afrikan Spir. I mention it because of his emphasis of the fundamental role of 'the law of identity'.

Epistemology

...Spir referred to his philosophy as "critical philosophy". He sought to establish philosophy as the science of first principles, he held that the task of philosophy was to investigate immediate knowledge, show the delusion of empiricism, and present the true nature of things by strict statements of facts and logically controlled inference. This method led Spir to proclaim the principle of identity (or law of identity, A ≡ A) as the fundamental law of knowledge, which is opposed to the changing appearance of the empirical reality.

Ontology

For Spir the principle of identity is not only the fundamental law of knowledge, it is also an ontological principle, expression of the unconditioned essence of reality (Realität=Identität mit sich), which is opposed to the empirical reality (Wirklichkeit), which in turn is evolution (Geschehen). The principle of identity displays the essence of reality: only that which is identical to itself is real, the empirical world is ever-changing, therefore it is not real. Thus the empirical world has an illusory character, because phenomena are ever-changing, and empirical reality is unknowable.

Religion and morality

Religion, morality and philosophy, have for Spir the same theoretical foundation: the principle of identity, which is the characteristic of the supreme being, of the absolute, of God. God is not the creator deity of the universe and mankind, but man's true nature and the norm of all things, in general *. The moral and religious conscience live in the consciousness of the contrast between this norm (Realität) and empirical reality (Wirklichkeit). "There is a radical dualism between the empirical nature of man and his moral nature" and the awareness of this dualism is the sole true foundation of moral judgment.

I suppose, extrapolating on this a little, that the only real occurences of 'is' are those that can be denoted by the '=' sign, in, for example, mathematical formulae. In such contexts, x=x has a degree of certitude that can never be conferred on the declaration that an individual particular is such-and-such. For example, things that are named as part of a class of things ('this is an apple') are always subject to further qualification (like, what kind of apple, perhaps it's a replica, and so on.)

So this is why abstractions (such as signs and numbers) possess the intrinsic intelligibility that material particulars do not. I also think it is why the ability to grasp the meaning of the equals sign, that X=X, is essential to the formation of intelligible ideas and language.

Note the relationship between Spir's idea, and the notion of the reality of intelligibles, which is fundamental to pre-modern philosophy.

-------

* I can't help but notice the resemblance of this overall idea to neo-platonism.

The principle of identity displays the essence of reality: only that which is identical to itself is real, the empirical world is ever-changing, therefore it is not real. Thus the empirical world has an illusory character, because phenomena are ever-changing, and empirical reality is unknowable.

See. This is what happens when one hews to the principle of identity as some kind of metaphysical postulate. You have to leave the world behind. Religious tripe. All the more reason to be suspicious of metaphysical extrapolations with respect to it. The closer you get to Platonism, the more off-track one is.

Yeah and I'm pretty sure Nietzsche didn't see it that way either. That the empirical world is ever changing, is not a reason to conclude that it is not real... it's the other way arround. We should should be suspicious of knowledge taken to far on the back of axiomatic principles like the law of identity, because the world does in fact seem to be ever changing. Only when we freeze time does the law really hold up.

And I take that to be the question of the OP, what then is the justification of it? It can't be proven because it's the basis of the whole system, and only if things are frozen in time do they remain the exact same thing. So I think the justification is not because it's true, proven or follows from our experience of the world. The justification is utility, because it works... for our purposes. Things remain constant enough, or change slow enough, that we can assign categories and work with them. It's the best and only thing we have to attain some kind of knowledge about the world.

God is not the creator deity of the universe and mankind, but man's true nature and the norm of all things, in general

He clearly wants to replace religion by a simplistic and childish exercise in infinite regress, the kind of which has never produced anything of value or even worth knowing. His views are therefore worthless.

I think Popeye expressed things very well

Just extend his sentiments to "x" OR "apple" OR whatever :smile:

I also think it is why the ability to grasp the meaning of the equals sign, that X=X, is essential to the formation of intelligible ideas and language. — Wayfarer

Well, you have to start somewhere, I suppose.

What do you think of X=X+1 ? :gasp:

Only when we freeze time does the law really hold up. — ChatteringMonkey

Necessary truths are not ‘in time’. I think you're making the mistake of believing they're real in an objective sense, when they're actually transcendental or necessary truths - ‘true in all possible worlds’.

In matter-form dualism, the form is what makes particulars intelligible. It is also what brings order out of chaos, as matter in itself is unintelligible until it takes form. That is why, for classical dualism, the form or principle of a particular is what is real, as it is grasped intuitively by the intellect rather than mediated by sense. The form is also what a particular truly is, whereas this or that instance is accidental and temporal.

Don’t overlook the original impetus of philosophy was to identify an unchanging reality in the flux of experience.

He clearly wants to replace religion by a simplistic and childish exercise in infinite regress — alcontali

He was known as a philosopher, rather than a religious writer, although it seems very little of his work has been translated.

@Wayfarer is right, in natural language "A is B" can mean different things, depending on context. It can indicate class membership (people are animals), it serve as a reduction (people are [nothing other than] animals), and so on. In the case of this Spir fellow, what he has in mind would be more precisely called permanence, or more generally, invariance. That is not the same as the simple equality/identity used in logic and math. Such sloppy use of language has occasioned a lot of miserable sophistry (cf. Ayn Rand's abuse of the "principle of identity").

That is not the same as the simple equality/identity used in logic and math. — SophistiCat

Yes, in mathematics, the syntactic entailment of the "=" operator is defined -- axiomatized really -- for each different data type. For example, you will find an axiom for the meaning of equality in number theory and in set theory ("extensionality").

Even when you create your own custom data types, you will have to provide such definition by yourself. Otherwise, the system does not know how to evaluate equality.

This principle is reflected all the way down to practical applications in programming languages.

For example, you can turn standard tables into custom data types in the lua programming language. You can do this by attaching a metatable with functions, one of which is supposed to define what equality means.

__eq - Check for equality. This method is invoked when "myTable1 == myTable2" is evaluated, but only if both tables have the exact same metamethod for __eq. If the function returns nil or false, the result of the comparison is false; otherwise, the result is true. — Metatable Events, __eq section

Not one formal system "knows" what equality is supposed to mean, unless you explicitly tell it.

It's just a model, and the best one we have. Give me something better. You can try to poke holes but what I'm saying is solid. If you would seriously consider my points then you might see that your issues have already been addressed. Sloppy language is not the issue, and is equally as bad as poor reading comprehension skills.

Necessary truths are not ‘in time’. I think you're making the mistake of believing they're real in an objective sense, when they're actually transcendental or necessary truths - ‘true in all possible worlds’.

In matter-form dualism, the form is what makes particulars intelligible. It is also what brings order out of chaos, as matter in itself is unintelligible until it takes form. That is why, for classical dualism, the form or principle of a particular is what is real, as it is grasped intuitively by the intellect rather than mediated by sense. The form is also what a particular truly is, whereas this or that instance is accidental and temporal.

Don’t overlook the original impetus of philosophy was to identify an unchanging reality in the flux of experience. — Wayfarer

I don't think they are real in an objective sense, in fact I don't think the word true applies at all to 'necessary truths'. Truth value comes from verification, you check to see if a statement is true or not by looking. Logical necessity is only truth-preserving if you will, i.e. 'if the premise is true, then what follows logically is also true'.

This all seems completely backwards to me. Ideas are not real, precisely because they don't exist in time. That's what it means to be real, to exist in space and time. Only particulars exist, and universals are abstractions of those. They enables us to 'abstract away from reality' to gain more general applicable knowledge. But that knowledge is not reality itself. I mean, that's like saying the map is more real that the world it is based on.

And yes, the original impetus of philosophy was mistaken.... a mistake we struggled the next couple of millennia to get away from. The fact that we have a need for certainty or permanence is no reason to assume that that can be attained. There is no unchanging reality is what our senses tell us.

This all seems completely backwards to me. — ChatteringMonkey

Of course. You're modern.

Yeah I guess I was born that epoch.

Of course. You're modern. — Wayfarer

A snide reply doesn't really deserve an answer. But I will say this, Socrates and his followers weren't exactly classical either, they were just wrong. They represented a break from and a decline for ancient classical Greece. Only then did they turn away from the sensual to the abstract, when Greece (Edit: Athens I should say) was already in decline.

There is no unchanging reality is what our senses tell us. — ChatteringMonkey

It's interesting to reflect on Buddhism in this regard. Buddhism famously says that everything is anicca, impermanent, and that there is no permanent essence, substance (in the philosophical sense) or self. The principle is that all of the testimony of the 'five heaps' (skandhas) which are the aggregates of sense + the organ of mind (manas) does not contain anything which is self, which is permanent, and which is not intrinsically unsatisfactory (dukkha).

However - this is always the case, it is utterly invariable - which I think is interesting. I mean, it's not as if the Buddha's teaching was true in India circa 450 b.c.e. but has since been superseded by something else. No, what Buddhists teach is that it's always true but that people fall away from being able to grasp it due to the predominance of ignorance in the 'dharma-ending age'.

So there too, I aver, there is actually a quest for what is imperishable, what does not pass away, the deathless, the not-subject-to corruption. But the way this is conveyed in Buddhist philosophy is by way of negation, by showing that everything that is sensed and perceived 'arises depending on causes and conditions'. When that is abandoned, which is precisely the aim of the Buddhist discipline of renunciation, only then is 'the deathless, the unconditioned, the unborn' realised.

And I wonder if there is not some resonance between that principle, and Spir's axiom that 'the empirical is unknowable.'

Russell argued that this is a primitive proposition that must be assumed true without proof.

Why, you ask, we do that? Well, it is hard to see what alternative we have in regards to X =X. Is it possible for a thing to not be itself?

I don't think you're on the same page. The tautology: X=X comes from our biological capacity to grasp identity. It is a representation/model of one way in which humans understand and navigate the world. THE way really. — Mac

I don't see how X=X represents how we identify things. This is essentially saying Mac=Mac. What does this tell me that I don't already know? If I were to ask, "What is Mac?" and you reply Mac=Mac, then is that all there is to Mac? Or does Mac=Human being + English speaker + X + Y, etc.? Identity, in the way you are using the term (that has nothing to do with the mathematical property of identity so I don't understand why you're using a different mathematical property to argue your point) is a way of taking all of your Xs, Ys and Zs and putting it under another symbol, "Mac", to make communicating all of your Xs, Ys and Zs more efficient.

When it comes to being you, you are much more than a name, right? You're more than just some scribbles on a screen. Scribbles on a screen are observed and interpreted just as your body and behavior are. I can distinguish scribbles from bodies. Names are not bodies and their behaviors. Names refer to, or are about, those bodies and their behaviors. So, identity is not a case of x=x, it is a case of x=a + b + c. My identity would be y = a + b + d, and so on.

"an apple is an apple", but why?

Intuition.

Some things just make innate sense, we say they are self-evident. That is the answer you are looking for. There is no deeper, more specific, or more meaningful reason.

I do not see anyone trying to prove x=x

Because traditionally it has been taken as one of the _starting assumptions (premises), which are called axioms or postulates.

https://en.m.wikipedia.org/wiki/Law_of_thought

Aristotle, for example, considered it to be the primary axiom for deriving the very concept of truth and falsity, thus consequently upon which all the rest of logic depends.

Aren't axioms, self-evident assumptions? If so, when can we accept self-evident beliefs, just when they are practical? Do we have to analyse the relation between truth and practicality then?

Yes, axioms are self-evident assumptions, but being self-evident is supposed to upgrade its status from “mere assumption” to “something better”.

To be practical in the useful sense of that word entails reliability and consistency, which really is ‘validity’ in the most meaningful, i.e. practical sense.

But by practical I do not just mean necessarily useful, I mean our ability to interact with physical reality on a common ground where we proclaim ‘objectivity’ for any given claim, and thereby confirming, or at least increasing our confidence in axioms and hypotheses.

The proposition is simply: A thing resembles itself. The question is, "what is the proof?"

An apple is an apple can refer to a whole fruit category or species, meaning apples generally share common properties that differentiate them from strawberries, crocodiles, and everything else. It is really about similarity rather than identity.

More pragmatic is referring to actual apple out there in the world, and then identity has to do with location. The proof is empirical observation that only one physical particle can exist at any given point in space at any given point in time, which basically means any actual or physical thing, as opposed to virtual or imaginary thing, can only be what it is when it is, i.e. it can only be itself and nothing else.

I see your problem now. But once again, I understand this is a philosophy forum, and in that vein it is true that we look too deeply sometimes, especially with such kinds of ontological questions.

You are pretty much correct but why is x=x a problem? Sure x = a number of combined variables. But that also means that x is a particular thing. And no other thing is that particular thing so in making this distinction we must also accept that x is unique; x is itself. Which, again, can be a culmination of other variables, maybe infinite amounts. It is easier and still works formulaically to use the shorthand, x=x. If x were not equal to itself then a harder question would arise: "Does x exist independently of itself?"

My biggest question to you is "why should x=x be something you don't already know?" That's the point. I brought biology into the conversation because of its relevance to corresponding mathematical models. x=x is obvious to us because we evolved for it to be. Otherwise we would not have survived in the same way.

The Law of Identity states that a certain thing is identical to itself...

It also states that a certain thing is different to every other thing, in a way. But in any case it is vague and ambiguous statement, especially since there is indeed more than one valid context in which the law can, and has been interpreted.

Words, being ontologically virtual, a form of embedded information, like all the other imaginary entities from mental realm are not identical with things they represent, and a single word can represent multiple things, or even change meanings depending on various factors.

In this context physical entities, in contrast to those abstract ones, actually do stand for what they are. And in that sense we can say physical entities are identical to their ontological manifestation or existence within space and time, as opposed to be a representation or pointer to something else.

...and I ask why

Semantically it is about coherence, and the law states that during a reasoning the meaning of any term must remain constant. It originated in this context. Here, the reason why is because otherwise conversation would be meaningless.

Mathematically it is about interchangeability, and the law states algebraic manipulation of variables around equal sign preserves equation validity as long as both sides compute the same value. In this case the reason why is because it makes no difference to the calculation result.

Physically it is about persistence, and the law states something does not cease to exist or become something else for no reason, it remains to be that which it was, uniquely defined by its spatial location and geometrical morphology at any given point in time. The answer to the why question here is because we observe it to be so.

This last paragraph above I guess is the context you are asking about, and of course quantum mechanics has its own take on the fundamental nature of existence, but then I do not hear anyone is saying QM makes sense or follows rules of logic, so I suppose we can simply ignore it.

Yes, axioms are self-evident assumptions — Zelebg

Many axioms aren't self-evident in any fashion.

For example, take the axiom of regularity in ZFC set theory. The reason why it is there, is because in 1917 Dmitry Mirimanoff started writing lengthy rants on the existence of sets that are not "well founded". Imagine that you define:

A = { A }

This construction is not stable, because it is equal to:

A = { { A } }
A = { { { A } } }
and so on
...

So, it is not allowed to construct a set that looks like A = { A }. Therefore, the axiom of regularity is some kind of syntactic bug fix.

The same holds true for the axiom of specification. It prevents unrestricted comprehension which can otherwise cause Russell's paradox. Hence, it is another bug fix.

The language, i.e. the set notation itself, along with the language of first-order logic allow for specifying contradictions. Hence, one way to alleviate the problem -- but not really to solve it completely -- is to add rules (as axioms) that stamps out the most obvious ones.

So, axioms are also mere syntax restrictions for the language in which the theory is being elaborated. That type of axioms cannot be self-evident in any fashion, because you will first have to discover the problems that you will need to fix, by actually using the language.

Furthermore, what is there self-evident about the other axioms in ZFC?

For example, the axiom of power set. They happened to need power sets for Cantor's infinity theorems, but they could not guarantee that these power sets always exist. So, they ended up axiomatizing their existence. Problem solved. Is that kind of origin for an axiom, something to be considered "self-evident"? I don't think so ...

:up:

If you’re fine with Buddhism’s “nothing is permanent” being a permanent truth, why then is a modern western philosophy that says the same not permanent truth enough for you?

Because, Western philosophy has no equivalent to Nirvāṇa, clearly.

Ideas are not real, precisely because they don't exist in time. That's what it means to be real, to exist in space and time. Only particulars exist, and universals are abstractions of those. They enables us to 'abstract away from reality' to gain more general applicable knowledge. But that knowledge is not reality itself. I mean, that's like saying the map is more real that the world it is based on. — ChatteringMonkey

To get the conversation back on track - the problem here is that metaphysics, generally, is the discipline of reckoning 'what must be the case' in order for things to exist in time and space. 'Metaphysics anticipates the general structures of reality by formulating the way our knowing operates.  Science actually works out the explanation of the data by a never-ending process of research.' ~Lonergan.

So, such things as logical principles, scientific laws, mathematical objects, are all essential to empirical science, but they don't necessarily exist in time and space either. Rather, they form part of the architecture of reason, by means of which judgements about time and space are arrived at.

That is why, for instance, there are many intense and unresolved metaphysical arguments about (for instance) the meaning of quantum physics, whether there are multiple universes, whether life and mind are ultimately reducible to organic chemistry - and so on. And only certain aspects of many of these arguments comprise objects that exist in time and space. Empirical theories need to be validated against observational evidence - although even that is now being disputed - but metaphysical postulates cannot.

My remark about you being modern, is not a pejorative, but it was a bit snide. But it is natural for us moderns to think that ideas exist 'inside' the mind, or a 'product' of the brain. We carry around this world-picture comprising us, as intelligent subjects, who model or represent the world in our highly-evolved forebrain, largely along darwinian lines. But we don't see that this model is also a construction in the sense understood by critical philosophy (such as Spir's). We are conditioned into naturalism and scientific realism by consensus, and it's often hard to question.

Many axioms aren't self-evident in any fashion.

It would be interesting to hear if there are any non-self-evident axioms outside abstractions in the field of mathematics.