ground, bricks, walls, ceiling, windows, and a door altogether make a set, which is the house. — javi2541997

I would think of those as aspects of the house, not members of the house. I wouldn't think of a house as being a set. There are sets of aspects of a house. But that set is not a house. A house is something you live in and pay a mortgage on. You don't live in a set and pay mortgage on a set. There is a housing market and a housing shortage; there is not a sets market and a sets shortage.

Set consisting of three balls colored red, white and blue. They also have differing weights. What is THE order? Just curious.
— jgill

The order is how items are organised with one another based on a specific attribute. The only distinguishing feature is that they are spherical. The weight and colours are only accessories. The set would be spheres, and the order would be the three balls. Right? — javi2541997

If the attribute is color then there are six orderings based on that attribute:

red ball, white ball, blue ball
red ball, blue ball, white ball
blue ball, red ball, white ball
blue ball, white ball, red ball
white ball, red ball, blue ball
white ball, blue ball, red ball

There is not just one ordering that we can call "THE" ordering.

If the attribute is size, and it is not the case that there are two balls with the same size, then there are six orderings based on that attribute:

the largest ball, the middle sized ball, the smallest ball
the largest ball, the smallest ball, the middle sized ball
the middle sized ball, the largest ball, the smallest ball
the middle sized ball, the smallest ball, the largest ball
the smallest ball, the largest ball, the middle sized ball
the smallest ball, the middle sized ball, the largest ball

There is not just one ordering that we can call "THE" ordering.

Do you see that?


By the way, this pertains to linear orderings (aka 'total orderings'). There are other kinds of orderings, especially partial orderings, but here the context is linear orderings.

The crank writes, "[TonesInDeepFreeze claims] that the elements of a set may be concrete objects."

If the elements cannot be concretes and can't be abstractions, then what can they be?

Or does the crank reject even the notion of sets and elements?

Is a rock not a concrete? If a rock is not a concrete, then what is an example of a concrete?

If there is no such set that is the set of rocks on my table, then what are examples of sets?

/

The crank writes, "The sense of humour leaves the head sophist [TonesInDeepFreeze] exposed, revealing no control over the inclination to equivocate."

I dare not ask what in the world that is supposed to mean.

By the way, though I am a magnitude of light years away from being a sophist, I would rather be merely a sophist than a crank, since being a crank includes being a sophist and a lot worse too.

I would think of those as aspects of the house, not members of the house. I wouldn't think of a house as being a set. There are sets of aspects of a house. But that set is not a house. — TonesInDeepFreeze

If that isn't a house, what set are you talking about? Assume they are all 'aspects' of a set called furniture. We could agree on that. But what is the sense of doing those things separately? All of the 'aspects' I mentioned in my example follow a common logic. They'll end up in construction. A house or building. I can't envision a house without a wall or a ceiling as structural elements. Otherwise, this type of construction would be unsustainable. Perhaps I am misunderstanding the concepts of "set," "order," "members," and so on. I am aware of my limited understanding on the subject. But I still believe they are members a house.

Do you see that? — TonesInDeepFreeze

Yes, I do. I never claimed there was one and only “THE” order. I referred to the balls in the example of jgill because that was what I thought when trying to use logic. But I hadn’t in mind only one ‘ordering’.

If that isn't a house — javi2541997

I didn't say that a house is not a house. I said a house is not a set.

But what is the sense of doing those things separately? — javi2541997

What things separately?

I can't envision a house without a wall or a ceiling as structural elements. — javi2541997

Nor can I. That doesn't entail that a house is a set. Again, a house is a thing you live in. You don't live in a set; you live in a house.

Perhaps I am misunderstanding the concepts of "set," "order," "members," — javi2541997

Sets:

S = {the door of the house, the roof of the house, the floor of the house ... the balcony of the house}.

S is not a house. It is a set whose members are features of the house.

Members:

The members of S are the features of the house.

Order:

<door, roof, floor ... balcony> is one order

<floor, balcony, door ... roof> is another order

What is the case, is that "X=X" is an ambiguous and misleading representation of the law of identity. This is because "=" must mean "is the same as", to represent that law, but it could be taken as "is equal to". — Metaphysician Undercover

I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about.

Notice that in the axiom of extensionality it is taken to mean "is equal to". Therefore when Tones takes "X=X" to be an indication of the law of identity there is most likely equivocation involved. — Metaphysician Undercover

I don't think so. I don't agree with you.

So, do you recognize, and respect the fact that group theory is separate from, as a theoretical representation of, the objects which are said to be members of a specified "group"? — Metaphysician Undercover

Yes, as ichthyology is different from any particular fish or school of fish or class of fish.

And, I'm sure you understand that just like there is a theoretical representation of the group, there is also a theoretical representation of each member of the group. In set theory therefore, there is a theoretical "set", and also theoretical "elements". — Metaphysician Undercover

I don't know what you mean by "theoretical" elements. The integers form a group under addition. Is the number 5 theoretical? Well it's abstract, as numbers are. What of it?

So when Tones says that a set may consist of concrete objects, this is explicitly false, because the set is the theoretical representation, and the elements of the set are theoretical representations as well. Through such false assertions, Tones misleads people and earns the title of sophist. — Metaphysician Undercover

Is concrete different or the same as physical? 5 is a concrete mathematical object, I guess. "Concrete" is not a term of art in this context, although there's a thing called a concrete category.

You're making up your own definitions of words and arguing with me about them. I am lost.

When Tones speaks about the set "George, Ringo, John, Paul", these names signify an abstract representation of those people, as the members of that set, the names do not signify the concrete individuals. You, Fishfry, have shown me very clearly that you know this. So there is an imaginary "George", "Ringo" etc., which are referred to as members of the set. The imaginary representation is known in classical logic as "the subject". We make predications of the subject, and the subject may or may not be assumed to represent a physical object. Comparison between what is predicated of the subject, and how the object supposedly represented by the subject appears, is how we judge truth, as correspondence. — Metaphysician Undercover

Too deep for me. I take Tones's point that = in set theory derives from = in the underlying logic. I have no problem with that and it was perfectly obvious as soon as he drew my attention to it.

What is important to understand in mathematics, is that the subject need not represent an object at all. It may be purely imaginary, like your example Cinderella. This allows mathematicians to manipulate subjects freely, without concern for any "correspondence" with objects. Beware the sophist though. I believe that when the sophist says that the members of a set may be abstractions, or they may be concrete objects, what is really meant if we get behind the sophistry, is that in some cases the imaginary, abstract "element", may be assumed to have a corresponding concrete object, and sometimes it may not. Notice though, that in all cases, as you've been insisting in discussions with me, the elements of the sets are abstractions, as part of the theory, and never are they the actual physical objects. Failure to uphold this distinction results in an inability to determine truth as correspondence. And that is the effect of Tones' sophistry — Metaphysician Undercover

Ok you should take this up with Tones. You failed to convince me that I am a victim of anyone's sophistry.

I'll return to the schoolkids example briefly to tell you why I didn't like it. — Metaphysician Undercover

Jeez man you already told me you don't like it, so I stopped using it. To me it's a good illustration of how a collection may be ordered in many different ways.

Using that example made it unclear whether "schoolkids" referred to assumed actual physical objects, or imaginary representations. That's why "real-world analogies" are difficult and misleading. The names, "George", "Paul", etc., appear to refer to real-world physical objects, and Tones even claims that they do, but within the theory, they do not, they are simply theoretical objects. If we maintain the principle that the supposed "schoolkids" are simply imaginary, then they have no inherent order unless one is stipulated as part of the rules for creating the imaginary scenario. Set theory ensures that the elements have no inherent order, but this also ensures that the elements are imaginary. — Metaphysician Undercover

Sets are mathematical abstractions. I don't know what you mean by "imaginary," which is a term of art in math referring to complex numbers with real part 0.

This is wrong, and where Tones mislead you in sophistry. A set is not identical to itself by the law of identity. The set has multiple contradictory orderings, and this implies violation of the law of identity. — Metaphysician Undercover

No you are as wrong as can be about that.

We allow that "a thing", a physical object has contradictory properties with the principle of temporal extension. At one time the thing has a property contradictory to what it has at another time, by virtue of what is known as "change", and this requires time. But set theory has no such principle of temporality, and the set simply has multiple (contradictory) orderings. — Metaphysician Undercover

You have been misunderstanding this point for years, and I surely have nothing new to say on the topic.

As I said, the reference was to the identity of indiscernibles, not the law of identity. You recognize that these two are different. The proof was not by way of the law of identity. If you still believe it was, show me the proof, and I will point out where it is inconsistent with the law of identity. — Metaphysician Undercover

I'd walk you through the Wiki page on the axiom of extensionality, but this also is something I've done for years with you, to little productive effect.

We agree on this very well. The principle we need to adhere to, is that this is always an "abstraction game". If we start using names like "Ringo" etc., where it appears like the named elements of the set are concrete objects, then we invite ambiguity and equivocation. And if we assert that the elements are concrete objects, like Tones did, this is blatantly incorrect. — Metaphysician Undercover

You must be misrepresenting what Tones said, since he made his point with me; and after that, I found several clear references supporting his point.

The three fundamental laws of logic, identity, noncontradiction, and excluded middle, are inextricably tied together. Therefore one cannot discuss identity without expecting some reference to the other two. There has been some philosophical discussion as to which comes first, or is most basic. Aristotle seemed to believe that noncontradiction is the most basic, and identity was developed to support noncontradiction. — Metaphysician Undercover

Aristotle thought bowling balls fall down because they're "like the earth," and fire goes up because it's "like the air." But never mind that. You are swimming in murky logical waters that have absolutely nothing to do with anything I'm saying.

What C.S. Peirce noticed, is that if we allow abstract objects to have "identity" like physical objects do, as Tones seems to be insisting on, then necessarily the validity of the other two laws is compromised. Instead of denying identity to abstract objects, as I do in the Aristotelian tradition of a crusade against sophistry, Peirce sets up a structure outlining the conditions under which noncontradiction, and excluded middle ought to be violated. — Metaphysician Undercover

I'm ill-equipped to argue Peirce and Aristotle with you. I don't think your points bear on set theory.

You are missing the point. The law of identity refers explicitly to things, "a thing is the same as itself". A "set" is explicitly a group of things. Therefore when you say X = X, and X is a set, rather than a thing, then "=" does not signify identity by the law of identity. — Metaphysician Undercover

A set is not a "thing?" A set is a thing in set theory. It's an abstract thing, to be sure. But it's still a thing.

Right, this is the point. "Time", or temporal extension allows that a thing may have contradictory properties, at a different time, yet maintain its identity as the same thing, all the while. This is fundamental to the law of identity. Without time (as in mathematics), the multiple orderings of a set, which Tones referred to, are simply contradictory properties. That is a good example of the issue Peirce was looking at. — Metaphysician Undercover

No, orderings are not "contradictory properties." Technically, an order on a set is another set, namely the set of pairs (x,y) for which we mean to denote that x < y in the ordering. The ordering is distinctly and noticeably separate from the set it applies to.

Fine, but can you respect the fact that "equal" does not imply "identical", despite the sophistical tricks that Tones is so adept at. — Metaphysician Undercover

That distinction has no meaning or relevance in my understanding of the world. "equal" and "the same as" are entirely synonymous. I do take your point that 2 + 2 = 4 does not mean that they are the same as strings. We've been over this many times, as have many philosophers. The morning star and the evening star are the "same object," (which turns out to be a planet and not even a star) but they have different senses. What of it, this is not news to anyone.

No, that's simply wrong. A particular apple is a physical object. A set is an abstraction. An instance of an apple is a physical object. Your supposed "instance" of a set is an abstraction, a concept. The two are not analogous, and I argue that this is a faulty, deceptive use of "instance". — Metaphysician Undercover

Would you agree that "number" is a general abstraction and that 5 is a partcular instance of number? Isn't that the most commonplace observation ever?

An instance is an example, and understanding of concepts or abstractions by example does not work that way. — Metaphysician Undercover

5 is a terrific example of a number. One of the best I know.

Assume the concept "colour" for example. If I present you with the concept "red", this does not provide you with an instance of the concept "colour". — Metaphysician Undercover

It doesn't? Red is not an instance of the concept of color? How do you figure that?

An instance of the concept "colour" would be the idea of colour which you have in your mind, or the idea of colour which I have in my mind, expressed through the means of definition. — Metaphysician Undercover

Incoherent. Red is a color. Red is an instance of the concept of color.

Each of those would provide you with an example of the concept of "colour", an instance of that concept. The concept "red" does not provide you with an example of the concept of "colour". — Metaphysician Undercover

What??

Nor does a specific "set" provide you with an example or instance of the concept "set". — Metaphysician Undercover

Of course it does.

What you are saying in this case is completely mixed up and confused. — Metaphysician Undercover

Same back atcha.

We do have significant disagreement concerning your claim to a proof that "X=X", when X signifies a set, means that X is the same as itself by virtue of the law of identity. You have not provided that proof in any form which I could understand. — Metaphysician Undercover

I ain't draggin' your butt through the Wiki page on extensionality again. That was a very dispiriting experience the last time I did it. However, I'll point you at the relevant sentence.

"The axiom given above assumes that equality is a primitive symbol in predicate logic."

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

What things separately? — TonesInDeepFreeze

All the things I previously referred to. The ‘objects’ or ‘elements’ that constitute a house: walls, ceiling, windows, door, etc.

Again, a house is a thing you live in. You don't live in a set; you live in a house. — TonesInDeepFreeze

Yes, yes. I understand that I live in a thing, but my point was different. I tried to explain that the ‘thing’ is based on different elements. Without these elements or 'objects', the principal thing (the house) is senseless, in my humble view. Maybe I was wrong in using those concepts in a confusing way. Yet I think we both agree that the house is senseless without furniture, unless you are minimalist. But even a minimalist house needs walls, a door, and a ceiling. Therefore, these three elements are necessarily elements of the house.

Order:

<door, roof, floor ... balcony> is one order

<floor, balcony, door ... roof> is another order — TonesInDeepFreeze

I see. Thanks. But then I wonder: what is the point of that order, or does it arise spontaneously? Obviously not. The house is what they relate to.

The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals. — TonesInDeepFreeze

Oh. In that case @Metaphysician Undercover is right and you are making a point I can't agree with.

I take your point about set equality as expressed in the Wiki page on extensionality, which says, "The axiom given above assumes that equality is a primitive symbol in predicate logic."

If you mean something else, we're back to square one.

The ‘objects’ or ‘elements’ that constitute a house: walls, ceiling, windows, door, etc. — javi2541997

The same set can be specified in two different ways:

{the door, the floor, the roof ... the balcony} [fill in '...' with all the other features of the house .]

{x | x is a feature of the house}

Without these elements or 'objects', the principal thing (the house) is senseless — javi2541997

Okay.

these three elements are necessarily elements of the house. — javi2541997

In casual conversation, the word 'elements' can be used that way. But if we are talking in a focused context about sets, 'elements' refers to members of a set. And the house is not a set. Sure, in some informal way, we could stretch the meaning of 'set' so that in some view a house is a set. But in a focused sense of 'set', a house is not set, just as a rock is not a set.

If anything at all were a set - a house, zebra, rock, cloud - then 'set' wouldn't have any special meaning. You could point to a stop sign and say, "Hey look at that set over there, the stop sign". But that's not the common notion of 'set'.

In casual conversation, the word 'elements' can be used that way. But if we are talking in a focused context about sets, 'elements' refers to members of a set. And the house is not a set. — TonesInDeepFreeze

Ah. Well, I think I have to agree with you. My arguments were based on casual and informal examples, and I can’t go further than that. Thanks for your explanations.

I think those examples and a common informal context are okay. They suggest that, for example, a rock is not a set. My point is that it would only be a far stretch of the notion of 'set' that would permit taking a rock to be a set.

The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals.
— TonesInDeepFreeze

you are making a point I can't agree with.

I take your point about set equality as expressed in the Wiki page on extensionality, which says, "The axiom given above assumes that equality is a primitive symbol in predicate logic."

If you mean something else, we're back to square one. — fishfry

'=' is primitive.

But there is more to say.

So indeed, let's go back to square one:

'=' is primitive in logic (first order logic with equality, aka 'identity theory').

And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals

Then set theory adds its axiom:

Axy(Az(zex <-> zey) -> x=y) ... extensionality

Now we ask how we derive:

(zex & x=y) > zey

Answer: from the indiscernibility of identicals. Indeed the above is an instance of the indiscernibility of identicals, where P(x) is zex.

But there is more to say. — TonesInDeepFreeze

I don't think I'm going to get involved in the details at this point.

So indeed, let's go back to square one: — TonesInDeepFreeze

"Let's not and say we did," as the saying goes.

'=' is primitive in logic (first order logic with equality, aka 'identity theory'). — TonesInDeepFreeze

Aha. "Identity theory" is first order predicate logic with equality. Is that your own terminology? Nothing in the Wiki disambiguation page for identity theory refers to it

And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals

Then set theory adds its axiom:

Axy(Az(zex <-> zey) -> x=y) ... extensionality

Now we ask how we derive:

(zex & x=y) > zey

Answer: from the indiscernibility of identicals. Indeed the above is an instance of the indiscernibility of identicals, where P(x) is zex. — TonesInDeepFreeze

I'm going to pass on engaging with this. Just don't have the inclination at the moment.

But one query. If you are doing first order logic, how do you quantify over all propositions P? Maybe I shouldn't ask.

"Identity theory" is first order predicate logic with equality. Is that your own terminology? — fishfry

I stated explicitly several times that that is what I mean by 'identity theory'. I recall having seen the term used professionally before, and so I adopted it a long time ago, but I would have to dig to find citations. I like it, because it is a first order theory about one certain predicate that is indeed the identity predicate.

If someone says "I'm talking about blahblah theory'" and they tell me the axioms, then I don't quarrel with them about it. I know the axioms so I know precisely what is meant by 'blahblah theory'.

If you are doing first order logic, how do you quantify over all propositions P? — fishfry

We quantify over them in the meta-theory not in the object theory.

That is what an axiom schema is.

For example (leaving out some technical details here:):

In first order PA the induction axiom schema:

For all formulas P:

(P(0) & An(P(n) -> P(Sn))) -> An P(n)

In set theory, the axiom schema of separation:

For all formulas P:

AzExAy(y e x <-> (y e z & P(y)))


Those are statements in the meta-theory that describe an infinite set whose members are all axioms that are in the object-theory.

An interesting point is that while we can express the indiscernibility of identicals as a first order schema, we can express the identity of indiscernibiles as a first order schema if and only if there are only finitely many operation and predicate symbols.

It's an interesting exercise to try to express the identity of indiscernibiles as a first order schema with a language of infinitely many non-logical symbols. You'd think you'd just reverse the indiscernibility of identicals. But when you try, it doesn't work! If I'm not mistaken, one of the famous logicians proved it can't be done.

/

Another nice thing: Identity theory can be axiomatized another way, courtesy of Wang:

For all formulas P:

Ax(P(x) <-> Ey(x=y & P(y)))

From that we can derive both the law of identity and the indiscernibility of identicals.

I stated explicitly several times that that is what I mean by 'identity theory'. — TonesInDeepFreeze

You never said that LOL!

I recall having seen the term used professionally before, and so I adopted it a long time ago, but I would have to dig to find citations. I like it, because it is a first order theory about one certain predicate that is indeed the identity predicate. — TonesInDeepFreeze

Now that I know what you mean, it's helpful.

If someone says "I'm talking about blahblah theory'" and they tell me the axioms, then I don't quarrel with them about it. I know the axioms so I know precisely what is meant by 'blahblah theory'. — TonesInDeepFreeze

I see your point. But I don't always catch your meaning from your symbology.

I still have it in my queue to go back to your recent post about the indiscernibles. Maybe if you could make your point in words. I thought the = of set theory is the = from the underlying logic. But now you say it's not. So I'm confused again. If you could explain it clearly in a sentence or two I'd find it helpful

And I know that if I make this request, and you give me a response and I don't related to it, that's frustrating to you. Maybe there's a happy medium of explanatory level.

Those are statements in the meta-theory that describe an infinite set whose members are all axioms that are in the object-theory. — TonesInDeepFreeze

Ok. I see your point. Lately wishing I'd paid attention in logic class.

An interesting point is that while we can express the indiscernibility of identicals as a first order schema, we can express the identity of indiscernibiles as a first order schema if and only if there are only finitely many operation and predicate symbols.

It's an interesting exercise to try to express the identity of indiscernibiles as a first order schema with a language of infinitely many non-logical symbols. You'd think you'd just reverse the indiscernibility of identicals. But when you try, it doesn't work! If I'm not mistaken, one of the famous logicians proved it can't be done.
— TonesInDeepFreeze

Sadly this is all over my head. Maybe I'll crack open a logic book. If I could only dispatch a clone.

Another nice thing: Identity theory can be axiomatized another way, courtesy of Wang:

For all formulas P:

Ax(P(x) <-> Ey(x=y & P(y)))

From that we can derive both the law of identity and the indiscernibility of identicals. — TonesInDeepFreeze

That looks interesting.

The substance of these questions has been before you repeatedly and you make no substantive answer. — tim wood

Please allow me to clarify, if I wasn't clear enough for you last time. You and I have a completely different understanding of the nature of "a relation". We could not even find grounds to start any agreement, to converse. Consequently you'll understand "relation" in your way, and I'll understand "relation" in my way. Since "order" is a specific type of relation, any discussion about order, between us, will be rife with misunderstanding. Furthermore, I have no inclination to stoop to your level, and utilize your meaning of "relation", so that you might actually understand me, because I see it as nothing but childish closed mindedness.

I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about. — fishfry

This is exactly the problem, failing to distinguish between "same as" and "equal to". Because you do not believe that there is a distinction to be made here, you will not notice the effects of such a failure, and you will insist that it doesn't bear on anything you care about. Insisting that it doesn't bear on anything you care about will allow you to be mislead, even tricked by intentional deception (as you were by the sophist's employment of "identity of indiscernibles"), and you may never ever even notice it.

Here is a simple example of where the difference bears in a substantial way, though I am sure there are more complex examples. In quantum physics, a quantum of energy is emitted as a photon, and an equal quantum may be detected as a photon. Since these two quanta of energy are equal, they are said to be "the same" photon. That is the mathematician's use of "same", equal quanta implies one quantum, a photon. By the law of identity "same" implies temporal continuity, such that the photon exists, with that identity, for the entire period between emission and detection. Equivocation between these two senses of "same" inclines some people to believe that the photon exists, as the same "particle" for the entire period of time between emission and detection. However, the electromagnetic energy is observed to exist as waves in the meantime.

This produces significant theoretical problems. Some claim a contradictory wave/particle duality theory, in which the energy travels as both waves and as particles at the same time. Furthermore, since the photon of energy emitted is assumed to be "the same photon" as the photon detected, and it's path cannot be determined, it is claimed to have multiple paths all at the same time. All of this sort of problem is due to equivocation of "same". The mathematical "same", an equal quantum of energy is emitted and detected, is confused with "same" by the law of identity, to conclude that a distinct quantum (particle) of energy, known as the photon, has continuous existence between the time of emission and the time of detection.

You say that this issue doesn't bear on anything you know or care about, but until you recognize and understand the issue you'll never know how it bears. Furthermore, I saw how the head sophist, persuaded you to see a mathematical axiom differently, through reference to the identity of indiscernible, so I know that it really does bear on things that you care about.

No, orderings are not "contradictory properties." Technically, an order on a set is another set, namely the set of pairs (x,y) for which we mean to denote that x < y in the ordering. The ordering is distinctly and noticeably separate from the set it applies to. — fishfry

What is said about a thing is distinct from the thing itself. Contradiction is not in the thing itself, it is in what is said about the thing. To say that a thing has contradictory orderings is contradiction. The contradiction "is distinctly and noticeably separate from the [thing] it applies to".

That distinction has no meaning or relevance in my understanding of the world. "equal" and "the same as" are entirely synonymous. — fishfry

You are in denial, just like the sophist. "Equal" means to have the same value within a system of valuation, "same" means identical, not different. Notice that "equal" is a qualified sense of "same" the "same value", meaning identical value, whereas "same" refers to identity itself without such qualification. Two distinct things are said to be equal, being judged according to a specified value system. Two distinct things are not the same. Please tell me that you understand this difference.

Would you agree that "number" is a general abstraction and that 5 is a partcular instance of number? Isn't that the most commonplace observation ever? — fishfry

This is colloquial vernacular insufficient for logical rigour. The proper classification is like this. The abstraction "number" is more general, and the abstraction "5" is more specific, just like "animal" is more general, and "human being" is more specific, or "colour" is general and "red" specific. Neither is a "particular instance".

One might however say that there is a particular instance of the abstraction "5", and the abstraction "number", in your mind, and another particular instance in my mind. But that would be an ontological stance which would be denying common Platonism. Platonists would say that what I just called particular instances, are really just parts of one unified concept "5".

Red is not an instance of the concept of color? How do you figure that? — fishfry

Fishfry, do you not understand what "instance" means? Here, from OED, "an example or illustration of". How do you think that a specific colour, red, is an example or illustration of the concept of colour? Red cannot exemplify "colour", because all the other colours are absent from it. That's why we go from the more general to the more specific in the act of explaining. Referring to the more specific abstraction, "red" is an instance of "specifying", it is not an instance of "colour".

"The axiom given above assumes that equality is a primitive symbol in predicate logic." — fishfry

How do you propose that this indicates that equal implies "same as"? I cannot follow your association. What the head sophist calls "identity theory" is simply an axiom of identity which is inconsistent with the law of identity. The sophist dictates that "=" means identical to, and this is the first principle of the sophistry referred to as "identity theory".

You and I have a completely different understanding of the nature of "a relation". We could not even find grounds to start any agreement, to converse. Consequently you'll understand "relation" in your way, and I'll understand "relation" in my way. Since "order" is a specific type of relation, any discussion about order, between us, will be rife with misunderstanding. — Metaphysician Undercover

And I have repeatedly made it clear that you can have your beliefs, your understandings, all day long no complaints from me.

But you insist that these beliefs of yours constitute facts about the world and then refuse any substantive engagement about them as facts, dodging, avoiding, and evading. You hold a relation is an independently existing immaterial real thing, explicitly not an idea or product of mind, yet you offer no account of its existence or of how you know it exists. You say that of groups of things, there is one and one only order of them, their natural order (dependent on their circumstance), but you give neither account nor explanation, nor answer direct questions about how any of that works or what any of it means.

And you wax long on everything else but substance when substance would be cleaner, shorter, and presumably conclusive.

You're the man who insists that 2+2=5. Which is fine for you in your sandbox. But as an adult it is incumbent on you to recognize you're not in your sandbox, and to stop acting petulantly as if you were a child in one. Now, I had thought you were attempting to salvage a little dignity by quitting the discussion. Alas, not so.

Just above I offered some questions, which so far you have ducked. Why not give them a try?

I stated explicitly several times that that is what I mean by 'identity theory'.
— TonesInDeepFreeze

You never said that LOL! — fishfry

I stated the axioms of identity theory in multiple posts. Not funny, but true.

I thought the = of set theory is the = from the underlying logic. But now you say it's not. — fishfry

I did not say that it's not.

I'll say again:

First order logic with identity provides:

(1) law of identity (axiom)

(2) indiscernibility of identicals (axiom schema)

(3) interpretation of '=' as standing for the identity relation (semantics)

Set theory takes (1) - (3) and adds:

(4) extensionality (axiom)

@fishfry

Nearly all of these text symbols are quite common:


~ ... it is not the case that

-> ... implies

<-> ... if and only if

& ... and

v ... or

A ... for all

E ... there exists a/an

E! ... there exists a unique

Axy ... for all x and for all y [for example]

if P(x) is a formula, then, in context, P(y) is the result of replacing all free occurrences of x with y [for example]

= ... equals

< ... is less than

<= ... is less than or equal to

> ... is greater than

>= ... is greater than or equal to

+ ... plus

- ... minus

* ... times

/ ... x divided by y

^ ... raised to the power of

! ... factorial

e ... is an element of

0 ... the empty set (also, zero)

w ... the set of natural numbers [read as 'omega']

{x | P} ... the set of x such that P [for example]

{x y z} ... the set whose members are x, y and z [for example]

<x y> ... the ordered pair such that x is the first coordinate and y is the second coordinate [for example]

(x y) ... the open interval between x and y [for example]

(x y] ... the interval between x and y, including y, not including x [for example]

[x y) ... the interval between x and y, not including x, not including y [for example]

[x y] ... the closed interval between x and y [for example]

U ... the union of

P ... the power set of

/\ ... the intersection of

x u y ... the union of x and y [for example]

x n y ... the intersection of x and y [for example]

x\y ... x without the members of y [for example]

|- ... proves

|/- ... does not prove

|= ... entails

|/= ... does not entail

PA ... first order Peano arithmetic

S ... the successor of

# ... the Godel number of

Z ... Zermelo set theory

ZC ... Zermelo set theory with the axiom of choice

ZF ... Zermelo-Fraenkel set theory

ZFC ... Zermelo Frankel set theory with the axiom choice

Z\I ... Zermelo set theory without the axiom of infinity

(Z\I)+~I ... Zermelo set theory with the axiom of infinity replaced by the negation of the axiom of infinity

Z\R ... Zermelo set theory without the axiom of regularity

ZF\R ... Zermelo-Fraenkel set theory without the axiom of regularity

ZFC\R ... Zermelo Frankel set theory with the axiom choice without the axiom of regularity

p ... possibly

n ... necessarily

when needed for clarity, ' ' indicates an expression not its referent ('Sue' is a name, Sue a person)

This is exactly the problem, failing to distinguish between "same as" and "equal to". — Metaphysician Undercover

It's not a problem to me. I suspect, but have no supporting evidence, that it's not a problem even for most philosophers. It's a problem for you, and I hope you can get it resolved so that it no longer troubles you.

Because you do not believe that there is a distinction to be made here, — Metaphysician Undercover

They're two phrases or words for the same thing. I concede that you have some deep or perhaps pseudo-deep reason to make a distinction, but you haven't explained it to my satisfaction.

you will not notice the effects of such a failure, and you will insist that it doesn't bear on anything you care about. — Metaphysician Undercover

Is it your contention that if I understood this failure, I would suddenly arise and go over to the math department at the nearest university and give them a piece of my mind? Or renounce my heresy, do penance, confess to a priest? Or what, exactly, would you like me to do?

Insisting that it doesn't bear on anything you care about will allow you to be mislead, even tricked by intentional deception (as you were by the sophist's employment of "identity of indiscernibles"), and you may never ever even notice it. — Metaphysician Undercover

You're cracking me up. I find your prose very funny tonight. You are going on about this but making no point at all.

Here is a simple example of where the difference bears in a substantial way, though I am sure there are more complex examples. In quantum physics, — Metaphysician Undercover

AHA!!!!!!! After berating me about the playground, and getting me to stop using real-life examples, you whip out an example from physics. But physics is not math.

Now explain this to me ONCE AND FOR ALL. Are we talking about pure math and set theory? Or are we talking about the physical world of time, space, energy, quantum fields, and bowling balls falling towards earth?

You can not have it both ways.

a quantum of energy is emitted as a photon, and an equal quantum may be detected as a photon. Since these two quanta of energy are equal, they are said to be "the same" photon. That is the mathematician's use of "same", equal quanta implies one quantum, a photon. By the law of identity "same" implies temporal continuity, such that the photon exists, with that identity, for the entire period between emission and detection. Equivocation between these two senses of "same" inclines some people to believe that the photon exists, as the same "particle" for the entire period of time between emission and detection. However, the electromagnetic energy is observed to exist as waves in the meantime. — Metaphysician Undercover

The physics complaint department is across the street. I'm the math complaint department. In math, "equals" and "the same" and "is identical to" are synonymous. I can't help you with your complaints about physics.

And you see, having BERATED ME ABOUT THE PLAYGROUND, you now give me yet another physical example. But I have already agreed not to use physical analogies or examples anymore, because physical things are different than abstract things.

So your physics example has no bearing on mathematics. In any event, photons are just excitations in the electromagnetic field. It's far from clear what a "thing" or "object" is in physics these days.

This produces significant theoretical problems. Some claim a contradictory wave/particle duality theory, in which the energy travels as both waves and as particles at the same time. Furthermore, since the photon of energy emitted is assumed to be "the same photon" as the photon detected, and it's path cannot be determined, it is claimed to have multiple paths all at the same time. All of this sort of problem is due to equivocation of "same". The mathematical "same", an equal quantum of energy is emitted and detected, is confused with "same" by the law of identity, to conclude that a distinct quantum (particle) of energy, known as the photon, has continuous existence between the time of emission and the time of detection. — Metaphysician Undercover

You rejected my playground story and now you're resorting to examples in physics. Please stay on topic, You are wasting your keystrokes talking about the wrong thing.

Photons are not sets. I have no idea what physicists mean by "same," "equal," or "identical." I doubt they do either, they don't bother themselves with philosophy these days.

You say that this issue doesn't bear on anything you know or care about, — Metaphysician Undercover

When I'm talking about the foundations of math, of course. When I arrive home in the evening, it makes quite a big difference to me if I return to the same residence or just one that's "equal" to it in value.

You are quite the sophist tonight yourself.

but until you recognize and understand the issue you'll never know how it bears. Furthermore, I saw how the head sophist, persuaded you to see a mathematical axiom differently, through reference to the identity of indiscernible, so I know that it really does bear on things that you care about. — Metaphysician Undercover

You haven't said a single thing of substance. You've given an example from physics when the subject is math. You've said nothing. As Truman Capote once said of a book he didn't like, "That's not writing. That's typing."

What is said about a thing is distinct from the thing itself. Contradiction is not in the thing itself, it is in what is said about the thing. To say that a thing has contradictory orderings is contradiction. The contradiction "is distinctly and noticeably separate from the [thing] it applies to". — Metaphysician Undercover

Well that certainly clears things up.

You are in denial, just like the sophist. "Equal" means to have the same value within a system of valuation, "same" means identical, not different. Notice that "equal" is a qualified sense of "same" the "same value", meaning identical value, whereas "same" refers to identity itself without such qualification. Two distinct things are said to be equal, being judged according to a specified value system. Two distinct things are not the same. Please tell me that you understand this difference. — Metaphysician Undercover

In physics? In playgrounds? Could be, for all I know, to the extent that your word salad communicated anything at all. In math? No. You don't understand how math works, and you continually demostrate that.

This is colloquial vernacular insufficient for logical rigour. The proper classification is like this. The abstraction "number" is more general, and the abstraction "5" is more specific, just like "animal" is more general, and "human being" is more specific, or "colour" is general and "red" specific. Neither is a "particular instance". — Metaphysician Undercover

When it comes to colors, or colours if you prefer, I admit I'm on shaky ground. I don't know if if philosophers consider red an instance of color, though I think it is.

But when it comes to math. I am sure. The set of real numbers is most definitely a particular set in the universe of sets, and is an instance of the general concept of set, or the universe or sets, or the category of sets,

One might however say that there is a particular instance of the abstraction "5", and the abstraction "number", in your mind, and another particular instance in my mind. But that would be an ontological stance which would be denying common Platonism. Platonists would say that what I just called particular instances, are really just parts of one unified concept "5". — Metaphysician Undercover

You finally said something interesting. Is the 5 in your mind the same as the 5 in my mind? I think so, but I might be hard pressed to rigorously argue the point.

So what do you say? Are the 5 in your mind and the 5 in my mind the same? Identical? Equal? Curious to know. You did interest me with this example.

Fishfry, do you not understand what "instance" means? Here, from OED, "an example or illustration of". How do you think that a specific colour, red, is an example or illustration of the concept of colour? Red cannot exemplify "colour", because all the other colours are absent from it. That's why we go from the more general to the more specific in the act of explaining. Referring to the more specific abstraction, "red" is an instance of "specifying", it is not an instance of "colour". — Metaphysician Undercover

Is an apple an instance of fruit? Apples don't have a peelable yellow skin. 'Splain me this point. By this logic, nothing could ever be a specific instance of anything, since specific things always differ in some particulars from other things in the same class.

How do you propose that this indicates that equal implies "same as"? — Metaphysician Undercover

Equal is the thing being formally defined or referenced from logic. "Same as" is a colloquial usage with no formal definition.

I cannot follow your association. What the head sophist calls "identity theory" is simply an axiom of identity which is inconsistent with the law of identity. The sophist dictates that "=" means identical to, and this is the first principle of the sophistry referred to as "identity theory". — Metaphysician Undercover

You demean only yourself and nobody else to continually refer to another member of this forum by a pejorative.

I stated the axioms of identity theory in multiple posts. Not funny, but true. — TonesInDeepFreeze

Not true, but funny.

I did not say that it's not.

I'll say again:

First order logic with identity provides:

(1) law of identity (axiom)

(2) indiscernibility of identicals (axiom schema)

(3) interpretation of '=' as standing for the identity relation (semantics)

Set theory takes (1) - (3) and adds:

(4) extensionality (axiom) — TonesInDeepFreeze

I'm sure you're right. Just a little beyond my awareness.

Nearly all of these text symbols are quite common: — TonesInDeepFreeze

Not quite my point, but thanks.

I stated the axioms of identity theory in multiple posts. Not funny, but true.
— TonesInDeepFreeze

Not true, but funny. — fishfry

Not not true, but not funny.

Not quite my point, but thanks. — fishfry

At least, if you are ever interested in a formula, but you don't know the use of the symbol, there you have it. Of course, if you're not interested in formulas, though they are the most exact and often the most concise communication, then I can't help that.

At least, if you are ever interested in a formula, but you don't know the use of the symbol, there you have it. Of course, if you're not interested in formulas, though they are the most exact and often the most concise communication, then I can't help that. — TonesInDeepFreeze

Symbols need explanatory words to go with them. This is probably more true in math than in logic.