The problem is that in many instances, like in "4+4=8", it cannot signify the same object — Metaphysician Undercover

Yes, it can! If you followed by tribe example, you'll see that you can do that! consider 4 to be the name of the set of all four member sets, now you join any member of that set with a member of it that is disjoint form it, and the result of all such unions would be 8 member sets that would be collected in a different set named as 8. It is very similar to the tribe condition. If you have the patience (which I agree is difficult) to follow the whole example I gave, you'll see the analogy. It would be solved in a very nice manner through conceiving numbers as denoting "sets", well actually sets of sets.

Of course the concept of having sets of sets is not a nice concept and not easy at all, but it can be interpreted in hierarchical labeling of collections, but that's another story.

as I said I'm not good with symbols, so I just get lost trying to figure out what you're saying. — Metaphysician Undercover

"Lost in symbols, hey!", me too really, I wonder if one can can get rid of that symbolic approach to mathematics and use instead of them understandable words within some rigorous language.

I don't think we can talk of sets here, because set theory already premises that "4" in one set refers to the same object as "4" in another set, and this is the false premise which I am trying to expose. So to be talking of sets is to already assume what you are trying to prove (begging the question).

Anyway, I'll try again to decipher your example. I think that the problem with the example is that once the symbol represents a group (tribe), then the individual members of the group loose their identity. So you say "a tribe is married to another tribe", when in reality each member of one tribe is married is to a member of another. This allows you, as a mathemagician, to set up a sort shell trick, where the actual thing under the shell, meaning the persons being referred to by the tribe name, is hidden.
Consider this:

Now we have the situation: A || B to mean tribe A is married to tribe B (according to rules above).
Now this is a predicative formulation, why, because A||B is a "proposition", it something that can be true or false, and the symbol || is denoting a "binary relation", so it is a "predicate" symbol. — Zuhair

In this case, what it is which makes "tribe A is married to tribe B", a proposition which is either true or false, is the individual marital status of the individual people. Without this there is no truth or falsity to the proposition.

So you proceed to hide the status of the individuals by saying that the proposition is either true or false. Now it appears like the status of the individuals is irrelevant, because the only relevant thing is the truth or falsity of the proposition. But in reality the truth or falsity of the proposition is still dependent on the status of the individuals. Therefore we must consider the status of the individuals.

Notice that we can have the situation were tribe S can marry itself!!! so we can have S || S
Notice that S occurred twice in the proposition "S || S" but still it denotes ONE object, although this object is a totality of many individuals, however that whole of many individuals is considered here as one object. So repeated occurrence of the symbol symbol in an expression doesn't denote different denotation, no here S repeatedly occurred in "S || S" but it still carries the same denotation, namely tribe S. — Zuhair

So this situation, in which tribe S is married to tribe S, if we consider the status of the individuals, must be analyzed. Remember the conditions, a male must be married to a female, so a person could not marry oneself. And, if the 50 girls of tribe S married the 50 boys of tribe S, there would only be 50 marriages, but in the case of tribe A being married to tribe B, there would be 100 marriages. This is exactly the quantitative difference I am talking about. If both the 4s in "4+4=8" represented the same object, then there would not be eight here, there would only be four. This is exactly what when S is married to S, there is not 100 marriages, only 50. But when A marries B there is 100 marriages. That's a substantial difference in the quantity of marriages.

P(S||S) = Q

Now we have two distinct occurrence of the symbol S on the left, but still it has the SAME denotational coverage! Both symbols of S denote the same object that is " TRIBE "S" ". — Zuhair

So this is not really true. S is married to S really denotes that the females of tribe S are married to the males of tribe S, which is substantially different from what is denoted by A is married to B.. So one S represents the males, while the other S represents the females. You have divided S into the subgroups MS and FS, and you ought to say that MS is married to FS. And now the mathemagician's shell trick of equivocation has been exposed. You claim that the same thing lies under each S, but in reality half of tribe S is under one S, and the other half of tribe S is under the other S. This is the only way to speak of S being married to S. This is verified from the fact that Q, the progeny of this union, is only half of C, the progeny of the union of A and B.

ou have divided S into the subgroups MS and FS, and you ought to say that MS is married to FS. And now the mathemagician's shell trick of equivocation has been exposed. You claim that the same thing lies under each S, but in reality half of tribe S is under one S, and the other half of tribe S is under the other S. This is the only way to speak of S being married to S. This is verified from the fact that Q, the progeny of this union, is only half of C, the progeny of the union of A and B. — Metaphysician Undercover

No, the laws of the country doesn't specify a tribe of one gender, tribes can only be named if they have 50 women and 50 men. Notice the definition of marriage between tribes doesn't say what's the total number of marriages, so although you have 50 marriages between tribe S and itself, and 100 marriages between tribes A and B when they are different, still both cases are concealed by the laws, and both receive the same description of being "married tribes". The other point is that for the case of S and S each couple would given birth to 2 children one male and one female, and that would make the progeny tribe made of 100 people 50 men and 50 women and so would constitute another tribe according to the rules of the country, which is tribe Q. While each married couple of tribes A and B only give birth to ONE child, but totally they'll have equal amount of girls and boys. That's how the country breeds! Those are fixed game rules. So yes there are differences even in how the resulting tribe came into existence upon marriage of the tribes, but still the rules of the country are insensitive to those difference and thus collects them under the same parcel. So the nutshell is that we'll have the same treatment of S married to S as of when A is married to B, despite the inner differences.

Of course there would be some hidden details no doubt, but the point is that there are indeed hidden difference, but since the definitions involved are blind to those differences they would pass the same. Like when we say for example "MAN" this denotes a lot of grown up males, but there are still many differences but all fall under the same SHELL.

So definitely sets, tribes, etc.. do conceal differences, that's the point of them really.

No, the laws of the country doesn't specify a tribe of one gender, tribes can only be named if they have 50 women and 50 men. Notice the definition of marriage between tribes doesn't say what's the total number of marriages, so although you have 50 marriages between tribe S and itself, and 100 marriages between tribes A and B when they are different, still both cases are concealed by the laws, and both receive the same description of being "married tribes". — Zuhair

You're missing the point of the criticism. What is "concealed", is the fact that half of S is married to the other half of S. In the case of A and B, all of A is married to all of B. So "S is married to S" does not mean the same thing as "A is married to B", because each "S" in "S is married to S" only represents half the entire original tribe of "S", whereas each of "A" and "B" remain consistent in representing the entirety of the tribes. Do you get the hint of equivocation in what "S" represents?

There is inconsistency in the application of the rules for what the symbols stand for. "S" is used to stand for the entire tribe of 50 men and 50 women, as stipulated at the beginning. But in "S is married to S", according to the example "A is married to B", the two "S"s must stand for different groups (tribes) which are married to each other. So, one S represents the women of the original S, and the other S represents the men of the original S. In other words, there is equivocation in the meaning of "S". Do you see this? "S is married to S" doesn't mean the whole tribe is married to itself, as consistency with "A is married to B" would imply, it actually means that half the tribe is married to the other half. Therefore each S in this case signifies half the tribe, whereas "S" was originally used to represent the whole tribe.

Of course there would be some hidden details no doubt, but the point is that there are indeed hidden difference, but since the definitions involved are blind to those differences they would pass the same. Like when we say for example "MAN" this denotes a lot of grown up males, but there are still many differences but all fall under the same SHELL. — Zuhair

The hidden difference is the difference in what "S" signifies. But that difference qualifies as equivocation, so the example is invalid.

So imagine there are four chairs, and we represent those four chairs with the symbol "4". — Metaphysician Undercover

We're not talking about chairs. Four chairs over here are different than the four chairs over there.

Once again you are avoiding the question. We are talking about 4 + 4 = 8. You claim the two instances of '4' represent or stand for or refer to or mean two different things. I categorically deny that. I have repeatedly challenged you to explain that remark and you deflect by talking about chairs. You have no argument. You got confused by your grade school teacher and you can't get out of that psychological box.

You have claimed that in ZFC things are claimed to be equal that are not identical. I have categorically denied that and challenged you to provide an example. You have repeatedly failed to do so.

You have claimed that mathematicians use the word equality when they really mean congruence, equivalence, or isomorphism. I have categorically denied that (with certain well-understood casual figures of speech in particular contexts) and challenged you to provide a specific example. You have repeatedly failed to do so.

You have no argument but you have your misunderstandings and a lot of words and handwaving.

I ask you to introspect on the point that if you can't come up with specific examples, perhaps you don't understand your own ideas as well as you think you do.

We're not talking about chairs. Four chairs over here are different than the four chairs over there. — fishfry

You asked for an example, so I gave it. What objects do the two 4s in "4+4=8" refer to in the example, if not the group of chairs here, and the other group of chairs over there?

Once again you are avoiding the question. We are talking about 4 + 4 = 8. You claim the two instances of '4' represent or stand for or refer to or mean two different things. — fishfry

No, you're not paying attention fishfry. I very specifically made a distinction between what the two 4s refer to, or stand for, and what they mean. They each mean a very similar thing, that there is a group of four objects representedby the symbol, but they refer to, or stand for, distinct things, like the two distinct groups of four chairs.

I implore you, please try to understand this. It's as if you refuse to distinguish between what a word means and what a word refers to. Here's an example. I might talk about my "computer". Do you know what "computer" means? Do you understand, that what "computer" means is something completely different from the object referred to when I talk about my computer? What it means is something completely different from what it stands for or refers to. These two are categorically different and to conflate them is a category mistake. That's the distinction I'm trying to make when I say there is a difference between what "4" means, and what the two 4's each refer to in the expression "4+4=8".

You have claimed that mathematicians use the word equality when they really mean congruence, equivalence, or isomorphism. — fishfry

That's ridiculous. I've repeated over and over again, that mathematician use "equality" to mean equality as defined by the axiom employed. And, the axioms do not define "equality" as identity. It is you who keeps making the incorrect assertion that mathematicians use "equality" to mean identity.

I ask you to introspect on the point that if you can't come up with specific examples, perhaps you don't understand your own ideas as well as you think you do. — fishfry

I gave you so many specific examples, like using 4+4=8 to recognize that putting two groups of four chairs together makes eight chairs. and also the very the act of counting things. Are you unable to read or something?

No, you're not paying attention fishfry — Metaphysician Undercover

I came here tonight to append a note to my previous post, which I composed in my mind before I saw this ... ahem ... remark of yours. It changes nothing. Here is what I wanted to say:

* I apologize if I sound strident. And I don't want to be strident. But in truth I've been trying to leave this conversation for a while. I stated my intention twice already and weakened.

On my part I'd be so happy to simply agree to disagree. Over time I'll go over your posts with an open mind. I would like to understand your point of view and in the process I might well learn something. I'm much more of an open-minded fellow than I sometimes appear. Make an argument I can understand, and I may well agree with you.

But we're talking past each other. I hereby agree to disagree with your point of view as expressed in this thread. We're talking past each other and no productive dialog is occuring.

* Ok that's pretty much what I wanted to write. But then I saw the quoted text ... and whether it's true or not, it comes off as having a bit of an edge to it. So neither of us is attaining our highest selves here. Let's let this go and meet again perhaps in some other thread.

But you are right in fact. I am not paying the slightest attention to your argument. That's another sign I should depart the thread. I'm making my arguments and you are making yours but nobody has gotten any more enlightened in many a post. So I'm out, unilaterally. I do not concede the point but I'm all outta ammo.

And yeah when I show up to apologize for sounding strident, I click on your post and I see your little snark and of course I don't read another thing. Becoming a self-fulfilling prophecy.

I do promise and commit to reading through your posts in much more detail at my leisure. I will also pick up the conversation between you and @Zuhair, which I haven't read but can see that it's an exploration of your ideas by someone who seems to at least know what you are talking about.

Bottom line I have no idea what you're talking about. But it's cool. Peace.

n other words, there is equivocation in the meaning of "S". Do you see this? "S is married to S" doesn't mean the whole tribe is married to itself, as consistency with "A is married to B" would imply, it actually means that half the tribe is married to the other half. Therefore each S in this case signifies half the tribe, whereas "S" was originally used to represent the whole tribe. — Metaphysician Undercover

Honestly I failed to see the "equivocation" you are referring to. "S" represents the WHOLE tribe, it represent all 50 woman and 50 men, i.e. it represents the collection of 100 persons, 50 of which are women and 50 of which are men, and this meaning remained consistently throughout the application, it NEVER changed at all. So I don't see any equivocation at all.

"50 men of tribe S are married to 50 women of tribe S,
AND
50 women of tribe S are married to 50 men of tribe S."

this completes all the required conditions for fulling "marriage" between tribes per the rules of that country. So accordingly the proposition S || S (i.e. S is married to S) is true.

But you are right in fact. I am not paying the slightest attention to your argument. — fishfry

Bottom line I have no idea what you're talking about. — fishfry

I assume then, that you still do not understand the distinction I made between what a symbol means, and what it refers to, or stands for. Perhaps if you read up on the kind/token distinction, that will help you.


When you say "S is married to S", it is quite clear that one S represents the fifty men and the other the fifty women. But you claim S represents the whole tribe. Hence the charge of equivocation.

50 men of tribe S are married to 50 women of tribe S,
AND
50 women of tribe S are married to 50 men of tribe S." — Zuhair

Your use of "AND" as a conjunction between the two expressions above provides the necessary ambiguity for your equivocation. "S is married to S" can refer to one situation only. Yet you use two distinct expressions. Since you allow that "S is married to S" represents the two distinct situations expressed above, the charge of equivocation is justified.

Your use of "AND" as a conjunction between the two expressions above provides the necessary ambiguity for your equivocation. "S is married to S" can refer to one situation only. Yet you use two distinct expressions. Since you allow that "S is married to S" represents the two distinct situations expressed above, the charge of equivocation is justified. — Metaphysician Undercover

Not it is NOT justified! Because we are using the "AND" in the GENERAL case of definition of marriage between any tribes A,B (whether A, and B are the same tribe or not), the general rule is:

IF
[50 men of tribe A are married to 50 women of tribe B
AND
50 women of tribe A are married to 50 men of tribe B]
THEN
A || B

I only applied that rule to the case of tribe S, where 50 men of them are married to 50 women of them and 50 women of them are married to 50 men of them as well. Just substitute S instead of A and S instead of B, and you get the conclusion S || S. No equivocation at all.

Not it is NOT justified! Because we are using the "AND" in the GENERAL case of definition of marriage between any tribes A,B (whether A, and B are the same tribe or not), the general rule is:

IF
[50 men of tribe A are married to 50 women of tribe B
AND
50 women of tribe A are married to 50 men of tribe B]
THEN
A || B — Zuhair

You are refusing to acknowledge the equivocation in your use of "AND" in the rule. In the case of "A is married to B", quoted above, "AND" is used as a conjunction between two phrases which refer to two distinct sets of circumstances. In the case of S is married to S, "And" would refer to two distinct descriptions of the same set of circumstances.

So, in case (1), of "A is married to B", you have situation Y ( "50 men of tribe A are married to 50 women of tribe B") "AND", situation Z ("50 women of tribe A are married to 50 men of tribe B").
But in the case (2) of "S is married to S", you have the situation X, with two different descriptions of X ('50 men are married to 50 women', "AND" '50 women are married to 50 men').

See, in case (1) you are saying there is situation Y, "AND" situation Z. In case (2) the conjunction "AND" joins two descriptions, saying of the situation X, this description "AND" that description are true. Therefore there is equivocation in the your use of "AND", which is unacceptable for a "rule", one says 'there is situation Y "AND" situation Z', while the other says 'this "AND" that are true of situation X'.

Just substitute S instead of A and S instead of B, and you get the conclusion S || S. No equivocation at all. — Zuhair

Making such a substitution alters the meaning of "AND" Therefore the example employs equivocation.

I assume then, that you still do not understand the distinction I made between what a symbol means, and what it refers to, or stands for. Perhaps if you read up on the kind/token distinction, that will help you. — Metaphysician Undercover

You can assume what you like. What's true is that I've given up interacting with you. Your persistent rudeness reflects badly on you. I'm fully aware of the subject matter that you claim I'm ignorant of. You persistently avoid engaging with anything I write. You're a nasty piece of work. Unpleasant.

The symbol '4' represents the multiplicity/quantifier of a set of elements. The quantifier of a set, removes all attributes of the elements, color, gender, age, etc., i.e. any identity.
Thus '4' is a reference set to match one to one to an unknown set to determine its 'size' or quantity.

Thus '4' is a reference set to match one to one to an unknown set to determine its 'size' or quantity. — sandman

OK, now what happens if we remove the "unknown set" which is matched to the reference set, so that we can just deal with the reference set itself? Is the reference set an object itself, does it contain objects, or what does it mean to be a "quantifier"?

The reference set, eg. the set of integers, is a mental construct, used in the process of counting, a practical convenience. Counting is the most fundamental process of measurement, the answer to 'how many'. The nature/identity of the elements is a matter of definition, what attributes must the elements have to be a member of a set.

The reference set, eg. the set of integers, is a mental construct, used in the process of counting, a practical convenience. Counting is the most fundamental process of measurement, the answer to 'how many'. The nature/identity of the elements is a matter of definition, what attributes must the elements have to be a member of a set. — sandman

So, in relation to the subject of this thread, is a mental construct, "the set of integers" for example, properly called "an object"? If so, then we have an infinite object, which seems incoherent because objects are known as objects by understanding their boundaries. If a mental construct is not an "object", then what exactly is "the reference set"? If we use the reference set for comparison, making a one to one relation in the act of counting for example, and if the reference set doesn't exist as an object, how does it exist?

It is a real physical object as it exists in the mind, a neural pattern, in the field of medicine. The problematic term is ‘infinite’. The mind has no experience with things without boundaries (the definition of that term). In the field of mathematics, ‘infinity’ is not a number or quantifier. It’s a relation for a set without a limit.
A more meaningful adjective for the set of integers is extendible. Using Peano type axioms of formation, we can always make a larger integer, but never a largest integer.
This suggests it is the process, not the object that is without limit. Since the symbol ‘1’ for the unit represents an immaterial abstraction, we can use as many as needed from an inexhaustible supply. The set of integers will manifest itself as a finite set for as long as the forming process continues. As you said, human thought is only familiar with things having boundaries. We can only measure things with boundaries.
Our world is based on abstractions of the mind, since we can’t comprehend the reality of it.

You are refusing to acknowledge the equivocation in your use of "AND" in the rule. — Metaphysician Undercover

Yes, I'm refusing this. "AND" here is "logical conjunction", it specifically means a function from the truth value of each statement linked by "AND" to the truth value of the whole statement in such a manner that the truth value of the whole statement (i.e. the two statements linked by "AND", and "AND" itself) is positive (i.e. is true) if and only if both statements linked by "and" have positive truth values. So "AND" here has a specific role assigned to it, and that role serves as its meaning. It has nothing to do with the imagined mixed roles you are speaking about. So back again, all of what our rule is saying is that the following:

IF the statement (50 men of tribe A are married to 50 women of tribe B) is TRUE
AND the statement (50 women of tribe A are married to 50 men of tribe B) is TRUE
THEN the statement "tribe A is married to tribe B" is TRUE.

AND is specifically the logical conjunctive article, nothing more nothing less. Now let apply this to tribe S were 50 men of tribe S are married to 50 women of tribe S, we have the antecedent:

The statement (50 men of tribe S are married to 50 women of tribe S) is TRUE
AND
The statement (50 women of tribe S are married to 50 men of tribe S) is TRUE.

Then by the above rule it follows that

tribe S is married to tribe S.

There is no equivocation whatsoever!

What I'm speaking about is simple blind following of the rules, nothing more nothing less. "AND"
is given a constant meaning throughout applications of it whether A is B or whether A is not B.

No equivocation!

This suggests it is the process, not the object that is without limit. — sandman

Actually, it suggests that the "set of integers" is not an object. And, the fact that the interval between two integers can be divided indefinitely indicates that numbers are not objects.

IF the statement (50 men of tribe A are married to 50 women of tribe B) is TRUE
AND the statement (50 women of tribe A are married to 50 men of tribe B) is TRUE
THEN the statement "tribe A is married to tribe B" is TRUE.

AND is specifically the logical conjunctive article, nothing more nothing less. Now let apply this to tribe S were 50 men of tribe S are married to 50 women of tribe S, we have the antecedent:

The statement (50 men of tribe S are married to 50 women of tribe S) is TRUE
AND
The statement (50 women of tribe S are married to 50 men of tribe S) is TRUE. — Zuhair

You've just inverted the equivocation by insisting that "AND" means the same thing in each case. If we are to conclude, by following the example as a "rule", that tribe S is married to tribe S, there will be equivocation in the use of "married". The phrase "is married to" will mean something different in "tribe A is married to tribe B", from what it means in "tribe S is married to tribe S".

What this demonstrates is that the "rule" is faulty. If the rule is followed, there is produced a conclusion which is consistent with the "rule". But there is inconsistency between what "married" refers to in the rule, and what "married" refers to in the conclusion. Equivocation is created by following the rule. Therefore the so-called rule ought not be taken as a rule, rather it should be taken as something which could lead one into deception.

he phrase "is married to" will mean something different in "tribe A is married to tribe B", from what it means in "tribe S is married to tribe S". — Metaphysician Undercover

No! you are confusing matters. Notice my original statement:

RULE: For every tribe A for every tribe B (A || B if and only if for every male a of A there is one female b of B such that: a m b, and for every woman a of A there is one male b of B such that: a m b). — Zuhair

Notice the "if and only if", the above statement is a DEFINITION of "||". Notice that it was symbolized by another symbol from "m" which was given to marriage between individual.

Marriage between tribes (symbolized by ||) has NO meaning by itself, it is just a string of letters, the country gave it a meaning by the statement after the "if and only if" above. So you cannot say it leads to equivocation of meaning or anything like that, because its meaning is understood to be fully traceable to the specifications building it posed by the rule, in other ways that rule is a DEFINITIONAL RULE. Without it you have no meaning of tribal marriage at all.

In those rigid kinds of definitions, there is no room for equivocation or the alike. These are strict rule following machinery. Equivocation is out of question here.

Notice the "if and only if", the above statement is a DEFINITION of "||". Notice that it was symbolized by another symbol from "m" which was given to marriage between individual.

Marriage between tribes (symbolized by ||) has NO meaning by itself, it is just a string of letters, the country gave it a meaning by the statement after the "if and only if" above. So you cannot say it leads to equivocation of meaning or anything like that, because its meaning is understood to be fully traceable to the specifications building it posed by the rule, in other ways that rule is a DEFINITIONAL RULE. Without it you have no meaning of tribal marriage at all. — Zuhair

The point is that the symbol "||" refers to a different situation in S||S than it does in A||B. Therefore the rule produces ambiguity in the use of that symbol, and the possibility of equivocation. If we assume that "||" has the same meaning in each case, we are deceived by equivocation. Therefore the rule is a faulty rule, and ought not be accepted.

In those rigid kinds of definitions, there is no room for equivocation or the alike. These are strict rule following machinery. Equivocation is out of question here. — Zuhair

Very clearly this is not true. Your "DEFINITION of '||" allows that two very distinct and different types of situations are referred to by "||". Therefore ambiguity is inherent within the definition itself. Your "DEFINITIONAL RULE" is ambiguous, and equivocation in the conclusion follows from that ambiguity.

Yes, you are proceeding with "strict rule following machinery", but there is ambiguity in the rule itself, the definition of the term. And this ambiguity produces equivocation in your professed conclusion, that "S||S" describes a situation similar to "A||B". That the two different situations are similar is an illusion created by the ambiguity inherent within the definition of "||".

'S married to S' is not defined. It would allow same sex marriage.

As you said, human thought is only familiar with things having boundaries. — sandman

What is the 'closure' of the concept of an integer? I'm sympathetic to intuition-ism and constructivism, but I don't agree about this familiarity only with boundaries. The concept of 'for each' is quite natural to us. A proof by induction does help us intuit a truth about an infinite set of numbers.

This isn't to deny that certain problems can crop up. But finitism, for instance, has its own problems.

The point is that the symbol "||" refers to a different situation in S||S than it does in A||B. Therefore the rule produces ambiguity in the use of that symbol, and the possibility of equivocation. If we assume that "||" has the same meaning in each case, we are deceived by equivocation. Therefore the rule is a faulty rule, and ought not be accepted. — Metaphysician Undercover

S||S is a particular case of A||B; also C||D when C, D are disjoint tribes is also a particular case of A||B.
To complicate the situation we may even allow for asymmetric gender partial Overlaps between tribes, like in saying there are tribes K, L where 20 men are shared between tribes K,L and say 12 woman are shared between tribes L,K, still we can get the same rule applicable to them! So there is a spectrum of possible overlaps, all those would be cases of A||B. Of course in each specific case there will be additional features that discriminate this case from others, for example the particularities of the tribes themselves, also the particularity of the number of actual marriages between the tribes, etc.. all of these doesn't matter, since they all meet the definition of ||. This is like variation in particularities of objects fulfilling a predicate, for example the predicate "is a circle", now not all circles are really a like, they might vary in their size for example, in their colors, etc.., that doesn't affect them all being circles. No equivocation at all. Similarly the relationship || between tribes has strict definition, and whenever that definition is met, then the relationship holds between the respective tribes, variations in particularities of individual actualization of that relationship are immaterial as immaterial is the size of the circle in meeting the definition of a circle. A circle is a circle whether its big or small in size, similarly A||B holds whenever tribes A,B fulfill the definition of ||, whether the actual marriages between the two tribes is 100, or 50, (or any other number in case of partially overlapping tribes). No equivocation at all. Equivocation might arose only when || is APPLIED in a manner that doesn't depend on the mere definition of it, that confuses different applications of || and attributes the same consequence to these as if they were the same, but that's something that has to do with APPLICATION of ||, and actually with a kind of non-careful application, i.e. an erroneous application of the relationship ||, it has nothing to do with the mere definition of || at all.

The whole matter began when I wanted to coin a relation that can exist between something and itself other than the identity relation! So the relation || as I defined in the example can occur between a tribe and itself, and also can occur between distinct tribes, so its not the identity relation. As far as the "application" of relation ||, there is no equivocation at all.

So identity is not the ONLY relation that can occur between something and itself.

But

Identity is the ONLY relation that can ONLY occurs between something and itself.

S||S is a particular case of A||B; also C||D when C, D are disjoint tribes is also a particular case of A||B. — Zuhair

I'll grant you that as true. But the point is that there is ambiguity as to what "||" signifies. So, we must be careful not to equivocate.

This is like variation in particularities of objects fulfilling a predicate, for example the predicate "is a circle", now not all circles are really a like, they might vary in their size for example, in their colors, etc.., that doesn't affect them all being circles. No equivocation at all. — Zuhair

That's right, there need not be any equivocation if we respect the differences. The equivocation would only occur if we say that one circle is "the same" (as specified by the law of identity) as the other, on account of them each being a circle.

Similarly the relationship || between tribes has strict definition, and whenever that definition is met, then the relationship holds between the respective tribes, variations in particularities of individual actualization of that relationship are immaterial as immaterial is the size of the circle in meeting the definition of a circle. — Zuhair

That definition of "||" is not so strict as you seem to think. You describe it as "the relationship || between tribes", as if it is necessarily a relationship between a plurality of "tribes". Yet in the case of S||S it does not represent a relationship between tribes, but the internal relations of one tribe. Do you apprehend this difference? This is the difference between what is internal to an object (part of the object), and external to the object (not part of the object).

We name an object, "X" for example, and we can describe relationships between this object and other objects. Or, we can describe relationships internal to that object, as relations of the parts, but these relations are not relations between the object X and other objects. The former is properly relations of the object, the latter is relations of the parts (as objects themselves).

Now, you conflate these two with your definition. You say that if the parts of tribe A have a specified relation with the parts of tribe B, we can call that a relation between the objects named by A and B. This is not true though, there is no such relation between two objects "A" and "B" being described. There is only a group of parts (as objects themselves), with specified relations between these elements. This is very evident in the case of S||S. There is not two objects "S", with relations between those two objects, there are relations between elements and a rule which produces the claim of two objects named S with a relation between those two objects. It appears to me, like you may have some system for naming a particular group of parts by names such as "A", "B", and "S", but this assignment might be completely random, so we cannot say that these names refer to objects. They refer to groupings of elements, the elements themselves being objects

Therefore you have some random, or principled groupings of objects, and these groups of objects have received names, like "S". These groups are not objects, but artificial groups. You also have the members of the groups, which are the real objects that exist in certain relationships with each other. Your system for naming the groups as objects is ambiguous because it doesn't distinguish between internal relationships (relations between members of the group) and external relationships (relationships between a member of one group and a member of another group). Because of this failure, to distinguish the internal from the external, which is an essential aspect of an object, your groupings cannot be understood as objects. Very clearly in the case of S||S the two S's cannot be understood as signifying the same object with a relationship to itself. It really signifies that the objects which are the members of group S have the necessary relations with each other, to apply the rule, and say S||S.

The rule is ambiguous because it allows that internal relations between the members of a group are treated in the same way as relations between members of one group and members of another group. This annihilates, negates, or renders insignificant, the boundaries of the group, leaving the groupings as meaningless

The whole matter began when I wanted to coin a relation that can exist between something and itself other than the identity relation! So the relation || as I defined in the example can occur between a tribe and itself, and also can occur between distinct tribes, so its not the identity relation. As far as the "application" of relation ||, there is no equivocation at all.

So identity is not the ONLY relation that can occur between something and itself.

But

Identity is the ONLY relation that can ONLY occurs between something and itself. — Zuhair

Let's be specific with the terms here. I said that in the case of "4+4=8", if "4" is to represent an object, each of the two 4's must represent a distinct object. You gave S||S as an example of a case where each of the two S's represents the same object. Now I've demonstrated that "S" does not represent an object at all, but an artificial grouping, which cannot be apprehended as an object due to an inability to distinguish which elements are internal to (part of) the group, and which elements are external to (not part of) the group. The artificial grouping does not follow the rules of having a meaningful boundary, which is necessary for an object.

Now, it appears like an artificial grouping can be created which does not follow the rules of what "an object" is. The law of identity, what you call here "the identity relation", applies specifically to objects. If you can create an artificial group, which is not an object, it's quite clear that this artificial group might not follow the rules of what an object is. So your example really shows nothing, because your artificial group is not consistent with "object", therefore the identity relation is not applicable to your artificial group.

That definition of "||" is not so strict as you seem to think. You describe it as "the relationship || between tribes", as if it is necessarily a relationship between a plurality of "tribes". — Metaphysician Undercover

Hmmm.. I see the confusion here, OK, when I said a relationship between tribe(s), I only meant that each of its arguments is a tribe. That's all. It doesn't indicate plurality. It doesn't indicate that those arguments must be distinct from each other. For example identity is a relationship between object(s), it doesn't mean that the arguments of identity are distinct objects, of course not.

As regards false grouping argument of yours, and that tribes are not objects etc.... I object to this argument. A tribe is a well specified entity, it refers to the totality of specified individuals. And in the example I've outlined that each group of 50 men and 50 women that go and register themselves in the registry of the country as a tribe, then those would be called a tribe, so a tribe in this case is the totality of all those individual objects so registered. This is a well specified entity. Now whether || is not sensitive to "internal" relationships within a tribe versus "external" relationships, and that this would blur up the boundaries of tribes, etc.. argument of yours, this is not correct. Yes definitely || is not sensitive to internal x external relations between tribes, however this doesn't entail that the tribes are not well defined entity, it only means that the relation || is not sensitive to boundaries of tribes, that doesn't mean that the tribes don't have clear cut boundaries. Clearly each tribe is a well defined entity and what is external to it is very well demarcated, it is what is not a registered member in it, and what is a member of it and what is not, is well defined in the registry of that country. So each tribe is SHARPLY demarcated, and in that sense it is indeed an object, although a plural kind of object rather than a singular kind. It is this insensitivity of || to boundaries of tribes that cause it to be able to occur inside a tribe and for other tribes to be in-between them (outside each of them to the other), yes that's what cause it to be a relationship that can be between something and itself for some objects and also at the same time can occur between something and other things for some objects.

So we do have S||S, each of S represents the SAME tribe (which is indeed an object), and yet || is not a relationship of identity! So the occurrence of a same symbol on either side of a relation symbol doesn't entail that each occurrence must stand for distinct object.

Can't elaborate on my response. There is no human experience with 'infinite' unbounbded/without limit entities. Cantor was an illusionist, who fooled many people. That's it.

Can't elaborate on my response. There is no human experience with 'infinite' unbounbded/without limit entities. Cantor was an illusionist, who fooled many people. That's it. — sandman

Well the though police aren't going to kick down your door for thinking so. Nor will they harass you for denying the theory of special relativity.

Personally I think you are caught up in a conspiracy theory here. And conspiracy theory (one of the great if disorganized religions of our time) seduces by presenting itself (deceptively) as the opposite of credulity. You imply that basically all mathematicians are fooled. This is a belief in something highly unlikely that you are unwilling to justify except in terms of an amateur's hunch. What I think you're missing is that all of this metaphysical jazz is what you are bring to the situation. Math is dry and technical. It is philosophically agnostic. Individual mathematicians may have metaphysical beliefs, but those beliefs don't play a role in proofs.

Cantor didn't prove something 'magical' about the 'real world,' even if he himself thought so. Or at least mathematicians are not at all bound to experience Cantor's work like that.

It doesn't indicate that those arguments must be distinct from each other.

...

A tribe is a well specified entity, it refers to the totality of specified individuals. — Zuhair

OK, since a "tribe" is meant to be an object, you are defining "object" with these principles. Accordingly, an object is a whole, composed of the totality of specified parts. And, there may be an overlap between the parts of one object and another, such that the parts of one object may also be the parts of another object.

Now, I suggest that you are missing a very important (essential) aspect of an "object" in your definition. For the existence of an object, it is necessary that the parts exist in specific relationships to one another. This is why your definition, and consequent "rule" is ambiguous, as I've said. An "object" is not a random collection of parts, in any random relations, but the parts must exist in a very specific way in relation to each other, in order to constitute an entity. Those relationships are essential to the existence of any object, such that when the specified relations cease to exist, the object ceases to exist, despite the fact that the totality of parts may continue to exist. In other words, "totality of specified individuals" is insufficient for "well specified entity", and your claim that "a tribe is a well specified entity" is absolutely false. It is this ambiguity (lack of definition) in the relationships between parts of your so-called "entity" which allows for your mathemagistic sophistry.

This is a well specified entity.

...

Clearly each tribe is a well defined entity... — Zuhair

As explained, this is absolutely false. Your entire argument above, relies on the truth of this false assertion, which you seem to think that if you repeat it enough it will magically become true.

Cantor was an illusionist, who fooled many people. That's it. — sandman

Very true, that's why I like to call people like Cantor mathemagicians.

Individual mathematicians may have metaphysical beliefs, but those beliefs don't play a role in proofs. — Eee

Yes, those metaphysical beliefs clearly do play a role in mathematical proofs because they are entrenched in the axioms, as foundational support for those axioms. And Cantor is a good example. What is at issue here is how we conceive of an "object".