In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. — aletheist
However, we can point at a collection of three apples and say both "that is 2+1 apples" and "that is 3 apples." — aletheist
Moreover, we can substitute "2+1" for "3" in any proposition without changing its truth value or in any equation without changing its result. What should we conclude from this? — aletheist
TonesinDeepFreeze has been asserting that "2+1" denotes the same object as "3" does, in a similar way. They very clearly each signify something different. The only attempt by Tones, to support this conclusion with a premise, was a vague reference to extensionality. — Metaphysician Undercover
we do not have the premises required to conclude that they denote the same object. Therefore your conclusion that they denote the same object is fallacious. — Metaphysician Undercover
Fair enough, thanks. Indeed, an extensional context corresponds to denotation (object), which is the same for "Henry Fonda" and "the father of Peter Fonda"; while an intensional context corresponds to signification (interpretant), which is different for the two signs as I have acknowledged all along.That's correct in an extensional context, but not in an intensional context: — TonesInDeepFreeze
The "Fonda" example was provided as an argument for the truth of it. — Metaphysician Undercover
I am treating "2+1" and "3" as signs here, and I already acknowledged that their signification is different. At issue is whether their denotation is different. What "+" represents in isolation is irrelevant, all that matters here is that I can point to the same group of items and truthfully say both "that is 2+1 apples" and "that is 3 apples."That's not true, because the operation signified by "+" is not evident in the group of three apples, so it is not a true representation of "2+1". It is just a representation of "3". — Metaphysician Undercover
I actually might do exactly that, if I were teaching them basic addition such as 2+1=3.If you were teaching children you would not show them a group of three apples and tell them this is 2+1. — Metaphysician Undercover
Of course not, but we can conclude (in an extensional context) that they denote the same thing.We cannot conclude that because the expressions can be substituted, they signify the same thing. — Metaphysician Undercover
Indeed, but as the Fonda example has brought to light, @Metaphysician Undercover apparently confuses denotation and signification. The result is wrongly denying that two different expressions signifying different interpretants can nevertheless denote the same object.Ordinary mathematics regards '2+1' and '3' as having the same denotation — TonesInDeepFreeze
I can see Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda." — fishfry
"equal" is assigned according to some system of judgement, so only the properties deemed significant within that system are accounted for, and this is insufficient for the conclusion of "the very same object". — Metaphysician Undercover
There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound. — Metaphysician Undercover
Therefore the argument that "the father of Peter Fonda" denotes the same thing as "Henry Fonda" is a fallacious argument, by means of begging the question. The argument relies on assuming the conclusion. — Metaphysician Undercover
(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.
(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do. — TonesInDeepFreeze
Ordinary mathematics regards '2+1' and '3' as having the same denotation, because we prove
2+1 = 3
In general, for any terms T and S, we infer
T = S
when we prove it and then we may say that T and S have the same denotation. — TonesInDeepFreeze
Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms. — TonesInDeepFreeze
Of course in natural language and everyday discourse there may be vagaries that make definitive determinations difficult or impossible. So if you want to demand a context in which there are no vagaries, then of course all bets are off with natural language usage. So to proceed with understanding certain principles, of course we must assume, for sake of discussion, some context in which we are not thwarted by such vagaries as you mention. — TonesInDeepFreeze
Similarly, of course I take it for granted that we already understand that 2+1 = 3, either by proof or by common mathematical knowledge. It is from that understanding that we then observe that '2+1' and '3' denote the same number. — TonesInDeepFreeze
But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number. — TonesInDeepFreeze
The claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person is not an argument! It is a conclusion. It is a conclusion from the premise (however it has been established) that Henry Fonda is the father of Peter Fonda. No one every suggested otherwise! — TonesInDeepFreeze
I am treating "2+1" and "3" as signs here, and I already acknowledged that their signification is different. At issue is whether their denotation is different. What "+" represents in isolation is irrelevant, all that matters here is that I can point to the same group of items and truthfully say both "that is 2+1 apples" and "that is 3 apples." — aletheist
Indeed, but as the Fonda example has brought to light, Metaphysician Undercover apparently confuses denotation and signification. The result is wrongly denying that two different expressions signifying different interpretants can nevertheless denote the same object. — aletheist
(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.
(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do.
— TonesInDeepFreeze
I told you already, extensionality provides a false premise. — Metaphysician Undercover
extensionality provides a false premise — Metaphysician Undercover
When a human being judges two distinct things as having the same properties, and says therefore that they are equals, this does not make them into the same thing. — Metaphysician Undercover
The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing. — Metaphysician Undercover
When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness. — Metaphysician Undercover
proving that two things are equal does not imply that they are the same (share all properties) — Metaphysician Undercover
If in reality, language use is filled with vagaries, and we want to discuss the truth about language use, then we need to account for the reality of those vagaries. To assume a context without vagaries as your prerequisite premise for proceeding toward an understanding of certain principles of language use, is simply to assume a false premise. — Metaphysician Undercover
In the case of Henry Fonda, we have observed with our senses, the very object being referred to. In the case of numbers we have not observed any such objects. — Metaphysician Undercover
You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. — Metaphysician Undercover
Claiming a denotation when there is only meaning, — Metaphysician Undercover
Do you understand the fallacy of "begging the question", assuming the conclusion? — Metaphysician Undercover
you [altheist] and Tones are the ones confusing denotation and signification. — Metaphysician Undercover
Good, and they also both denote the same object with "did," which is the relation of doing. However, they presumably denote different objects with "X" and "Y," although since these are variables it is conceivable that they could also denote the same object--for example, the activity of exercise.I can see how "I did X", and "I did Y", both refer to the same object with "I", but each signify something different. — Metaphysician Undercover
We already went over this with "Henry Fonda" and "the father of Peter Fonda." These signs both denote the same object despite signifying different interpretants because it happens to be a fact that Henry Fonda is the father of Peter Fonda.How do you come up with this idea that two phrases which signify something completely different actually denote the same object. — Metaphysician Undercover
As someone once said ...Clearly, in our use of mathematics there is signification without denotation. — Metaphysician Undercover
Do you understand the fallacy of "begging the question", assuming the conclusion? — Metaphysician Undercover
Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again. — TonesInDeepFreeze
Rather, we infer they share all properties from having first proved that they are equal. — TonesInDeepFreeze
We don't judge two things are equal. — TonesInDeepFreeze
No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics. — TonesInDeepFreeze
You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion. — TonesInDeepFreeze
Good, and they also both denote the same object with "did," which is the relation of doing. — aletheist
We already went over this with "Henry Fonda" and "the father of Peter Fonda." These signs both denote the same object despite signifying different interpretants because it happens to be a fact that Henry Fonda is the father of Peter Fonda. — aletheist
Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again.
— TonesInDeepFreeze
Do you know the law of identity? It states that a thing is the same as itself. It says nothing about equality or equivalence. That two things are equal is a human judgement. — Metaphysician Undercover
Rather, we infer they share all properties from having first proved that they are equal.
— TonesInDeepFreeze
See, no strawman. You prove that they are equal (human judgement), then you infer from this, that they are the same. — Metaphysician Undercover
We don't judge two things are equal. — TonesInDeepFreeze
When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness.
— Metaphysician Undercover
We don't judge two things are equal. We judge that two terms refer to the same thing. And, of course, such judgements may be mistaken due to human error. — TonesInDeepFreeze
You very clearly stated "having proved that they are equal". — Metaphysician Undercover
The indiscernibility of identicals does not provide the principle required for substituting equal things. — Metaphysician Undercover
Things are judged to be equal not on the basis that they are indiscernible. — Metaphysician Undercover
what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. — Metaphysician Undercover
I really can't see how a relation is an object. — Metaphysician Undercover
That is because you evidently have an extremely narrow definition of "object" and refuse to accept how that word is defined as a technical term within the discipline of semeiotic. Anything whatsoever that is denoted by a sign is the object of that sign.I really can't see how a relation is an object. — Metaphysician Undercover
Not at all, do some research into semeiotic (also called "semiotic" or "semiotics") and you will learn that what I have been discussing is a well-established field of study. I did not anticipate having to get into this level of detail when I offered the simple observation that "Henry Fonda" and "the father of Peter Fonda" denote the same object, which is utterly uncontroversial within that field.I think you are making things up as you go. — Metaphysician Undercover
There is no contradiction because denotation and signification are not synonymous, they correspond to different functions of a sign. Again, what a sign denotes is its object and what a sign signifies its interpretant. The object of a term is whatever it stands for, while the object of a proposition is the collection of objects denoted by the terms that serve as its subjects. The interpretant of a sign is whatever it conveys about its object, and thus is usually what we have in mind when we talk about the meaning of that sign.On top of this you allow that two phrases might signify different things, yet denote the very same thing. This indicates very clearly that there are contradicting interpretations of the same phrases. One interpretation says that they are different, the other that they are the same. Yet you allow that the contradicting interpretations are both correct. — Metaphysician Undercover
So get these straight already:
(1) My explanation runs in this order:
Determine equality, then it is justified to assert that the terms denote the same.
(2) Equality implies indiscernibility. I did not opine one way or the other whether indiscernibility implies equality. — TonesInDeepFreeze
I know about identity vastly more than you do. And your reply merely repeats your own thesis. And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement. — TonesInDeepFreeze
Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different. — TonesInDeepFreeze
First we determine (by proof or whatever method) that 2+1 is 3. — TonesInDeepFreeze
Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle. — TonesInDeepFreeze
What I said:
equality -> indiscernibility. — TonesInDeepFreeze
But you keep saying that I say:
indiscernibility -> equality
even after I've told you that is not what I say. — TonesInDeepFreeze
Don't reverse the direction of my conditionals. — TonesInDeepFreeze
In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning. — TonesInDeepFreeze
That is because you evidently have an extremely narrow definition of "object" and refuse to accept how that word is defined as a technical term within the discipline of semeiotic. Anything whatsoever that is denoted by a sign is the object of that sign. — aletheist
Not at all, do some research into semeiotic (also called "semiotic" or "semiotics") and you will learn that what I have been discussing is a well-established field of study. — aletheist
The object of a term is whatever it stands for, while the object of a proposition is the collection of objects denoted by the terms that serve as its subjects. — aletheist
It is not "my" definition, it is the well-established definition within the discipline of semeiotic.So I see your definition as completely misguided because it's not conducive for understanding. — Metaphysician Undercover
Then we can stop wasting each other's time.I do not agree with the fundamental principles of that proposed field of study. — Metaphysician Undercover
On the contrary, in my experience you routinely dismiss things out of hand, simply because they fail to conform to your peculiar, narrow, dogmatic definitions of terms.I am not one to dismiss things off hand, without some understanding of the fundamental principles. — Metaphysician Undercover
You also say things like this so that it sounds like you know what you are talking about when you really have no idea.This appears to involve a fallacy of composition. — Metaphysician Undercover
And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement.
— TonesInDeepFreeze
If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds. — Metaphysician Undercover
Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different.
— TonesInDeepFreeze
You said that from a judgement of equality you can infer that they are the same. I'll quote for the third time:
"Rather, we infer they share all properties from having first proved that they are equal."
You are clearly arguing that if they are equal then they are the same. — Metaphysician Undercover
you cannot use the indiscernibility of identicals to support your claim that they are identical. — Metaphysician Undercover
But you keep saying that I say:
indiscernibility -> equality
even after I've told you that is not what I say.
— TonesInDeepFreeze
This is the only way that the principle of indiscernibility could be used to support your claim that equality means the same as. — Metaphysician Undercover
Are you and I the same because we are equal? — Metaphysician Undercover
you have no special definition for "equal". — Metaphysician Undercover
to define that sense of "equality" with "value" — Metaphysician Undercover
In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning.
— TonesInDeepFreeze
See, you admit right here, that you only concern yourself with a part of what "2+1", and what "1" refer to. — Metaphysician Undercover
I am not one to dismiss things off hand, without some understanding of the fundamental principles. — Metaphysician Undercover
But since what I am looking for is an indication that 2+1 really is the same thing as 3, — Metaphysician Undercover
what I am looking for is an indication that 2+1 really is the same thing as 3 — Metaphysician Undercover
You believe that equality holds based on human judgement. That doesn't entail that I said that equality or identity does. It's a strawman to represent me as saying something I did not say. — TonesInDeepFreeze
Of course if 2+1 and are equal 3 then they are the same. — TonesInDeepFreeze
Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility. — TonesInDeepFreeze
Still yet again, I am not using indiscernibility to support that 'equal' means 'identical' or that 'equal' means 'the same'. Stop mixing up what I've said and then representing your own mixed up version as if my own. — TonesInDeepFreeze
In ordinary mathematics, 'equal' is not defined but rather is a primitive. It is the sole primitive of first order identity theory. In that context 'equal' and 'identical' are two words for the same undefined primitive. — TonesInDeepFreeze
And again the two different terms '2+1' and '3' refer to the same object. They have the same reference. However, of course, they do not have the same sense. — TonesInDeepFreeze
That is rich from someone who dismisses approaches in ordinary mathematics while insisting on remaining ignorant of understanding their fundamental principles or even reading a single page in a book or article about the subject. — TonesInDeepFreeze
I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here. — fishfry
'S' stands for the successor operation.
def: 1 = S0
def: 2 = 1+1
def: 3 = 2+1
The proof in this case is utterly trivial, from the definition of '3'. — TonesInDeepFreeze
I asked you for an instance of equality which is not a human judgement. You didn't give me one. — Metaphysician Undercover
In case you're having a hard time to understand, I see this as very clearly false. You and I are equal, as human beings, but we are not the same. — Metaphysician Undercover
You seem to think that numbers are somehow special — Metaphysician Undercover
necessarily the same — Metaphysician Undercover
we can proceed to say that a thing is indiscernible from itself — Metaphysician Undercover
2+1 is discernible from 3 — Metaphysician Undercover
how in hell are you supporting this obviously false assumption that "if 2+1 and are equal 3 then they are the same"? — Metaphysician Undercover
The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. — Metaphysician Undercover
signified — Metaphysician Undercover
mathematicians use "=" to relate two distinct expressions with distinct meanings — Metaphysician Undercover
I cannot understand the fundamentals because they are unsound. — Metaphysician Undercover
All I see is "=" here. Where's the proof that "=" means the same as? — Metaphysician Undercover
Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet. — TonesInDeepFreeze
The case is that you can't read. I replied about the notion of human equality many many posts ago, and you skipped recognizing my reply, and I even mentioned a little while ago again that I had made that reply and you skipped that reminder too! — TonesInDeepFreeze
I have not said that numbers are special regarding denotation. — TonesInDeepFreeze
I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic. — TonesInDeepFreeze
'2+1' and '3' have different senses but not different denotations. No matter how many times I point out the disctinction between sense and denotation, and even after I linked you to an Internet article about it, you keep ignoring it. — TonesInDeepFreeze
Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness. — TonesInDeepFreeze
Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models. — TonesInDeepFreeze
Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet.
— TonesInDeepFreeze
That itself is a judgement, that these unwritten equalities are equalities. Clearly equality remains a human judgement. See "equal" is a human concept. To say that there are equalities which humans haven't discovered, is to already judge them as equalities. — Metaphysician Undercover
all you did was assert that equality in mathematics is more precise than equality in other subjects. — Metaphysician Undercover
I have not said that numbers are special regarding denotation.
— TonesInDeepFreeze
This is exactly what you are saying. By insisting that "equal to" in the case of numbers means 'denotes the same object', you are saying that numbers have some special quality which can make two distinct but equal things into the same thing. You are claiming that numbers have a special status which makes equal things into the same thing. — Metaphysician Undercover
You don't have to use the word "necessarily", to mean it. — Metaphysician Undercover
When you say that being equal implies that they are the same, you refer to a logical necessity — Metaphysician Undercover
Otherwise it would be false to say 'if they are equal then they are the same'. — Metaphysician Undercover
I've already explained to you how you do not have the premise required to say that "2+1" denotes the same object as "3", when the two signify different things ("have different senses"). — Metaphysician Undercover
"The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". — Metaphysician Undercover
You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. — Metaphysician Undercover
The same thing is the case with "2+1" and "3". They signify different things (have different senses). Now, you do not have the required premise to conclude that they denote the same object. — Metaphysician Undercover
Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness.
— TonesInDeepFreeze
You mean that false premise? — Metaphysician Undercover
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