In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. — aletheist

That's correct in an extensional context, but not in an intensional context:

Suppose Alice doesn't know that Henry Fonda is the father of Peter Fonda. Then consider these two sentences:

(1) Alice knows that Henry Fonda is Henry Fonda.

(2) Alice knows that Henry Fonda is the father of Peter Fonda.

But (1) is true (since Alice knows that identity is reflexive), while (2) is false even though (2) just substitutes 'the father of Peter Fonda' for 'Henry Fonda'.

'knows that' creates an intensional context in which substitution may not preserve truth values.

However, we can point at a collection of three apples and say both "that is 2+1 apples" and "that is 3 apples." — aletheist

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". If you were teaching children you would not show them a group of three apples and tell them this is 2+1.

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

We might say that expressions which signify equal value can be substituted, within that value system . We cannot conclude that because the expressions can be substituted, they signify the same thing. They are only "the same" in relation to that value assigned to them.

If I need assistance, and Tom, Dick, or Harry, will do, each having equal value for the task, I ask for Tom, Dick, or Harry, as they are interchangeable in relation to this value. They each make "the task will be done" true. This does not mean that each of them is the same thing as each other.

You're lying about me again:

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

(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.

Stop lying about me.

Again on this point:

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

Again, this is conflating denotation is evidence, proof, demonstration and knowledge.

We conclude that two different names denote the same object by proving they do.

'2+1' and '6-3' have the same denotation. Of course, to legitimately assert that '2+1' and '6-3' requires first having grounds for the assertion, such as mathematical proof of the equation:

2+1 = 6-3.

That's what mathematicians do; they prove formulas, including equations. When the equation is proven, then we are justified in claiming that '2+1' and '6-3' denote the same object.

That's correct in an extensional context, but not in an intensional context: — TonesInDeepFreeze

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.

The "Fonda" example was provided as an argument for the truth of it. — Metaphysician Undercover

The example was given not so much as an argument but as an illustration for you to understand a basic idea.

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.

As I mentioned before, this is the case both by ordinary mathematical practice and as made rigorous in mathematical logic. That's just a fact about certain conventions in ordinary mathematics. Whether ordinary mathematics should use that convention is a separate issue, of which I have not taken a position except to point out that alternatives are complicated.

A natural language example, such as the Fonda example, doesn't prove anything about mathematics, but, as I mentions, it illustrates the general principle.

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 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."

If you were teaching children you would not show them a group of three apples and tell them this is 2+1. — Metaphysician Undercover

I actually might do exactly that, if I were teaching them basic addition such as 2+1=3.

We cannot conclude that because the expressions can be substituted, they signify the same thing. — Metaphysician Undercover

Of course not, but we can conclude (in an extensional context) that they denote the same thing.

Ordinary mathematics regards '2+1' and '3' as having the same denotation — TonesInDeepFreeze

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.

I can see Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda." — fishfry

What he doesn't understand is that denotation is only one part of meaning. There is both denotation, which is extensional, and sense. 'Henry Fonda' and 'the father of Peter Fonda' denote the same thing. But indeed the they have different senses.

Ordinary mathematics deals with terms extensionally. For mathematics to deal with terms also with regard to sense requires a more complicated linguistic/logical system. As I mentioned there are proposals for such systems, but they are beyond ordinary mathematics.

"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

You have it backwards again. Mathematics does not prove that objects are equal by showing they share all properties. Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms.

Also, mathematical theories are in mathematical languages and regard models with objects and their relevant properties for particular areas of mathematical interest. In mathematics, one is free to state other languages, theories and models in which other properties are relevant besides those in some previous treatment.

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

That is ridiculously captious and sophomoric. It is deserves all three tropes: red herring, blowing smoke, and grasping at straws.

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.

Moreover, of course, in this context, we are assuming that there has been ample evidence that Henry Fonda is the father of Peter Fonda, and that we can agree on that for the purpose of the illustration regarding denotation of the names.

Again for about the fifth time: When I claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person, I don't claim that I have shown that Henry Fonda is the father of Peter Fonda. That's not the point, and only someone pretty obtuse would miss this. To show that Henry Fonda is the father of Peter Fonda is a matter of empirical inquiry. Of course in this context I assume that we take it for granted that we know empirically (or by whatever means of such common knowledge) that Henry Fonda is the father of Peter Fonda, and on that basis, we make the linguistic observation that the names 'Henry Fonda' and 'the father of Peter Fonda' denote the same person.

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.

For about the dozenth time, get it straight:

First we find out that 2+1 = 3, and then we may justifiably claim that '2+1' and '3' denote the same number.

/

The philosophy of language does take on issues with problematic, indeterminate, equivocal, conflicting, temporally complicated, and paradoxical denotation. But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number.

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

This is at the heart of it.

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!

You blame others for fallacies in arguments they never made! (I.e., variation on straw man.)

(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. False premises produce unsound conclusions, which do not prove anything. 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. The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing.

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

Clearly you, (and extensionality in general) have this backward. If we start with the law of identity, "a thing is the same as itself", as a fundamental premise, and we compare this with equality, which is a property that we assign to things, you ought to see this. 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. We may not be able to account for all the potential differences between them, and thereby over look some, making a faulty judgement of "same".

So, we can truthfully say that a thing will be judged to be equal to itself, but we cannot truthfully say that things which are judged to be equal are the same thing. Therefore, when you say that T=S, you say that T has the same value as S within that system of judgement, and this means that the symbols have the same meaning within that system, but it does not mean that they denote the same object.

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

This precisely, is the false premise. Being equal is a human judgement, and being equal does not imply being the same. We can proceed the other way, and say that being the same implies being equal, but we cannot proceed from being equal to an implication of being the same. This is because two distinct things can be judged as equal, when they are not the same. Therefore proving that two things are equal does not imply that they are the same (share all properties). It only implies that they share the properties by which they are judged to be equal. And there is your false premise.

This is a basic fact of the way that we use signs and symbols. We use the symbol "2" here, and we use it later in some other application. These are two distinct instances of that symbol, they are not the same thing. However, they have an equality in what they signify. Each distinct instance of using that similar symbol signifies "the same value". This means that the two instances have the same meaning. It does not mean that they denote the same object.

So, proving that two distinct yet similar instances of a symbol "2", have an equal value, only proves that they have the same meaning. It does not prove that they denote the same object.

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

This is counterproductive. 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. Therefore by adopting such a position we proceed toward a misunderstanding rather than an understanding.

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

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. You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. Obviously, there are no such objects, the numerals have meaning dependent on the context of usage, just like any other symbols. They do not denote any objects, and your so-called understanding is actually a misunderstanding.

But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number. — TonesInDeepFreeze

Claiming a denotation when there is only meaning, is a false premise.

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

Do you understand the fallacy of "begging the question", assuming the conclusion?

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

To be clear, what is at issue is whether there is a denotation at all (when denote is defined as you do). Read the above.

The "+" is not irrelevant, it must be accounted for in your interpretation. You cannot simply leave words out of a phrase, in your interpretation, to make it say what you want it to say, or denote what you want it to. I really do not see any logic to your claim that two expressions can have distinct significations, yet denote the same object. I can see how "I did X", and "I did Y", both refer to the same object with "I", but each signify something different. Since each expression signifies something completely different, if we replace what is signified with "denoting an object" as you seem inclined to, then we do not come up with the same object. How do you come up with this idea that two phrases which signify something completely different actually denote the same object. I would call that contradiction.

As I stated clearly in the last post. A group of three apples does not truthfully represent "2+1". If that's not obvious to you, go back to grade school and find out how they represent "2+1".

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

Actually it has become very clear now, that you and Tones are the ones confusing denotation and signification. Clearly, in our use of mathematics there is signification without denotation. You and Tones are seeing an object denoted by "2+1", when there is none. That is misinterpretation.

(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

Whatever views you have about the distinction between extension and intension, and between denotation and sense, I gave you more than "vague reference" about them.

extensionality provides a false premise — Metaphysician Undercover

You may think you've shown a false premise, but you haven't.

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

I never said that we infer that distinct things are equal, let alone that they are equal due to having the same properties. You're resorting to strawman again.

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

Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again.

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.

proving that two things are equal does not imply that they are the same (share all properties) — Metaphysician Undercover

No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics.

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

Over and over you swing this "false premise" charge like a crudely made cudgel. It's mere assertion.

You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion.

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

It was my point that the Fonda example involves first making an empirical determination. On the other hand, mathematics is deductive and axiomatic.

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

Of course, if numbers are not at least abstract objects, then they cannot be referred to as objects. Then '2' has no denotation to a number. Go ahead and formulate your mathematics that way if you like, but you offer no formulation or even hint of one. And, I also mentioned that if we confine our attention to mathematics at the pure formula level, then object and denotation don't even need to be mentioned.

Claiming a denotation when there is only meaning, — Metaphysician Undercover

Denotation is part of the meaning.

Do you understand the fallacy of "begging the question", assuming the conclusion? — Metaphysician Undercover

I understand it better than you. And you've not shown I am question begging. Moreover, I showed exactly how I am not question begging, and you skipped that. This is like your claim that I was inconsistent - you never showed an inconsistency.

you [altheist] and Tones are the ones confusing denotation and signification. — Metaphysician Undercover

I never used the term 'signification'.

blank post

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

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.

How do you come up with this idea that two phrases which signify something completely different actually denote the same object. — 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.

Clearly, in our use of mathematics there is signification without denotation. — Metaphysician Undercover

As someone once said ...

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

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.

And you said:

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. Let me put it simply, proving that they share one property, "are equal" does not prove that they share all properties. You need another premise, which states that equal things are the same thing. But we know that premise is false because we see all sorts of equal things (equal volume, equal weight, etc.) which do not make two things the same. Equality is not sufficient for a judgement of same.

We don't judge two things are equal. — TonesInDeepFreeze

You very clearly stated "having proved that they are equal". Therefore you do judge that they are equal, that's what proving is, providing the justification for judgement.

No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics. — TonesInDeepFreeze

The indiscernibility of identicals does not provide the principle required for substituting equal things. Things are judged to be equal not on the basis that they are indiscernible. Clearly, as altheist agrees, and what ought to be obvious to you, what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. Therefore it is false to claim that the principle of the indiscernibility of identicals supports such a substitution. It does not.

You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion. — TonesInDeepFreeze

Clearly mathematics has an extremely vague notion of identity, one not consistent with the law of identity, allowing that similar things which are judged to be equal may be substituted as if they are the same thing.

Good, and they also both denote the same object with "did," which is the relation of doing. — aletheist

I really can't see how a relation is an object. I think you are making things up as you go.

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

This explains nothing. Words like "did" signify something. But you insist instead, that they denote an object. But you also allow that they signify things as well, and denote objects at the same time. 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.

You mangle nearly everything.

(1) Claiming I've said things when I did not say them.

(2) Screwing up the direction of my explanation so that your representation of my explanation is not my explanation.

(3) Reversing the direction of conditionals.

(4) Ignore explanations and decisive points and instead keep repeating yourself past them.

(5) Ignore distinctions explicitly stated.

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

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.

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

I said to stop claiming I've said things I did not say.

We don't judge two things are equal. — TonesInDeepFreeze

And now taking me out of context. Here is the context:

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

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.

You very clearly stated "having proved that they are equal". — Metaphysician Undercover

You're totally mixed up as to the order of the statements in my explanation. Again:

First we determine (by proof or whatever method) that 2+1 is 3. From that determination we are justified in claiming that '2+1' and '3' refer to the same number.

Don't screw up the sequence of statements in my explanation to thus mangle it.

The indiscernibility of identicals does not provide the principle required for substituting equal things. — Metaphysician Undercover

Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle.

Things are judged to be equal not on the basis that they are indiscernible. — Metaphysician Undercover

Again, you got it backwards! For the third time I've told you, I did not claim to infer equality from indiscernibility. I claimed to infer indiscernibility from equality.

What I said:

equality -> indiscernibility.

But you keep saying that I say:

indiscernibility -> equality

even after I've told you that is not what I say.

Don't reverse the direction of my conditionals.

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

There is both denotation and sense. Substitutivity holds as rule only for denotation. I was the one who gave the early example in this thread where substitutivity fails when the context is not extensional.

Meanwhile, if you don't recognize the use of substitutivity in even just basic math, then you can't do math.

I really can't see how a relation is an object. — Metaphysician Undercover

You can't see because you ignore mathematics. A relation is set of tuples. That set is an object.

You are ignorant of the view in which meaning has at least two components: denotation and sense.

Denotation is only part of the meaning of a term.

In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning. And, as I've pointed out a few times already, if you wish to have mathematics that concerns also sense, or the intensional aspect, then you are welcome to formulate such mathematics or to look up formulations that have been given by other mathematicians and philosophers of mathematics.

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 think you are making things up as you go. — 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.

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

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.

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.

(3) Substitutivity holds in extensional contexts, and it may fail in intensional contexts.

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

Equality is insufficient for a judgement of "same". That's very simple, clear, and obviously true, from all the instances where equal things are not the same thing. I'll expound on this below, but you ought to respect this principle instead of trying to deny it, and insist that equal things are necessarily the same thing.

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

If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds.

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.

First we determine (by proof or whatever method) that 2+1 is 3. — TonesInDeepFreeze

Obviously this is false, 2+1 is equal to 3 but it is not the same as.

Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle. — TonesInDeepFreeze

I explained why the indiscernibility of identicals does not support your assertion. Obviously 2+1 is not indiscernible from 3. Therefore you cannot use the indiscernibility of identicals to support your claim that they are identical.

What I said:

equality -> indiscernibility. — TonesInDeepFreeze

This is false because what "2+1" signifies is very clearly discernible from what "3" signifies. There are two numbers denoted, 2 and 1, while "3" only denotes one number. We've been through this countless times already and you are in denial of the truth. Admit the fact, 2+1 is not indiscernible from 3.

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. So I assumed that this is what you meant. The other way, the way you claim to be using it, would work if it were true, but it is clearly a false premise. Equal is not sufficient for indiscernibility. That's obvious from all the cases of equal things which are discernible.

Don't reverse the direction of my conditionals. — TonesInDeepFreeze

I assumed you were trying to make a sound argument. However, you've now corrected my to show that you are simply using a false premise. You admit now that your premise is that if things are equal they are the same. Therefore I'll take you back to what I asked days ago. Are you and I the same because we are equal? You have no special pleading now, for a special sense of "equal", which is supported by "indiscernible", because you've just admitted that you support "indiscernible" with "equal". By turning this around you have no special definition for "equal".

And to define that sense of "equality" with "value" doesn't help you because all senses of "equality" rely on a judgement of value. Quantitative value is no more special than moral value as an indicator as to whether or not two things are the same. The value which we assign to a thing is not a thing's identity.

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. Therefore you ignore the other aspects, which are clearly different from each other, and you proceed to claim that what they represent is identical. What this really means is that they are the same in some aspect, but not in every aspect, so it is false to claim that they are the same.

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

Sorry aletheist, but I must inform you that technical definitions narrow down a word's definition. That is because a broader definition allows for ambiguity. So your attempt to broaden the definition of "object", is not at all an attempt at a technical definition. It's an attempt to create ambiguity, which might be useful for the creation of vagueness and equivocation. So I see your definition as completely misguided because it's not conducive for understanding.

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

I have, and I do not agree with the fundamental principles of that proposed field of study. It appears to be lost in ambiguity and category mistake. This opinion which I have, you might be able to detect. I am not one to dismiss things off hand, without some understanding of the fundamental principles.

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

This appears to involve a fallacy of composition. And I think this is why your way of looking at "2+1" appears so incorrect to me. You say that "2+1" signifies something which is other than what "3" signifies, yet "2+1" denotes the same object as "3". You make this conclusion of denoting the same object through a fallacy of composition, concluding that the attributes of the parts within the statement "2+1" can be summed up into a collection, to make an object with the exact same attributes as 3.

So I see your definition as completely misguided because it's not conducive for understanding. — Metaphysician Undercover

It is not "my" definition, it is the well-established definition within the discipline of semeiotic.

I do not agree with the fundamental principles of that proposed field of study. — Metaphysician Undercover

Then we can stop wasting each other's time.

I am not one to dismiss things off hand, without some understanding of the fundamental principles. — 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.

This appears to involve a fallacy of composition. — 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.

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

This is another instance of imposing your view as if it entails something I said that I did not say. 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.

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've mixed up two different issues, and got me wrong on both of them.

Issue 1: You claimed it is question begging to say that '2+1' and '3' denote the same object.

Of course if 2+1 and are equal 3 then they are the same. But what you claimed was that I was begging the question by saying that '2+1' and '3' have the same denotation. And I explained that it is not question begging since we infer that '2+1' and '3' have the same denotation from first determining that 2+1 equals 3.

Issue 2: You claimed that I said that indiscernibility implies identity.

But I did not. I said that identity implies indiscernibility.

There are three principles of identity/indiscernibility:

(1) If identical then indiscernible.

(2) If Indiscernible then identical.

And we may combine for:

(3) Identical if and only if indiscernible.

All I said is (1).

And again, not question begging.

you cannot use the indiscernibility of identicals to support your claim that they are identical. — Metaphysician Undercover

You have a cognitive problem that prevents you from discussing this without getting grievously mixed up about it. You keep reversing the direction of implication.

Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility.

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

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.

Are you and I the same because we are equal? — Metaphysician Undercover

I answered that many posts ago! Again you argue by just skipping past many key points in the replies to you.

you have no special definition for "equal". — Metaphysician Undercover

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.

to define that sense of "equality" with "value" — Metaphysician Undercover

And I don't define any sense of 'equality' with 'value'.

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

Not so much what I concern myself with personally, but rather what ordinary mathematics concerns itself with.

I mentioned the extensional vs intensional distinction many many posts ago. 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. Again, yes, ordinary mathematics is extensional and concerns only the denotation part and not the sense part. I referred you to the Stanford philosophy encyclopedia article that discusses this. I said many posts ago that you may also consider formulations in which intensionality is considered.

I am not one to dismiss things off hand, without some understanding of the fundamental principles. — Metaphysician Undercover

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.

But since what I am looking for is an indication that 2+1 really is the same thing as 3, — Metaphysician Undercover

I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here.

what I am looking for is an indication that 2+1 really is the same thing as 3 — Metaphysician Undercover

'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'.

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

I asked you for an instance of equality which is not a human judgement. You didn't give me one. That's probably because you understand that such a thing is ridiculous.

Of course if 2+1 and are equal 3 then they are the same. — TonesInDeepFreeze

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. Therefore we cannot conclude that if two things are equal they are the same. You seem to think that numbers are somehow special, so that if they are equal they are necessarily the same. I'm waiting for you to attempt to justify this belief, which to my understanding is demonstrably false.

Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility. — TonesInDeepFreeze

Right, so the law of identity states that a thing is identical to itself. That's identity. Then we can proceed to say that a thing is indiscernible from itself, and this is consistent. Now, 2+1 is discernible from 3, so how do we conclude that they are the same?

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

Then how in hell are you supporting this obviously false assumption that "if 2+1 and are equal 3 then they are the same"?

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

The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. Therefore your "undefined" primitive is a violation of the law of identity. If the right and left side of the equation signified the exact same thing, as required by the law of identity and if equal signifies identical, then all equations would read like "X=X", or "Y=Y", or some other way of saying that the very same thing is represent on the right and the left. However, mathematicians use "=" to relate two distinct expressions with distinct meanings, which clearly do not signify the exact same thing.

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

This is the contradiction which altheist was trying to impose on me. If "2+1" signifies something different from "3", then it is impossible that what they denote is the same object by way of contradiction. If they are supposed to be signifying different predications, so contradiction is avoided, then no object is denoted, just two distinct predications without a subject, predicated of nothing.

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

That's nowhere near as bad as someone who routinely applies mathematics without recognizing the falsity of fundamental principles. I cannot understand the fundamentals because they are unsound. Contradiction or falsity make understanding impossible. But accepting contradiction, or falsity and proceeding to apply these principles is self-deception and misunderstanding.

I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here. — fishfry

Talk about begging the question. That's what your so-called proof did.

'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

All I see is "=" here. Where's the proof that "=" means the same as?

I asked you for an instance of equality which is not a human judgement. You didn't give me one. — Metaphysician Undercover

I don't recall you asking me such a question. If you did, then please link to the post where you asked it so that I can see the context. Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet. And I don't see any relevance to what I've said about equality.

Meanwhile, there are many decisive points I have raised that you have skipped.

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

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!

You seem to think that numbers are somehow special — Metaphysician Undercover

I have not said that numbers are special regarding denotation.

necessarily the same — Metaphysician Undercover

I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic.

we can proceed to say that a thing is indiscernible from itself — Metaphysician Undercover

Yes we can, but that alone is not the principle of the indiscernibility of identicals.

2+1 is discernible from 3 — Metaphysician Undercover

'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.

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

Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness.

The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. — Metaphysician Undercover

You've not shown any inconsistency.

signified — Metaphysician Undercover

If I recall, I have not used the word 'signify'. Again, a term has both a denotation and a sense.

mathematicians use "=" to relate two distinct expressions with distinct meanings — Metaphysician Undercover

Yes, since meaning includes both denotation and sense. I explained to you probably more than half a dozen times already that ordinary mathematics concerns itself only with denotation and that if you want to have sense handled also, then you need a more complicated framework.

I cannot understand the fundamentals because they are unsound. — Metaphysician Undercover

You've never even read page 1 in a book on the foundations of mathematics. So of course you can't understand anything about it.

All I see is "=" here. Where's the proof that "=" means the same as? — Metaphysician Undercover

Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models.

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.

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

And I will continue to skip it because all you did was assert that equality in mathematics is more precise than equality in other subjects. The point being that in no subject does "equal to" mean "the same as", not even mathematics. As I explained the left side does not signify the same thing as the right. And your assertion of the precision of mathematics still doesn't get you to the point of being the same. Equal to, and the same as, are distinct conceptions.

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.

.

I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic. — TonesInDeepFreeze

You don't have to use the word "necessarily", to mean it. When you say that being equal implies that they are the same, you refer to a logical necessity which dictates that if they are equal then they are necessarily the same. Otherwise it would be false to say 'if they are equal then they are the same'.
Which of course, is obviously false to say, because that necessity is based in a false premise.

'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

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").

Remember your example? "The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". "Henry Fonda" also denotes a particular person. Again, that is the "sense". Now, you do not have the premise required to validly conclude that these two persons, indicated by those two senses, are the same person. 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. You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. Likewise, you can stipulate that "2+1" denotes the same object as "3", but that's simply begging the question. You are creating the premise required to support your desired conclusion, and that's a fallacy.

So no matter how many times you assert that despite the fact that "2+1" and "3" mean something different, they denote the same object, you have not produced a valid argument to prove this. All you've produced is the false premise that if they are equal then they are the same.

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?

Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models. — TonesInDeepFreeze

Shouldn't we call this what it really is, the method of the false premise?

Over many posts, you keep telling me what I think or said, and you're wrong. You're a bane.

And you claimed that you asked me a question I didn't answer. I then asked you to link me to that, because I don't recall you asking me a question I did not answer recently. Please link me to it.

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

What you just said, whatever its merit, doesn't vitiate anything I've said.

all you did was assert that equality in mathematics is more precise than equality in other subjects. — Metaphysician Undercover

Formal languages, including the language of identity theory, are more precise than natural languages. But the point I made was not so much about precision but that 'equality of human beings' in the sense of equal rights or whatever is a very different meaning of 'equality' in mathematics.

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

Wrong. I'm not saying any of that.

You don't have to use the word "necessarily", to mean it. — Metaphysician Undercover

True, but I don't mean it.

When you say that being equal implies that they are the same, you refer to a logical necessity — Metaphysician Undercover

Nope. I am not bringing the notion of logical necessity into play.

Otherwise it would be false to say 'if they are equal then they are the same'. — Metaphysician Undercover

That's a non sequitur.

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

You don't explain. You assert and then argue fallaciously.

"The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". — Metaphysician Undercover

You said it both denotes and is its sense. Denotation and sense are different.

You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. — Metaphysician Undercover

One can stipulate premises and then infer conclusions. That is not question begging. Also, we don't have to stipulate that Henry Fonda is the father of Peter Fonda, since we can arrive at that claim by empirical or historical evidence.

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

We prove that 2+1 = 3. Then we prove that '2+1' and '3' have the same denotation by the method of models. I've told you that about a half dozen times now.

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

You've not shown any false premise in the method of models.

Now, please link me to the post in which you claim you asked me a question I did not answer.