Why, A=a Is Not Valid

[Owen]

It's my opinion that:

x=x is an axiom, a belief, of modern logic.

We presume that any proper name can be a value of x.
For example; John Smith = John Smith, 2=2, Mars=Mars, etc.

But, Vulcan=Vulcan is false, and, Pegasus=Pegasus is false.
(because they do not exist)

Proper names, such as Vulcan, that do not refer are a part of our language.

The axiom x=x does not include non-referring proper names as values of x.

We implicitly presume, in logic, that all proper names refer to existent objects.
Proper names from fiction or science, that do not refer, are not values of our variables at all.

Quine: "to be is to be a value of a variable"
"no entity without identity"

Objects described by contradictory predicates do not exist either, and are not values of x in x=x.

Descriptions are values of our variables only if they do exist, only if they do refer.

Examples:
(the present king of France)=(the present king of France) is false.
(that which is not equal to itself)=(that which is not equal to itself) is false.
(the whole number between 2 and 3)=(the whole number between 2 and 3) is false.

E!x <-> x=x. x exists, if and only if, it is self identical.
(for all x, named or described)

Things that do not exist are not self identical, they have no primary predicates that are true of them at all.
We talk about them via secondary truths.
We cannot say what they are, but, we can say what they are not.
That is to say, non-referring names are not meaningless!

Another reason why the axiom x=x should be rejected is that,
necessarily(x=x), follows.
That is, there are no contingent identities and there are no contingent existences. (E!x <-> Ey(x=y))

|-. x=y -> [](x=y)
|-. E!x -> [](E!x)

The axiom (x=x) is OK for logical/mathematical objects but it fails for empirical objects.
It works in mathematical logic but it fails in philosophical logic.

(George Bush)=(George Bush), is true.
It is necessarily true that: (George Bush)=(George Bush), is false.
(George Bush)exists, is true.
It is necessarily true that: (George Bush)exists, is false.

Clearly, we need to distinguish between 'factual truth' and 'logical truth' ...they are not the same!

I have a problem with the conclusion that, necessarily everything exists, do you?

Owen

[slayer]

Owen

Leaving aside our disagreement about truth, of which you say that truth is only that which can be proved, which is counterintuitive at the very least, I want to comment on what you say here about x=x.

The axiom x=x merely states that x, whatever it is, is identical to x. But since x is a variable, x=x doesn't express a proposition until we substitute in names for both instances of 'x.' But the fact that the same name can denote different objects makes it that x=x isn't necessarily true; we need to stipulate that it is true.

You say: (George Bush)=(George Bush), is true.

slayer -- Well, I don't see why "George Bush" denotes the same George Bush in both instances. If you say that George Bush = George Bush is true, then we know that both instances of "George Bush" denote the same person.

It's also not a settled question in philosophy whether any statement containing a non-denoting name or non-denoting definite description (because the object does not exist) would therefore make the identity statement, or any statement containing the name, false. It's an open question whether it would get a truth value at all.

For example. Is the following true? The man standing in front of me (when there is obviously nobody standing in front of me) is yawning right now. Well, I think our intution would have us say that that's neither true nor false, because there is no man in front of me which to ascribe the property of yawning to. There are cases on both sides is all I'm saying.

What isn't being distinguished, which is why I think you side with Godel regarding our previous discussion, is between something existing necessarily and something necessarily existing. If it's contingently true that George Bush exists in this world, then George Bush necessarily exists in this world. That's not to say that George Bush's existence was a matter of necessity (existing necessarily).

Owen: The axiom (x=x) is OK for logical/mathematical objects but it fails for empirical objects.
It works in mathematical logic but it fails in philosophical logic.

slayer -- The reason why x=x works in mathematical logic is because the names used in substituting for 'x' only have one denotation, hence x=x will always be true. But the names we use to substitute in for 'x' in philosophical logic can denote more than one object, so we need 'is true' here.

slayer

[Owen]

slayer:
The axiom x=x merely states that x, whatever it is, is identical to x. But since x is a variable, x=x doesn't express a proposition until we substitute in names for both instances of 'x.'

The authors of Principia Mathematica, Russell and Whitehead, would disagree with your understanding.
Many propositions of PM do not substitute names for free variables, nor do they have quantifiers over them.

For example: theorem *208. p -> p, asserts, for anyproposition p: p implies p.

*10.1 |- Ax(Fx) -> Fy

"I.e. what is true in all cases is true in any one case."


We can deal with the theorems of logic without using any specific individual names at all, if we are given the quantifiers.

e.g. Ay(Ax(Fx) -> Fy), Ay(Fy -> Ex(Fx)).

The distinction between 'an arbitrary constant' and 'a variable', is not clear to me, how about you?

slayer: But the fact that the same name can denote different objects makes it that x=x isn't necessarily true; we need to stipulate that it is true.

Within the context of logic, all names refer uniquely.
(the one and only x such that x=y) = y, is a theorem, for all y.

All names refer to unique objects and all unique objects have only one name. Of course, they may have many different descriptions.

x=x for all x, is stipulated true, by axiom, it is assumed true in classical logic.

slayer: You say: (George Bush)=(George Bush), is true.
Well, I don't see why "George Bush" denotes the same George Bush in both instances. If you say that George Bush = George Bush is true, then we know that both instances of "George Bush" denote the same person.

It is intuitively understood that X has a unique reference or meaning, if X exists.

slayer: It's also not a settled question in philosophy whether any statement containing a non-denoting name or non-denoting definite description (because the object does not exist) would therefore make the identity statement, or any statement containing the name, false. It's an open question whether it would get a truth value at all.

Not for me!

I don't agree with your summary here.

The present king of France exists, is not meaningless; it has sense.
We can demonstrate why it is false.

We can say: The present king of France exists, or, The present king of France doe not exist, .. is tautologous.

The present king of France exists, is a well-formed-formula.
e.g. (!@#$%) is meaningless.

How can this be true if 'the present king of France' is meaningless.

Further, if all statements about non-referring terms are false (i.e. meaningful) then how can you say that both (the present king of France exists) is false, and, (the present king of France does not exist) is false?

slayer: For example. Is the following true? The man standing in front of me (when there is obviously nobody standing in front of me) is yawning right now. Well, I think our intution would have us say that that's neither true nor false, because there is no man in front of me which to ascribe the property of yawning to.

Not so, within Russell's theory of descriptions.

The man standing in front of me (when there is obviously nobody standing in front of me) is yawning right now, is false, because there is no man in front of me which to ascribe the property of yawning to.

(The man standing in front of me (when there is obviously nobody standing in front of me)) has no properties.

Do you want to say that all statements about non-denoting terms are meaningless or that they are false?

"There are cases on both sides is all I'm saying."

Yes, there are three ways of dealing with non-referring terms according to R. Carnap, Meaning and Necessity, page 38.

1) Hilbert-Bernays: non-denoting descriptions are meaningless in all contexts. That is, statements containing them are neither true nor false.

2) Russell: all primary statements about them are false.

3) Frege-Quine-Canap-Montague: non-referring things exist but, they are equal to: 0 (zero for Frege), {} (the null set for Quine),
a* (the null object for Carnap), that which is not equal to itself (for Montague).

I reject 1), as in this post.
2) applies to philosophical logic and mathematical logic, imo.
3) applies to only to mathematical logic.


We can incorporate both Russell and Frege-Quine methods within the same classical sytem of logic, in this way. (ix: means the x such that)

Russell: D1. G(ix:Fx) defined Ey(Ax(x=y <-> Fx) & Gy).

Frege-Quine: D2. G(ix|Fx) defined,
Ey(Ax(x=y <-> Fx) & Gy) v ~ExAx(x=y <-> Fx) & G{}.

Russell's method is the only one that makes sense when dealing with physical individuals and classes and numbers etc..


slayer: What isn't being distinguished, which is why I think you side with Godel regarding our previous discussion, is between something existing necessarily and something necessarily existing.

I agree here, if we are dealing with described objects.

[](E!(ix:Fx)) <-> ([]E!) (ix:Fx), is not valid

[](E!(ix:Fx)) ->. [](E!(ix:Fx)) <-> ([]E!) (ix:Fx), is valid.


slayer: If it's contingently true that George Bush exists in this world, then George Bush necessarily exists in this world. That's not to say that George Bush's existence was a matter of necessity (existing necessarily).


For given objects: [](E!x) <-> ([]E!)x, is true for all objects x.

Necessarily(G. W. Bush exists) <-> (G. W. Bush) necessarily exists, is true.


Owen: The axiom (x=x) is OK for logical/mathematical objects but it fails for empirical objects.
It works in mathematical logic but it fails in philosophical logic.

slayer -- The reason why x=x works in mathematical logic is because the names used in substituting for 'x' only have one denotation, hence x=x will always be true. But the names we use to substitute in for 'x' in philosophical logic can denote more than one object, so we need 'is true' here.

Not for me.
The values of our variables, in philosophical logic, are uniquely existing things, that include empirical objects.

Owen

[slayer]

Hello Owen,

Owen: The authors of Principia Mathematica, Russell and Whitehead, would disagree with your understanding.
Many propositions of PM do not substitute names for free variables, nor do they have quantifiers over them.

For example: theorem *208. p -> p, asserts, for anyproposition p: p implies p.

slayer -- I don't see where the disagreement is. 'p --> p', for any proposition (which is the key phrase) p: p implies p. This is compatible, if not directly supports, what I'm saying about x = x and p --> p, for that matter -- they don't express propositions until we substitute in for the variables.

Owen: It is intuitively understood that X has a unique reference or meaning, if X exists.

slayer -- Fine, but this doesn't address my point. I concede that X has a unique reference, but there are two of them ('X', that is). You might be saying that both instances of X are untuitively thought to refer to the same object, and I agree that this is intuitive, which is why I gave reasons for why I thought it could be otherwise. That is, why "George Bush" in both instances didn't have to refer to the same George Bush, namely because names can denote different objects (at least outside of mathematical logic).

Owen: I don't agree with your summary here.

The present king of France exists, is not meaningless; it has sense.
We can demonstrate why it is false.

slayer -- Well, I never said anything about 'the'present king of France', nor anything about all statements including a non-denoting term or description being meaningless. In fact, I only argued that sometimes it seems that in such cases the statement is false, and for different cases the statement should receive no truth value.

But now let me ask you: How would you demonstrate that 'the present king of France exists' is false without begging the question?

Owen: Do you want to say that all statements about non-denoting terms are meaningless or that they are false?

slayer -- Again, I never contended this. And, yes, Russell had a way of making such statements that included non-denoting terms 'false.' He manages it by assigning primary scope such that it becomes a question of existence, not a question about having the property of yawning,which would get secondary scope. But I never contended that Russell didn't have a way of doing this. There are other scenarios though that Russell didn't envision which he had no way of settling. Furthermore, Russell's apparatus allows us to set up the statements that way, but the apparatus doesn't have the power to determine which way the statement should actually be taken. And the way they should be taken is what is under contention. So Russell can't help you here.

Owen: Not for me.
The values of our variables, in philosophical logic, are uniquely existing things, that include empirical objects.

slayer -- I thought variables were place-holders only? Specifically, they're not names or denote anything. When did this change in logic?

sincerely,

slayer

[Owen]

Hello Owen,

Owen: The authors of Principia Mathematica, Russell and Whitehead, would disagree with your understanding.
Many propositions of PM do not substitute names for free variables, nor do they have quantifiers over them.

For example: theorem *208. p -> p, asserts, for anyproposition p: p implies p.

slayer -- I don't see where the disagreement is. 'p --> p', for any proposition (which is the key phrase) p: p implies p. This is compatible, if not directly supports, what I'm saying about x = x and p --> p, for that matter --

"..they don't express propositions until we substitute in for the variables."

Hi slayer,

Our dissagreement is that some writers do not use substitution instances (names of particular individuals) in their expression of logical propositions (theorems).

For example: Both Russell and Carnap assert the propositions
1. x=x, 2. Fy -> ExFx.

Free x in 1, and, free y in 2, act like pronouns in natural languages.
No substitution of names for the free variables is used or needed.


Owen: It is intuitively understood that X has a unique reference or meaning, if X exists.

slayer -- Fine, but this doesn't address my point. I concede that X has a unique reference, but there are two of them ('X', that is). You might be saying that both instances of X are untuitively thought to refer to the same object, and I agree that this is intuitive, which is why I gave reasons for why I thought it could be otherwise. That is, why "George Bush" in both instances didn't have to refer to the same George Bush, namely because names can denote different objects (at least outside of mathematical logic).

Objects with no, or more than one representation (name) are not unique.
Logic only deals with unique names.
Proper names always name the same object.

But now let me ask you: How would you demonstrate that 'the present king of France exists' is false without begging the question?

It is granted that: there is no king of France at the present time.
Therefore there cannot be a unique king of France either!

(There is no present king of France. & The present king of France is unique.) is a contradiction.

~Ex(x is the present king of France) & Ey(Ax(x=y <-> x is the present king of France) is contradictory.

Owen

[slayer]

Hey Owen

Sorry, I asked about how you would show that 'the present king of France exists' is false without begging the question, when I intended to ask how you'd do that for 'the present king of France is bald."

And it is with this example, and similar other ones, which the question of scope determines what we're answering. And nothing about Russell's system allows us to choose which is the correct interpretation, but instead his system allows us to interpret it such that we can give it (the itnerpretation we've chosen) a truth value of false. Of course, like I suggested, to say that this particular interpretation is the correct one would be to beg the question.

I'm the present king of France, by the way,

slayer

[Owen]

And it is with this example, and similar other ones, which the question of scope determines what we're answering. And nothing about Russell's system allows us to choose which is the correct interpretation, but instead his system allows us to interpret it such that we can give it (the itnerpretation we've chosen) a truth value of false. Of course, like I suggested, to say that this particular interpretation is the correct one would be to beg the question.

I'm the present king of France, by the way,

slayer
----------------------------------------------------

No-one can be the present king of France!
It does not exist.

With or without 'scope'...

The present king of France is bald, is defined:

Pemise 1. Ey(Ax(x=y <-> x is a present king of France) & y is bald).

Premise 2. ~Ex(x is a present king of France)

3. Ey(Ax(x=y <-> (x is a present king of France & ~Ex(x is a present king of France)) & y is bald).

(x is a present king of France & ~Ex(x is a present king of France)), is contradictory for all x.

Therefore 3 becomes..

4. Ey(Ax(x=y <-> contradiction) & y is bald)

5. Ey(Ax~(x=y) & y is bald),

by, (x=y <-> contradiction) <-> ~(x=y).

6. Ey(~Ex(x=y) & y is bald)

But, Ex(x=y) is a theorem. See: *13.15 of Principia.

Therefore ~Ex(x=y) is a contradiction.

7. EyAx(contradiction & x is bald).

8. (contradiction & x is bald) <-> contradiction.

9. EyAx(contradiction) <-> contradiction.

therefore,

10. Ey(Ax(x=y <-> x is a king of France) & y is bald) is logically false!

~Ey(Ax(x=y <-> x is a king of France) & y is bald), is logically true.

There is no property that the present king of France has, including the property of Existence or of self-identity.

Owen

[a random hack]

hey owen :)
could you do one of those logic thingies for 'slayer is an idiot'?
please?:D

[sahyo]

:lol:

[sonrisa]

first he claims to be Superman, now he's claiming to be the present king of France? Random, the po boy sounds delusional to me! :goodlaugh:

[Owen]

hey owen
could you do one of those logic thingies for 'slayer is an idiot'?
please?

--------------------
perhaps... :)
========


Hi random hack,

Yes, I think so, :) .. if he persists in claiming that he is the present king of France after reading the logic thingy above.


Owen

[a random hack]

:lol:

[slayer]

Here's something you can't show is true using your version of logic, Owen.

The claim that truth is that which corresponds to reality. The reason why you can't do it, is because you're confused about 'Truth' and 'true in this system of logic.'

You've already explained how "God exists" could be true in one system, depending on what you assign to R, and how "God does not exist" could be true in another system, again, depending on what you assing to R.

Which system then is supposed to describe the world accurately? By 'accurately' I only mean 'as the world is.' The fact of the matter is that you have no way of determining which system accurately describes the world. And it's not because nobody knows whether God does or does not exist, but it's that you don't subscribe to a correspondence theory of truth, because you think Truth is synonymous with 'true in this system of logic' or 'that which can be proven.' But Truth is determined with respect to reality, not with respect to logic.

You may know the machinery of logic, but you're philosophically very inept.

still the present king of France,

slayer

PS. When you realize that I can claim that I'm the present king of France and yet you can't show -- with your definition of Truth -- that that's not how the world is, then you'll understand something about philosophy.

[sonrisa]

and still delusional :shakehead:
sad really....

Random, click here & check out these lyrics (http://www.lyricsmp3.net/D19000/kk18749.htm)


these too! (http://www.lyricsmp3.net/D19000/kk18751.htm)


the present king of france? :D

[Owen]

slayer: still the present king of France,

Apparently you do not understand the statement "There is no present king of France".

Your arrogance and insulting manner is not interesting to me.

Evidently the gang here is correct about you!

I give up! Have a nice day.

Owen

[slayer]

Owen,

Let me first apologize for my tone to you previously, it was unjustified.

Second, let me articulate the point I was trying to make -- especially with "I'm the present king of France."

Now, your comment about me not understanding is very wrong, and hence it too sounds very arrogant. You'll come to realize after this post that I very much understood the statement 'there is no present king of France.' I'm familiar with its philosophical beginnings and with the fact that France currently does not have a king; that is, it's not a monarchy. And hopefully you might also realize the consequences of your position that the Truth is only that which can be proven (I think I've paraphrased you fairly).

I claimed that I was the present king of France in order to make a point. And the point is this. Let's say 'R' means 'slayer is the present king of France.' You would also give another symbolic sentence the meaning that 'there is no present king of France.' Fine, but after you logically showed that R was false, then what could you say about the world? If you say that you've shown that the world isn't as I say it is, when claiming to be the present king of France, then I would respond that your proof didn't demonstrate anything of the kind, because you've arbitrarily assigned a truth value to 'there is no present king of France.' If it's not arbitrary, then you do in fact, counter to what you claim, subscribe to a correspondence theory of truth. That is, the truth (or falsehood) of 'there is no present king of France' is determined by the facts -- by how the world is.

This was the point of my claim that I'm the present king of France.

No, my insulting manner is not interesting, but neither is your casual dismissal of my previous post, which was philosophically interesting.

the present king of Arrogance,

slayer

[sonrisa]

the present king of Arrogance,


not delusional today!!

:P

[rich]

Originally posted by sonrisa@Apr 21 2004, 01:56 AM
the present king of Arrogance,


not delusional today!!

:P
the present king of Arrogance,

sonrisa posted:

not delusional today!!

:P

QUOTE]Not the King, Just an essense of Arrogance.

[sonrisa]

essence of Arrogance-- is that something like Eau de Arrogance?

B)