Why, A=A is not valid.

[Witt]

A=A or x=x is an axiom, a belief, of 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, where Vulcan is the planet (hypothesized by astronomers) in our solar system that is needed to explain via Newtonian physics, the unusual orbit of Mercury.

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 or math, 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, ie. 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.

Where 'E!' is the existence predicate:
E!x <-> x=x. That is, 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 truths at all.
We talk about them via secondary truths, we can say what they are not.
Things that do not exist, are not meaningless, i.e. it makes sense to say, the existent present king of France does not exist.


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 exists, if and only if, there is some existent object that x is.

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

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

George Bush exists, is true.
Necessarily George Bush exists, is false.

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

Is a revision of 'Identity theory' in order, I think so, do you?

Witt

[xorbie]

I am not sure what exactly this means for philosophy. Any logical proof still relies on the x=x principle, in fact all of logic relies on it. You can't really have anything without that.

I do not follow you when you say that if x=x is allowable, then necessarily everything exists. It just means that anything that exists is equal to itself. It is important to note that "=" is a function in itself that takes two parameters. The function does not take parameters that don't exist, or that are faulty.

]We presume that any proper name can be a value of x.[/B]

I don't presume that at all. It depends on what you define "=" as. If you define it as "having the same number of letters" than I would not let x=2. If you define x as numerically equal, I would not let any string in.

Perhaps I am looking at this too much from a programmers pov (though I am not one), but this does not seemt o be a problem to me.

[Witt]

xorbie:
I am not sure what exactly this means for philosophy.

What we count as existent or non-existent, does indeed, set limitations to our philosophies.

xorbie: Any logical proof still relies on the x=x principle, in fact all of logic relies on it. You can't really have anything without that.

Wrong and double wrong!
Much of logic is independent of the identity predicate.

For example: propositional logic, syllogistic logic, Boolean logic, predicate logic ...

Description theory, set theory, etc. (mathematics) does require the identity predicate.

Why do I need identity theory to prove that '~(p & ~p)' ?


xorbie:
I do not follow you when you say that if x=x is allowable, then necessarily everything exists.

Leibnitz's Law states: x=y -> (Fx <-> Fy).

x=y -> (nec(x=x) <-> nec(x=y)),

but, nec(x=x) is true ..if we allow x=x.

Therefore: x=y -> nec(x=y), is a theorem. (for all things x and y)

xorbie: It just means that anything that exists is equal to itself. It is important to note that "=" is a function in itself that takes two parameters. The function does not take parameters that don't exist, or that are faulty.

That, is the question.
If we presume that all values of the variable x exist, then clearly x=x.

That is to say x=x means: for all existent x's, x=x. !

e.g. (the present king of France)=(the present king of France), does not apply.


We presume that any proper name can be a value of x.

xorbie: I don't presume that at all.
It depends on what you define "=" as.

Yes, if we define identity as in Russell (1910 PM):
x=y defined AF(Fx <-> Fy), then we must grant that x=x.

But, we do not need that definition when empirical values apply.

I submit: x=y defined E!x & E!y & AF(Fx <-> Fy).

In which case, x=x is not valid.
That is to say, x=x <-> E!x.

xorbie:
If you define it as "having the same number of letters" than I would not let x=2. If you define x as numerically equal, I would not let any string in.

Identity is defined as having the same predications.

Identity is not defined in first order predicate logic, it is primitive there.

Witt

[xorbie]

I suppose... but I just don't know.

[paul30]

I agree with Witt.

Except for math (which is really just a system of tautologies), there are hardly any areas in our life in which two things are precisely alike.

Even subatomic particles are probably not EXACTLY alike, although this would be hard to prove.

Names are ways of assimilating really dissimilar things, and lead to all kinds of falsehoods.

Herclitos said, "We never step into the same river twice."

I think this is correct. A profounder observation is that of the Buddhists that "I"--what I think of as my "self"--do (or does) not exist. I am simply an emptiness with a multiplicity of relations.

The same is true of all "things" that we see. Our seeing is an arbitrary sorting of what is basically an undifferentiated continuum.

[mhc]

Except for math (which is really just a system of tautologies), there are hardly any areas in our life in which two things are precisely alike.
But A=A is NOT two things. There is only one "thing".
A=A means A is identical to itself. It means "This pen is the same thing as this pen" (said while holding up a single pen). Tautological, no doubt, but as true as true can get.

And Witt--
How can you say that Vulcan IS a hypothetical planet, and then say that "Vulcan" does "not refer". You just referenced it!

[Theli]

I've always thought of A=A as an hypothesis, for us to accept the conclution A we must first accept the premise A.
Does it really say that the conclution is true, or does it say that A is true under the condition placed by the premise?
Wouldn't a non-hypothetical tautology look like this:
P1: A
P2: A=A
C: A

As far as A=A is concerned it says nothing about reality, just like 5+5=10 says nothing about reality either.

[Witt]

paul30:
Except for math (which is really just a system of tautologies), there are hardly any areas in our life in which two things are precisely alike.

I would say there are no areas in our lives in which two different things are equal.
That they are different demonstrates that they are not equal.
Identity entails that all properties of one apply equally to the other.

But, in language, we often do grant two different names to refer to the same object.
For example: Cicero=Tully, Everest=Chomolungma, Morning star = Evening star, the current president of the US = G. W. Bush, etc.

Even in mathematics, eg. (1+1)=2, each term refers to the same object.

What is equal is what these names or descriptions denote, if they do denote.

x=y -> (E!x & E!y)
Identity entails existence.

Witt

[mhc]

So all possible things necessarily exist?

[Theli]

I would say there are no areas in our lives in which two different things are equal.
That they are different demonstrates that they are not equal.
Language is rather vague when it comes to identity, as you point out but I don't see why the condition "=" must have such precise values on both sides.

An example:

1. My shirt is blue. The sky is blue. My shirt has the same color as the sky.
2. The sky has a slightly brighter blue than my shirt. My shirt has not the same color as the sky.
Wich of those two is incorrect if any, and where does the error lie? (assuming my shirt is slightly darker than the sky)

[Witt]

mhc:
And Witt--
How can you say that Vulcan IS a hypothetical planet, and then say that "Vulcan" does "not refer". You just referenced it!

No I did not. The object referred to by the name "Vulcan" does not exist. But, the name "Vulcan", or the description of Vulcan does exist.

'Vulcan IS a hypothetical planet' is a description of a presumed object which may or may not exist. I referenced the name not what the name names.

Being described as a hypothetical planet does not assure its existence.

If we can make a primary assertion about Vulcan, eg. Vulcan rotates, then we have proof of its existence.
But, secondary predications do not entail existence.

Witt

[mhc]

If all possible things exist, then no non-existent things are possible,
so no things do not exist.
:)

[Witt]

mhc: So all possible things necessarily exist?

Why do you conclude this?

It may be possible that someone has 10M heads, but it is hardly necessary.

Witt

[Witt]

An example:

Theli:
1. My shirt is blue. The sky is blue. My shirt has the same color as the sky.
2. The sky has a slightly brighter blue than my shirt. My shirt has not the same color as the sky.
Wich of those two is incorrect if any, and where does the error lie? (assuming my shirt is slightly darker than the sky)


(the color of my shirt) = (the color of the sky), means they are the same color.

(the color of my shirt) is different from (the color of the sky), means they are not the same color.

(assuming my shirt is slightly darker than the sky), 1 is false and 2 is true.

Witt

[BDS]

My shirt is blue and the sky is blue does not imply that my shirt and the sky are the same color. It merely states that they both fall into the same category of color.

After all, a Great Dane and a Chihuahua are both "dogs". Does this mean they are the same?

So the truth or falsity of #1 is based on what we mean by "the same color". If we mean "the same species of color" (in the way that the Great Dane and Chihuahua are the same species of animal, i.e. dogs) then #1 is true, not false, and #2 is false, not true. There's no contradiction here, it's just a matter of how we use words.

[exnihilo]

To answer xorbie's question, to put it simply, thinking about language and it's fundamental iterability in this way allows for the recognition of the components of play, ambiguity and openess inherent in any linguistic system. One of the greatest faults of logic is that this recognition is never made and language is presented as tied to the same strict rules that govern logical and mathematical thinking, which it clearly does not. The fact is from a finite number of letters and limited number of words--although extremely large, not infinite--we can produce an infinite number of meanings, associations, statements and intentions. Jacques Derrida has written much on this subject, check out Dissemination and Writing and Difference for many great essays regarding the critical importance of this issue. A familiarity with the work of Ferdinand de Saussure would also prove helpful in this area as well.

[pmurray]

"A is A" almost never applies in the real world. It's a fiction that philosophers adopt, to make philosophy possible.

"My shirt is blue"
"The sky is blue"
"my shirt is the same colour as the sky".

Not at all. Or at least, only within a given tolerance. Only for a given value of "is".

[jofo]

This is basically what Alfred Korzybski said... there can obviously never be an equality between a word and a thing. Once you start confusing them, you're in deep water, because you stop evaluating your sensory input.

[Farren]

Witt,

I must admit to not entirely following some of your later reasoning on this thread, but the following problem with your initial reasoning strikes me:

If an identity term automatically implies a physical or valid object then the use of that term as a reference is invalid in language or logical statement

So

"The present king of France" translates as *error* &
"The Klingon warship" translates as *error* &
"#%$#@" translates as *error*

in which case *error* = *error*, which is similar to saying 0=0 in numeric terms. Nothing is indicated or stated, but both sides have the single equivalent quality of absence.

Clearly, however "The present king of France" and "The Klingon Warship" do have seperate meanings when shared between individuals. They have (implicit) fictional meaning.

If this is allowed then

"The fiction referenced by the Klingon Warship" = "The fiction referenced by the Klingon Warship" is also clearly true.

In either scenario I don't see an invalidation of the axiom "A=A"

If A has the qualities "Implied Existence" and "Name" and the value of "Implied Existence" is false, then A still equals A.

[Witt]

Farren:

"The present king of France" translates as *error* &
"The Klingon warship" translates as *error* &
"#%$#@" translates as *error*

in which case *error* = *error*, which is similar to saying 0=0 in numeric terms. Nothing is indicated or stated, but both sides have the single equivalent quality of absence.
-----------------------------------------------------------

Hi Farren,

Your translations won't do.

1. "The present king of France" cannot exist, because there are no kings of France at the present.
2. "The Klingon warship" does not exist, because it is fictional ..by definition fictional things do not exist.
3. "#%$#@", is gibberish and it is not a well formed anything.

1a. (the present king of France)=(the present king of France), is contradictory.

2a. (the Klingon warship)=(the Klingon warship), has no meaning at all outide of the story (it too is fiction).

3a. (#%$#@)=(#%$#@), has no possible meaning anywhere.

None of these expressions can be compared to 0=0.

0=0 is meaningful and tautologous, it does not represent: nothing=nothing ..which is clearly contradictory, ie. nothing does not exist.

Farren:
Clearly, however "The present king of France" and "The Klingon Warship" do have seperate meanings when shared between individuals. They have (implicit) fictional meaning.

The present king of France, does not have fiction meaning at all!

Farren:
If A has the qualities "Implied Existence" and "Name" and the value of "Implied Existence" is false, then A still equals A.

That "A" is a name does not imply reference.

The planet Vulcan does not exist even though "Vulcan" is a proper name.

I can provide a logical proof that: The present king of France exists is contradictory, if you are interested.

Witt

[Witt]

Theli:
I've always thought of A=A as an hypothesis, for us to accept the conclution A we must first accept the premise A.
Does it really say that the conclution is true, or does it say that A is true under the condition placed by the premise?
Wouldn't a non-hypothetical tautology look like this:
P1: A
P2: A=A
C: A

If A is a proposition, then you are correct.

(A & (A=A)) -> A, is tautologous.

But, the usual interpretation of A=A is, all objects are self identical, or, everything is equal to itself.
Propositions are then not values of the variable A.
For example: Theli=Theli, 2=2, 2=root(4), etc., are instances of A=A.


Theli:
As far as A=A is concerned it says nothing about reality, just like 5+5=10 says nothing about reality either.

My point about A=A is that it is true if and only if the object A already exists.

Witt

[conkermaniac]

So do you believe that i = i?

[Witt]

conkermaniac:
So do you believe that i = i?

Of course, if you mean +sqrt(-1).

There are an endless number of different numbers such that: x^2=-1.

The square root of (-1) does not exist among: integers, rational numbers, irrational numbers, real numbers.
But, it does exist among: complex numbers, hypercomplex numbers, ie. any number of dimension >1.

Witt

[Theli]

(the color of my shirt) = (the color of the sky), means they are the same color.

(the color of my shirt) is different from (the color of the sky), means they are not the same color.

(assuming my shirt is slightly darker than the sky), 1 is false and 2 is true.
So you are saying that a logical claim cannot be vague and correct at the same time?
They are both "blue" aren't they? Does the word "blue" describe a very specific frequency of light or is it a rather vague description?
I would say that both claims are true, it depends on how precise you define a color. There is no absolute measurement to seperate two colors.
My point about A=A is that it is true if and only if the object A already exists.
Is A really an object though?
Whatever you place beside the "=" can only be a concept, an interpretation of an object, and interpretations are always vague.



BDS...
After all, a Great Dane and a Chihuahua are both "dogs". Does this mean they are the same?
Same what excacly?

[Witt]

Witt: My point about A=A is that it is true if and only if the object A already exists.

Theli: Is A really an object though?


A represents an object. It cannot be the object that it represents.

Theli: Whatever you place beside the "=" can only be a concept, an interpretation of an object, and interpretations are always vague

(Theli)=(Theli), is not vague.
2=2, is not vague.
These are precise statements.

x=y, is defined as: Fx <-> Fy, for every predicate F.

Why do you think they are vague??

Witt

[Clutch]

Witt,

Vacuous names are an issue in the philosophy of language. But in logic -- or rather, in most logics -- there is an explicit assumption "up front" that the constants have referents, ie, for any constant 'c', E!x(x=c).

This is of course an idealization, akin to assumptions of frictionless surfaces in physics. Logicians are perfectly aware of this, and have commented on it in detail. Why would it show that A=A is invalid? (I'm assuming that you're not distinguishing "not valid" from "invalid".) Logical validity is defined for a logic, after all; no logician thinks that that the A's in the formula stand for absolutely all things known loosely as names in natural language.

The fact is, one would be very hard pressed to come up with a logician who thinks that logic models, or ought to model, natural language in its entirety. It is rather a matter of regimenting fragments of a language in order to clarify and better apply some intuitive canons of good reasoning. Such intuitions are not univocal, and are context-dependent, and hence there are many, many different logics -- each offering a different sort of regimentation, under different kinds of idealization. While there is much disagreement as to the comparative value and generality of these various logics, there is virtually no disagreement on the question of whether any logic could or should recover the expressiveness of natural language.

[Witt]

Clutch: Vacuous names are an issue in the philosophy of language. But in logic -- or rather, in most logics -- there is an explicit assumption "up front" that the constants have referents, ie, for any constant 'c', E!x(x=c).

Yes, this is Quine's point of view.

"No entity without identity", everything exists .. necessarily, etc.


But, I do not agree.

Vacuous names, those names which do not refer, do not refer to existent objects.

But, the empty set, that set which has no members, does exist!
It does refer.

It seems clear to me that the empty set IS included in (all sets).

First order predicate logic does function with the defect that every instance of the variable x exists.

That is to say, Quine is correct ..everything exists!

But, surely 'that which is and is not' cannot exist ??

What do you think?

Witt

[Clutch]

"No entity without identity", everything exists .. necessarily, etc."No entity w/o identity" -- ie, no positing an entity without giving some identifying properties -- has nothing to do with vacuous names. "Pegasus" is a vacuous name with perfectly good identity conditions.
But, I do not agree. Vacuous names, those names which do not refer, do not refer to existent objects.With whom are you disagreeing? Everyone agrees with what you just wrote; it's a definition of "vacuous name". The question is whether the philosophical issues about vacuous names are a problem for logic. They are not. Logicians are aware of this vexed question, but idealize away from it -- leaving it to philosophy of language. Just as one can idealize away from complexities of friction or relativistic effects for the purpose of focussing on other factors.

In short, the natural language notion of a name includes some that do not refer. The logical notion of a name restricts itself to those that do refer. And subject to that restriction, A=A is, plausibly, valid.

[Witt]

quote:
--------------------------------------------------------------------------------
"No entity without identity", everything exists .. necessarily, etc.
--------------------------------------------------------------------------------

Clutch:
"No entity w/o identity" -- ie, no positing an entity without giving some identifying properties -- has nothing to do with vacuous names. "Pegasus" is a vacuous name with perfectly good identity conditions.

That flying horse of Greek mythology, has no possibility of existence. (because of the concept of Myth)

Pegasus = Pegasus, is clearly false.

That which is a flying horse, within the context of Greek mythology, is equal to anything .. is beyond mysterious!

Witt

[Xeno]

Does A=A necessarily imply possible existence of A or merely equality of the concept A?

That is, saying A=A does not guarantee an existent with the properties A, only that a thing classified as A will have the exact properties of another thing classified as A.

[exnihilo]

Originally posted by pmurray
"A is A" almost never applies in the real world. It's a fiction that philosophers adopt, to make philosophy possible.

"My shirt is blue"
"The sky is blue"
"my shirt is the same colour as the sky".

Not at all. Or at least, only within a given tolerance. Only for a given value of "is".

No, this position is only found in analytic philosophy, which is a very limited and increasingly irrelevant branch of philosophy. Such an equation has been rejected in philosphy since Nietzsche and illustrated by the ultimate failure of Russell and Whitehead's project. Thus, such a statement would be much more at home in the realm of mathematics and science rather than in philosophy proper which is now dominated by anti-foundational principles rather than classically logical/binary ones.
Consequently, a=a has little to do with what philosophy is all about, nor does it "make philosophy possible" as you erroneously claim. Perhaps, you should go read some philosophy before making such unwarranted and spurious assertions.

[pmurray]

Originally posted by Farren
Clearly, however "The present king of France" and "The Klingon Warship" do have seperate meanings when shared between individuals. They have (implicit) fictional meaning.

If this is allowed then

"The fiction referenced by the Klingon Warship" = "The fiction referenced by the Klingon Warship" is also clearly true.


But I'm talking about the fictional "Warbird", and you are talking about the fictional "Predator".

Or let's take a more comon fiction: Mickey Mouse. Surely we are talking about the same thing? Not at all. You are talking about happy memories hrom your childhood, and I am talking about an obviously racist take on the "happy nigger".

[Clutch]

Examples like "Ted is tall" are not instances of the schema X=X, and are not relevant to the thread. They are cases of predication as opposed to identity.

[Clutch]

Originally posted by exnihilo
Perhaps, you should go read some philosophy before making such unwarranted and spurious assertions. <Irony-meter shatters>

[conkermaniac]

Originally posted by Witt
conkermaniac:
So do you believe that i = i?

Of course, if you mean +sqrt(-1).

There are an endless number of different numbers such that: x^2=-1.

The square root of (-1) does not exist among: integers, rational numbers, irrational numbers, real numbers.
But, it does exist among: complex numbers, hypercomplex numbers, ie. any number of dimension >1.

Witt Then why should i be any different than the present king of France? I noticed that you mentioned that i does not exist among real numbers (which is the basis of real-world numbers), yet it does exist when we extend our definition of numbers to include complex numbers. I believe that the same can be said of the present king of France. It only appears to generate an error, just as sqrt(-1) appears to generate an error. However, if we only broaden our environment to include, say, and alternate universe, then there can be a present king of France, and therefore, the two values can be considered equal.

[Theli]

Theli: Whatever you place beside the "=" can only be a concept, an interpretation of an object, and interpretations are always vague.

Witt:
(Theli)=(Theli), is not vague.
Ofcourse it is, the word "Theli" says absolutely nothing about me. I could drasticly change either physically or mentally and the word would still apply to me. No concept in the world describes an object precisely. So why should logic wich is based on these concepts require absolute precision?
2=2, is not vague.
2=2 is an exception though as it is not a concept describing an object.

[Xeno]

A=A is a comparative operator of properties.

If we take A=The present king of France, there are numerous properties along with this assignment, including:

- A is a king.
- A is male.
- A lives currently.
- A is currently a king.
- It is possible A does not exist.

Taking all these properties into account, A=A is true for all meanings of the word "is".

Similarly, if we take A="That which cannot equal itself", it will have the following properties:

- A cannot be equated to itself

In which case A=A is true when comparing the properties of itself.

[Farren]

Xeno that's exactly what I'm thinking.

I take Witts point about "The present King of France" and the Klingon warship" being different, but...

"The present king of France" = "The present king of France"

is a reference failure on both sides of the equation. There are two possible ways of understanding this statement.

1. Assuming the reference is true

"Were there a present king of france,

the present king of france would be the present king of france"

This is functionally similar as allowing the identity to refer to "a thing that could but does not exist"

2. Assuming the identity is nonsense

"This sentence, 'The present king of france is the present king of france', means nothing"

In this second case the entire sentence is rendered invalid by the term used, so "the present king of france is the present king of france" doesn't yield

A=A

it yields

*Unable to interpret this term* = *Unable to interpret this term*

which is not the same thing. Another way of putting this is to say if we are assuming

- that identity implies existence
- In the case of a klingon warship a fiction is implied so the existence of the fiction is required
- In the case of the present king of france a fiction is not implied so the existence of an actual king of france is required
- That A is a valid identity

Then "The present king of france" is not a valid identity, and "The present king of france is the present king of france" does not render down to

A=A

In either scenario, I can't see A=A failing to be accurate.

(forgive my inability to use formal predicate calculus)

[Theli]

For us to say that A=A is true, the information provided with the definition of A must be true, and the information not provided with the definition must be irrelavent to the point.

[PJPSYCO]

Again, with the same post(I know you changed the title). Do you learn nothing?

A=A only applies to ideas. It is used to reject nonsencical ideas.

Words can refer to more than one thing. A=A is not word with a meaning = same word with a different meaning. It is idea is same idea. If A=B and A=C then B must equal C or you have two different ideas that are not both A. The argument must be redefined to A=B and C=D or something to that effect.

The only application to objects is that if you say that your object is defined by idea B then by idea C. I can conlude that your object is either two objects or nonexistent, and you are confused.

[Witt]

Xeno:

A=A is a comparative operator of properties.

If we take A=The present king of France, there are numerous properties along with this assignment, including:

- A is a king.
- A is male.
- A lives currently.
- A is currently a king.
- It is possible A does not exist.
-----------------------------------------

At the present time, it is not the case that:

- A is a king.
- A is male.
- A lives currently.
- A is currently a king.

And, it is not possible that A exists at this time.

At some other time, they were/are true.

Xeno:
Taking all these properties into account, A=A is true for all meanings of the word "is".

The description is equal to the description, is true.
But, the object described is not equal to the object described.

Names and descriptions have reference, if and only if, they do exist.

Xeno: Similarly, if we take A="That which cannot equal itself", it will have the following properties:

- A cannot be equated to itself

In which case A=A is true when comparing the properties of itself.
-------------------------------

The x such that ~(x=x), has no properties at all.
To posses a property is to exist.
(assuming that properties are confirmable predicates)

It does not have the property of non-existence either.
To say that A has non-existence is to say, it is not the case that A does have existence.

The property exists but there is no object that has that property.

Witt

[Witt]

Farren:

Xeno that's exactly what I'm thinking.

I take Witts point about "The present King of France" and the Klingon warship" being different, but...

"The present king of France" = "The present king of France"

is a reference failure on both sides of the equation. There are two possible ways of understanding this statement.

1. Assuming the reference is true

"Were there a present king of france,

the present king of france would be the present king of france"
---------------------------------


Yes, If there is a present king of France then it is equal to itself, but not otherwise.


Farren:

This is functionally similar as allowing the identity to refer to "a thing that could but does not exist"

2. Assuming the identity is nonsense

"This sentence, 'The present king of france is the present king of france', means nothing"
------------------------------

False. The present king of France is The present king of France, is meaningful ..it is contradictory, ie. it is not meaningless.


Farren:

In this second case the entire sentence is rendered invalid by the term used, so "the present king of france is the present king of france" doesn't yield

A=A

it yields

*Unable to interpret this term* = *Unable to interpret this term*

which is not the same thing. Another way of putting this is to say if we are assuming

- that identity implies existence
- In the case of a klingon warship a fiction is implied so the existence of the fiction is required
- In the case of the present king of france a fiction is not implied so the existence of an actual king of france is required
- That A is a valid identity

Then "The present king of france" is not a valid identity, and "The present king of france is the present king of france" does not render down to

A=A

In either scenario, I can't see A=A failing to be accurate.
(forgive my inability to use formal predicate calculus)
----------------------------------


Yes, A=A fails, that is to say, it is not true.
It is not a value of the variable A in, A=A.

Ordinary language seems to work just as well.

Witt

[Witt]

The square root of (-1) does not exist among: integers, rational numbers, irrational numbers, real numbers.
But, it does exist among: complex numbers, hypercomplex numbers, ie. any number of dimension >1.

Witt
--------------------------------------------------------------------------------
conkermaniac:

Then why should i be any different than the present king of France? I noticed that you mentioned that i does not exist among real numbers (which is the basis of real-world numbers), yet it does exist when we extend our definition of numbers to include complex numbers. I believe that the same can be said of the present king of France. It only appears to generate an error, just as sqrt(-1) appears to generate an error. However, if we only broaden our environment to include, say, and alternate universe, then there can be a present king of France, and therefore, the two values can be considered equal.

I agree with you here. We do need to specify our domain of discourse every time.

When A does exist, within our universe of discourse, then A=A is true. When A does not exist, within our universe of discourse, A=A is false.

If we include described objects, as values of the variabe A, then it fails. For example, that which is not self-identical does not exist in any universe.

The description does exist, but, what it describes does not exist.

Witt

[Clutch]

Originally posted by Witt
When A does exist, within our universe of discourse, then A=A is true. When A does not exist, within our universe of discourse, A=A is false.This is not quite right, since it's not that the schema is obviously false when vacuous names are admitted, but rather that it's false for instances of non-referring descriptions. It's a further (and contentious) question whether names are to be identified with descriptions, so it's a contentious question whether the schema is false for instances of vacuous names as well. (As opposed to being trivially true, perhaps.)

But this is all familiar. Russell (and Russell and Whitehead) long ago explained why 'The F = the F' is not an instance of a logical law -- in short, because it will be false in the event that there exists no F.

Nothing you've written here contradicts any received view among logicians.

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
When A does exist, within our universe of discourse, then A=A is true. When A does not exist, within our universe of discourse, A=A is false.
--------------------------------------------------------------------------------

Clutch:

This is not quite right, since it's not that the schema is obviously false when vacuous names are admitted, but rather that it's false for instances of non-referring descriptions.

A=A is false in both cases.

Clutch:
It's a further (and contentious) question whether names are to be identified with descriptions, so it's a contentious question whether the schema is false for instances of vacuous names as well. (As opposed to being trivially true, perhaps.)

It is a theorem that: (the x such that x=y) = y.
That is, all names can be expressed as a description.
Quine went so far as to say that he can eliminate names and variables by this procedure. See: Methods of Logic, page 283.

Clutch:
But this is all familiar. Russell (and Russell and Whitehead) long ago explained why 'The F = the F' is not an instance of a logical law -- in short, because it will be false in the event that there exists no F.

It is correct to say (the x:Fx)=(the x:Fx) is not an instance of x=x.

PM states the theorem: Exists(the x:Fx) <-> (the x:Fx)=(the x:Fx)
see, *14.28 page175.

Note also that: since, (the x:x=y)=y, then E!(the x:x=y) <-> E!y.

Which Russell claims cannot be said.

"It would seem that the word "existence" cannot be significantly applied to subjects immediately given: i.e. not only does our definition give no meaning to "E!x," but there is no reason in philosophy to suppose that a meaning of existence could be found which would be applicable to immediately given subjects."

Principia Mathematica page 175.

Witt

.

[Clutch]

Originally posted by Witt
A=A is false in both cases.I take it this is meant as a preliminary to some argument, and not as an argument itself.

Clutch:
It's a further (and contentious) question whether names are to be identified with descriptions, so it's a contentious question whether the schema is false for instances of vacuous names as well. (As opposed to being trivially true, perhaps.)

It is a theorem that: (the x such that x=y) = y.
That is, all names can be expressed as a description.Huh? This is completely wrong. Assuming that 'y' is a name (it is a useful convention to use "early" letters a, b, c, for names and later letters for variables) what it shows is that a description and a name can co-refer. So what? Who on earth ever denied this? The question -- let's read it again -- was whether names are to be identified with descriptions. Whether all a name is, is a kind of description. That's what you'd need to show, in order to use arguments about the failure of A=A under descriptions as a means of proving its failure under vacuous names. But you show nothing of the sort -- indeed, nothing of any sort -- simply by asserting that a name and a description can pick out the same object.
Quine went so far as to say that he can eliminate names and variables by this procedure. See: Methods of Logic, page 283.Yes, Quine did say this. And, as I've pointed out, this was contentious, and did not admit of anything like a proof. Do you have one? Ever hear of this Kripke fellow, who has these hugely influential (but also contentious) arguments against taking names as descriptions?
Clutch:
But this is all familiar. Russell (and Russell and Whitehead) long ago explained why 'The F = the F' is not an instance of a logical law -- in short, because it will be false in the event that there exists no F.

It is correct to say (the x:Fx)=(the x:Fx) is not an instance of x=x.

PM states the theorem: Exists(the x:Fx) <-> (the x:Fx)=(the x:Fx)
see, *14.28 page175.Hence my observing that, inasmuch as what you're saying makes sense, R&W said it long ago.
Note also that: since, (the x:x=y)=y, then E!(the x:x=y) <-> E!y.

Which Russell claims cannot be said.
Russell is right, and your theorem is ill-formed. It moves from using 'y' as a name to using it as a variable.

[peripeteia]

Hello everyone..I m enjoying following your little thread and its quite good..But I do have a question.. Clutch you stated: “The question -- let's read it again -- was whether names are to be identified with descriptions. Whether all a name is, is a kind of description. That's what you'd need to show, in order to use arguments about the failure of A=A under descriptions as a means of proving its failure under vacuous names”.

Im just curious is this discussion strictly limited to the philosophy of lang..or can one bring up certain critiques leveled by poststructuralists and postmodernists specifically dealing with A=A thingy and the relation between names and its referentials i.e., descriptions? Let me say something quick[since you also used Quine and Krepke—both are very good I might add]..In language, identity is not the congruent overlay of a signifier[in this case what the ‘name’ ‘is’ or denotes] on itself. It is the internal negation of other elements in the system[‘name’ is seen as a system], a difference that fixes an identity..Which also delineates the non nomological character of names which can do a bummer in using symbolic logic..

Just adding my two cents worth..


Have a great weekend everyone..

[Clutch]

Originally posted by peripeteia
Hello everyone..I m enjoying following your little thread and its quite good..But I do have a question.. Clutch you stated: “The question -- let's read it again -- was whether names are to be identified with descriptions. Whether all a name is, is a kind of description. That's what you'd need to show, in order to use arguments about the failure of A=A under descriptions as a means of proving its failure under vacuous names”.

Im just curious is this discussion strictly limited to the philosophy of lang..or can one bring up certain critiques leveled by poststructuralists and postmodernists specifically dealing with A=A thingy and the relation between names and its referentials i.e., descriptions? Let me say something quick[since you also used Quine and Krepke—both are very good I might add]..In language, identity is not the congruent overlay of a signifier[in this case what the ‘name’ ‘is’ or denotes] on itself. It is the internal negation of other elements in the system[‘name’ is seen as a system], a difference that fixes an identity..Which also delineates the non nomological character of names which can do a bummer in using symbolic logic. Peripeteia, you can bring up what you like! I'm not sure I understood everything you wrote, but I certainly agree that identity cannot be a "congruent overlay", since that presupposes two distinct things -- an overlayer and an overlaid. Whereas identity is the relation in which a thing stands to itself.

Pleased to make your acquaintance



(Heh-heh-heh. Over-laid. As if that's possible.)

[Witt]

Witt:
It is a theorem that: (the x such that x=y) = y.
That is, all names can be expressed as a description.

Clutch:
Huh? This is completely wrong. Assuming that 'y' is a name (it is a useful convention to use "early" letters a, b, c, for names and later letters for variables)

Useful yes, necessary no.
As early as chapter *9 of PM, R & W assert as a theorems:
|-: AxFx -> Fy, and |-: Fy -> ExFx.

Later in *13 and *14 they use quantifiers over constant values such as a, b, c, as well as variables x, y ,z.

E.g: AxEc(x=c), AxEy(x=y), etc.

Fy is read, Fy is true for any y.
AxFx is read, Fx is true for all x.

Witt:
Note also that: since, (the x:x=y)=y, then E!(the x:x=y) <-> E!y.
Which Russell claims cannot be said.

Clutch:
Russell is right, and your theorem is ill-formed. It moves from using 'y' as a name to using it as a variable.


((the x:x=a)=a) -> E!(the x:x=a) <-> E!a, has the same meaning as,
((the x:x=y)=y) -> E!(the x:x=y) <-> E!y, has the same meaning as,
Ay{((the x:x=y)=y) -> E!(the x:x=y) <-> E!y}.

And, they are each Well-Formed theorems, (they follow from Leibnitz's Law).

Constants, like (a, b, c, etc.), are not essential to symbolic logic at all.


Witt

[Kenneth]

sophistry...

the question of existence is irrelevant.

God = God

is true, whether you believe in him or not. Because both entities in the equation are subjects. both are identical, which is why this equation is called an identity.

Things that are the same in all things, are the same. [Euclid, Pos. 3]

If God--before the equal sign--does not exist, the God--after the equal sign--also does not exist. So A=A is valid.

[Witt]

Kenneth:

sophistry...

the question of existence is irrelevant.

God = God

is true, whether you believe in him or not. Because both entities in the equation are subjects. both are identical, which is why this equation is called an identity.

Things that are the same in all things, are the same. [Euclid, Pos. 3]

If God--before the equal sign--does not exist, the God--after the equal sign--also does not exist. So A=A is valid.
------------------------------------------------------------

Hi Ken,

Names or descriptions that do not refer are not the subject of any true statement.
Non-existent things have no primary predications at all, including self-identity.

If God does not exist, (God=God) is false ..necessarily.
There is nothing that can be said truly about non-existent things.
We can only say, with truth, what they are not!

That which is defined or described by a contradictory predication cannot exist.

Proof:

The present king of France does not exist, because there is no king of France at this time.

Surely, if there are no king of France then there cannot be a unique king either, can there?

(the present king of France)=(the present king of France), means,

There is something such that it is king of france if and only if it is unique and it's equal to that thing.

Ey(Ax(x=y <-> x is king of France) & y=y)

But, x is king of France ..is not satisfied by any existent object at this time.
ie. x is king of France is false for all x's.

Therefore, Ey(Ax(x=y <-> x is king of France) & y=y) <-> Ax(x=y <-> contradiction) & y=y).

Since Ay(y=y) is an axiom of classical logic.

Ey(Ax(x=y <-> contradiction) <-> EyAx~(x=y)

But, ~AyEx(x=y) is a contradiction. ie. AyEx(x=y) is a theorem.

Therefore, ~(the present king of France)=(the present king of France), can be asserted.

For example,
If we assume that God has the quality of omni-benevolence then (in the presence of evil) that description of God is contradictory.

Do you have a non-contradictory definition of God?

God=God (or God exists), is false ..if the description of it is contradictory!

Witt

[John Page]

Originally posted by Witt
Names or descriptions that do not refer are not the subject of any true statement.
Non-existent things have no primary predications at all, including self-identity.
Witt:

....but names or descriptions refer in the first instance to the relevant mental concept. Thus I can argue god is a word that refers to a word that has no proven physical corollary.

Even when you use the expression "non-existent thing" that expression refers to the concept of something that has no referent.

As to identity, that is an abstract/mental phenomenon which in logic is assumed to be unique. In reality, I believe that when we refer to the identity of something, we are refering to an instance of a form. I am wondering what you really mean by "self-identity".

Cheers, John
P.S. Now to look for that other thread we were exchanging posts on...

[John Page]

Originally posted by Witt
Do you have a non-contradictory definition of God?
God is a mental concept used by human beings a) to explain events for which there is no other reason, b) as an anchor for moral frameworks, c) to permit societies to work for long term gains (e.g. be good and you'll get a nice afterlife).

Cheers, John

[tronvillain]

Anyone else finding this "proof" amazingly unimpressive? Just about everyone I expect.

What does "the present king of France" refer to? Either it refers to a hypothetical or fictional entity, or it refers to nothing and is simply gibberish. Take your pick, but in both cases A=A holds:

the hypothetical present king of France = the hypothetical present king of france

the meaningless string of words = the meaningless string of words

So Witt, unless you can offer some other explanation of what "the present king of France" refers to, you do not appear to have anything to talk about. Remember, if you say "It does not refer to anything", that is the same as saying "It is a meaningless string of words."

[Witt]

tronvillain: What does "the present king of France" refer to? Either it refers to a hypothetical or fictional entity, or it refers to nothing and is simply gibberish. Take your pick, but in both cases A=A holds:

No, it does not.

tronvillainthe: hypothetical present king of France = the hypothetical present king of france

Is false!

tronvillain: the meaningless string of words = the meaningless string of words

It is false.

tronvillain:
So Witt, unless you can offer some other explanation of what "the present king of France" refers to, you do not appear to have anything to talk about. Remember, if you say "It does not refer to anything", that is the same as saying "It is a meaningless string of words."



Nonsense!

qwerty, is not a well formed anything, it is gibberish.
!@#$%, is also not a well formed anything, it too is gibberish.

The present king of France, is not gibberish, because we can say that it does not exist!

The present king of France, is bald .. is false
The present king of France, is a pink elephant ..is false.
The present king of France, is a number, ..is false.

There is no thing that 'the present king of France' is.

(!@#$%) is wise, has no possibility of sense!

Witt

[Keith Russell]

Good afternoon.

Words are only accurate when they refer to concepts, and only true when the concepts (to which the words refer) accurately correspond to reality.

I view 'A is A' as a conceptual statement. To me, it means that we recognize that a 'thing' is by definition finite; that we recognize a thing as a thing by the separation, the difference between that thing, and all other things.

It means that a thing is only what it is, whatever it is--

--and that it is not whatever it isn't.

K

[John Page]

Originally posted by Witt
There is no thing that 'the present king of France' is.

Doesn't it fall into the class of things that exist in the mind but for which there is no possibility of it existing in real life (given that France is presently a republic)?

Look at the quadrants in this chart (http://www.reconciliationism.org/reality.htm) , the "present king of france" is a set of words that refers to a thing that is imaginary and known (i.e. in the bottom right) but that cannot possibly refer to something that is physical (known or unknown).

Cheers, John

[tronvillain]

Witt:
tronvillain: the hypothetical present king of France = the hypothetical present king of france

Is false!
Exactly how is it false? It appears to be true for any given hypothetical present king of France, or in other words for any specific king who might exist if present day France had an existing monarchy. It is simply saying that a logically possible but nonexist entity is identical to itself.
tronvillain: the meaningless string of words = the meaningless string of words

It is false.
Exactly how is it false? The string of words the+present+kind+of+France appears to be the very same string of words as the+present+king+of+France. Would you care to demonstrate that it is not?
tronvillain:
So Witt, unless you can offer some other explanation of what "the present king of France" refers to, you do not appear to have anything to talk about. Remember, if you say "It does not refer to anything", that is the same as saying "It is a meaningless string of words."



Nonsense!

qwerty, is not a well formed anything, it is gibberish.
!@#$%, is also not a well formed anything, it too is gibberish.

The present king of France, is not gibberish, because we can say that it does not exist!

The present king of France, is bald .. is false
The present king of France, is a pink elephant ..is false.
The present king of France, is a number, ..is false.

There is no thing that 'the present king of France' is.

(!@#$%) is wise, has no possibility of sense!
I agree that "the present king of France" is not gibberish, but that is because I consider it to refer to a hypothetical individual: someone who does not actually exist, but who is logically possible (like a fictional character). As far as I can tell, that is the only alternative to the position that "the present king of France" is a meaningless string of words, and since you have not offered any alternative I assume you do not see any alternatives either. Oh, and it is not necessarily true that "The present king of France, is bald .. is false" since it is logically possible that a present king of France could be bald.

If you assert that "There is no thing that 'the present king of France' is, do you also assert that "There is no thing that 'the number five" is? If you do, why do you not attempt to proove that 5=5 is false? Perhaps because you realize that the huge flaw in your position would be too obvious then. ?

[Witt]

Good afternoon.

Keith Russell:
Words are only accurate when they refer to concepts, and only true when the concepts (to which the words refer) accurately correspond to reality.

Yes, all words are conceptual.
Words are never true. Only statements have the possibility of truth.

Keith Russell:
I view 'A is A' as a conceptual statement. To me, it means that we recognize that a 'thing' is by definition finite; that we recognize a thing as a thing by the separation, the difference between that thing, and all other things.

What does 'finiteness' have to do with existence?

That there is a difference between a and b shows that they are not equal.

Keith Russell:
It means that a thing is only what it is, whatever it is--

--and that it is not whatever it isn't.
-----------------------------------------------


Your remark is tue, iff, It exists!


Witt

[John Page]

Originally posted by Witt
That there is a difference between a and b shows that they are not equal.
That there is a difference between a and a shows that they are not equal.

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
That there is a difference between a and b shows that they are not equal.
--------------------------------------------------------------------------------


John:
That there is a difference between a and a shows that they are not equal.

There cannot be a difference between a and a.

Witt

[John Page]

Originally posted by Witt
There cannot be a difference between a and a.
Then how come there are two of them?

I know this response may seem trivial, and I have a feeling that you'll come back saying only their form is identical. If the latter is the case, then, where does this thing called "form" reside? Each of our minds store the form of "a" in order to determine the truth of existence of an "a" when we experience it. However, not all a's are identical and not all forms of a are identical.

This covers some ground we've kicked before - see here from STanford's Plato site "For example, if properties are abstract objects, then the property of being abstract should itself exemplify the property of being abstract. In various passages throughout his dialogues Plato appears to hold that Forms (which are often taken to be his version of properties) participate in themselves. Indeed, this claim serves as a premise in what is known as his Third-Man Argument which, he seems to think, may show that the very notion of a Form is incoherent (Parmenides, 132ff). Russell's paradox raises more serious worries about self-exemplification. It shows that any account which allows properties to exemplify themselves must be carefully formulated if it is to avoid paradox (a polite word for inconsistency). "

No forms are universal, they exist multiplicitly in minds and we share our understanding of them intersubjectively.

Cheers, John

[Witt]

Witt :
There cannot be a difference between a and a.

John:
Then how come there are two of them?

There is not, two of them at all.
They are the same symbol, ie. they represent the same, indeed the identical, object that a names.
It does not make sense to say there are two different a's, imo.

John:
I know this response may seem trivial, and I have a feeling that you'll come back saying only their form is identical.

Only the objects referred to are identical, not their descriptions or their names.

The form of a name is independent of the reference of that name.

(1+1) has a different form than 2, but both expressions refer to the same identical entity.

Witt

[John Page]

Originally posted by Witt
Witt :
There cannot be a difference between a and a.

John:
Then how come there are two of them?

There is not, two of them at all.
They are the same symbol, ie. they represent the same, indeed the identical, object that a names.
It does not make sense to say there are two different a's, imo.

There's definitely two and I haven't been drinking. They are different symbols of the same form and, having likeness in from to our human perception, can be used to denote that they refer to the "same thing".
Originally posted by Witt
Only the objects referred to are identical, not their descriptions or their names.
So a symbol/name/description cannot be an object? If so, what is your ontology?
Originally posted by Witt
The form of a name is independent of the reference of that name.
Agreed.
Originally posted by Witt
(1+1) has a different form than 2, but both expressions refer to the same identical entity.
:confused: So, within the field of decimal aritmetic, what is the "identical entity" to which they refer and how do you prove that?

Cheers, John

[Witt]

Witt :
There cannot be a difference between a and a.

John:
Then how come there are two of them?

Witt:
There is not, two of them at all.
They are the same symbol, ie. they represent the same, indeed the identical, object that a names.
It does not make sense to say there are two different a's, imo.

John:
There's definitely two and I haven't been drinking. They are different symbols of the same form and, having likeness in from to our human perception, can be used to denote that they refer to the "same thing".

a is the same symbol that a is, even though there is a first a and a second a wrt this line.

The predicates of a, are different from the predicates of "a".

Witt:
Only the objects referred to are identical, not their descriptions or their names.

John:
So a symbol/name/description cannot be an object?

Yes they can be objects. They are linguistic objects.
Letters, words, descriptive phrases, etc., are linguistic things.

"a" = "a", is true even if a=a is false.

For example: "Vulcan"="Vulcan" is true, but,
Vulcan=Vulcan, is false.

If we admit into our ontology non-referring names such as "Vulcan", then we need to define identity differently.

Instead of: x=y =df AF(Fx <-> Fy), we need,
x=y =df E!x & E!y & AF(Fx <-> Fy).

If either term or both do not exist then they are not equal.
Vulcan does not exist, therefore: ~(Vulcan=Vulcan).

Witt:
The form of a name is independent of the reference of that name.

John: Agreed.

Witt:
(1+1) has a different form than 2, but both expressions refer to the same identical entity.

John:
So, within the field of decimal aritmetic, what is the "identical entity" to which they refer and how do you prove that?

"2" is the name of the number 2, but there are many descriptions of that same number. (1+1), (3-1), sqrt(4), sqrt(2))^2, etc..

(1+1) is a described number, (ix:1+1=x).
That which is the sum of 1and1.

Proof of (1+1)=2 depends of the foundational axioms that you are using. Russell and Whitehead were deep into the second volume before they could prove it.

More recent publications have greatly simplified the logic required.

Witt

[John Page]

Originally posted by Witt
a is the same symbol that a is, even though there is a first a and a second a wrt this line.

First, there are 8 instances of the symbol a on that line. Second, they share identity through commonality of form. That form resides in your mind and my mind and communicated intersubjectively (etc etc as discussed before).
Originally posted by Witt
It does not make sense to say there are two different a's, imo.

Why not, there patently are, a1 and a2.
Originally posted by Witt
The predicates of a, are different from the predicates of "a".
Just using "a" doesn't make the issue go away, "a" is still referring to something. I think we use to refer directly to what the symbol a is refering indirectly to.
Originally posted by Witt
John:So a symbol/name/description cannot be an object?

Witt: Yes they can be objects. They are linguistic objects.
Letters, words, descriptive phrases, etc., are linguistic things.

Identified by their form and made unique by their position in spacetime? If not, how do you define the word "object"?
Originally posted by Witt
"a" = "a", is true even if a=a is false.

For example: "Vulcan"="Vulcan" is true, but,
Vulcan=Vulcan, is false.
:confused:
Originally posted by Witt
If we admit into our ontology non-referring names such as "Vulcan", then we need to define identity differently.

I disagree. Non referring names are meaningless w.r.t. the non-mental world but in mental reality they refer to a concept.
Originally posted by Witt
Instead of: x=y =df AF(Fx <-> Fy), we need,
x=y =df E!x & E!y & AF(Fx <-> Fy).

If either term or both do not exist then they are not equal.
Vulcan does not exist, therefore: ~(Vulcan=Vulcan).

But there is no thing that is absolutely identical to another thing - equality is only the result of comparing the forms of the sense data that belies their existence. Now, if you start tracking back along the sense data to the object, something that is completely imaginary will be instantiated within the mind/brain only.
Originally posted by Witt
Witt:
(1+1) has a different form than 2, but both expressions refer to the same identical entity.

John:
So, within the field of decimal aritmetic, what is the "identical entity" to which they refer and how do you prove that?

"2" is the name of the number 2, but there are many descriptions of that same number. (1+1), (3-1), sqrt(4), sqrt(2))^2, etc..

And the "identical entity" number 2 is where?
Originally posted by Witt
(1+1) is a described number, (ix:1+1=x).
That which is the sum of 1and1.

Proof of (1+1)=2 depends of the foundational axioms that you are using. Russell and Whitehead were deep into the second volume before they could prove it.

More recent publications have greatly simplified the logic required.
You misunderstood, sorry, how do you prove that (1+1) and 2 refer to the identical entity (in the field of decimal arithmetic). I'm still looking for this identical entity.

Cheers, John

[B612]

Attached File

[tronvillain]

Still no response to me Witt? Oh, and if you do respond, do not bother pointing out that the string of words are different because managed to mispell king. *chuckle*

Witt:
"a" = "a", is true even if a=a is false.

For example: "Vulcan"="Vulcan" is true, but,
Vulcan=Vulcan, is false.
John, I believe what Witt is doing here is using quotation marks to differentiate between logically possible but nonexistent objects and existent objects. It seems a little odd to me, but there you are.

Witt, it occurs to me that not only do you not attempt to disprove A=A using statements like "2=2" you also did not use statements like "a square circle=a square circle." You are only using something "an existent leprechaun=an existent leprechaun", but without explicitly stating the "existent." Without explicitly mentioning that people naturally think of the identity of a logically possible but nonexistent leprechaun, so when you pounce and point out that there are no such things as leprechauns you may appear for a moment to have found a significant problem (this has some elements in common with the ontological argument). Upon reflection, if leprechauns do not exist, then "an existent leprechaun" is as logically impossible as "a square circle": both are ultimately meaningless, essentially gibberish. Would you care to now argue that "a square circle=a square circle" disproves A=A?

[Witt]

tronvillain:
Still no response to me Witt? Oh, and if you do respond, do not bother pointing out that the string of words are different because managed to mispell king. *chuckle*

Please excuse my tardiness.
I'll bet I make more typos than you do ..haha.

Witt:

quote:
--------------------------------------------------------------------------------
"a" = "a", is true even if a=a is false.

For example: "Vulcan"="Vulcan" is true, but,
Vulcan=Vulcan, is false.
--------------------------------------------------------------------------------

tronvillain:
John, I believe what Witt is doing here is using quotation marks to differentiate between logically possible but nonexistent objects and existent objects. It seems a little odd to me, but there you are.

I believe it was Quine (Mathematical Logic) that introduced the notion of using quotation marks to express a name as opposed to expressing what is named.

For example: "Vulcan" has 6 letters is true, but, Vulcan has 6 letters is false.
"tronvillain" has 11 letters is true, but, tronvillain has 11 letters is false.
They are different 'types' of objects.

tronvillain:
Witt, it occurs to me that not only do you not attempt to disprove A=A using statements like "2=2" you also did not use statements like "a square circle=a square circle." You are only using something "an existent leprechaun=an existent leprechaun", but without explicitly stating the "existent." Without explicitly mentioning that people naturally think of the identity of a logically possible but nonexistent leprechaun, so when you pounce and point out that there are no such things as leprechauns you may appear for a moment to have found a significant problem (this has some elements in common with the ontological argument). Upon reflection, if leprechauns do not exist, then "an existent leprechaun" is as logically impossible as "a square circle": both are ultimately meaningless, essentially gibberish. Would you care to now argue that "a square circle=a square circle" disproves A=A?

If the description of a purported object is not contradictory then that object has possible existence.
Even possible(God exists) is true, if the definition of God is not contradictory.

"if leprechauns do not exist, then "an existent leprechaun" is as logically impossible as "a square circle":"

Of course, both terms do not refer to anything.
Their names/descriptions exist but what they refer to is vacuous.

Witt:
quote:
--------------------------------------------------------------------------------
tronvillain: the hypothetical present king of France = the hypothetical present king of france

Is false!
--------------------------------------------------------------------------------

tronvillain:
Exactly how is it false? It appears to be true for any given hypothetical present king of France, or in other words for any specific king who might exist if present day France had an existing monarchy. It is simply saying that a logically possible but nonexist entity is identical to itself.

A hypothetical object is one whose existence is presumed, ie, its existence must be possible.

But, the existence of a present king of France is not possible at this time.

quote:
--------------------------------------------------------------------------------
tronvillain: the meaningless string of words = the meaningless string of words

It is false.
--------------------------------------------------------------------------------

tronvillain:
Exactly how is it false? The string of words the+present+kind+of+France appears to be the very same string of words as the+present+king+of+France. Would you care to demonstrate that it is not?

The string of words, the+present+king+of+France, is equal to itself, ie. the description or the name exists but it does not have a reference or extension.

'the meaningless string of words' is a description of something which is meaningless. Any expression which is meaningless has no possible meaning, ie. there are no grammatical sentences that contain meaningless expressions. Including self-identity.
The present king of France, is not meaningless at all.
eg. The present king of France is bald, is false and not meaningless.

quote:
--------------------------------------------------------------------------------
tronvillain:
So Witt, unless you can offer some other explanation of what "the present king of France" refers to, you do not appear to have anything to talk about. Remember, if you say "It does not refer to anything", that is the same as saying "It is a meaningless string of words."

Nonsense!

qwerty, is not a well formed anything, it is gibberish.
!@#$%, is also not a well formed anything, it too is gibberish.

The present king of France, is not gibberish, because we can say that it does not exist!

The present king of France, is bald .. is false
The present king of France, is a pink elephant ..is false.
The present king of France, is a number, ..is false.

There is no thing that 'the present king of France' is.

(!@#$%) is wise, has no possibility of sense!
--------------------------------------------------------------------------------
tronvillain:
I agree that "the present king of France" is not gibberish, but that is because I consider it to refer to a hypothetical individual: someone who does not actually exist, but who is logically possible (like a fictional character).

It is not logically possible, at this time.
It is possible, of course, at some other time.

tronvillain:
As far as I can tell, that is the only alternative to the position that "the present king of France" is a meaningless string of words, and since you have not offered any alternative I assume you do not see any alternatives either. Oh, and it is not necessarily true that "The present king of France, is bald .. is false" since it is logically possible that a present king of France could be bald.

Not so. Because it is true to say there is no king of France at this time, it is also true to say there is not a unique king of France that is bald, either! There is no hypothetical possibility at this time.

The hypothetical planet Vulcan does exist within Newtonian physics, ie. it is possible there.
But, it is not possible under our current understanding of relativity physics.

tronvillain:
If you assert that "There is no thing that 'the present king of France' is, do you also assert that "There is no thing that 'the number five" is? If you do, why do you not attempt to proove that 5=5 is false? Perhaps because you realize that the huge flaw in your position would be too obvious then. ?

5=5 is tautologous. It is a theorem of arithmetic.
It cannot be proven false, within classical logic.
That it is true shows that 5 exists.

5=5 -> Ex(x=5)
5=5 -> EF(F(5))

Existence is defined: E!x =df EF(Fx).

E!(ix: x is presently king of France) is false, because x is king of France is false for all x's.

ie. G(ix: x is presently king of France) is false for all predicates G.
ie. ~EG(G(ix: x is presently king of France)), is asserted.
ie. The present king of France does not exist.
There is no primary predicate that it has.

Witt

[John Page]

Originally posted by Witt
I believe it was Quine (Mathematical Logic) that introduced the notion of using quotation marks to express a name as opposed to expressing what is named.
Then a=a is false, whereas "a"="a" is true because the forms of the name are equivalent.

Either:

1. a is the object, in which case it cannot be identical (in all respects) to any other a. (LOI)

Or:

2. a is not the object in which case we're just comparing forms.

IMHO one must devise an understanding that avoids predication upon predication - that road leading to infinite regress and the issues that are concomitant with Tarski's theory of truth (in which case we never find the true object) or to Russell's Antinomy through the self-reference of a class being a member of itself.

Cheers, John

[tronvillain]

Hello again Witt. Part of the problem with this discussion is that we are actually arguing multiple positions simultaneously, and there is always the danger of confusing them. Anyway, back to the discussion:
I believe it was Quine (Mathematical Logic) that introduced the notion of using quotation marks to express a name as opposed to expressing what is named.

For example: "Vulcan" has 6 letters is true, but, Vulcan has 6 letters is false.
"tronvillain" has 11 letters is true, but, tronvillain has 11 letters is false.
They are different 'types' of objects.
Ah, so the quotation marks simply indicate the name rather than something like "logically possible but nonexistent objects." Seems pretty reasonable, except for the few occasions in the same discussion when quotation marks might be used for another reason.
If the description of a purported object is not contradictory then that object has possible existence.
Even possible(God exists) is true, if the definition of God is not contradictory.

"if leprechauns do not exist, then "an existent leprechaun" is as logically impossible as "a square circle":"

Of course, both terms do not refer to anything.
Their names/descriptions exist but what they refer to is vacuous.
Ah, but that is exactly my point: if "Vulcan" (this may be one of those occasions I mentioned) refers "an existent additional planet" and you assert that "Vulcan does not exist" then what "Vulcan" refers to is vacuous. It is gibberish, possessing no more meaning than do the words "a square circle" and as a result presents no problem to A=A.
A hypothetical object is one whose existence is presumed, ie, its existence must be possible.

But, the existence of a present king of France is not possible at this time.
*shrugs* Then call it a "logically possible object" if you like: a "present king of France" is an individual who could potentially exist if present day France had a monarchy. He is no different than a fictional character, and I hope that you will not attempt to disprove A=A using "Sherlock Holmes=Sherlock Holmes." It then follows that "the present king of France is bald" is not necessarily false.
The string of words, the+present+king+of+France, is equal to itself, ie. the description or the name exists but it does not have a reference or extension.

'the meaningless string of words' is a description of something which is meaningless. Any expression which is meaningless has no possible meaning, ie. there are no grammatical sentences that contain meaningless expressions. Including self-identity.
The present king of France, is not meaningless at all.
eg. The present king of France is bald, is false and not meaningless.
Ah, but if you take the position that "the present king of France" refers to absolutely nothing, rather than to a logically possible but nonexistent individual, then it has no more meaning than "hall to calendar automobile running by elevator cadence" does. Or perhaps when you say "the description or name exists" you are acknowledging it as a logically possible but nonexistent object? If something does not exist, most people take the name to refer to the description, but apparently you do not. Anyway, if you take that position you avoid the "meaningless string of words" defense, but you leave yourself wide open to the "square circle" counterattack I have already given.

I think that addresses everything relevant in your last post.

[John Page]

Originally posted by John Page
.....the issues that are concomitant with Tarski's theory of truth (in which case we never find the true object)...
Please let me clarify what I meant before someone jumps on this. I'm talking about Tarski's requirement for a meta language to explain/describe truth using formal objective criteria. IMO, language contains no such thing and a completely objective way of defining truth, so following Tarski's approach one would need a meta-meta language to formally and objectively define truth w.r.t. the meta language and then a meta-meta-meta language ad nauseum.

Cheers, john

[mikev]

IMHO :banghead: "A="

[Witt]

mikev,

What does "A=" mean?

Witt

[spacer1]

John Page,
...so following Tarski's approach one would need a meta-meta language to formally and objectively define truth w.r.t. the meta language and then a meta-meta-meta language ad nauseum.
I was planning to create a new discussion on this very topic, because this is exactly the problem I had noticed with Tarski's truth. However, I just assumed that I must have been missing something, given my lack of philosophical training.

[Trylan]

!@#$% = !@#$%

Completely true. Since the symbols have no meaning attached, they are equal in having no meaning.

[Witt]

Trylan: !@#$% = !@#$%

!@#$% = !@#$%, is not a well-formed-fomula.

It is not a statement which has the possibility of truth or of falsity, within classical logic.

Of course, if you want to defy classical logic, lots of luck.

Witt

[tronvillain]

Well, it is true that the string of symbols on the left is the same as the string of symbols on the right, regardless of whether the string of symbols refers to anything.

"the string of symbols !@#$% = the string of symbols !@#$%" is true.
"the string of symbols !@#$% = the string of symbls ?&*%!#" is false.

And "the string of symbols" would usually be assumed by anyone presented with something like "!@#$% = !@#$%"

[mikev]

Originally posted by Witt
mikev,

What does "A=" mean?

Witt


Witt,

I was trying to say (well, sortof~) that IMHO `no outcome of any sum will ever equal 'what is true' or 'truth' even... `

*I started on this 'big little story' trying to explain my vision on the subject but I'm afraid my english is not good enough to express my true meaning... I'm sorry I flushed it after writing it ..and so did not think it would have been a valuable post.. leaving me with this lame reply.. sorry.. I will try again later on..



Regards,

mikev :)

[Witt]

quote:
--------------------------------------------------------------------------------
Originally posted by Witt
mikev,

What does "A=" mean?

Witt
--------------------------------------------------------------------------------

mikev:

Witt,

I was trying to say (well, sortof~) that IMHO `no outcome of any sum will ever equal 'what is true' or 'truth' even... `

*I started on this 'big little story' trying to explain my vision on the subject but I'm afraid my english is not good enough to express my true meaning... I'm sorry I flushed it after writing it ..and so did not think it would have been a valuable post.. leaving me with this lame reply.. sorry.. I will try again later on..
------------------------------------------------------------------------------------

Hi mikev,

The expression "A=" is interesting.
Quine uses it to try and eliminate names in favor of described names. ie. (the x such that: (A=)x) = A.

(A=)x, is read: A=x.


I am sure that 'good english' is not a requirement for posting in this forum.
Most people here are very understanding about english as a second language.

Give it a go, I believe that you may have concerns that could be very interesting for many here.

mikev:
IMHO `no outcome of any sum will ever equal 'what is true' or 'truth' even...

Do you mean arithmetic sum (+) or do you mean logical sum (or)?

Could you expand on this point?

Witt

[Quantum Ninja]

Witt, I think what you're doing is trying to show that A=A is not always a sound argument. But logicians are only concerned with validity, not soundness. For instance, consider the following argument:

If fish are mammals, then the earth is flat.
Fish are mammals.
Therefore, the earth is flat.

That's a completely valid argument, even though the premises are pure bullshit (AKA false). When determining validity, we assume the premises are true, regardless of whether they are or are not.

I think the same applies when using inference rules. We assume that the individual constants are referring to existent objects, regardless if they actually exist or not. When someone says, A=A, you need to make the assumption that A exists. For instance, you are correct in saying that the present King of France does not exist, but if he were to exist, then the present King of France would equal the present King of France. Just like when we assume the premises true while determining validity, we assume the objects referred to by individual constants exist while applying inference rules.

All logic breaks apart, not just identity introduction (A=A), if you work under the conditions that the objects referred to by the individual constants don't exist. For instance, either the present King France is bald or the present King of France is not bald doesn't make very much sense if we assume he doesn't exist. Even though this statement is tautologous, it's basically meaningless if we suppose the present King of France does not exist. We need to assume the present King of France exists for the logic to apply (i.e. the hidden premise that the "present King of France exists" is true). So even though A=A may not always be sound, it is always valid.

[mikev]

Hi Witt! *and ofcourse everyone on this forum :)

This has always riddled me;
just that to me, (in respect to 'honest reason') it's an even more abstract principle to 'binary' find solution to 'what is 'true', wich seems commonly accepted. (therefor often frustrating me) formulating 'solution' even feels as if one doesn't see or respect truth or even 'chooses' to not accept 'truth' 'as is'; where "A=" should be enough to deal with, to 'try and comprehend' in 'true form'. Where 'Albert Einstein' calls 'his' "energy= equasion" relative.. relative to what? (spare me the (semi-valid) answer) Or the point where 'Dr. John Nash' did not agree with himself over...

"energy is. energy is everything, everything is energy." everything cannot be an X nor a Y. It just 'is' ..saying "= = true". and leaves only a singular 'truth' and so no formula is even needed to see/accept. "1=you, 2=me, 3=someone else, 4=more than" to me says: "3=1 : all > or < is irrelevant for no answer is 'needed'". If there's no problem, no solution is 'required'. requiring (re: validity) should be 'true', not a necessity to define.

I'm sorry I don't want to kill dialogue here.. on the contrary.. then again.. It seems dialogue, hence philosophy, often only is accepted if the binary principles most of us reason with/upon, are 'respected'. That to me feels unreal and immoral.

Don't get me wrong. I do respect 'try'
but when truth is 'tried' I think it's defying it's very nature.
You know: "live the question"; "experience nature, embrace what 'IS'" rather than seeking (dis)believe over relativities.

Example;
One can distinguish one color by another 'only' if there are more colors to compare the one color to define with...
then; that is physics. 'Truth' cannot be a physical 'something'. Can one imagine a color non existant to 'socalled proven' science? I know I can, but since I cannot physically (re-)create it I cannot prove it? But that doesn't mean it IS not to me. (re: respect). The whole concept of 'proof' undermines 'what is true'.


In reply to: "Witt: Do you mean arithmetic sum (+) or do you mean logical sum (or)?"
'Conditioned Logical assumption' I guess.. my point is in 'conditioned' (rudimentary~, evolutionary~ :: still 'less honest' than is possible)

--
re: 'good english'; it's just that im not sure of 'context and interpretation'. Also I'm dyslectic. (I think Speaking and Writing really is poor to express one's true meaning by (better alternatives may be; painting, making music, sculpturing and so on) As where art may overcome the often ingored evolution of the way humanity is communicating these days, and still ignoring 'honest reason'. Dyslexia is yet another good example; It's defined as 'malfunction' rather than 'a gift'. Also it has nothing to do with words or letters... (re: word-thinking/image-thinking) Who is to honestly define truth by formulae..?!

sheesh.. sorry to say but I dont feel as if I said what I wanted to say.. but in respect to dialogue I figured I'ld post this anyway. Maybe this way we can help to understand ourselves and widen horizons.. I had to start replying somewhere.. :) I guess a reply would be very helpful to me to sharpen my view for i know im missing some relevant points here..


Have a good one! Thanks!


Regards,

mikev

[Witt]

Quantum Ninja:

Witt, I think what you're doing is trying to show that A=A is not always a sound argument. But logicians are only concerned with validity, not soundness. For instance, consider the following argument:

If fish are mammals, then the earth is flat.
Fish are mammals.
Therefore, the earth is flat.

That's a completely valid argument, even though the premises are pure bullshit (AKA false). When determining validity, we assume the premises are true, regardless of whether they are or are not.
-------------------------------------


(fish are mammals) -> (the earth is flat),
&
(fish are mammals),
-> (the earth is flat).

This argument has the logical form: 1. ((p -> q) & p) -> q, and,
it is logically true for all values of truth or falsity that p or q may have. That is, the form is tautologous, or valid. It is true for all propositions p and q.
Your example is an instance of that logical form.

We do not 'need' to assume that the premises are true.
If ((p -> q) & p) is true, then, ((p -> q) & p) -> q is true, follows.
If ((p -> q) & p) is false, then ((p -> q) & p) -> q is true, follows.


Quantum Ninja:
We assume that the individual constants are referring to existent objects, regardless if they actually exist or not. When someone says, A=A, you need to make the assumption that A exists.

I agree, in classical first order logic, (x=x) is true for all x's ..by axiom. That is to say, (x exists) is true, for all x's.
The existence predicate is equivalent to the self-identity predicate, ie. E!x <-> x=x, for all x's.
(x exists) can be defined in first order logic as Ey(x=y), ie.
x exists, means, there is some existent thing that x is.

Ax(x exists) is an assumption of classical logic, because Ax(x=x) is an axiom.


Quantum Ninja:
For instance, you are correct in saying that the present King of France does not exist, but if he were to exist, then the present King of France would equal the present King of France.

Yes, E!x <-> x=x, is true even when x does not exist.
eg. E!(ix: ~(x=x)) <-> (ix:~(x=x))= (ix:~(x=x)), is true.

E!(ix:Fx) <-> (ix:Fx)=(ix:Fx), is a theorem of first order classical logic.

Quantum Ninja:
Just like when we assume the premises true while determining validity, we assume the objects referred to by individual constants exist while applying inference rules.

Not so, we assume the objects referred to by individual constants exist .. in virtue of the implicit axiom AxE!x, ie. Ax(x=x).
In 'free' logic Ax(x=x) is not true, but, Ex(x=x) and Ex~(x=x) are both true.

Quantum Ninja:
All logic breaks apart, not just identity introduction (A=A), if you work under the conditions that the objects referred to by the individual constants don't exist. For instance, either the present King France is bald or the present King of France is not bald doesn't make very much sense if we assume he doesn't exist. Even though this statement is tautologous, it's basically meaningless if we suppose the present King of France does not exist. We need to assume the present King of France exists for the logic to apply

I don't agree here.
As you say, either the present King France is bald or the present King of France is not bald, is tautologous. And therefore, it is not meaningless!
We do not assume that the present king of France does exist at all, indeed, we know that it does not exist.

The present king of France exists, is false.
(The present king of France)=(The present king of France), is false.
Ey((The present king of France)=y), is false.

Quantum Ninja:
(i.e. the hidden premise that the "present King of France exists" is true). So even though A=A may not always be sound, it is always valid.

There is no such hidden premise, as I see things.

Witt

[tronvillain]

*sigh* Apparently I must keep repeating myself.

Witt:The present king of France exists, is false.
(The present king of France)=(The present king of France), is false.
Ey((The present king of France)=y), is false.
Sorry, but that simply does not follow. What do you mean when you say "the present king of France"? Obviously you are referring to a logical possibility "an individual who would exist if present day France was a monarchy" or something to that effect. If that is all you mean when you say "the present king of France", then while you are justified in saying "The present king of France exists, is false" you are not justified in saying "(The present king of France)=(The present king of France), is false." After all, saying that is no different than saying "(Sherlock Holmes)=(Sherlock Holmes), is false.

Or perhaps when you say "(The present king of France)=(The present king of France), is false." you really mean "(The present king of France who exists)=(The present king of France who exists), is false." Of course, that is easily countered by pointing out that "the present king of France who exists" is complete gibberish since present day France is not a monarchy ("the present king of France" has meaning, "the present king of France who exists" does not).

:rolleyes:

[Witt]

tronvillain:

*sigh* Apparently I must keep repeating myself.

Witt:

quote:
--------------------------------------------------------------------------------
The present king of France exists, is false.
(The present king of France)=(The present king of France), is false.
Ey((The present king of France)=y), is false.
--------------------------------------------------------------------------------


Sorry, but that simply does not follow. What do you mean when you say "the present king of France"? Obviously you are referring to a logical possibility "an individual who would exist if present day France was a monarchy" or something to that effect. If that is all you mean when you say "the present king of France", then while you are justified in saying "The present king of France exists, is false" you are not justified in saying "(The present king of France)=(The present king of France), is false." After all, saying that is no different than saying "(Sherlock Holmes)=(Sherlock Holmes), is false.

Or perhaps when you say "(The present king of France)=(The present king of France), is false." you really mean "(The present king of France who exists)=(The present king of France who exists), is false." Of course, that is easily countered by pointing out that "the present king of France who exists" is complete gibberish since present day France is not a monarchy ("the present king of France" has meaning, "the present king of France who exists" does not).
----------------------------------------------------



Witt:

That which is defined or described by a contradictory predication cannot exist.

Proof:

The present king of France does not exist, because there is no king of France at this time.

Surely, if there are no king of France then there cannot be a unique king either, can there?

(the present king of France)=(the present king of France), means,

There is something such that it is king of france if and only if it is unique and it's equal to that thing.

Ey(Ax(x=y <-> x is king of France) & y=y)

But, x is king of France ..is not satisfied by any existent object at this time.
ie. x is king of France is false for all x's.

Therefore, Ey(Ax(x=y <-> x is king of France) & y=y) <-> Ax(x=y <-> contradiction) & y=y).

Since Ay(y=y) is an axiom of classical logic.

Ey(Ax(x=y <-> contradiction) <-> EyAx~(x=y)

But, ~AyEx(x=y) is a contradiction. ie. AyEx(x=y) is a theorem.

Therefore, ~((the present king of France)=(the present king of France)), can be asserted.
--------------------------------------

*sigh*

Let me try to show the proof in a different way.

It is a known fact that, there cannot be a king of France at this time.
~Ex(x is king of France) can be asserted.

T1. x is king of France, is contradictory, for any x

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

1. Exists(the present king of France) <-> Ey((Ax(x=y <-> x is presently king of France) & Exists(y)), by D1.

2. Exists(the present king of France) <-> Ey((Ax(x=y <-> contradiction) & Exists(y)), by T1.

3. Exists(the present king of France) <-> Ey(Ax~(x=y) & Exists(y)), by (p <-> contradiction) <-> ~p.

4. Exists(the present king of France) <-> Ey(~Ex(x=y) & Exists(y)), by Ax~Fx <-> ~ExFx.

5. Exists(the present king of France) <-> Ey(contradiction & Exists(y)), by the theorem Ex(x=y).

6. Exists(the present king of France) <-> (contradiction), by (p & contradiction) <-> contradiction.

7. ~(Exists(the present king of France)),
by (p <-> contradiction) <-> ~p.

Q.E.D.

~((the present king of France)=(the present king of France)),
and,
~Ex((the present king of France)=x),
are proven in the same way.

Witt

[John Page]

Originally posted by Witt
Ey(Ax(x=y <-> x is king of France) & y=y)

But, x is king of France ..is not satisfied by any existent object at this time.
ie. x is king of France is false for all x's.

Witt:

Excuse me butting in, but your equivocation about x seems clear.

You have defined "x is the King of France" as not being satisfied by any object at this time. However, you then go on to conclude that x is the King of France is false for all x's!!

Two observations:
1. The King of France did exist at some times and must therefore be some x's.
2. You have discussed the "King of France" as a logical possibility. Well then, that is exactly what the object is, a mental object which in fact does exist that is defined in such a way that it has no physical corollary (because, for example, France is presently a Republic). To determine the truthfulness of "the existence of the present King of France" we compare in our minds how such a thing could be so and, being unaware of the restoration of Louis the 69th, we say that such a person is imaginary only.

Cheers, John

[Witt]

quote:
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Originally posted by Witt
Ey(Ax(x=y <-> x is king of France) & y=y)

But, x is king of France ..is not satisfied by any existent object at this time.
ie. x is king of France is false for all x's.

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Witt:

Excuse me butting in, but your equivocation about x seems clear.

You have defined "x is the King of France" as not being satisfied by any object at this time. However, you then go on to conclude that x is the King of France is false for all x's!!

Two observations:
1. The King of France did exist at some times and must therefore be some x's.
2. You have discussed the "King of France" as a logical possibility. Well then, that is exactly what the object is, a mental object which in fact does exist that is defined in such a way that it has no physical corollary (because, for example, France is presently a Republic). To determine the truthfulness of "the existence of the present King of France" we compare in our minds how such a thing could be so and, being unaware of the restoration of Louis the 69th, we say that such a person is imaginary only.
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(x is the King of France) is not satisfied by any object at this time, if and only if, (x is the King of France) is false for all x's.

~Ex(x is the present king of France) <-> Ax~(x is the present king of France), is an instance of the logical theorem: ~ExFx <-> Ax~Fx.

Where is your assumed equivocation.

That there were kings of France in the past, and, that there might be a king of France in the future, does not alter the known truth that there cannot be a king of France now.

The description does exist but the person who is king of France at the present time does not exist.

It is not a fictional character.

Witt

[tronvillain]

Interesting. Except that you completely failed to answer my question. What do you mean when you say "the present king of France"? Give me a direct answer to that question!

Now, as I see it there are two possible answers (if one of these is not your answer then please explain what your answer actually is) you can give:

1)"What I mean when I say 'the present king of France' is something along the lines of 'an individual who would exist if present day France was a monarchy' or in other words I am referring to a hypothetical individual."

2)"What I mean when I say 'the present king of France' is 'the present king of France who exists."

Of course, neither of these two answers present a problem for A=A:

1) If that is all you mean when you say "the present king of France", then while you are justified in saying "The present king of France exists, is false", you are not justified in saying "(The present king of France)=(The present king of France), is false." After all, saying that is no different than saying "(Sherlock Holmes)=(Sherlock Holmes), is false." A hypothetical individual is essentially a fictional character.

2) If that is what you mean when you say "the present king of France", then you do not really mean anything since if no present king of France exists then "the present king of France who exists" refers to nothing and therefore has no more meaning than a string of letter like "xydfmb." I trust that you will not attempt to disprove A=A with "xydfmb=xydfmb." *chuckle*

Either way, your proofs fall apart at their very first step. Now, answer the question.

[Witt]

tronvillain: Interesting. Except that you completely failed to answer my question. What do you mean when you say "the present king of France"? Give me a direct answer to that question!

'The present king of France' has no meaning. It has no extension at all. Is that direct enough for you?
The description 'the present king of France' cannot exist at this time.
The description 'the present king of France' attempts to refer to an existent person, but it does not (now) refer to any-thing at all.

"*sigh* Apparently I must keep repeating myself."

tronvillain:
Now, as I see it there are two possible answers (if one of these is not your answer then please explain what your answer actually is) you can give:
1)"What I mean when I say 'the present king of France' is something along the lines of 'an individual who would exist if present day France was a monarchy' or in other words I am referring to a hypothetical individual."

A king of France, or, the king of France, refers to an existent person only if France is a monarchy.
When France is not a monarchy then obviously there cannot be a king of France, nor can there be the king of France.
It cannot be a hypothetical individual, at this time.
It is a hypothetical object at some future date.
A king is possible, only when kingship is possible.

tronvillain:
2)"What I mean when I say 'the present king of France' is 'the present king of France who exists."

The present king of France, does not exist.
The existent present king of France, does not exist.

tronvillain: Of course, neither of these two answers present a problem for A=A:

Apparently you still do not understand the meaning of,
(the present king of France)=(the present king of France).

(ix:Fx)=(ix:Fx) means Ey(Ax(x=y <-> Fx) & y=y), by definition!
That is, (the present king of France)=(the present king of France)
means, Ey(Ax(x=y <-> x is presently king of France) & y=y).

Since, Ey(Ax(x=y <-> x is presently king of France) & y=y), is contradictory, we can assert: ~((the present king of France)=(the present king of France)).

tronvillain: 1) If that is all you mean when you say "the present king of France", then while you are justified in saying "The present king of France exists, is false", you are not justified in saying "(The present king of France)=(The present king of France), is false."

Wrong. It is a theorem of classical logic with identity that:
~E!(ix:Fx) <-> ~((ix:Fx)=(ix:Fx)).

See: Principia Mathematica, Russell and Whitehead, *14.28, page 184.
See: Introduction to Symbolic Logic, Rudolf Carnap, page 144.
See: Methods of Logic, W. V. Quine, page 274.
etc.

tronvillain: 2) If that is what you mean when you say "the present king of France", then you do not really mean anything since if no present king of France exists then "the present king of France who exists" refers to nothing and therefore has no more meaning than a string of letter like "xydfmb." I trust that you will not attempt to disprove A=A with "xydfmb=xydfmb." *chuckle*

I agree that, xydfmb=xydfmb is gibberish, and it is neither true nor false.

(The present king of France)=(The present king of France), Is false.

Why do you want to identify nonsense with falsity?

tronvillain: Either way, your proofs fall apart at their