Witt

Is that which is not equal to itself...
...equal to itself or not?

1. In virtue of meaning, it cannot be equal to itself.

2. By the definition of identity..x=y defined Fx<->Fy, for all F..it must be equal to itself.

(the x:~(x=x))=(the x:~(x=x)) defined
F(the x:~(x=x)) <-> F(the x:~(x=x)) for all F.

But, F(the x:~(x=x)) <-> F(the x:~(x=x)) is tautologous for all F..therefore it is equal to itself, isn't it?

What do you think about this puzzle?

Witt

onceuponapriori

"Is that which is not equal to itself...
...equal to itself or not?"

I'd have to say that the question is nonsensical, as it violates the Law of Identity.

Witt

"Is that which is not equal to itself...
...equal to itself or not?"

onceuponapriori:
I'd have to say that the question is nonsensical, as it violates the Law of Identity.

It is a well-formed-fomula, therefore it is not nonsense.

Frege, Carnap, Montague, etc. claims that (the x:~(x=x)) exists and that it equals the null object.

Each claim, (the x:~x=x)=(the x:~(x=x) is a theorem.


Russell states that (the x:~(x=x)) does not exist and that:
~((the x:~(x=x))=(the x:~(x=x))) is a theorem.

It appears to depend on how we define the definite article 'the'.

How do you define (the x such that Fx)?

Witt

Scrambles

This is a rehash of the last thread on identity.

On Denoting


'it is not always false of x that x is not identical with itself and that "if y is not identical to itself then y is identical with x" is always true of y'

'it is not always false of x that x is not identical with itself' fails.



Scrambles

Scrambles

Also, if "that which is not equal to itself" is allowed to exist, then all predicates applied to it apply to it, so it is equal to itself. But by definition it is not equal to itself. This violates the law of contradiction.

Allowing it to be the null object is a little artificial. In that case "that which is not equal to itself" means something totally different to what it denotes.


Scrambles

Witt

Scrambles:
'it is not always false of x that x is not identical with itself and that "if y is not identical to itself then y is identical with x" is always true of y'

'it is not always false of x that x is not identical with itself' fails.

??

~Ax(~(x=x)) <-> Ex(x=x). I agree that x=x for some x.

The law of identity states: x=x for all x.
If proper names are to represent the values of the variable x, then:
(all x:x=x) -> (Scrambles=Scrambles & 2=2 & Vulcan=Vulcan ...)

But, Vulcan=Vulcan is false. (because it does not exist)

Therefore ~(all x: x=x), i.e. there is some value of x that is not self-identical, eg. Vulcan, God, Pegasus, the x such that ~(x=x), the present king of France, etc.

Witt

Scrambles

'it is not always false of x that x is not identical with itself' is equivalent to,
(Ex)(~(x = x)).

If you can apply predicates to Vulcan it certainly is true (because of course P(Vulcan) <---> P(Vulcan) for any predicate). And I think that you believe it is correct to apply predicates to Vulcan.

Russell (sort of) says that 'Vulcan=Vulcan' is false because it does not exists. But 'Vulcan=Vulcan' in this case is short-hand for:

(Ex)(Vulcan(x) & (y)[Vulcan(y) --> x=y] & x=x).

The statement above is false because Vulcan(x) is never satisfied (where Vulcan(x) means that x is the planet necessary under Newtonian mechanics). In other words 'Vulcan' is a
denoting phrase which does not denote any value of x. Neither does 'the present king of France' which, I believe, Russell makes abundantly clear in his essay.

So basically, if you really want to state 'Vulcan = Vulcan', it is nonsensical because Vulcan does not refer to anything. Russell has a way of making statements like this make sense, which results in the statement above.

Do you have some vested interest in doing away with identity?



Scrambles

Big Spoon

Sounds to me like you're evoking Russell's paradox.

Witt

Big Spoon: Sounds to me like you're evoking Russell's paradox.

Russell's paradox involves the membership predicate.

That which is not equal to itself, does not.

Witt

John Page

Yes, but Vulcan=Vulcan is either false (as you interpret above), a contradiction (Vulcan(x)=Vulcan(y)), or a tautological axiom (Vulcan). Now what did Aristotle say....?
Nonsensical "Vulcan" = Vulcan(<null>) or an empty set if you wish or arbitrary form.

On the link to Russell's Descriptions you gave, he states that the third type of denoting is "ambiguous". He gives the example of "man" - but this is just a form like any other denoted thing! (Note: Even what one might claim to be a specific unique instance of a thing is always with reference to something else. For a face to appear before our eyes it must persist over time, for example).

I prefer that a name denotes a form and there may be many experiences that fit that form. We may intersubjectively agree or come to understand the form(s) which a name may denote - which is why context is so important in language.

Now, the concept that we now have of form is of something permanently etched against which we can compare our experiences. However, forms are not inviolate so while one may make an assumption like "It is true that all A's are equal" if we wander out of the context a different A may creep in. The reason I mention that forms can vary is to anticipate using A or some other symbol to denote form and yell "Eureka" here is our reference value - this doesn't work because there may be many instances of a form of a thing (at least one per mind I guess). The reason we're having this thread is a need to indulge in the social activity of agreeing what the hell we're talking about.

IMO, our forms are templates, axiomatic concepts stored in our mind/brains. If it were not so, we would need small nameplates attached to everything in reality (external to the mind) so we knew their names and what they looked like.

Cheers, John