John Page

Scrambles:

Can you direct me to how to understand the notation used here?

Thanks, john

Scrambles

John Page,

(ix : phi(x)) is (the x such that phi(x))

Any expression psi(ix : phi(x)) is taken as
(Ex)[(y)(phi(y) <-> y=x)] & psi(x).


Scrambles

Witt

Can you direct me to how to understand the notation used here?
Thanks, john

IMO, Scrambles' original reference, On denoting - Bertrand Russell, is the best source of Russell's theory of descriptions.

Also:
Russell's Introduction to Mathematical Philosophy, chapter XIV.
Principia Mathematica, chapter *14.
Quine, Methods of Logic, chapter 43.
Carnap, Introduction to Symbolic Logic, etc..

The key idea is Russell's 'contextual definition'.

His claim is that: (the x such that Fx) can only be known through a context.

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

G(ix:Fx) <-> Ey(Ax(x=y <-> Fx) & Gy).
G(ix:Fx) <-> Ey(Ax(x=y -> Fx) & Ax(Fx -> x=y) & Gy).
G(ix:Fx) <-> Ey(Fy & Ax(Fx -> x=y)) & Gy).
G(ix:Fx) <-> (ExFx & AxAy((Fx & Fy)-> x=y) & Ax(Fx -> Gx)).
G(ix:Fx) <-> (ExFx & AxAy((Fx & Fy)-> x=y) & Ex(Fx & Gx)).

If (ix:Fx) can be shown to exist, i.e. to be unique, then it is a value of the individual variable. That is to say it is an object.

Due to the special treatment required for descriptions, eg. AxGx -> G(ix:Fx) is not valid unless (ix:Fx) exists, many authors simply avoid descriptions unless they are known to exist, in which case they are treated as objects.

They are 'complete' symbols if they are unique!

Witt

Primal

Oh Witt, you are right, that was Peter. My mistake. So Peter what did you mean?

Primal

That's not logical deduction but mere axiom accepted at face value. Also I don't think you have properly stated the law of noncontradiction, it simply says:

A cannot equal non-A. I don't see how that is equivocation.

John Page

I was pointing out that the process of deduction requires equivocation, not the LNC itself. Logical deduction is defined as the process of reasoning by which a conclusion necessarily follows from its premises.

Let "Socrates is a man. All men are mortal. Socrates is mortal" be the propositions A, B and C.

One therefore looks for inconsistencies between A, B and C. In going through this deductive process I am saying that one must necessarily examine the two cases C is T or C is ~T. This is equivocation of C.

The way I see it (which may be wrong, of course) is that we knowingly "lie" to ourselves about C (by implication under the laws of propositional logic because C cannot both be true and false for the same circumstances) in order to be sure of its actual value.

Cheers, John

John Page

OK, this seems to permit substitution where the subject is deemed to have common qualities with the entity representing it (even though we know they're not identical). This is consistent with the propsed existence of a Comparison/Detection process through which representational entities are formed (necessary for us to propose "Let x=y" for example.)
But the object will never be the variable, the variable is the one or more descriptions required to identify a thing-in-itself that is admitted into the epistemological realm through a Comparison/Detection process.
I can copy a symbol, so how can it be unique?

I understand the need to treat a thing-in-itself (object) differently than a description. The issue is, I feel, how do we know an (ix:Fx) exists and, if so, how can we show that it is not just a description.

Cheers, John

Witt

Witt:

The key idea is Russell's 'contextual definition'.

His claim is that: (the x such that Fx) can only be known through a context.

D1. G(the x:Fx) defined Ey(Ax(x=y <-> Fx) & Gy).
....
If (ix:Fx) can be shown to exist, i.e. to be unique, then it is a value of the individual variable. That is to say it is an object.

Witt: Due to the special treatment required for descriptions, eg. AxGx -> G(ix:Fx) is not valid unless (ix:Fx) exists, many authors simply avoid descriptions unless they are known to exist, in which case they are treated as objects.

They are 'complete' symbols if they are unique!
-----------------------------------------------------------


John: I can copy a symbol, so how can it be unique?

You seem to be confusing a name with what the name refers to.

x=x means the object named x is identical to itself.
"x"=x is false, i.e. the object named is not its name.

We are concerned about the object's uniqueness not the name's uniqueness.

3=3 means the number represented by the numeral (name) 3 is equal to itself. That the left hand side is in a different position on the page than the right hand side is irrelevant, they both represent the same object.

1+2=4-1=3 all talk about the same identical object..the number 3.

John: I understand the need to treat a thing-in-itself (object) differently than a description. The issue is, I feel, how do we know an (ix:Fx) exists and, if so, how can we show that it is not just a description.

(ix:Fx) exists, iff, there is one and only one x such that Fx.
(ix:Fx) exists, iff, (x=y <-> Fx) for all x's.

The reference of a described object is the object referred to, if any.
The sense of a described object is the predicate used to describe it.

If the described object has no reference, eg. the present king of France, then it only refers to its sense.

(the present president of the US)=(the previous govenor of Texas)=(the husband of Laura Bush)=(G. W. Bush), all name the same individual (unique) object.

Non-existent descriptions have sense but no reference.

The object referred to by a contradictory predication does not exist!

We do refer to an object by an existent description or by its name.

Witt

Primal

John please explain what you mean by A,B and C. And T, ~T.

Like give an example using normal language.

Also if you could, would you not use any such symbols at all. I find it somewhat hard to follow when you do that instead of using plain English.

John Page

Witt:

Thanks again. I think I'm understanding the rules, but I'm having trouble making them fit with reality.
This is the terminology that threw me initially. Surely a symbol is something that is used to refer to something else (in the pictogram sense, perhaps) and even in information theory a symbol is a unit of information i.e. it implicitly represents something else.

Now, let me suggest that we only know of things through something called the mind. Our mind can be used to apply the system of logic to things that it knows about - some of which may be objects and others no.

I'm having rather an entertaining conversation with Tyler Durden here who is using someone else's text to argue nominalism.

It strikes me that a set is defined by its criteria for membership and that the set itself is a real object contained within the mind. example, there is stuff outside the mind, some of which we might call an instance of a chair. The truthfulness of the statement "A chair has four legs" has no bearing upon the existence of any chair - it needs to be tested against the criteria laid down (in the mind) for something to be considered a chair.

To expand a little further, a set as defined here is a purely mental entity and is a collection of criteria against which sense impressions (not necessarily raw sense impressions) are tested. For example a mind might determine if an object is a chair by examining whether a human body could reasonably rest on it (functional criteria) and whether it had legs (substance criteria).

Going back to your OP, then, A represents an abstract entity with A'ness. You have the concept of A in your head, I have the concept of A in my head. When we agree A=A, then, we are intersubjectively agreeing the concept A. This being the case, the identity of A is not unique because there are many copies of it.

For these reasons I am uncomfortable with the description theory - (ix:Fx) definitely exists but its form can only be mental.

Cheers, John