Date: Nov 28, 2012 10:13 AM Author: Dan Christensen Subject: Re: PREDICATE CALCULUS 2 On Nov 28, 1:35 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> On Nov 28, 2:26 pm, Dan Christensen <Dan_Christen...@sympatico.ca>

> wrote:

> > On Nov 27, 8:59 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

>

> > > There are 2 ALLs which is more complicated but you can format it as a

> > > SUBSET using a cartesian product of the 2 X values with a common Y.

>

> > > isfunction(r) <- ALL(Y1) ALL(Y2) r(X,Y1)^r(X,Y2) -> Y1=Y2

>

> > There is more to functionality than this. I may not fully understand

> > your unusual notation (PROLOG?), but it would seem you have left out

> > the requirement that FOR ALL elements of some domain set, THERE EXISTS

> > a unique image in a codomain set. (This is where I think quantifiers

> > become indispensable).

>

> > > {(Y1,Y2)|r(X,Y1)^r(X,Y2)} C {(Y1,Y2)|r(X,Y1)^r(X,Y2)->Y1=Y2}

>

> > [snip]

>

> > The same comment applies... I think.

>

> > Anyway, I am still waiting for proofs of the following:

>

> > 1. {(x,y) | x in S, y=x} is a function mapping the set S onto itself

> > 2. {x | ~x in x} cannot exist

> > 3. {x | x=x} cannot exist

>

> > You really need to address these fundamental results.

>

> > For what it is worth, and from what little I know about PROLOG, it

> > doesn't seem to be capable of all that is required to do mathematical

> > proofs in general. It may be able to model some interesting and useful

> > aspects of predicate logic and set theory, but, for your purposes,

> > important pieces of the puzzle seem to be missing.

>

>

> Nope, this is exactly the definition of function.

>

> {(Y1,Y2) | r(X,Y1)^r(X,Y2)} C {(Y,Y) | r(X,Y)}

>

> which simply guarantees only 1 Y value for any X value.

>

This is a common mistake. According to this erroneous view, every set

{(x,y)} is a function for any objects x and y. The functionality of a

set of ordered pairs is always defined in terms of a domain and

codomain set. Here is a typical formal(ish) definition of a function

from Wiki (my comments in []'s):

"A function f from X [the domain of f] to Y [the codomain of f] is a

subset of the Cartesian product X × Y subject to the following

condition: every element of X is the first component of one and only

one ordered pair in the subset.[3] In other words, for every x in X

there is exactly one element y such that the ordered pair (x, y) is

contained in the subset defining the function f."

http://en.wikipedia.org/wiki/Function_(mathematics)#Definition

Example: Let X=Y={0,1}.

Then {(0,0), (1,1)} is function from X to Y, while {(0,1)} is not.

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com