
Re: PREDICATE CALCULUS 2
Posted:
Nov 28, 2012 10:13 AM


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

