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Re: PREDICATE CALCULUS 2
Posted:
Nov 28, 2012 2:41 PM
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On Nov 29, 1:13 am, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:
> On Nov 28, 1:35 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
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> > 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:
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> > > > 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.
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> > > > isfunction(r) <- ALL(Y1) ALL(Y2) r(X,Y1)^r(X,Y2) -> Y1=Y2
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> > > 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).
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> > > > {(Y1,Y2)|r(X,Y1)^r(X,Y2)} C {(Y1,Y2)|r(X,Y1)^r(X,Y2)->Y1=Y2}
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> > > [snip]
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> > > The same comment applies... I think.
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> > > Anyway, I am still waiting for proofs of the following:
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> > > 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
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> > > You really need to address these fundamental results.
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> > > 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.
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> > Nope, this is exactly the definition of function.
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> > {(Y1,Y2) | r(X,Y1)^r(X,Y2)} C {(Y,Y) | r(X,Y)}
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> > which simply guarantees only 1 Y value for any X value.
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> 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):
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> "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
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> Example: Let X=Y={0,1}.
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> Then {(0,0), (1,1)} is function from X to Y, while {(0,1)} is not.
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Semantics escapes you!
{(0,1)} is a function.
You can return the domain of r(X,Y)
with another predicate.
Herc
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