?? Functional Dependency Question ??

On Oct 22, 7:20 pm, JOG wrote:
> On Oct 22, 6:12 am, David BL wrote:
>
> > > Bob Badour wrote:
>
> > > > ... If you changed the
> > > > question slightly to "When there are no purple parts, which suppliers
> > > > supply all of the purple parts?" then the answer would be "all of them".
>
> > Reminds me of first year Uni, when my maths lecturer asked us whether
> > every hippopotamus in the room was wearing pink panties.
>
> Which is it then logicists?
>
> Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ] = false
>
> Ax [ inRoom(x) ^ isHippo(x) -> Wearing(x, pink panties) ] = true

I'm no logicist but I think only the second is a faithful translation.

Otherwise, wouldn't statements like

every frog is an animal

be deemed false?
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Bob Badour wrote:
....
>
> I think the answer is actually "none of them". If you changed the
> question slightly to "When there are no purple parts, which suppliers
> supply all of the purple parts?" then the answer would be "all of them".
>

Maybe a useful principle to follow when phrasing db queries out loud is
to preface all possible parameters with one of the adjectives "all" or
"any".
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paul c wrote:
> Bob Badour wrote: ...
>>
>> I think the answer is actually "none of them". If you changed the
>> question slightly to "When there are no purple parts, which
>> suppliers supply all of the purple parts?" then the answer would be
>> "all of them".
>>
>
> Maybe a useful principle to follow when phrasing db queries out loud
> is to preface all possible parameters with one of the adjectives
> "all" or "any".

or "none of" which might not add any capability but which might make the
phrasing more amenable. People who are prone to long sentences, me
included, must remember that the rm is what it is, as they say, and no
more, so we are on safer ground if we try to talk in language that
parallels what it is capable of. There are many honourable modes of
language and abstractions of thought that are important to us, but the
RM claims to be useful for organizing only a small part.


A quote I like from SICP:

The acts of the mind, wherein it exerts its power over simple ideas, are
chiefly these three: 1. Combining several simple ideas into one compound
one, and thus all complex ideas are made. 2. The second is bringing two
ideas, whether simple or complex, together, and setting them by one
another so as to take a view of them at once, without uniting them into
one, by which it gets all its ideas of relations. 3. The third is
separating them from all other ideas that accompany them in their real
existence: this is called abstraction, and thus all its general ideas
are made.

John Locke, An Essay Concerning Human Understanding (1690)
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On Oct 22, 7:20 pm, JOG wrote:
>
> Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ] = false

Is this actually the case? There doesn't appear to be any defined set
over which the universal quantification is defined, so I think the
left hand side is meaningless not false.

I think a meaningful universal quantification must be able to be
written in the form

Ax [ P(x) -> Q(x) ]

where { x | P(x) } is a well defined set.

We can write

Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]

as

Ax [ true -> inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]

So it seems that we must define P(x) = true, but then { x | P(x) } is
the set of all things which is meaningless.
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On Oct 23, 2:23 am, David BL wrote:
> On Oct 22, 7:20 pm, JOG wrote:
>
>
>
> > Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ] = false
>
> Is this actually the case?  There doesn't appear to be any defined set
> over which the universal quantification is defined, so I think the
> left hand side is meaningless not false.
>
> I think a meaningful universal quantification must be able to be
> written in the form
>
>     Ax [ P(x) -> Q(x) ]
>
> where { x | P(x) } is a well defined set.
>
> We can write
>
>     Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> as
>
>     Ax [ true ->  inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> So it seems that we must define P(x) = true, but then { x | P(x) } is
> the set of all things which is meaningless.
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Apologies for double click posting.

On Oct 23, 2:23 am, David BL wrote:
> On Oct 22, 7:20 pm, JOG wrote:

> > Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ] = false
>
> Is this actually the case? There doesn't appear to be any defined set
> over which the universal quantification is defined, so I think the
> left hand side is meaningless not false.
>

While you're right, could you not read that first statement as:
Ax [Exists(x) -> inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]

How do you like them apples! Either way, false or meaningless, its
clearly not what the lecturer was saying so the second statement was
the banker:
Ax [inRoom(x) ^ isHippo(x) -> Wearing(x, pink panties) ]

which is true (material implication is always true if the antecedent
is false). Hence, as far as formal logic is concerned all the hippos
in the room are indeed wearing pink panties. And blue panties too in
fact. And no panties as well...

> I think a meaningful universal quantification must be able to be
> written in the form
>
> Ax [ P(x) -> Q(x) ]
>
> where { x | P(x) } is a well defined set.
>
> We can write
>
> Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> as
>
> Ax [ true -> inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> So it seems that we must define P(x) = true, but then { x | P(x) } is
> the set of all things which is meaningless.

Why is the "set of all things" meaningless?
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On Oct 27, 9:36 am, JOG wrote:
> Apologies for double click posting.
>
> On Oct 23, 2:23 am, David BL wrote:
>
> > On Oct 22, 7:20 pm, JOG wrote:
> > > Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ] = false
>
> > Is this actually the case? There doesn't appear to be any defined set
> > over which the universal quantification is defined, so I think the
> > left hand side is meaningless not false.
>
> While you're right, could you not read that first statement as:
> Ax [Exists(x) -> inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]

Assuming that someone has properly defined these relations, and in
particular Exists(x) (or equivalently its extension is well defined),
that translation would presumably be false.


> How do you like them apples! Either way, false or meaningless, its
> clearly not what the lecturer was saying so the second statement was
> the banker:
> Ax [inRoom(x) ^ isHippo(x) -> Wearing(x, pink panties) ]
>
> which is true (material implication is always true if the antecedent
> is false). Hence, as far as formal logic is concerned all the hippos
> in the room are indeed wearing pink panties. And blue panties too in
> fact. And no panties as well...
>
>
>
>
>
> > I think a meaningful universal quantification must be able to be
> > written in the form
>
> > Ax [ P(x) -> Q(x) ]
>
> > where { x | P(x) } is a well defined set.
>
> > We can write
>
> > Ax [ inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> > as
>
> > Ax [ true -> inRoom(x) ^ isHippo(x) ^ Wearing(x, pink panties) ]
>
> > So it seems that we must define P(x) = true, but then { x | P(x) } is
> > the set of all things which is meaningless.
>
> Why is the "set of all things" meaningless?

An axiom of set theory states that it is always meaningful to define a
subset of a given set according to some well defined predicate.

ie the axiom states that given set S, and predicate f, { x in S |
f(x) } is a well defined set.

In particular we can take the subset of the extension of P(x) defined
as the set of all sets that are not members of themselves, leading to
Russell's paradox.
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