Re: Probable Influence/Causation Versus Algebraic Geometry/Topology 2: PI vs Scheme
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 10 Feb 2006 23:50:31 -0800
From Osher Doctorow mdoctorow@xxxxxxxxxxx
But isn't (A-->B) just another set equal to A' U B, the union (and/or)
of A' (the complement of A or the part of the Universe outside A)
and/or B, where A, B are sets?
(A-->B) is a set, but it's a rather curious type of set just as the
Universe is a rather curious type of set. Its closest relative, in fact
"almost an isomorph or homomorph" is the logical conditional/logical
implication (a-->b) where a, b are propositions.
Intuitively, (a-->b) means "if a then b". "Rigorously", it is defined
as:
1) (a-->b) = ~(a ^ ~b) = ~a V b
where ~ is negation ("not"), ^ is conjunction ("and"), V is disjunction
("and/or"). Notice the analogy:
2) (A-->B) = (AB' )' = A' U B
Many scientists and mathematicians have a dislike of logic because in
philosophy logic is often taught in an unmathematical form which is
either over-wordy or both over-wordy and under-wordy and often
obscuring or leaving out interconnections (a lot of over-wordy people
become under-wordy when testing others, both consciously and
subconsciously at different times).
I'd recommend teaching logic literally from (1) and (2) above applied
for example to physics.
But to return to (a-->b), it differs enormously from a, b, ~a, a ^ b, a
V b. Looking first at ~a, "not a", let's say that a is the
proposition: A tree with green leaves exists. Then ~a says: A tree
with green leaves does not exist. If b is "John is a man," then a V b
is: either a tree with green leaves exists or John is a man or both.
And a ^ b is: both a tree with green leaves exists and John is a man.
But now look at (a-->b). This says: if a tree with green leaves
exists, then John is a man. This ties together a and b much more
closely than ~a, a V b, and a ^ b. It indicates a "dependency" of b
on a which is very close to that in conditional probability and
Probable Influence/Causation (PI). True, it is not terribly far from a
^ b which says that a and b occur together, but since a and b need not
occur together if a depends on b, it goes beyond a ^ b. It does have
a ^ b as a special case.
The closest thing to a ^ b is correlation, just as Probable Correlation
is a special case of PI and in fact Probable Correlation is P(X<-->Y)
in which set intersections are analogous to logical conjunction. Recall
that A <--> B is defined as (A-->B)(B-->A) where adjacent parentheses
are intersected, and this reduces to AB U A' B'.
This relationship between Correlation and conjunction and intersection
and (Probable) Causation is completely missed in philosophical logic,
in mainstream mathematical logic, in mainstream probability-statistics,
etc.
Osher Doctorow
.
- Follow-Ups:
- References:
- Prev by Date: Probable Influence/Causation Versus Algebraic Geometry/Topology 2: PI vs Scheme
- Next by Date: Re: The can rolling on the traincar. yet another gedanken fun thread.
- Previous by thread: Probable Influence/Causation Versus Algebraic Geometry/Topology 2: PI vs Scheme
- Next by thread: Re: Probable Influence/Causation Versus Algebraic Geometry/Topology 2: PI vs Scheme
- Index(es):
Relevant Pages
|
