Pei Wang wrote:
Charles,
What you said is correct for most formal logics formulating binary
deduction, using model-theoretic semantics. However, Edward was
talking about the categorical logic of NARS, though he put the
statements in English, and omitted the truth values, which may caused
some misunderstanding.
Pei
On 10/7/07, Charles D Hixson <[EMAIL PROTECTED]> wrote:
Edward W. Porter wrote:
So is the following understanding correct?
If you have two statements
Fred is a human
Fred is an animal
And assuming you know nothing more about any of the three
terms in both these statements, then each of the following
would be an appropriate induction
A human is an animal
An animal is a human
A human and an animal are similar
It would only then be from further information that you
would find the first of these two inductions has a larger
truth value than the second and that the third probably
has a larger truth value than the second..
Edward W. Porter
Porter & Associates
24 String Bridge S12
Exeter, NH 03833
(617) 494-1722
Fax (617) 494-1822
[EMAIL PROTECTED]
Actually, you know less than you have implied.
You know that there exists an entity referred to as Fred, and that this
entity is a member of both the set human and the set animal. You aren't
justified in concluding that any other member of the set human is also a
member of the set animal. And conversely. And the only argument for
similarity is that the intersection isn't empty.
E.g.:
Fred is a possessor of purple hair. (He dyed his hair)
Fred is a possessor of jellyfish DNA. (He was a subject in a molecular
biology experiment. His skin would glow green under proper stimulation.)
Now admittedly these sentences would usually be said in a different form
(i.e., "Fred has green hair"), but they are reasonable translations of
an equivalent sentence ("Fred is a member of the set of people with
green hair").
You REALLY can't do good reasoning using formal logic in natural
language...at least in English. That's why the invention of symbolic
logic was so important.
If you want to use the old form of syllogism, then at least one of the
sentences needs to have either an existential or universal quantifier.
Otherwise it isn't a syllogism, but just a pair of statements. And all
that you can conclude from them is that they have been asserted. (If
they're directly contradictory, then you may question the reliability of
the asserter...but that's tricky, as often things that appear to be
contradictions actually aren't.)
Of course, what this really means is that logic is unsuited for
conversation... but it also implies that you shouldn't program your
rule-sets in natural language. You'll almost certainly either get them
wrong or be ambiguous. (Ambiguity is more common, but it's not
exclusive of wrong.)
Well, truth values would allow one to assign probabilities to the
various statements (i.e., the proffered values plus some uncertainty),
but he specifically said we didn't know anything else about the terms,
so I don't see how one can go any further. If you don't know what a
human is, then knowing that Fred is one doesn't tell you anything about
his other characteristics.
So when you have two statements about Fred, you "know" the two
statements, but you don't know anything about the relationship between
them except that their intersection is non-empty. Since it was
specified that we didn't know anything about them, Fred could be a line,
and human could be vertical lines and animal could be named entities.
For fancier forms of logic (induction, deduction, etc.) you need to have
more information. Most forms require that there be at least a partial
ordering available, if not several. Many modes of reasoning require
that a complete ordering be available. (It doesn't need to be an
ordering that guarantees that every iteration will end up with a member
of the set...consider the problem of stepping through a hash table...you
can do it, but you'll get lots of empty cells, and you can't predict the
order. What you can predict is complete coverage. This is an
importantly useful characteristic. It lets you check "for all" assertions.
I'll admit I haven't read your papers on NARS, but I don't see how that
could obviate these "primitive" characteristics. You can't do induction
without an ordering. Deduction doesn't require an ordering, but it
requires rules of inference. Simple assertions don't require rules of
inference, but do require assertion...which generally means a verb
(possibly understood). This is why "if x then y" is often translated
into English as "x implies y", but a better translation might be "x
implies y, but I'm not asserting x".
I.e., two children of the same parent can be expected to have similarities.
P.S.:
ABDUCTION INFERENCE RULE:
Given S --> M and P --> M, this implies S --> P to some degree
I.e., two children of the same parent can be expected to have
similarities (in the context of inheritance...they will at least be
similar to the extent that they inherited the same characteristics).
INDUCTION INFERENCE RULE:
Given M --> S and M --> P, this implies S --> P to some degree
I.e., two parents of the same child can be expected to have similarities
(in the context of inheritance). This one seems dubious, but to the
extent that it's true then one should also expect "P-->S to some degree".
If I look a parents and their children, this seems reasonable...though
the "to some degree" is quite unpredictable. OTOH, if I look at object
classes, it seems to fail completely. It's quite surprising to find
induction appear to be less certain than abduction. Either I'm not
properly understanding what is meant (Well, I did mention that I hadn't
read the original papers), or perhaps this needs a bit more thought. It
seems very sensitive to context. I also note that I can't relate this
definition easily to the meaning of induction used in the phrase in
"mathematical induction". Or to electrical induction.
OTOH, it's certainly true that if two parents are related through a
child, one can expect, at minimum, for them to be members of closely
related species. ... I feel uncomfortable with calling that piece of
reasoning induction, however. Model-consistency seems a better phrase.
(I.e., I have a model of the world, and in that model only closely
related species can engender offspring. N.B.: I am aware of model
violations, where, e.g., microbes can cause insects or mammals to
engender offspring...so I have a more detailed model to account for
that.) This is clearly a much more complex process than your proposed
simple rule...but I'm not certain that "induction" is an appropriate
term. I have a model for how other forms of induction work, and this
doesn't appear to fit into it. (OTOH, it's a rather loose model, and if
this usage became well-established, it would probably adjust. But the
adjustment would, for at least a while, feel unnatural.)
Still, utility rules. If this rule is useful, then it's a valid rule.
I may be unhappy with the name that it was given, and may feel that it
appears unduly context sensitive, but I'm trying to apply it in the more
general space of reasoning, rather than within the context of your
proposed system. And it's quite plausible that as program objects
become more complex, then it will be more difficult to perform multiple
inheritance between distantly related objects. (One might consider why
so many computer languages have opted for single inheritance with
interfaces. It might be a consequence of this rule [which I still don't
want to call induction].)
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