higher-order KIF and CG's
Dan Schwartz <schwartz@iota.cs.fsu.edu>
Date: Sat, 6 Nov 93 01:21:34 -0500
From: Dan Schwartz <schwartz@iota.cs.fsu.edu>
Message-id: <9311060621.AA01638@iota.cs.fsu.edu>
To: interlingua@ISI.EDU
Subject: higher-order KIF and CG's
Dear Fritz Lehman,
This is in response to your request for "a few more examples of
higher-order or mixed-order statements". Here follows a brief
description of two such systems, both having two distinct semantic
levels, and both being in fact a multivalent logic at the first level
and a bivalent logic at the second level. I'll leave it to the KIF/CG
aficionados to determine whether these can be expressed in their
systems.
First is a portion of my doctoral dissertation, completed in 1981, and
finally published as [1] in 1987. It's aim was to formalize as much as
I could of the logic associated with L.A. Zadeh's semantics of fuzzy
sets. To illustrate, the language allows propositions like
middle-aged(x) == ~old(x) & ~young(x)
where middle-aged, old, and young are interpreted as fuzzy subsets of a
universe of ages, so that, for an individual age, a, the truth value of
say
middle-aged(n-a)
('n-a' being the name of age a) is the degree of membership of age a in
the fuzzy set associated with middle-aged. Then, as is customary in
fuzzy sets theory, the connectives ~, &, and V are interpreted for
instantiated formulas exactly as in Lukasiewicz logic, i.e., as 1-, min,
and max.
As a result, the connective == becomes what Zadeh referred to as
"semantic equivalence" (which is closely related to something known in
the literature on multivalent logics as "strong equivalence"). To wit,
the relations on either side of the == are semantically equivalent if,
under all possible instantiations, both sides receive the same truth
value.
Considering that two relations either are, or are not, semantically
equivalent, this leads to a bivalent logic for talking about the
properties of semantic equivalence. For example,
p(x)==q(x) &-dot q(x)==r(x) ->-dot p(x)==r(x)
expresses transitivity of semantic equivalence (the dotted connectives
are second level). I was able to develop a semantically complete
axiomatization of this system, using a rather obscure theorem from set
theory. Semantic completeness can be seen to depend crucially on this
theorem, to the extent that semantic completeness holds for theories
with countably infinite languages, but fails for uncountable theories.
The languages allow fuzzy relations of any arity, but do not include
quantifiers.
Second is a logic recently published as [2], which aims at capturing
such reasoning as
*Most* birds can fly.
Tweety is a bird.
-----------------------------------
It is *likely* that Tweety can fly.
and
*Usually*, if something is a bird it can fly.
Tweety is a bird.
---------------------------------------------
It is *likely* that Tweety can fly.
Here the task was to devise a language which allows for expressing the
implicit relationships between quantification (all, most, many, few,
etc.), likelihood (certainly, likely, uncertain, unlikely, etc.) and
usuality (always, usually, sometimes, seldom, etc.). All such modifiers
are interpreted probabalistically. To illustrate, let M="most x",
C="certainly', and L="likely". Then the first syllogism above would be
expressed as
M C (bird(x)->can_fly(x))
C bird(Tweety)
-----------------------------
L can_fly(Tweety)
and the first line, e.g., would be "true" if the probability of the
proposition bird(x)->can_fly(x) falls within some specified range, say
[2/3,1], where this is computed as the conditional probability that
something can fly, given that it is a bird. (Similarly, negation,
conjunction, and disjuction have probabalistic interpretations at this
level.) Thus lower-level (unmodified) propositions have degrees of
truth, given as probabilities, and upper-level propositions (using
modifiers) are either true or false.
>From the standpoint of the upper level, the three lines of the above
syllogism are all atomic formulas, and logical combinations of such
involving higher-leveled versions of not, or, and, implies, and iff are
allowed. There is also an intermediate form of implies, introduced to
avoid nesting of inferences at the lower level. For example, letting
A="all x", the formula
MC(p(x)->q(x))<--->A(Cp(x)-->Lq(x))
says
For most x it is certain that p(x) implies q(x)
iff
for all x, if it is certain that p(x) then
it is likely that q(x).
The -> is interpreted as above (as conditional probability), the <--->
is the classical bivalent iff, and the --> is the intermediate implies,
also interpreted as conditional probability. The resulting logic
validates all such syllogisms and formulas as above, and is aimed at
devising a new approach to nonmonotonic reasoning. It has not yet been
axiomatized.
[1] D.G. Schwartz, Axioms for a theory of semantic equivalence. Fuzzy
Sets and Systems, 21 (1987) 319--349.
[2] D.G. Schwartz, Toward a logic for fuzzy syllogisms. Proceedings of
the Second IEEE Conference on Fuzzy Systems, San Francisco, CA, March
31--April 1, 1993, pp. 71--75.
************************************************************************
Daniel G. Schwartz Office 904-644-5875
Dept. of Computer Science, MC 4019 CS Dept 904-644-2296
Florida State University Fax 904-644-0058
Tallahassee, FL 32306-4019 Home 904-385-7735
U.S.A. schwartz@cs.fsu.edu
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