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Re: [ontolog-forum] Constraint Solving versus Inferencing

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Sat, 11 Oct 2014 02:14:17 -0400
Message-id: <5438CAB9.3090107@xxxxxxxxxxx>
I just wanted to clarify some points in this thread.    (01)

> For a constraint to be expressed in Boolean logic, aren't you then
> moving out of the realm of propositional logic and into the realm
> of first order predicate logic?    (02)

No.  Boolean logic is usually used as a synonym for propositional logic.    (03)

But it's important to note that Boolean algebra is actually more general
than propositional logic, since George Boole himself applied the algebra
to several different domains:    (04)

  1. If the letters represent propositions, you get propositional logic:
     1 represents truth, 0 falsity, AxB is AND, A+B is OR, -A is NOT.
     Peirce observed that if A implies B, the truth value of A must be
     less than or equal to the truth value of B.  Therefore, A≤B can
     be used to represent "If A, then B."    (05)

  2. But GB also used the letters to represent a Boolean algebra of
     monadic predicates:  1 is the predicate that is true of everything,
     0 is true of nothing, AxB is the predicate that is true of
     everything for which A and B are both true, etc.    (06)

  3. GB also used the letters to represent sets:  1 is the universe
     of discourse, 0 is the empty set, A+B is the union of A and B,
     AxB is the intersection, and -A is the complement of A.  But
     Boole's version of set theory was actually a version of mereology:
     he did not distinguish a single value x from the set {x}.    (07)

Description logics take advantage of points #2 and #3 by treating
a class a pair:  a set and a predicate that is true of everything
in the set.  The same Boolean operators apply to both the sets and
the predicates that determine them.    (08)

> Even if you only use equality (=), and non-equality ('=)
> to express your constraints, don't those two act as predicates?    (09)

Well, yes.  But you don't get predicate calculus without quantifiers.
If you only have constants as the arguments of the predicates, the
expressive power is limited to propositional logic.    (010)

You can have constraint logic programming (CLP) with just equality.
If you extend the notation to support inequalities, the solution
to a CLP problem may consist of a family of possible options.    (011)

John    (012)

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