Computability and Logic

PROBLEMS 301
if there are rational numbers r i that if taken for the x i would make each inequality
in the set come out true (with respect to the usual addition operation and order
relation on rational numbers). Show that there is a decision procedure for the
coherence of finite sets of inequalities.
24.5 In sentential logic the only nonlogical symbols are an enumerable infinity of
sentence letters, and the only logical operators are negation, conjunction, and
disjunction ∼, &, ∨. Let A 1 ,...,A n be sentence letters, and consider sentences
of sentential logic that contain no sentence letters, but the A i , or equivalently, that
are truth-functional compounds of the A i . For each sequence e = (e 1 ,...,e n )
of 0s and 1s, let P e be (∼)A 1 & ... &(∼)A n , where for each i, 1 ≤ i ≤ n, the
negation sign preceding A i is present if e i = 0, and absent if e i = 1. For present
purposes a probability measure μ may be defined as an assignment of a rational
number μ(P e ) to each P e in such a way that the sum of all these numbers is 1.
For a truth-functional combination A of the A i we define μ(A) to be the sum
of the μ(P e ) for those P e that imply A, or equivalently, that are disjuncts in
the full disjunctive normal form of A). The conditional probability μ(A\B) is
defined to be the quotient μ(A & B)/μ(A) ifμ(A) ≠ 0, and is conventionally
taken to be 1 if μ(A) = 0. For present purposes, by a constraint is meant an
expression of the form μ(A) § b or μ(A\B) § b, where A and B are sentences
of sentential logic, b a nonnegative rational number, and § any of ,≥.
A finite set of constraints is coherent if there exists a probability measure μ
that makes each constraint in the set come out true. Is the set of constraints
μ(A\B) = 3/4,μ(B\C) = 3/4, and μ(A\C) = 1/4 coherent?
24.6 Show that there is a decision procedure for the coherence of finite sets of
constraints.