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which implies that if q is true then either q or r or s is true, but this does not
mean (q ∨ r) only is true.
A simpler but interesting example that illustrates the scope of mistakes
in double resolution is given below. Let us consider the following clauses:
¬ p ∨ q
and ¬q ∨ p
Resolving these twice yields a null clause, which is always false. But
the above system comprising of { p→ q, q→p} implicates p→p and q → q
by chain rule, which in no way supports the falsehood of the resulting clause
after resolution [3].
5.12 Semi-decidability
A logic is called decidable if there exists a method by which we can correctly
say whether a given formula is valid or invalid. Readers may remember that
validity of a formula α means satisfiability of the formula for all possible
interpretations. A sound and complete proof method is able to prove the
validity of a formula [3]. But if the formula is invalid, the proof procedure (by
resolution principle or otherwise) will never terminate. This is called semidecidablity.
FOL is semi-decidable, as it is unable to prove the invalidity of a
formula.
5.13 Soundness and Completeness
The issues of soundness and completeness of the resolution principle for
propositional logic have already been discussed in a previous section. This
section discusses these issues for predicate logic. To prove the completeness
of the resolution theorem of predicate logic, the following definitions and
theorems are presented in order.
Definition 5.15: The Herbrand Universe (HS) for a given set of clauses S
is defined as the set of all possible ground terms, constructed by replacing the
variables in arguments of functions by the same or other functions or
constants, so that they remain grounded (free from variables) after substitution.
It is to be noted that HS is an infinite set [9].
For example, suppose that there exists a single clause in S, given by
Q( X, f (X, a) ) Λ P( X, a) → R (X, b)