Let L be the first-order language with a single predicate S(p;q), meaning “p sha

Question

Let L be the first-order language with a single predicate S(p;q), meaning “p shaves q.” Assume a domain of people. a. Consider the sentence “There exists a person P who shaves every one who does not shave themselves, and only people that do not shave themselves.” Express this in L. b. Is this sentence satisfiable? Explain your answer. c. Convert the sentence in a. to clausal form. d. Construct a resolution proof to show that the clauses in c. are inherently inconsistent. (Note: you do not need any additional axioms.)

Explanation / Answer

“There exists a person P who shaves every one who does not shave themselves, and only people that do not shave themselves.”

p q person(p) person(q) (¬ S(q,q) S(p,q))

3.Convert the sentence in a. to clausal form.

1st order formula:

pqperson(p)person(q)(¬S(q,q)S(p,q)) (1)

pqperson(p)person(q)(¬S(q,q)S(p,q))(S(p,q)¬S(q,q)) (2)

remove implication:

pqperson(p)person(q)(S(q,q)S(p,q))(¬S(p,q)¬S(q,q)) (3)

skolemize off the existence:

qperson(P)person(q)(S(q,q)S(P,q))(¬S(P,q)¬S(q,q)):{p=P} (4)

drop universal qualifier:

person(P)person(q)(S(q,q)S(P,q))(¬S(P,q)¬S(q,q)) (5)

d. Construct a resolution proof to show that the clauses in c. are inherently inconsistent.

The bove CNF forms empty clause which is false

The logic is not satisiable

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