Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 357

The empty clause is derived, hence the argument is valid.. a First we check ifAimpliesB.

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Answers and Solutions to Selected Exercises 333

Now we transform the formulas ∀ x(P(x)→ B(x)), ∃ x(P(x)∧ W(x)),

∀ x(P(x)→ E(x)), and¬∃ x(W(x)∧ B(x)∧ E(x)) to clausal form:

C1={¬ P(x), B(x)} C2 ={ P(c)}

C3={ W(c)} C4 ={¬ P(x), E(x)}

C5={¬ W(x), ¬ B(x), ¬ E(x)}

for some Skolem constantc Now, applying the Resolution rule successively,

we get

C6 =Res(C3, C5) ={¬ B(c), ¬ E(c)} MGU[c/x]

C7 =Res(C4, C6) ={¬ P(c), ¬ B(c)} MGU[c/x]

C8 =Res(C1, C7) ={¬ P(c)} MGU[c/x]

C9 =Res(C2, C8) ={}

The empty clause is derived, hence the argument is valid

(f) Using predicatesY(x) for “xis yellow,”P(x) for “xis a plonk,” andQ(x) for “xis a qlink,” we can formalize the argument as follows:

¬∃ x(Y(x)∧ P(x)∧ Q(x)), ∃ x(P(x)∨ ¬ Q(x))

We next transform the formulas¬∃ x(Y(x)∧ P(x)∧ Q(x)),

∃ x(P(x)∨ ¬ Q(x)), and¬∃ x ¬( Y(x)∧ Q(x)) to clausal form:

C1={¬ Y(x), ¬ P(x), ¬ Q(x)} C2={ P(c), ¬ Q(c)}

for some Skolem constantc Now, applying the Resolution rule successively, we get

C5=Res(C1, C2) ={¬ Y(c), ¬ Q(c)} MGU[c/x]

C6=Res(C3, C5) ={¬ Q(c)} MGU[c/x]

The empty clause is derived, hence the argument is valid

4.5.6 We formalizeAandB in the domain of all men, usingP(x) for “xis happy”

andQ(x) for “xis drunk” as follows:

A:=¬∃ x(P(x)→ Q(x))

B :=∃ yP(y)∧ ¬∃ zQ(z)

(a) First we check ifAimpliesB Clausification ofAand¬ B:

C1 :={ P(x), ¬ Q(x)}, C2 :={¬ P(y)}, C3 :={ Q(s1)},

wheres1 is a new Skolem constant