Midterm Exam1.pdf

文档介绍：Math 557 – Midterm Exam #1
September 30, 2005
1. Find a sentence in prenex normal form which is logically equivalent to
(∀x ∃y Rxy) ⇒¬ ∃x P x.
2. Using the predicates Bx (“x is a barber in Podunk”), Cx (“x is a
citizen of Podunk”), and Sxy (“x shaves y”), translate the following
argument into a sentence of the predicate calculus.
Every barber in Podunk is a citizen of Podunk. Any barber in
Podunk shaves exactly those citizens of Podunk who do not
shave themselves. Therefore, there is no barber in Podunk.
Use an unsigned tableau to test this argument for logical validity.
3. Using the predicate Rxy (“x is an ancestor of y”), translate the follow-
ing argument into a sentence of the predicate calculus.
Every ancestor of an ancestor of an individual is an ancestor
of the same individual. No individual is his own ancestor.
Therefore, there is an individual who has no ancestor.
Is this argument valid? Justify your answer by means of an appropriate