# 1. The logical operator ? is read if and only if. P ? Q is defined as being equivalent to ( P.

1. The logical operator _ is read if and
only if. P
_ Q is
defined as being equivalent to (P _ Q) _ (Q _ P).
Based on this definition, show that P _ Q is
logically equivalent to (P _ Q) _ (P _ Q):

a. By using truth tables.

b. By a series of substitutions using the identities
on page 51.

2. Prove that implication is transitive in the
propositional calculus, that is, that ((P _ Q) _ (Q _ R)) _ (P _ R).

3. a. Prove that modus ponens is sound for
propositional calculus. Hint: use truth tables to enumerate all possible interpretations.

b. Abduction is an
inference rule that infers P

1. The logical operator _ is read if and
only if. P
_ Q is
defined as being equivalent to (P _ Q) _ (Q _ P).
Based on this definition, show that P _ Q is
logically equivalent to (P _ Q) _ (P _ Q):

a. By using truth tables.

b. By a series of substitutions using the identities
on page 51.

2. Prove that implication is transitive in the
propositional calculus, that is, that ((P _ Q) _ (Q _ R)) _ (P _ R).

3. a. Prove that modus ponens is sound for
propositional calculus. Hint: use truth tables to enumerate all possible interpretations.

b. Abduction is an
inference rule that infers P from P _ Q and Q. Show
that abduction is not sound (see Chapter 8).

c. Show modus tollens ((P _ Q) _ _ Q) _ _ P is sound.

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