Question

1. For each of the following propositions construct a truth table and indicate whether it is a tautology (i.e., it’s always true), a contradiction (it’s never true), or a contingency (its truth depends on the truth of the variables). Also specify whether it is a logical equivalence or not. Note: There should be a column for every operator. There should be three columns to show work for a biconditional.

a) (P Λ ¬Q) ⇔ ¬(P ⇒ Q)

b) (¬? V¬?) ⇔ ¬(P Λ Q)

c) (P V Q) Λ ( ¬(? Λ Q) Λ (¬?))

d) (P ⇒ (Q Λ R)) ⇔ ((P ⇒ Q) Λ (Q ⇒ R))

e) (P ⇒ (Q ⇒ R)) ⇔ ((P ⇒ Q) ⇒ R) f) ((P V R) ⇒ (Q V S)) ⇒ ((P⇒ Q) Λ (R ⨁ S))

Answer 1

a)

Tautology

P	Q	¬Q	(P Λ ¬Q)	(P ⇒ Q)	¬(P ⇒ Q)	(P Λ ¬Q) ⇔ ¬(P ⇒ Q)
0	0	1	0	1	0	1
0	1	0	0	1	0	1
1	0	1	1	0	1	1
1	1	0	0	1	0	1

b)

Tautology

P	Q	¬Q	¬P	(¬? V¬?)	(P Λ Q)	¬(P Λ Q)	(¬? V¬?) ⇔ ¬(P Λ Q)
0	0	1	1	1	0	1	1
0	1	0	1	1	0	1	1
1	0	1	0	1	0	1	1
1	1	0	0	0	1	0	1

c)

contradiction

P	Q	P V Q	? Λ Q	¬(? Λ Q)	¬?	¬(? Λ Q) Λ (¬?)	(P V Q) Λ ( ¬(? Λ Q) Λ (¬?))
0	0	0	0	1	1	1	0
0	1	1	0	1	1	1	0
1	0	1	0	1	0	0	0
1	1	1	1	0	0	0	0

d)

contingency

P	Q	R	Q Λ R	P ⇒ (Q Λ R)	P ⇒ Q	Q ⇒ R	((P ⇒ Q) Λ (Q ⇒ R))	(P ⇒ (Q Λ R)) ⇔ ((P ⇒ Q) Λ (Q ⇒ R))
0	0	0	0	1	1	1	1	1
0	0	1	0	1	1	1	1	1
0	1	0	0	1	1	0	0	0
0	1	1	1	1	1	1	1	1
1	0	0	0	0	0	1	0	1
1	0	1	0	0	0	1	0	1
1	1	0	0	0	1	0	0	1
1	1	1	1	1	1	1	1	1

e)

contingency

P	Q	R	Q ⇒ R	P ⇒ (Q ⇒ R)	P ⇒ Q	(P ⇒ Q) ⇒ R	(P ⇒ (Q ⇒ R)) ⇔ ((P ⇒ Q) ⇒ R)
0	0	0	1	1	1	0	0
0	0	1	1	1	1	1	1
0	1	0	0	1	1	0	0
0	1	1	1	1	1	1	1
1	0	0	1	1	0	1	1
1	0	1	1	1	0	1	1
1	1	0	0	0	1	0	1
1	1	1	1	1	1	1	1

f)

contingency

P	Q	R	S	P V R	Q V S	(P V R) ⇒ (Q V S)	P⇒ Q	R ⨁ S	(P⇒ Q) Λ (R ⨁ S)	((P V R) ⇒ (Q V S)) ⇒ ((P⇒ Q) Λ (R ⨁ S))
0	0	0	0	0	0	1	1	0	0	0
0	0	0	1	0	1	1	1	1	1	1
0	0	1	0	1	0	0	1	1	1	1
0	0	1	1	1	1	1	1	0	0	0
0	1	0	0	0	1	1	1	0	0	0
0	1	0	1	0	1	1	1	1	1	1
0	1	1	0	1	1	1	1	1	1	1
0	1	1	1	1	1	1	1	0	0	0
1	0	0	0	1	0	0	0	0	0	1
1	0	0	1	1	1	1	0	1	0	0
1	0	1	0	1	0	0	0	1	0	1
1	0	1	1	1	1	1	0	0	0	0
1	1	0	0	1	1	1	1	0	0	0
1	1	0	1	1	1	1	1	1	1	1
1	1	1	0	1	1	1	1	1	1	1
1	1	1	1	1	1	1	1	0	0	0