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.
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))
c)
| P | Q | ((P ∨ Q) ∧ (¬(P ∧ Q) ∧ ¬P)) |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | F |
| T | T | F |
It is neither tautology nor a contradiction
d)
| P | Q | R | ((P → (Q ∧ R)) ↔ ((P → Q) ∨ (Q → R))) |
|---|---|---|---|
| F | F | F | T |
| F | F | T | T |
| F | T | F | T |
| F | T | T | T |
| T | F | F | F |
| T | F | T | F |
| T | T | F | F |
| T | T | T | T |
It is neither tautology nor a contradiction
e)
| P | Q | R | ((P → (Q → R)) ↔ ((P → Q) → R)) |
|---|---|---|---|
| F | F | F | F |
| F | F | T | T |
| F | T | F | F |
| F | T | T | T |
| T | F | F | T |
| T | F | T | T |
| T | T | F | T |
| T | T | T | T |
It is neither tautology nor a contradiction
f)
| P | Q | R | S | (((P ∨ R) → (Q ∨ S)) → ((P → Q) ∧ (R ⨁ S))) |
|---|---|---|---|---|
| F | F | F | F | F |
| F | F | F | T | T |
| F | F | T | F | T |
| F | F | T | T | F |
| F | T | F | F | F |
| F | T | F | T | T |
| F | T | T | F | T |
| F | T | T | T | F |
| T | F | F | F | T |
| T | F | F | T | F |
| T | F | T | F | T |
| T | F | T | T | F |
| T | T | F | F | F |
| T | T | F | T | T |
| T | T | T | F | T |
| T | T | T | T | F |
It is neither tautology nor a contradiction
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