Propositional Logic: Tautologies, Contradictions, and Contingencies

A tautology is a propositional formula that is always true, regardless of the truth values of its component propositions. In this case, the propositional formula "p ∨ ~p" is always true because it states that either "p" is true or "not p" is true, which is always the case.

A contradiction is a propositional formula that is always false, regardless of the truth values of its component propositions. In this case, the propositional formula "p ∧ ~p" is always false because it states that both "p" and "not p" are true, which is impossible.

A contingency is a propositional formula that is neither a tautology nor a contradiction. Its truth value depends on the truth values of its component propositions. In this case, the propositional formula "p → q" is a contingency because its truth value depends on the truth values of "p" and "q".

To determine whether the propositional formula is a tautology, a contradiction, or a contingency, we can construct a truth table. The truth table shows that the propositional formula is true in all possible cases, regardless of the truth values of "p" and "q". Therefore, the propositional formula is a tautology.

To determine whether the propositional formula is a tautology, a contradiction, or a contingency, we can construct a truth table. The truth table shows that the propositional formula is false in one possible case, when "p" is true and "q" is false. Therefore, the propositional formula is a contradiction.

To determine whether the propositional formula is a tautology, a contradiction, or a contingency, we can construct a truth table. The truth table shows that the propositional formula is true in some cases and false in other cases, depending on the truth values of "p" and "q". Therefore, the propositional formula is a contingency.

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p → q) ∧ (q → r) ≡ (p → r). Therefore, the logically equivalent form of the propositional formula is "(p → r)".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p ∨ q) ≡ (p ∧ q). Therefore, the logically equivalent form of the propositional formula is "p ∧ ~q".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p → q) ∨ (r → s) ≡ (p ∨ r) → (q ∨ s). Therefore, the logically equivalent form of the propositional formula is "(p ∨ r) → (q ∨ s)".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p ∧ q) → r ≡ p ∨ (q → r). Therefore, the logically equivalent form of the propositional formula is "p ∨ (q → r)".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalences: (p ∨ q) ≡ (p ∧ q) and (r → s) ≡ (r ∨ s). Substituting these equivalences into the propositional formula, we get: (p ∨ q) ∧ (r → s) ≡ (p ∧ q) ∧ (r ∨ s) ≡ p ∧ (q ∨ (r → s)). Therefore, the logically equivalent form of the propositional formula is "p ∧ (q ∨ (r → s))".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p → q) ∨ (r → s) ≡ (p ∨ r) → (q ∨ s). Therefore, the logically equivalent form of the propositional formula is "(p ∨ r) → (q ∨ s)".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalence: (p ∧ q) → r ≡ p ∨ (q → r). Therefore, the logically equivalent form of the propositional formula is "p ∨ (q → r)".

To find a logically equivalent form of the propositional formula, we can use logical equivalences. In this case, we can use the following logical equivalences: (p ∨ q) ≡ (p ∧ q) and (r → s) ≡ (r ∨ s). Substituting these equivalences into the propositional formula, we get: (p ∨ q) ∧ (r → s) ≡ (p ∧ q) ∧ (r ∨ s) ≡ p ∧ (q ∨ (r → s)). Therefore, the logically equivalent form of the propositional formula is "p ∧ (q ∨ (r → s))".