Determine which of the following are tautologies, contradictions, or contingencies: 1. (P → ¬P) → ¬P 2. (P → (q → r)) → ((P → q) → (P → r))
Answer (1)
In logic, propositions can be classified as tautologies, contradictions, or contingencies. Let's analyze each of the given propositions step-by-step.
(P \rightarrow \neg P) \rightarrow \neg P
This expression is evaluated based on the rules of logical implication and negation.
P \rightarrow \neg P is only true when P is false because if P is true, \neg P is false, making the entire implication false.
Thus, for (P \rightarrow \neg P) to be true, P must be false. If P is false, then \neg P is true.
Now consider the entire expression: If (P \rightarrow \neg P) is true, then \neg P is true no matter what. Hence, we check:
If P is true, (P \rightarrow \neg P) becomes false, leading the outer implication to be vacuously true because the antecedent is false.
If P is false, \neg P is true, so the entire implication holds.
Therefore, (P \rightarrow \neg P) \rightarrow \neg P is a tautology because it is true regardless of the truth value of P.
(P \rightarrow (q \rightarrow r)) \rightarrow ((P \rightarrow q) \rightarrow (P \rightarrow r))
We need to explore the conditions under which this statement is true.
P \rightarrow (q \rightarrow r) simplifies the condition to when either P is false or r follows from q (i.e., q \rightarrow r).
(P \rightarrow q) \rightarrow (P \rightarrow r) can fail if P is true, q is false, but r is true, violating q \rightarrow r.
The two expressions on both sides of the outer implication share the condition that if P is false, the implications are true due to the truth of the antecedents.
Thus, when examining the full truth table for this proposition, there will be some scenarios where the proposition is false.
Conclusion : This expression is a contingency because its truth value depends on the specific truth values assigned to P, q, and r.
In summary, the first expression is a tautology, and the second expression is a contingency.
