Final answer:
By creating a truth table, it is found that the logical statements ¬ (H ≡ R) and ¬ (R ⊃ ¬ H) are contradictory, meaning they cannot both be true at the same time, which makes them inconsistent with each other.
Step-by-step explanation:
We are tasked with evaluating whether the logical statements ¬ (H ≡ R) and ¬ (R ⊃ ¬ H) are consistent, inconsistent, logically equivalent, or contradictory by constructing an ordinary truth table. Let's define our symbols first:
- ¬ is the negation operator.
- ≡ represents the biconditional, meaning 'if and only if'.
- ⊃ is the conditional operator, meaning 'implies'.
To answer the question, we need the truth values of H and R to create a truth table. Once we have that, we will be able to compare the truth values of both expressions to check for consistency or contradiction.
After constructing the truth table, we'll find that for each corresponding pair of truth values for H and R, the truth values for ¬ (H ≡ R) and ¬ (R ⊃ ¬ H) cannot both be true at the same time. This means they are contradictory. To understand why let's look at each expression individually:
- ¬ (H ≡ R) negates the statement that H and R have the same truth value.
- ¬ (R ⊃ ¬ H) negates the statement that if R is true, then H is not true; this is equivalent to saying that R is true and H is true as well, which contradicts the negation of the biconditional.
In summary, ¬ (H ≡ R) and ¬ (R ⊃ ¬ H) are contradictory because they cannot both be true at the same time. This is in line with the principle of noncontradiction, which is one of the fundamental rules of logic.