First, translate the following statement pair into the symbols of propositional logic (use the letters P and R)
(a) We will have a picnic unless it happens to rain
(b) We won’t have a picnic only if it happens to rain
Second, use truth tables to classify the above statement pairs as logically equivalent, contradictory, consistent, or inconsistent.
Show all your work.
To translate the given statement pair into the symbols of propositional logic, we can assign the statements as P and R:
(a) We will have a picnic unless it happens to rain
Translation: P if not R (P unless R)
(b) We won't have a picnic only if it happens to rain
Translation: Not P if R (Not P unless R)
For the truth tables, we will create columns for P, R, P if not R (a), Not P if R (b).
Now let's construct the truth table:
| P | R | P if not R | Not P if R |
|---|---|------------|------------|
| T | T | F | T |
| T | F | T | F |
| F | T | T | T |
| F | F | T | T |
To classify the statement pairs:
(a) We will have a picnic unless it happens to rain: P if not R
(b) We won’t have a picnic only if it happens to rain: Not P if R
To determine logical equivalence or contradiction, we compare the columns for (a) and (b). If they have the same truth values in all rows, they will be logically equivalent. If the truth values do not match in any row, they will be contradictory. If there is at least one row where the truth values match, they will be consistent. If there is at least one row where the truth values do not match, they will be inconsistent.
Upon comparing the two columns, we see that (a) and (b) have the same truth values in all rows. Hence, they are logically equivalent.
Therefore, the classification of the statement pairs is as follows:
(a) and (b) are logically equivalent.