I have this question on quantifiers which I cannot figure out, any help is much appreciated!
Let P (x), Q(x), and R(x) be the statements “x is a clear explanation,” “x is satisfactory,” and “x is an excuse,” respectively. Suppose that the domain for x consists of all English text. Express each of these statements using quan- tifiers, logical connectives, and P (x), Q(x), and R(x).
a) All clear explanations are satisfactory.
b) Some excuses are unsatisfactory.
c) Some excuses are not clear explanations.
Best Answer
Hint for (a): a common error is to write this as $$\forall x\ (\,P(x) \wedge Q(x)\,)\ .$$ However, this means that for all text, it is a clear explanation and it is satisfactory. What you actually want to say is that for all text, if it is a clear explanation then it is satisfactory. Can you do this? (Another hint: you only have to change one symbol in the above incorrect ansswer.)