Express the quantification:
Existential(!)x P(x)
using universal quantifications, existential quantification and logical expressions. For sake of typing this question, I'm going to let "E" be the existential quantifier, "A" be the universal quantifier", and "^" means "and":
so far I have:
Ex(P(x) ^ Ay --> P(y))
But I have no idea if this is actually correct! I really appreciate your help!
Thanks!
To express the existential quantification "Existential(!)x P(x)" using the universal quantifier and logical expressions, you can use the following expression:
¬Ax ¬P(x)
Breaking it down:
1. ¬Ax: This represents the negation of the universal quantification. It means "For all x, P(x) is not true." This captures the negation of the given existential quantification.
2. ¬P(x): This represents the negation of P(x).
Combining them, ¬Ax ¬P(x) means "It is not true that for all x, P(x) is true." This is equivalent to saying "There exists an x such that P(x) is false," which is the logical equivalent of Existential(!)x P(x).
Your attempt at expressing the statement using the universal quantifier and implication is not quite correct. The expression Ex(P(x) ^ Ay --> P(y)) does not accurately capture the given existential quantification. However, the expression ¬Ax ¬P(x) does correctly represent it.
I hope this helps! Let me know if you have any further questions.