1.5 Quantifiers & Predicate Logic

Predicate Logic

Which of the following are valid?

Exercise 1

1. Quantifiers of the same type (existential or universal) can be reordered without altering the meaning of the statement.
2. For example, let $$Q(x, y)$$ be ($$y > x)$$ and suppose the domain is nonnegative integers. Then the left side asserts that for every nonnegative integer, there is a larger nonnegative integer, which is true. The right side asserts that there exists a nonnegative integer greater than every other nonnegative integer, which is false. Therefore, the implication as a whole is false, and the statement is not valid.
3. Suppose that the left side is true. Then there exists an $$x_0$$ such that for all $$y$$, $$R(x_0$$, $$y)$$ is true. Thus, for all $$y$$, there exists an $$x$$ (namely, $$x_0$$) such that $$R(x, y)$$ is true. Therefore, the right side is also true, and the statement is valid.
4. If it's not true that there is an element in the domain with property $$S$$, then every element in the domain must not have property $$S$$. Conversely, if every element in the domain does not have property $$S$$, it can't be true that some element in the domain has property $$S$$.