∀x[∀y(xRy)] states that everything in the universe bares the relationship R to everything else in the universe. If I switch the positions of x and y, then the formula still makes the same claim. ∀y[∀x(xRy)] switches x and y in the quantifiers but yet again the meaning remains the same.
∀x[∃y(xRy)] makes both a universal claim as well as an existential claim. It states that for any possible thing x in the universe, there exists another thing y such that x bares relation R to y. If R is taken to be the predicate <, then ∀x[∃y(x<y)] is true if our universe is the set of all natural numbers. Switching x and y yields ∀x[∃y(y<x)] which turns out to be false if our universe remains but is true if we expand our universe to be the set of all integers. Swapping x and y in the quantifiers produces ∀y[∃x(x<y)] which turns out to mean the same thing as ∀x[∃y(y<x)] because the variable associated with each quantifier is in the same position in the atomic formula.
∃x[∀y(xRy)] if R is taken to be =, then ∃x[∀y(x = y)]. This statement says at least one thing exists in the universe such that it is equal to everything else in the universe. This statement is true in a one element universe because that element is both something and everything.
∃x[∃y(xRy)] means that at least one x and at least one y exist such that x bares R to y. Replace R with ∈ and the formula claims that something is a member of something else.
No comments:
Post a Comment