114 Logic as a Tool • ∃xA(x) is true if there is an object a from the domain of S which, assigned as a value of x, satisfies the formula A Formally, S , v |= ∀xA(x) if S , v |= A(x) for some variable assignment v that is an x-variant of v 3.2.4.4 Computing the truth of first-order formulae We can now (at least theoretically) compute the truth of any first-order formula in a given structure for a given variable assignment, step-by-step, following the logical structure of the formula and applying recursively the respective truth condition for the main connective of the currently evaluated subformula It is not difficult to show that the truth of a formula in a given structure for a given variable assignment only depends on the assignment of values to the variables occurring in that formula That is, if we denote the set of variables occurring in the formula A by VAR(A) and v1 , v2 are variable assignments in S such that v1 |VAR(A) = v2 |VAR(A) , then S , v1 |= A iff S , v2 |= A Still, note that the truth conditions for the quantifiers given above are not really practically applicable when the structure is infinite because they require taking into account infinitely many variable assignments Evaluating the truth of a first-order formula in an infinite structure is, generally speaking, an infinite procedure Nevertheless, we can often perform that infinite procedure as finite by applying uniform (yet ad hoc) arguments to the infinitely many arising cases Example 92 Consider the structure N and a variable assignment v such that v (x) = 0, v (y ) = 1, and v (z ) = The following then holds ã N , v |= ơ(x > y ) • However, N , v |= ∃x(x > y ) Indeed, N , v [x := 2] |= x > y • In fact, N , v |= ∃x(x > y ) holds for any assignment of value to y , and therefore N , v |= ∀y ∃x(x > y ) • On the other hand, N , v |= ∃x(x < y ), but N , v ∀y ∃x(x < y ) Why? • What about N , v |= ∃x(x > y ∧ z > x)? This is false; there is no natural number between and • However, for the same variable assignment in the structure of rationals Q, we have that Q, v |= ∃x(x > y ∧ z > x) Does this hold for every variable assignment in Q? 3.2.5 Evaluation games There is an equivalent, but somewhat more intuitive and possibly more entertaining, way to evaluate the truth of first-order formulae in given structures This is done by playing