@Blake Winter logic can prove that a system of equations has or does not have a solution https://en.wikipedia.org/wiki/Satisfiability Both satisfiable and unsatisfiable formuale are 'real' synactic objects. In fact, many tautologies begin with an existential qualifier such as "for some x, x is a real number and x times x is 2" Hence logic can prove that something 'exists' in this sense, such as the square root of 2 in the real numbers. https://math.stackexchange.com/questions/1415235/prove-the-existence-of-the-square-root-of-2 ^^ these proofs can all be reduced to pure logic and they are what are called 'existence proofs' in mathematics.