A reduction theorem for predicate logic

Extract

It is well known that in prepositional logic not every propositional connective is definable in terms of equivalence nor indeed in terms of equivalence together with negation. It is the purpose of this note to show that in a certain sense this is, however, possible within predicate logic.

Let F₁ be the class of predicate logical formulae containing only the logical connectives ¬ (not), ∨ (or), E (exists), and F₂ the class of predicate logical formulae containing only the logical connectives ≡ (if and only if) and E.

Our object will be to find a method for reducing any formulae ϕ of F₁ to a formula ψ of F₂ such that ϕ is valid if and only if ψ is valid.

Note first that for every domain D containing at least two individuals we can for every n-place relation R defined over D find an n + 1-place relation S defined over D such that for all x₁, …, x_n ∈ D,

holds. For instance, let S(y, x₁, …, x_n) be the relation y = x₁ & R(x₁, …, x_n).

Now let ϕ be an arbitrary formula of F₁ and A₁, …, A₃ a complete list of predicate letters occurring in ϕ of degree g₁, …, g_i, respectively. Let ϕ′ be the formula obtained from ϕ by replacing every prime formula A_i(x₁, …, x_gi) by

Then it follows immediately from the previous observation that (a) ϕ is satisfiable in D if and only if ϕ′ is satisfiable in D.