The theorem ∃x.TRUE(x) of first order logic is often advanced as evidence of the claim that the standard axiomatization of the pure predicate calculus is a Logic of NonEmpty Universes, LNEU, on the ground that nothing can exist in the empty universe.
However it can equally well be argued that existential quantification is a projection operator, and how could projection render the valid formula TRUE(x) invalid? TRUE(x) is valid in every universe, vacuously so in the empty universe since there are no interpretations of x to be considered.
In this note I make the point that when the semantics of first order logic is taken to be Henkin, Monk, and Tarski's notion of a concrete cylindric algebra CCA, the tension between these two conflicting criteria for judging the theoremhood of ∃x.TRUE(x) is automatically resolved by taking the Boolean algebra of truth values to be the degenerate one-element Boolean algebra. In effect CCA automatically recognizes the inconsistency between the above criteria and realizes it concretely by identifying the truth values TRUE and FALSE.
Definition Let D be a set of individuals, the
domain, and let V be a set of variables.
A
concrete cylindric
algebra B over D and V is a subalgebra
B ⊆
2DV of
the Boolean algebra
of all V-ary relations on a domain D.
Remark. When V is empty, B = 2, otherwise when D is empty, B = 1, the degenerate Boolean algebra.
Let L be any first order language, with variables (both free and bound) drawn from a set X of variables.
To each subset V of X associate the set LV of formulas of L whose variables, free and bound, all belong to V.
The predicate symbols of L0 must all be zeroary, justifying the identification of L0 with the propositional fragment of L.
For each B over arbitrary D and V ⊆ X, let IB: LV → B interpret the formulas of LV in B subject only to condition T0 below.
Define the theory T(L) to consist of those formulas Φ of L such that every IB having Φ in its domain map Φ to the top element of B.
Condition T0: T(L) ∩ L0 consists of the propositional tautologies of the framework, classical in the case of CCA, but it could just as well be intuitionistic, quantum, linear, etc.
(This condition is extremely mild, and most plausible notions of predicate calculus can be expected to satisfy it.)
Proposition. T(L) is independent of whether the empty domain D = ∅ is allowed.
Proof. Allowing D = ∅ can only decrease T(L). There are two cases, both covered by the above remark.
V = ∅. This is just propositional calculus, with B = 2, for which condition T0 makes the choice of D irrelevant.
V ≠ ∅. Then B = 1, whence IB maps every formula to the top element of B, which cannot decrease T(L). ∎
This argument is applicable to any logical framework whose semantics can be defined in terms of a subalgebra of an algebra of the form ADV consisting of A-valued V-ary relations on a domain D.