Logic of ER I - Core Logic & Meta-Theory: A Conservative Predicate Extension of First-Order Linear Tense Logic

Existential Realism (ER) proposes a two-tier ontology: only the present exists, yet the past and the future are real. This thesis arises from the diagnosis that in debates on the ontology of time, “existence” and “reality” are often implicitly identified, thereby making presentism and eternalism appear as competing positions that merely disagree about the extension of a single ontological predicate. ER instead separates a narrow, present-bound predicate of existence from a broader notion of reality that allows non-present entities to count as real when, for instance, they existed in the past or will exist in the future, allowing these to participate in time-directed deductions. This paper develops the core system ERL (Existential-Realism Logic): a classical first-order linear temporal logic, in the Prior tradition, enriched with two distinguished unary predicates E (“exists now”) and R (“is real”). ERL is (i) syntactically defined with precision, (ii) given a model-theoretic semantics via Kripke-style ER-frames, (iii) axiomatized through a Hilbert-style system, and (iv) supported by metalogical results establishing soundness and outlining a canonical completeness program. Where strong completeness requires additional standard assumptions (such as Henkinization or generalized models), this is made explicit. Further central technical results include: conservativity over classical first-order linear temporal logic without E/R; an explicit Barcan profile (Barcan schemas hold for unrestricted quantifiers under constant domains, but may fail for existence-restricted quantifiers ∀ᴱ/∃ᴱ); and non-reducibility results (E is not definable, over the class of ER-models, using purely temporal operators; R cannot be forced as a definitional derivative of E together with temporal operators). The overall aim is to demonstrate that ER constitutes a logically stable and metalogically respectable position within established modal-temporal traditions, without unnecessarily over-engineering the formal system. Soundness is proven; completeness is outlined via a canonical-model strategy but left for full technical development in subsequent work.