This was in a conversation in which Chapter 1 of Arbitrary Reference in Logic and Mathematics had just been pasted in its entirety and summarized.

Imagine that you’re an analytic philosopher theistic apologist, and you’re convinced that the best way to support mathematics is for the principle of arbitrary reference to be supported by the ideal choice being made by God, which allows for realism rather than fictionalism about mathematics, which is more intuitive, etc. Introduce this idea and make this argument for God’s existence in a well developed academic style.

Arbitrary Reference and the Divine Choice: A Theistic Ground for Mathematical Realism

1. Introduction

Mathematical reasoning routinely proceeds by the magic locution: “Let $a$ be an arbitrary object of the domain.” The operative idea—call it the Principle of Arbitrary Reference (PAR)—is that any object in the relevant domain can be singled out for the purposes of deduction, even when the domain is uncountable and most of its members are indescribable. Contemporary proof theory and semantics quietly presuppose as much: natural–deduction rules for $\forall$ and $\exists$ deploy fresh parameters; Tarskian semantics evaluates formulas relative to arbitrary assignments from variables to individuals; game-theoretic semantics operationalizes quantifiers as ideal choices. If PAR fails, the familiar bridge from reasoning about “this one, wholly generic $a$” to universal conclusions collapses.

I propose a theistic metasemantics: arbitrary reference is grounded in an ideal act of choice in the divine intellect. God, a necessarily existent omniscient mind, fixes (without communicating to us) the determinate referents that our “arbitrary” parameters latch onto; our practice is veridical precisely because it rides on this prior, perfect act of selection. On this view, PAR is underwritten by a Principle of Arbitrary Choice (PAC) realized in God: for every object within any admissible domain, God can select and “baptize” it so that reasoning involving a fresh name or free variable is reference-secure. The result is mathematical realism without fictionalism’s make-believe or platonism’s brute metasemantics.

The positive upshot is twofold. First, theism offers a transparent account of how arbitrary reference can be both determinately referential and epistemically undisclosive (fixed but unknown). Second, because this account explains the modal robustness, intersubjective convergence, and normative authority of our inferential practices, it constitutes an abductive argument for the existence of God.

2. From PAR to PAC and the Metasemantic Demand

Two commonplace observations generate a metasemantic demand:

1. Deductive dependence. Natural–deduction rules ($\exists$-elimination via a fresh name $b$; $\forall$-introduction from a proof in which “$a$” does not occur in undischarged assumptions) are sound only if a parameter can function as a rigid, determinate name for some object that could have been any member of the domain. Merely schematic letters will not do, because the proof tracks the same item across steps.

2. Semantic dependence. Model-theoretic truth clauses evaluate $\forall x,\varphi(x)$ by quantifying over all assignments $g$ and insist that for each $d$ in the domain there is an assignment $g[x\mapsto d]$. This presupposes that every object is in principle referenceable; otherwise the truth-conditions miscarry.

Taken together, these yield a metasemantic constraint:

(MS) For logical practice to be objectively valid, there must exist a ground that makes possible and secures determinate reference to any object as needed for deduction and semantics.

Human stipulations, socio-linguistic conventions, or proof recipes alone cannot meet (MS). They are finite, fallible, and parochial; they also lack the modal strength to underwrite the necessity we attribute to logical rules. Nor does “brute primitiveness” of reference illuminate how determinacy, cross-agent coordination, and cross-world stability are obtained.

3. The Theistic Proposal: Divine PAC

Let DPAC be the following thesis:

(DPAC) For any admissible domain $D$ and any $d \in D$, God can (and does, when our practice calls for it) fix the referent of an arbitrary parameter so that it denotes $d$, without revealing which $d$ it is.

Think of this as a Kripkean baptism moved to the divine intellect. The arbitrariness concerns freedom from creaturely constraints, not indeterminacy of object: the selected item is unknown to us yet perfectly determinate to God. Our parameters and free variables successfully designate because they participate in, and are normatively constrained by, this prior divine act. The metasemantic regress stops, not in unexplained primitives, but in a necessary mind whose cognitive powers are adequate to the totality of mathematical objects and structures.

Two clarifications:

• Realism, not voluntarism. The proposal is intellectualist, not voluntarist: God does not legislate mathematical truth by will; rather, necessary mathematical truths are ideas in the divine intellect. DPAC concerns reference-fixing and our access, not the contingency of arithmetic.

• Abstracta and unnameables. The theistic ground makes harmless the fact that most reals are undescribable by us: God’s cognitive grasp is not limited by our languages; the epistemic opacity of the picked-out object to us coexists with its ontic determinacy to God.

4. Three Arguments for Theism from Arbitrary Reference

4.1 The Semantic Grounding Argument

1. PAR Premise. Sound deduction and standard semantics require that arbitrary reference to any domain object be possible and determinate (MS).
2. Anti-Bruteness Premise. If determinacy, coordination, and modal stability of arbitrary reference are fundamental to logic, they demand an explanatorily adequate ground; crude primitivism is an inferior stopping point for core normative phenomena.
3. Candidate Comparison. Non-theistic candidates—bare platonism (primitive links to abstracta), deflationary metasemantics, or structuralism without objects—either (a) make reference a brute accident, (b) cannot account for the normativity and necessity of quantificational practice, or (c) replace objects with schemas and thereby presuppose what they explain (that every instance can, in principle, be singled out).
4. Theistic Explanation. A necessarily existent omniscient mind can fix reference for any object, grounding determinacy, coordination, and modal stability non-brutely; this satisfies (MS) elegantly.
   Conclusion. Therefore, theism provides the best metasemantic explanation of PAR; ceteris paribus, we have abductive reason to affirm God’s existence.

4.2 The Argument from Mathematical Practice (Bayesian Sketch)

Let $T$ = theism; $N$ = a non-theistic alternative (platonism, structuralism, fictionalism). Let $S$ = the success profile of mathematical practice involving PAR: necessity, intersubjective convergence, cross-context stability, and applicability.

• $P(S\mid T)$ is high: if there is a necessary divine intellect grounding reference, we expect mathematics to display necessity and robust coordination.
• $P(S\mid N)$ is lower: either reference is brute (no prediction of normativity), or semantics rides on practice (no prediction of necessity), or objects evaporate (undermining PAR).
  By a standard likelihood comparison, $S$ confirms $T$ over $N$.

4.3 The Modal Necessity Argument

1. Logical rules governing quantifiers are necessary (not contingent habits).
2. If their correctness depends on the possibility of arbitrary reference, then the ground of that possibility must be modally robust—present in all possible worlds where logic holds.
3. Abstract realms lack agency to anchor normative uptake; human conventions are contingent.
4. A necessary divine intellect can be the ground of both objectivity and normativity here.
   Conclusion. Hence, the necessity of our quantificational practice points to a necessary divine mind.

5. Objections and Replies

Objection 1: Bare Platonism suffices. “Let there be a realm of abstracta; reference is a primitive link.”
Reply. This yields objects without metasemantics. It tolerates radical coordination failures, explains neither the normative grip of the rules nor their modal necessity, and halts inquiry at a primitive where the theistic view explains more for less.

Objection 2: Conceptual role semantics. “Inferential roles fix meaning; no extra metaphysics needed.”
Reply. Inferential roles presuppose the correctness of rules like $\forall$-intro; but those rules presuppose successful arbitrary reference. Moving the bump in the rug doesn’t remove it: the metasemantic question remains why such roles latch onto determinate objects across all admissible domains.

Objection 3: Fine’s arbitrary objects.
Reply. This introduces an ontology designed to mimic the practice while leaving its success mysterious. Moreover, arbitrary objects either collapse into schemas (losing determinacy) or become sui generis entities whose connection to ordinary domains is under-motivated. DPAC preserves ordinary ontology and supplies a unifying ground.

Objection 4: Euthyphro for mathematics. “Are mathematical truths true because God wills them?”
Reply. No: on divine conceptualism, necessary truths are eternally, necessarily known by God. They neither depend on a contingent will nor float free of mind altogether. DPAC concerns reference-fixing and access, not voluntarist legislation.

Objection 5: Quietism. “PAR is a workable practice; stop asking why.”
Reply. Quietism is philosophically available but explanatorily costly. Where an illuminating, unifying, modally robust explanation is on offer, methodological parity with other areas (e.g., science) favors the theistic hypothesis.

6. Conclusion

Arbitrary reference is not a dispensable rhetorical convenience; it is the hidden load-bearing beam of both our deductive rules and our model-theoretic truth conditions. If we refuse brute metasemantics and seek a ground commensurate with the determinacy, normativity, and necessity of our practice, the theistic picture—arbitrary reference as participation in a divine act of ideal choice—emerges as explanatorily superior. It preserves realism (mathematics is about mind-independent structures as conceived by a necessary mind), respects the epistemic character of arbitrariness (fixed yet unknown), and secures the unity of semantics, proof, and modality. In doing so, it furnishes a principled, academically respectable argument for the existence of God from the very success of our most austere and reliable reasoning.