Technical notes · Difference (Δ)

Δ as Ontological Axiom

This page gives a formally rigorous treatment of Difference (Δ) as the unique, non-derivable ontological primitive in Informational Ontology (IO). It assumes familiarity with first-order logic, axiomatic method, and basic set-theoretic notation.

Abstract

Informational Ontology begins from a single primitive: difference. The central claim is that to exist is to differ, and that difference is the only candidate for an ontological axiom that cannot be denied without presupposing it. We formalize this claim by introducing a primitive binary predicate Δ, stating existence and ineliminability axioms, and showing how Δ underwrites identity, relation, and information. The goal is not to model difference in a particular physical theory, but to show that in any world where "existence" is meaningful, some structure extensionally equivalent to Δ must be instantiated.

1. Preliminaries and Notation

We work in a first-order meta-language enriched with a minimal ontological vocabulary. The point is not to reduce Δ to set theory, but to make IO's foundational claim legible to analytic metaphysics and the philosophy of information.

U denotes a non-empty domain of discourse (the set of items that can, in principle, exist in a given world-model).
Δ(x, y) is a primitive binary predicate read as "x differs from y". Δ is not defined in terms of more basic notions.
= is the usual identity relation of first-order logic; we argue that its non-trivial use presupposes Δ.
Angle brackets like <x, y> belong to the meta-language (we use them to talk about ordered pairs); they are not part of the object language of IO.

The modest formal project here is to show that any ontological theory that allows talk of distinct items, states, or propositions must operate in a language that presupposes a relation extensionally equivalent to Δ.

2. The Axiom of Difference

Δ is primitive in two senses: it is logically primitive (not defined in terms of more basic concepts) and ontologically primitive (not grounded in any deeper feature of reality).

2.1 Informal Principle

(Δ-Informal) To exist is to differ. If nothing differed in any respect, nothing would exist in any respect.

2.2 Formal Axioms

A minimal formalization requires both instantiation and ineliminability.

Axiom Δ1 (Existence of Difference): ∃x ∈ U ∃y ∈ U such that Δ(x, y).

There exist at least two items in the domain that differ in some respect. We do not yet specify in virtue of what they differ; that belongs to later stages (Relation, Information, etc.).

Axiom Δ2 (Ineliminability in Assertion): For any well-formed formula ϕ, the performative act of asserting "¬∃x ∈ U ∃y ∈ U Δ(x, y)" presupposes at least one instantiation of Δ.

Δ2 is a meta-level constraint: any attempt to deny the existence of difference is self-undermining, because the act of formulating and tokening that denial already requires differences (between the assertion and silence, between symbols, between truth-values, etc.).

3. Non-derivability of Δ

Saying that Δ is ontologically primitive means that no alternative ontological starting point can be specified without implicitly appealing to difference. We can show this by considering generic "starting-point" predicates and revealing their dependence on Δ.

3.1 Generic Competing Primitives

Let P be any unary predicate proposed as an ontological primitive (e.g. "substance", "matter", "mind", "experience", "energy").

(P0) ∃x ∈ U such that P(x).

For P to be non-trivial, there must be at least two admissible configurations: items that are P and items that are not, or at least two P-instances that can be discriminated.

3.2 Δ-Prior Constraint

Proposition 1 (Δ-Prior Constraint). For any non-trivial ontological predicate P, if P individuates a domain in which distinct items, states, or properties can be meaningfully discriminated, then the language in which P is expressed presupposes an extensionally equivalent relation to Δ.

Sketch of proof. Assume P is non-trivial: there exist x, y ∈ U such that both satisfy P and are distinguishable. "Distinguishable" minimally means that there exists some predicate Q with Q(x) ∧ ¬Q(y) or ¬Q(x) ∧ Q(y). The possibility of formulating that contrast requires that we can say "x differs from y" in some respect; call this relation Δ_P. Even if Δ_P is not explicitly symbolized, the theory tolerates an interpretation where a binary predicate plays the role of "difference". That role is what Δ names in IO. Hence, Δ (or its functional equivalent) is prior to any candidate P. ◻︎

4. Self-refuting Denial of Difference

A central move in IO's axiomatics is that Δ cannot be coherently denied. This is a strengthened version of Δ2: the attempt to assert the absence of all difference destroys the very conditions of assertion.

(¬Δ) "There are no differences whatsoever."

If (¬Δ) were true, there would be no difference between the occurrence and non-occurrence of any token (including this sentence), no difference between truth and falsity, and no difference between asserting and not asserting (¬Δ). But any concrete token of (¬Δ) already discriminates:

this token from its absence,
this sentence from others,
its asserted content from its negation.

Thus, the use of (¬Δ) in a context C entails ∃x, y ∈ U such that Δ(x, y). The denial is pragmatically self-refuting.

5. Δ as Precondition for Identity, Relation, and Information

With Δ established as ineliminable, we can state its role as a precondition for more familiar structures that later IO stages develop.

5.1 Identity

The classical identity relation "=" is governed by axioms like reflexivity, symmetry, and transitivity. However, its non-triviality depends on the possibility of non-identity.

Proposition 2. If identity is non-trivial in a domain U (i.e. there exist at least two items that can fail to be identical), then Δ is presupposed.

Sketch. For x ≠ y to be meaningful, there must be some respect in which x and y can differ. The intelligibility of "=" in any non-degenerate model thus presupposes a binary relation extensionally equivalent to Δ. ◻︎

5.2 Relation as Structured Difference (Preview)

The next IO stage, R, treats relations as structure over differences. Once distinct items exist, there is the question of how they differ and how they stand to one another. Formally, we can represent the set of all instantiated differences as:

D = {<x, y> ∈ U × U : Δ(x, y)}.

Once D exists, we can study its structure (connectivity, clustering, paths, etc.). That structure is what IO calls relation. Informally:

Relation is difference plus pattern; information is relation plus constraint.

6. Refutation Conditions for Δ

IO is axiomatic, so it is important to say what would count as refutation. Δ is not a free assumption; it stands or falls with the possibility of a difference-free yet intelligible ontology.

Refutation criterion. To refute Δ, one would need to specify a primitive Q such that:

a complete description of a world W can be given using Q without employing any discriminations, and
reference to difference is eliminable from both the language describing W and the meta-language in which the theory of W is articulated.

As soon as a theory admits distinct states, propositions, truth values, or semantic roles, it is already operating in a Δ-structured space, even if the symbol "Δ" never appears.

7. Conclusion and Outlook

We have articulated Δ as the unique ontological primitive in IO by:

stating existence and ineliminability axioms for difference,
showing that any competing primitive P presupposes Δ (Δ-Prior Constraint),
demonstrating that global denial of difference is pragmatically self-refuting, and
identifying Δ as a precondition for identity, relation, and information.

The next technical module, R — Relation, will take Δ as given and formalize the move from mere difference to structured difference using graph-theoretic and topological tools. That structure, in turn, supports the information-theoretic treatment on the I — Information page.