If propositions are to be used in formal inference, they must not only be stable and separable; they must also be invariant under transformation. This final presupposition is the most subtle—and the most fragile.
Logic operates by transforming propositions. It substitutes, negates, quantifies, abstracts, and recombines them. For these operations to count as inference rather than mere alteration, something must remain unchanged across transformation. That something is invariance.
What logic requires to remain invariant
Formal logic presupposes that:
truth-relations survive permissible transformations,
inferential validity is independent of representational form,
replacing a proposition with an equivalent one preserves what matters.
These assumptions allow logic to treat inference as structure-preserving movement rather than creative intervention. Transformation is permitted precisely because it is assumed not to touch the constitutive relations that ground truth.
Transformation as structural neutrality
Within formal systems, transformations are defined syntactically: rules specify what may be rewritten, substituted, or derived. Invariance is guaranteed internally by design.
But this guarantee is conditional. It holds only insofar as the transformations track distinctions that are themselves stable, separable, and non-constitutive of meaning. Logic assumes that form can change while relational content remains intact.
From a relational-ontological perspective, this is a strong and non-trivial commitment.
When transformation alters constitution
There are domains in which transformation is not neutral.
Shifts in perspective, scale, or contextual embedding can change what a proposition is, not merely how it is expressed. What appears to be a harmless substitution may reconfigure the relational field that constituted the proposition in the first place.
In such cases, invariance fails. Logical equivalence no longer guarantees ontological equivalence. Inference rules continue to apply, but what they preserve is no longer what matters.
Logic does not break here; its presuppositions do.
Paradox and the failure of invariance
Many logical paradoxes arise not from instability or non-separability alone, but from failures of invariance.
Self-reference, diagonalisation, and semantic ascent involve transformations that feed propositions back into the conditions of their own constitution. The transformation is formally licensed, yet it alters the relational ground upon which truth-values depend.
Logic records this as paradox or undecidability. Relational ontology reads it as a violation of invariance assumptions.
Invariance vs. universality
Invariance is often mistaken for universality.
Because logical rules preserve validity across transformations within a system, it is tempting to assume that they apply without remainder across all domains. But invariance is always local to a regime of permissible transformation. Outside that regime, the same operations may be destructive rather than preservative.
Logical validity is therefore conditional: it holds where transformations respect relational constitution. It fails where they do not.
The perspectival nature of invariance
What counts as an invariant relation depends on perspective.
A transformation that is benign at one level of description may be constitutive at another. Logic has no internal resources for detecting this shift. It presupposes, rather than establishes, the stability of its invariants.
Invariance, like stability and separability, is a perspectival achievement—not a global guarantee.
Implications for the series
With stability, separability, and invariance in view, we can now see the full shape of the logical space in which formal inference succeeds.
In the next post, we will draw these presuppositions together to map the space of the logically possible: the constrained domain in which formal logic operates successfully, and beyond which its breakdowns become not failures, but guides to relational excess.
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