Logic, like mathematics, presupposes conditions that are rarely named but fundamentally necessary. The first of these is stability.
Stability in logic is not about rigid determinism or static truth. It is about the capacity for propositions, statements, and their interrelations to be held steady across cuts, construals, and applications of inference rules. Without stability, the scaffolding upon which logic operates cannot exist.
Logical stability as a precondition
Formal inference assumes that:
propositions can be instantiated consistently,
relations among propositions (implication, entailment, contradiction) can persist without collapsing,
repeated applications of inference rules yield predictable transformations of truth-values.
These assumptions are entry conditions, not consequences. They mark the domain in which logical operations can gain traction. Where stability fails, no inference, no matter how formally valid, can be guaranteed to hold.
Perspective and the fragility of consistency
From a relational-ontological perspective, stability is never absolute. It is always perspectival: a proposition is stable only relative to a construal that holds its content and relations intact. Changing the perspective—shifting the scale, context, or interpretation—can destabilise even the most elementary logical units.
This perspectival nature explains why classical logical paradoxes emerge. Consider self-referential propositions or systems with interdependent axioms. The instability is not a quirk of formalism but a reflection of relational dynamics that cannot be stabilised under the required cut.
Stability vs. truth
It is crucial to distinguish stability from truth. A proposition may be “true” in a relational sense—holding across some set of circumstances—but fail to be stable in the sense required for formal inference. Logic does not track truth per se; it tracks structured, stable relations among propositions. Where stability is violated, formal logical derivations may fail even if the underlying relation is meaningful and intelligible.
Where stability fails
Failures of logical stability manifest as:
paradoxes, contradictions, and undecidability,
sensitivity to interpretation or context shifts,
breakdown of inference chains that would otherwise hold.
These are not failures of reason. They are diagnostic signals marking the limits of structural capture. They tell us where relational reality refuses to submit to the structural assumptions that logic imposes.
Implications for the series
Understanding stability as a perspectival precondition allows us to read logical breakdowns differently. They do not imply the absence of intelligibility or rationality. They indicate where the domain of formal inference is exhausted.
In the next post, we will explore a second presupposition of formal logic: separability. Just as propositions must be stable, they must also be individuable and capable of being treated as independent relata within logical operations. Where separability fails, logic encounters a different but equally instructive boundary.
No comments:
Post a Comment