Stability alone does not secure the conditions under which formal logic can operate. Propositions may persist across construals and yet remain logically intractable. The missing condition is separability.
Formal logic presupposes that propositions can be individuated and treated as independent units—combined, negated, or inferred from without altering their identity. Where this presupposition fails, logic encounters a distinct and revealing limit.
Separability as an entry condition of inference
To reason formally is to assume that:
propositions can be isolated as discrete entities,
logical relations obtain between propositions rather than constituting them,
variation in one proposition does not retroactively alter what another proposition is.
These assumptions underpin the use of variables, predicates, truth-values, and inference rules. Without separability, there are no stable inputs for logical operations.
Propositions that are constituted by their relations
From a relational-ontological perspective, separability is never guaranteed.
There are cases in which propositions do not exist independently of their relations to other propositions, contexts, or perspectives. What a proposition is cannot be specified without reference to the network of relations in which it participates.
In such cases, attempting to isolate propositions for formal manipulation changes their content. The cut required to individuate them simultaneously dissolves the relational conditions that make them meaningful.
Logic falters here not because reasoning fails, but because the presupposition of independently identifiable propositions cannot be sustained.
Context-sensitivity and co-constitution
Many logical difficulties arise where propositions are deeply context-sensitive. Their truth, applicability, or even identity shifts with background assumptions, pragmatic frames, or interpretive stance.
Formal logic attempts to treat context as external—something that can be bracketed or parameterised. Where context is co-constitutive, this move fails. There is no proposition prior to context that logic can manipulate.
This is not a defect of logic. It is a boundary condition.
Separability and contradiction
Contradictions often signal failures of separability rather than failures of consistency.
When propositions cannot be cleanly individuated, it becomes possible for what appears to be a single proposition to participate in incompatible relational roles. Logic registers this as contradiction, but the underlying issue is ontological: the assumption that propositions can be treated as independent units has been violated.
Seen this way, inconsistency is not always something to be eliminated. It is sometimes a marker of non-separable relational structure.
Individuation revisited
Individuation in logic is often taken for granted: a proposition is simply a proposition. But individuation, like stability, is a perspectival achievement.
What counts as a single proposition depends on how cuts are drawn—what distinctions are enforced, what relations are foregrounded, and what background is suppressed. Where these cuts cannot be held fixed, propositions multiply, merge, or dissolve under analysis.
Logic presupposes individuation; it cannot generate it.
Where separability fails
Failures of separability manifest as:
paradoxes driven by self-reference or circularity,
context-dependent propositions that resist formal isolation,
logical systems that fragment into incompatible formalisms depending on how propositions are carved.
These are not anomalies to be engineered away. They are indicators that relational reality exceeds the structural assumptions of logic.
Implications for the series
Understanding separability as a presupposition clarifies why extending logic often requires multiplying logics rather than refining one universal system. Each logic secures separability under a different construal.
In the next post, we will examine a third and final presupposition: invariance. Logic requires not only stable and separable propositions, but propositions whose truth-relations survive transformation. Where transformation alters constitution, logic encounters its deepest limits.
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