---
slug: alternative_space_quantification
name: Alternative-Space Quantification
one_liner: "Evaluate a proposition not against the single actual case but by quantifying over a structured space of alternatives reached through an accessibility relation — the general form of which modal (necessity/possibility) reasoning is one species."
type: prime
schema_version: 2
status: density_v2_draft
categories: [epistemic, structural, linguistic]
origin_domain: logic
alternate_origin_domains: [computer_science, philosophy_of_language, decision_theory]
aliases: [quantification_over_worlds, accessibility_relation_reasoning, operator_over_alternatives, possible_worlds_semantics, structured_alternative_evaluation]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-05 generalization) — the proposed GENUS of `modal_reasoning` (lift: strip the *modality* frame — the alternatives need not be necessity/possibility worlds — up to any evaluation of a proposition by quantifying over an accessibility-structured space of alternatives). Discovered in the op-05 pilot sweep 2026-07-03. Tier B candidate (sonnet-leaning coinage-adjacent; `operator_over_alternatives` absent, `possible_worlds` absent; graded moderate). STATUS: HOLD pending owner keep/cut."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `modal_reasoning` and the named siblings (temporal logic over branching timelines, deontic reasoning over permitted worlds, epistemic logic over knowledge-accessible worlds); confirm it is a genuine established genus (Kripke-semantics abstraction) covering the input AND the siblings and not a Tier-B coinage or a restatement of `modal_reasoning`; draw a subsumption edge modal_reasoning→alternative_space_quantification + sibling edges at incorporation; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 5
  transfer_evidence: 3
  reasoning: 'The bare structure — a truth is assessed by ranging an operator over a set of alternatives connected to the actual point by an accessibility relation, so that "holds everywhere accessible" and "holds somewhere accessible" become the two evaluation modes — is a pure logical form (Kripke frame semantics) with maximal structural abstraction (5) and no dependence on any medium. Domain breadth is 4: the same frame is instantiated by temporal, deontic, epistemic, and doxastic logics, and by branching-time model checking in CS. Transfer evidence is a shade lower (3) because, although the mathematics unifies these, the instances live within formal-logic and formal-verification practice; the abstraction is real but its documented reach is largely to other *formal* systems rather than to empirical domains, and the candidate is Tier-B (coinage-adjacent), so the genus label itself needs owner confirmation.'
similarity_to_nearest_existing_prime:
  slug: modal_reasoning
  similarity: 0.65
structural_framed:
  label: structural
  aggregate: 0.18
  criteria: {vocab_travels: 0.85, evaluative_weight: 0.1, institutional_origin: 0.25, human_practice_bound: 0.1, import_vs_recognize: 0.3}
  rationale: 'The pattern is a pure formal-semantic structure — quantify an operator over an accessibility-structured alternative space — carrying essentially no evaluative content and no human-practice dependence; it is recognized as a bare relation (a Kripke frame) rather than imported as a doctrine. Its vocabulary travels cleanly across temporal, deontic, epistemic, and doxastic logics and into computer-science model checking, so it reads clearly structural. It has a named institutional origin (Kripke 1963) but as a mathematical form, not a value-laden practice.'
phase_b_passes_completed: []
---
# Alternative-Space Quantification

## Core Idea

Alternative-space quantification is the pattern of *evaluating a proposition not against the single actual case but by quantifying over a structured space of alternatives, where which alternatives count is fixed by an accessibility relation*. Bare structure: instead of asking "is P true here," one fixes a set of alternative points (worlds, states, times, scenarios), specifies for each point which others are *accessible* from it, and then evaluates P with an operator that ranges over the accessible set — "P holds at every accessible alternative" (a box/necessity-style reading) or "P holds at some accessible alternative" (a diamond/possibility-style reading). The constraint stripped from the input `modal_reasoning` is *the modality frame*: modal reasoning is the special case where the operators are read as necessity and possibility and the alternatives are metaphysical possible worlds. Lift away "the alternatives are possibility-worlds and the operators are must/might" and what remains is the general Kripke-frame move — range a quantifier over an accessibility-structured alternative space — of which alethic modality is one species.

The load-bearing anatomy has three parts. First, a *space of alternatives*: the points over which evaluation ranges. Second, an *accessibility relation*: a structure on that space fixing, from any point, which alternatives are reachable and thus relevant — and it is the *properties* of this relation (reflexive, transitive, symmetric, branching) that determine the logic's character. Third, a *quantifying operator*: a universal or existential ranging over the accessible set, turning "true here" into "true throughout" or "true somewhere reachable." The characteristic payoff is *uniform expressive power*: a single semantic machine handles must/might, always/eventually, obligatory/permitted, and known/believed just by re-interpreting the space and re-tuning the accessibility relation — one structure, many logics. The characteristic cost is *sensitivity to the frame*: results hinge entirely on how the alternative space and its accessibility relation are drawn, and a mis-specified relation (too permissive, wrongly transitive) silently changes what comes out valid, so the method's answers are only ever as trustworthy as an accessibility structure that is itself a modeling choice, not a fact read off the world.

Read structurally rather than as the semantics of necessity, the same move recurs across formal systems that never mention possibility at all. In *temporal logic*, evaluation ranges over a branching space of timelines, the accessibility relation is the flow of time, and the operators become "always" and "eventually" (this is the machinery of model checking in computer science). In *deontic logic*, the alternatives are the worlds where all obligations are met, and the operators become "obligatory" and "permitted." In *epistemic logic*, the alternatives are the states an agent cannot distinguish from the actual one, and the operator becomes "the agent knows." These siblings are structurally unlike the alethic input — a branching timeline is not a space of metaphysical possibilities, an information partition is not a modal accessibility of worlds — yet each is the identical skeleton: a structured alternative space, an accessibility relation, and an operator quantifying over what is reachable.

## Notes

**Provenance.** Produced by operator 05 (generalization / lift) as the proposed genus of `modal_reasoning`, reached by stripping the modality frame up to any evaluation-by-quantification over an accessibility-structured alternative space (the Kripke-frame abstraction). Surfaced in the op-05 pilot sweep on 2026-07-03. Claimed siblings: temporal logic over branching timelines, deontic reasoning over permitted worlds, epistemic logic over knowledge-accessible worlds. Manifest tier: B (`operator_over_alternatives` absent, `possible_worlds` absent; graded moderate).

**For owner evaluation (keep/cut).** This is a **Tier-B candidate and should be read as coinage-adjacent** — the *underlying* abstraction (Kripke frame semantics unifying alethic, temporal, deontic, and epistemic logics) is genuinely established mathematics, but the *prime label* "alternative-space quantification" is the proposer's naming of it and needs Kurt's confirmation that this is the concept the corpus wants and not a bespoke wrapper. The case FOR: the siblings are strong and genuinely non-alethic (temporal, deontic, epistemic logics really are the same Kripke machine with different frames), so the genus survives the sibling test on the merits; the unification is a real and load-bearing insight in logic and formal verification, and the corpus lacks it. The specific RISKS Kurt should weigh: (1) **restatement of the input** — similarity to `modal_reasoning` is high (~0.65); the whole keep case rests on whether the corpus treats temporal/deontic/epistemic reasoning as *distinct primes that share this genus*, or as already-covered under a broadly-construed `modal_reasoning`. (2) **coinage vs established name** — if the concept is kept, consider naming it after its established home (Kripke/possible-worlds/frame semantics) rather than the coined phrase. (3) **overlap with `counterfactual_reasoning` / `possibility`** — check adjacency; this prime is specifically the *quantify-over-an-accessibility-structured-space* machine, not counterfactual selection per se. Sharp keep/cut question: *does the corpus want the Kripke-frame genus as its own node above `modal_reasoning`, and if so, is "alternative-space quantification" the right name or should it be renamed to its established form?*

**At incorporation.** Re-run similarity against `modal_reasoning` (the input) plus any `counterfactual_reasoning`, `possibility`, `temporal_logic`, or `epistemic_logic` prime to confirm it is a genuine genus and not a restatement. If kept, resolve the naming (coined label vs established possible-worlds/frame-semantics name) before drafting v1. Draw a subsumption edge `modal_reasoning` → `alternative_space_quantification` and sibling edges to temporal / deontic / epistemic logic primes if present or added. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
