IndisputableMonolith.Foundation.LogicFromCost
Derives classical logical constraints from the Recognition cost functional: propositions carry a balance ratio, stable truth sits at ratio one (zero cost), and contradictions have strictly positive cost so cannot exist. Anyone tracing how RS obtains non-contradiction from J-cost rather than as a primitive axiom would cite this module. The argument is definitional setup plus positivity and zero-cost selection lemmas imported from the existence and ontology layers.
claimA propositional configuration pairs a claim with a balance ratio $r > 0$: $r = 1$ is stable truth (zero cost), $r \to 0$ is absence (false), and $r \to \infty$ is unstable over-assertion. Contradiction configurations have strictly positive cost and are impossible at zero defect; consistent configurations are those in the zero-cost regime.
background
Recognition Science treats existence and truth as selection outcomes of cost minimization under the unique J-cost, not as primitives. The Law of Existence states that $x$ exists if and only if its defect vanishes. Ontology predicates package that selection story operationally: RS-existence and RS-truth are zero-cost outcomes under J.
This module sits one layer above pre-logical cost (configuration values on the unit interval) and turns those cost facts into a propositional calculus. A propositional configuration is a proposition together with a ratio measuring balance: ratio one is perfectly balanced (true and stable), ratio near zero is absence (false), and unbounded ratio is unstable assertion.
Sibling structure introduces cost of a proposition, stability and instability predicates, contradiction configurations with their cost, and consistency as the zero-cost class. The local claim is that logical contradiction is cost-forbidden, so classical non-contradiction is forced by the cost foundation rather than postulated.
proof idea
Definition-heavy module, not a single theorem. It introduces propositional and contradiction configurations, assigns them costs via the pre-logical and J-cost apparatus, and proves positivity: contradiction cost is strictly positive. Zero-cost selection (from the Law of Existence and ontology predicates) then yields that logical contradictions are impossible and that zero-cost contradiction is forbidden. Consistency is the residual zero-cost class. Downstream consumers import these predicates as the logical floor under the forcing chain.
why it matters in Recognition Science
Supplies the bridge from cost ontology to logic: non-contradiction and consistency become theorems about positive cost, not axioms. The sole recorded consumer is UnifiedForcingChain, which claims all of T0 through T8 are forced inevitabilities from the cost foundation (Recognition Composition Law). Without a cost-derived ban on contradiction, that chain would still need a separate logical substrate.
In framework terms this is pre-T5 scaffolding for the absolute floor: once contradiction is cost-impossible and existence is zero defect, the unique J, the self-similar fixed point $\phi$, the eight-tick octave, and $D = 3$ can be forced without smuggling classical logic in by hand. It closes the gap between PreLogicalCost / LawOfExistence and the unified forcing narrative.
scope and limits
- Does not prove uniqueness of J or the RCL; those live upstream or in the forcing chain.
- Does not derive T5–T8, phi, eight-tick structure, or D = 3 inside this module.
- Does not give a full first-order proof system; only cost-level contradiction and consistency.
- Does not treat quantitative mass, alpha, or continuum physics claims.
- Does not replace Mathlib classical logic; it interprets contradiction via cost.
used by (1)
depends on (3)
declarations in this module (19)
-
structure
PropConfig -
def
prop_cost -
def
IsStable -
def
IsUnstable -
structure
ContradictionConfig -
def
contradiction_cost -
theorem
contradiction_positive_cost -
def
IsLogicalContradiction -
theorem
logical_contradiction_impossible -
theorem
zero_cost_contradiction_forbidden -
structure
ConsistentConfig -
def
consistent_cost -
theorem
consistent_zero_cost_possible -
theorem
consistent_minimum_cost -
theorem
logic_from_cost -
theorem
why_logic_is_real -
theorem
prelogical_boolean_fragment -
theorem
mp_from_cost_and_logic -
theorem
logic_from_cost_summary