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Phase 6: S5 decision procedure and the ontological arguments - #8
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- S5 labelled tableau for propositional modal inferences: NNF, global necessities, one witness world per possibility, non-branching rules first, 200k-step budget; invalid inferences get a Kripke countermodel with duplicate worlds removed - Plantinga's modal ontological argument, checked by the tableau - Gödel's ontological argument (Scott's version) in higher-order S5: definitions of God-likeness, essence and necessary existence; T1, the corollary, T2 and T3 fully proved in natural deduction; objections on positivity and modal collapse - Tests: K, T, 4, 5, B, the S5 collapse, invalid forms, countermodel shape, and the step budget on a pigeonhole instance Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_019WEmxyo9CRKSbjbEdCuJTG
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Stacked on #7. Implements Phase 6 of
ROADMAP.md.src/core/s5.cpp) decides unjustified propositional modal inferences. An invalid one is reported with a Kripke countermodel, e.g.counterexample with 2 worlds, in w0 (actual) A = false; in w1 A = true. It expands non-branching formulas first and stops after 200k steps, so hostile input can't hang it (tested on a pigeonhole instance).Evidence it isn't rubber-stamping: nine targeted mutations of the Gödel proof are each rejected at the right line. They include bringing a non-modalized formula into a
box, a non-fresh name, adefstep that drops a[],dia_eclaiming necessity, and a premise that is no longer necessitated.Modelling decisions, recorded in the roadmap's decision log:
[]...), as Gödel and Scott intend.Generated by Claude Code