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assume binary relation C. I. Lewis canonical model Chapter class of frames classical logic closed under subformulas completeness results consistent COROLLARY definition degenerate denote derivable ends the proof equivalence relation equivalent modulo exists extension of KE4 extension of S5 Filtration Theorem finite model property finitely axiomatizable frame w.d.e. Fundamental Theorem Hence Hintikka schema implies index frame index logic inference rule integer irreflexive kites Kripke frames Kripke's last cluster Lemma Lindenbaum's Lemma logically finite modal axiom modal logic model w.d.e. n c Nat natural number neighborhood semantics nondegenerate cluster normal extension normal logic p-morphism precedes succeeds proof of Theorem propositional functions propositional letter prove quasi-normal logic quasi-regular regular logic relational frame S4Grz S4Zem schemata Scott frames Scroggs set of formulas smallest SOBOCINSKI strict partial ordering submodel Suppose Theorem 3.7 transitive frames valid virtually last