The main application of canonical models are completeness proofs. Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does not work for arbitrary L, because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L. We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if X is valid in every frame which satisfies P, for any normal modal logic L which contains X, the underlying frame of the canonical model of L satisfies P.

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The main application of canonical models are completeness proofs. Properties of the canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does not work for arbitrary L, because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L.

We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if X is valid in every frame which satisfies P, for any normal modal logic L which contains X, the underlying frame of the canonical model of L satisfies P. A union of canonical sets of formulas is itself canonical.

It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and compact. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical Goldblatt , , but the combined logic S4. In general, it is undecidable whether a given axiom is canonical. We know a nice sufficient condition: H. Sahlqvist identified a broad class of formulas now called Sahlqvist formulas such that: a Sahlqvist formula is canonical, the class of frames corresponding to a Sahlqvist formula is first-order definable, there is an algorithm which computes the corresponding frame condition to a given Sahlqvist formula.

This is a powerful criterion: for example, all axioms listed above as canonical are equivalent to Sahlqvist formulas. A logic has the finite model property FMP if it is complete with respect to a class of finite frames.

In particular, every finitely axiomatizable logic with FMP is decidable. There are various methods for establishing FMP for a given logic. Refinements and extensions of the canonical model construction often work, using tools such as filtration or unravelling.

As another possibility, completeness proofs based on cut-free sequent calculi usually produce finite models directly. Most of the modal systems used in practice including all listed above have FMP. In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete wrt a class of modal algebras, and a finite modal algebra can be transformed into a Kripke frame.

As an example, Robert Bull proved using this method that every normal extension of S4. Kripke semantics has a straightforward generalization to logics with more than one modality.

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## Nozick on Knowledge *

Truth-tracking is meant to deal with a range of hard epistemological cases, including Gettier-style problems and Brain-in-the-Vat-style skeptical arguments. PART I: Truth-tracking Explained Nozick begins his project by taking for granted the usual first two conditions for knowledge: 1 p is true; 2 S believes that p. Apply 3 : If it were , S would mistakenly believe that it was Barn Case: S drives past a barn, and comes to believe that he has seen a barn. Had he been in poor health, however, he would not have visited, and relatives would have told Grandma Smith that he was well. Later, a fake story is planted, in which it is claimed that the dictator was not really assassinated.

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## The Analysis of Knowledge

Personal life[ edit ] Nozick was born in Brooklyn to a family of Kohenic descent. His mother was born Sophie Cohen, and his father was a Jew from the Russian shtetl who had been born with the name Cohen and who ran a small business. He was then educated at Columbia University A. In addition, at Columbia he founded the local chapter of the Student League for Industrial Democracy , which in changed its name to Students for a Democratic Society. That same year, after receiving his bachelor of arts degree in , he married Barbara Fierer. They had two children, Emily and David. The Nozicks eventually divorced and he remarried, to the poet Gjertrud Schnackenberg.

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## Robert Nozick

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## In Defense of Sensitivity: Nozick, Kripke, and Predicate Exclusivity

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