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Modality Possible Worlds

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Possibility and Necessity Triangles have three sides. The Wassermans have two kids.

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Possibility and Necessity It is necessary that: Triangles have three sides. It is possible that: The Wassermans don’t have two kids.

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De Dicto and De re It is necessary that God is good. God is necessarily good. A sentence expresses a de dicto modal statement iff the modal operator modifies the sentence. A sentence expresses a de re modal statement iff the modal operator modifies the predicate.

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Essential and Non-Essential Properties Object o is essentially F iff i.o is F; and ii.it is necessary that: If o exists, then o is F. Object o is contingently F iff i.o is F; and ii.it is possible that: o exists and o is not F.

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The Linguistic Mistake W e could have used the world ‘triangle’ to mean ‘square’. If we had done that, triangles would have had four sides, rather than three. So it’s not necessary that triangles have three sides.

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The Epistemic Mistake S uppose that someone asked you whether 329*747=254,763. Before checking the calculation, you would say that it’s possible that 329*747=254,763 and it’s possible that it doesn’t. But if it’s possible that it doesn’t equal 254,763 then it’s not necessary that it does.

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Possibility and Necessity It is necessary that: Triangles have three sides. It is possible that: The Wassermans don’t have two kids.

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Possibility and Necessity □ (Triangles have three sides.) ◊ ( The Wassermans don’t have two kids).

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Possibility and Necessity □ (Triangles have three sides.) ~ ◊ ~ (Triangles have three sides.) ◊ ( The Wassermans don’t have two kids). ~ □ ~(The Wassermans don’t have two kids).

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Possible Worlds A statement is something that is true or false. A set of statements is consistent iff all of the members could be true together. A set of statements is complete iff, for any statement S, either S or not-S is a member of the set. A possible world is a consistent and complete set of statements. The actual world (w a ) is the possible world consisting of all true statements.

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Possible Worlds Predicates ‘W’ = ‘is a possible world’ ‘T’ = ‘is true in’ Definitions □ p = ( x) (Wx Tpx) ◊ p = ( x) (Wx & Tpx) Colloquially Necessity is truth in all possible worlds Possibility is truth in some possible world

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Possibility and Necessity It is necessary that: Triangles have three sides. It is possible that: The Wassermans don’t have two kids. Triangles have three sides.

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Lesson 1.4 Equations and Inequalities Goal: To learn how to solve equations and check solutions of equations and inequalities.

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