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Today’s Topics

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  1. Today’s Topics Relational predicates Multiple quantification Expansions of relational predicates

  2. Relational Predicates • Assert relations that exist between objects • Always have at least 2 variables • Depending on the predicate, have or lack properties of • Symmetry • Transitivity • Reflexivity • The order of the variables in a relational predicate is crucial

  3. Symbolizing with multiple quantifiers • When symbolizing a quantified sentence with multiple quantifiers, it is frequently a good idea to paraphrase inward. • Work from the gross external structure of an English sentence toward the finer structures

  4. Any thief can pick any lock • (x)(If x is a thief, then x can pick any lock) • (x)(Tx  x can pick any lock) • (x)(Tx (y)(If y is a lock, then x can pick it)) • (x)(Tx (y)(Ly  Pxy))

  5. The order of quantifiers is significant • (x)( y)Lxy says “everybody loves somebody or other” • (x)(y)Lxy says, “there is at least one person who loves everyone” • (y)(x)Lxy says “somebody is loved by everyone” • (y)(x)Lxy says “everybody is loved by somebody or other” • In a 3 element universe {a, b, c} where Lab, Lba, Lbb, Lbc, and Lcc are true: • 1 is true, 2 is true, 3 is false and 4 is true

  6. Truth Functional Expansions of Formulas Using Relational Predicates • A truth functional expansion of a formula using relational predicates works just like a truth functional expansion of a formula using only monadic predicates.

  7. Consider the 2 element universe {a,b} • The truth functional expansion of • (x)(y)Lxy is: • [Laa v Lab]  [Lba v Lbb] • The truth functional expansion of • (x)(y)Lxy is: • [Laa  Lab] v [Lba  Lbb]

  8. Interpretations of Universes with Relational Predicates • Providing an interpretation of a universe using relational predicates is a bit more difficult than with monadic predicates. • We can provide an exhaustive list of all of the ordered sets of elements in the universe R • ab • ac • bc

  9. Alternatively, we can use a graphic representation of the extension of a predicate. • In the following diagrams, the circle represents the universe. • The numbers in the circles are the elements in the universe. • An arrow running from one number to another, 1 2, means that 1 bears the relation in question to 2, but not that 2 bears the relation to 1 unless there is another arrow, 2 1

  10. Is the formula true or false in the universe as interpreted? I

  11. It is FALSE. There is no single object which bears the B relationship to everything in the universe. 8 comes close, but 8 does not bear B to itself, so 8 fails to bear B to everything. • A small change in the diagram, however, makes the formula true in the universe.

  12. This formula is true in the universe as interpreted,

  13. Is the formula true or false in the universe as interpreted? II

  14. Is the formula true or false in the universe as interpreted? III

  15. Is the formula true or false in the universe as interpreted? IV

  16. Is the formula true or false in the universe as interpreted? V

  17. The formula in universe II is TRUE—every object in the universe bears B to some object or other. • The formula in universe III is TRUE—everything in the universe bears B to some identifiable thing (2) including 2. • The formula in universe IV is FALSE • The formula in universe V is TRUE.

  18. Interpretations with Multiple Predicates and Relations • x(Tx  y(Ly  Rxy)) • y(Ly  x(Tx  Rxy)) • x(Tx  y(Ly  Rxy)) • y(Ly  ~x(Tx  Rxy)) • x(Tx  y(Ly  ~Rxy))