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## Today’s Topics

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**Today’s Topics**Relational predicates Multiple quantification Expansions of relational predicates**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**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**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))**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**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.**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]**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**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**Is the formula true or false in the universe as interpreted?**I**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.**Is the formula true or false in the universe as interpreted?**II**Is the formula true or false in the universe as interpreted?**III**Is the formula true or false in the universe as interpreted?**IV**Is the formula true or false in the universe as interpreted?**V**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.**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))