Chapter 1 X Quantifiers and Quantified Statements

1. Chapter 1X Quantifiers and Quantified Statements

2. Definitions & Examples • There are two basic types of quantified statements • First type: says something about a every member of a group or class of things • For example All dogs are hounds • Second type: says that a group with a certain characteristic has at least one member • For example: Some dogs are wild • Example 2: There is a dog that is a pointer

3. Definitions & Examples • Quantified statements can be combinations of these types • For example: All dogs are not hounds and some dogs are wild

4. Keywords for Quantifiers • The adjective “All” is called a universal quantifier • It is sometimes replaced with “For All”, “For every” or “ For every • The adjective “Some” is called a existential quantifier • It is sometimes replaced with “There is”, “There exists”, or “ There is at least one”

5. Check this out… Stuff with universal quantifier • Each of the following statements is saying the same thing • In other words, each statement would have the same truth value • All dogs are hounds • For all dogs x, x is a hound • For every dog x, x is a hound • For each dog x, x is a hound • For all x, if x is a dog, then x is a hound • Statements 2, 3, 4, 5 may seem awkward but there are advantages to each

6. Important NOTES regarding Existential • It is customary to NOT read plural into the quantifier “Some” even though statement 1 above seems to suggest that more than on dog is a pointer • The phrase “For Some” should be avoided since it is not always clear whether or not the intended meaning is “All” or “Some”

7. NOW Check this out… Stuff with existential quantifier • Each of the following statements is saying the same thing • In other words, each statement would have the same truth value • Some dogs are hounds • There is a dog that is a pointer • There exists a dog that is a pointer • There is a t least one dog that is a pointer • There is an x such that x is a dog and x is a pointer

8. Terms & Predicates • Consider the statements: • 3 is a whole number • X is a whole number • In these examples, “is a whole number” is called a predicate • 3 is called a term • What else would be called a term?

9. Terms & Predicates CONTINUED • Now lets assign them letters • Let W denote the predicate “is a whole number”; thus W3 is read as “ 3 is a whole number” • What if it was Wx? • Wx is read as “x is a whole number” • What about ~W1/2 ? • ~W1/2 is read as “1/2 is not a whole number?” • How is the statement “y is not a whole number” be written as? • ~Wy

10. Symbolic Representation of Quantified Statements • We previously translated statements into symbolic form, well kind of the same thing except no P1’s or P2’s here, there are special symbols • “(A x)” will replace and be read as “for all x, or for every x” • “( Ǝx)” will replace and be read as “ there is an x, or there exists an x”

11. Lets try it • Consider the statement: All tigers are cats • Let T be the predicate “is a tiger” • Let C be the predicate “ is a cat • You can use the symbols to rewrite the statement in the following ways: • “For all x, if x is a tiger, then x is a cat” • “For all x, ( Tx Cx)” • “(Ax)(Tx Cx) • The last statement is the symbolic form of the original statements

12. NOW ONE with an Existential • Consider the statement: Some quadrilaterals are squares • This is logically equivalent to • There exists an x such that x is a quadrilateral and x is a square • Let Q be the predicate “is a quadrilateral” • Let S be the predicate “ is a square” • You can use the symbols to rewrite the statement in the following ways: • “There exists an x such that Qx ^ Sx • “(Ǝx)(Qx^ Sx) • The last statement is the symbolic form of the original statements