Test 4 Review For test 4, you need to know: Definitions the recursive definition of ‘formula of PL’ atomic formula of PL sentence of PL bound variable free variable

Test 4 Review You need to know: The kinds of formulas of PL How to identify formulas of each kind How to break down a formula identifying its main logical operator its immediate subformula or subformulas any main logical operators of its immediate subformulas and its subformulas, and so forth and remember that for any formula P of PL, and remember that for any formula P of PL, P is itself a subformula

Test 4 Review Identify the subformulas of: (  y) [(Iay  Iya) & ~Gyy] Main logical operator Kind of formula Immediate subformula Main logical operators (if any) of it or them Immediate subformulas (if any) of it or them The formula itself

Test 4 Review (  y) [(Iay  Iya) & ~Gyy] Main logical operator: (  y) Kind of formula: quantified formula Immediate subformula: (Iay  Iya) & ~Gyy MLO: & Immediate subformulas: Iay  Iya and Gyy Left formula’s MLO is  and immediate subformulas are lay and lya. They and Gyy are atomic formulas.

Test 4 Review The subformulas of (  y) [(Iay  Iya) & ~Gyy]: (  y) [(Iay  Iya) & ~Gyy] (Iay  Iya) & ~Gyy Gyy (Iay  Iya) Iay Iya

Test 4 Review You need to be able to: Symbolize English sentences into PL using a provided symbolization key If appropriate, provide a symbolization using a universal quantifier and one using an existential quantifier for a sentence. Construct a symbolization key appropriate to a provided set of sentences

Test 4 Review (new slide) Equivalent forms: 1. All As are Bs. (  x) (P  Q)~ (  x) (P & ~Q) 2. Some As are Bs. (  x) (P & Q)~(  x) (P  ~Q) 3. Some As are not Bs. (  x) (P & ~Q)~(  x) (P  ~B) 4. No As are Bs. (  x) (P  ~Q)~(  x) (P & Q)

Test 4 Review To provide a symbolization key for a set of sentences, you need to: Specify a UD Provide formulas for all predicates Provide constants for names and definite descriptions. The UD can be as general as ‘Everything’ or as specific as ‘the jelly beans in this jar’. Consider the sentences, the predicates you’ll need, etc.

Test 4 Review Construct a symbolization key for the following set of sentences: ‘Witches are made of wood.’ ‘All things made of wood float.’ ‘Witches float.’ ‘Joan is a witch.’ UD: Everything Wx: x is a witch Mx: x is made of wood Fx: x floats j: Joan

Symbolizing English sentences in PL UD: the positive integers Lxyx is larger than y Sxyx is smaller than y Ex:x is even Ox:x is odd Dx:x is evenly divisible by 2 a:2 b:3

UD: the positive integers Lxy: x is larger than y Sxy: x is smaller than y Ex: x is even Ox: x is odd Dx: x is evenly divisible by 2 a: 2 b: 3 Even positive integers are evenly divisible by 2. No odd positive integer is evenly divisible by 2. If 3 is evenly divisible by 2, 3 is even. Any positive integer not evenly divisible by 2 is odd. 2 is smaller than 3, but 2 is even. 3 is not larger than itself, but it is larger than 2.

How changing the UD makes a difference UD: the positive integers and UD: the positive integers and Charles Ix:x is a positive integer Lxyx is larger than y Sxyx is smaller than y Ex:x is even Ox:x is odd Dx:x is evenly divisible by 2 a:2 b:3 c: Charles

Changing the UD makes a difference Even positive integers are evenly divisible by 2. (  x) [(Ix & Ex)  Dx] or ~(  x) [Ix & Ex) & ~Dx] No odd positive integer is evenly divisible by 2. ~(  x) [(Ix & Ox) & Dx] or (  x) [Ix & Ox)  ~Dx] Any positive integer not evenly divisible by 2 is odd. (  x) [Ix & ~Dx)  Ox] or ~(  x) [(Ix & ~Dx) & ~Ox]

Changing the UD makes a difference No positive integer is larger than Charles. ~(  x) (Ix & Lxc) (  x) (Ix  ~Lxc) Charles is larger than some positive integer. (  x) (Ix & Lcx) ~(  x) (Ix  ~Lcx)

1 UD:Marbles Rx:x is red Gx:x is green Bx:x is blue Sx:x is a shooter Cx:x is a cat’s eye 2 UD: The marbles being used by Sarah and Mark Mx:x is a marble Rx:x is red Gx:x is green Bx:x is blue Sx:x is a shooter Cx:x is a cat’s eye

Symbolize each of the following sentences using UD 1, and then using UD 2: 1. None of the marbles is red but some are shooters. 2. No red marble is a blue marble. 3. Some marbles are red or cat’s eyes, but no marble is both. 4. Some but not all of the marbles are blue. 5. Some of the marbles are cat’s eyes and some are shooters, but none of the marbles is red. 6. All the marbles that are shooters are red.