Test 4 Review

1
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

2
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, P is
  itself a subformula

3
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

4
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 formulas MLO is ? and immediate
  subformulas are lay and lya. They and Gyy are
  atomic formulas.

5
Test 4 Review

• The subformulas of (?y) (Iay ? Iya) Gyy
• (?y) (Iay ? Iya) Gyy
• (Iay ? Iya) Gyy
• Gyy
• (Iay ? Iya)
• Iay
• Iya

6
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

7
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)

8
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 youll
  need, etc.

9
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

10
Symbolizing English sentences in PL

• 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

11

• 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.

• 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

12
How changing the UD makes a difference

• UD the positive integers and Charles
• Ix x is a positive integer
• 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
• c Charles

13
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

14
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)

15

• 1
• UD Marbles
• Rx x is red
• Gx x is green
• Bx x is blue
• Sx x is a shooter
• Cx x is a cats 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 cats eye

16

• Symbolize each of the following sentences using
  UD 1, and then using UD 2
• None of the marbles is red but some are shooters.
• No red marble is a blue marble.
• Some marbles are red or cats eyes, but no marble
  is both.
• Some but not all of the marbles are blue.
• Some of the marbles are cats eyes and some are
  shooters, but none of the marbles is red.
• All the marbles that are shooters are red.