Test 4 Review

For test 4, you need to know:For test 4, you need to know:

DefinitionsDefinitions

the recursive definition of ‘formula of the recursive definition of ‘formula of PLPL’’

atomic formula of atomic formula of PLPL

sentence of sentence of PLPL

bound variablebound variable

free variablefree variable

Test 4 Review

You need to know:You need to know:

The kinds of formulas of The kinds of formulas of PLPL

How to identify formulas of each kindHow to identify formulas of each kind

How to break down a formula identifyingHow to break down a formula identifying

its main logical operatorits main logical operator

its immediate subformula or subformulasits immediate subformula or subformulas

any main logical operators of its immediateany main logical operators of its immediate

subformulas and its subformulas, and so forthsubformulas and its subformulas, and so forth

and remember that for any formula and remember that for any formula PP of of PLPL, , P is itself a subformula

Test 4 Review

Identify the subformulas of:

(y) [(Iay Iya) & ~Gyy]Main logical operatorKind of formulaImmediate subformulaMain logical operators (if any) of it or themImmediate 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: UD: the positive integersthe positive integers

LxyLxy x is larger than yx is larger than y

SxySxy x is smaller than yx is smaller than y

Ex:Ex: x is evenx is even

Ox:Ox: x is oddx is odd

Dx:Dx: x is evenly divisible by 2x is evenly divisible by 2

a:a: 22

b:b: 33

UD: the positive UD: the positive integersintegers

Lxy: x is larger than yLxy: x is larger than ySxy: x is smaller than ySxy: x is smaller than yEx: x is evenEx: x is evenOx: x is oddOx: x is oddDx: x is evenly divisible Dx: x is evenly divisible

by 2by 2a:a: 2 2b:b: 3 3

Even positive integers are Even positive integers are evenly divisible by 2.evenly divisible by 2.

No odd positive integer is No odd positive integer is evenly divisible by 2.evenly divisible by 2.

If 3 is evenly divisible by If 3 is evenly divisible by 2, 3 is even.2, 3 is even.

Any positive integer not Any positive integer not evenly divisible by 2 is evenly divisible by 2 is odd.odd.

2 is smaller than 3, but 2 2 is smaller than 3, but 2 is even.is even.

3 is not larger than itself, 3 is not larger than itself, but it is larger than 2. but it is larger than 2.

How changing the UD makes a difference

UD: UD: the positive integers and the positive integers and Charles

Ix:Ix: x is a positive integerx is a positive integer

LxyLxy x is larger than yx is larger than y

SxySxy x is smaller than yx is smaller than y

Ex:Ex: x is evenx is even

Ox:Ox: x is oddx is odd

Dx:Dx: x is evenly divisible by 2x is evenly divisible by 2

a:a: 22

b:b: 33

c: c: CharlesCharles

Changing the UD makes a difference

Even positive integers are evenly divisible by 2.Even positive integers are evenly divisible by 2.

((x) [(Ix & Ex) x) [(Ix & Ex) Dx] or Dx] or

~(~(x) [Ix & Ex) & ~Dx] x) [Ix & Ex) & ~Dx]

No odd positive integer is evenly divisible by 2.No odd positive integer is evenly divisible by 2.

~(~(x) [(Ix & Ox) & Dx] orx) [(Ix & Ox) & Dx] or

((x) [Ix & Ox) x) [Ix & Ox) ~Dx] ~Dx]

Any positive integer not evenly divisible by 2 is Any positive integer not evenly divisible by 2 is odd.odd.

((x) [Ix & ~Dx) x) [Ix & ~Dx) Ox] or Ox] or

~(~(x) [(Ix & ~Dx) & ~Ox]x) [(Ix & ~Dx) & ~Ox]

Changing the UD makes a difference

No positive integer is larger than Charles.No positive integer is larger than Charles.

~(~(x) (Ix & Lxc)x) (Ix & Lxc)

((x) (Ix x) (Ix ~Lxc) ~Lxc)

Charles is larger than some positive integer.Charles is larger than some positive integer.

((x) (Ix & Lcx)x) (Ix & Lcx)

~(~(x) (Ix x) (Ix ~Lcx) ~Lcx)

11

UD:UD: MarblesMarbles

Rx:Rx: x is redx is red

Gx:Gx: x is greenx is green

Bx:Bx: x is bluex is blue

Sx:Sx: x is a shooterx is a shooter

Cx:Cx: x is a cat’s x is a cat’s eyeeye

22

UD: The marbles being used UD: The marbles being used by Sarah and Markby Sarah and Mark

Mx:Mx: x is a marblex is a marble

Rx:Rx: x is redx is red

Gx:Gx: x is greenx is green

Bx:Bx: x is bluex is blue

Sx:Sx: x is a shooterx is a shooter

Cx:Cx: x is a cat’s eyex is a cat’s eye

Symbolize each of the following sentences using Symbolize each of the following sentences using UD 1, and then using UD 2:UD 1, and then using UD 2:

1.1. None of the marbles is red but some are None of the marbles is red but some are shooters.shooters.

2.2. No red marble is a blue marble.No red marble is a blue marble.

3.3. Some marbles are red or cat’s eyes, but no Some marbles are red or cat’s eyes, but no marble is both.marble is both.

4.4. Some but not all of the marbles are blue.Some but not all of the marbles are blue.

5.5. Some of the marbles are cat’s eyes and some Some of the marbles are cat’s eyes and some are shooters, but none of the marbles is red.are shooters, but none of the marbles is red.

6.6. All the marbles that are shooters are red.All the marbles that are shooters are red.