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INTRO LOGICINTRO LOGICDAY 24DAY 24

Derivations in PLDerivations in PL33

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OverviewOverview

Exam 1 Sentential Logic Translations (+)

Exam 2 Sentential Logic Derivations

Exam 3 Predicate Logic Translations

Exam 4 Predicate Logic Derivations

6 derivations @ 15 points + 10 free points

Exam 5 very similar to Exam 3

Exam 6 very similar to Exam 4

Exam 1 Sentential Logic Translations (+)

Exam 2 Sentential Logic Derivations

Exam 3 Predicate Logic Translations

Exam 4 Predicate Logic Derivations

6 derivations @ 15 points + 10 free points

Exam 5 very similar to Exam 3

Exam 6 very similar to Exam 4

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Predicate Logic RulesPredicate Logic Rules

OTilde-Universal-Out

OUniversal-Out

UDUniversal Derivation

OTilde-Existential-Out

OExistential-Out

IExistential-In

today

day 1

day 2

today

day 2

day 1

4

Rules Already Introduced – Day 1Rules Already Introduced – Day 1

OLD name

–––––

OLD name

–––––

a name counts as OLD precisely if it occurs

somewhereunboxed and uncancelled

O I

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Rules Already Introduced – Day 2Rules Already Introduced – Day 2

NEW name

–––––

NEW name

:

:

a name counts as NEW precisely if it occurs

nowhere unboxed or uncancelled

O UD

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Rules to be Introduced TodayRules to be Introduced Today

OTilde-Universal Out

OTilde-Existential-Out

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Tilde-Universal Out (Tilde-Universal Out (O) O)

is any variable

is any (official) formula

not everyone is H

––––––––––––––

someone is not H

xHx––––––xHx

––––––

example

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Tilde-Existential Out (Tilde-Existential Out (O) O) is any variable

is any (official) formula

no one is H

––––––––––––––

everyone is un H

xHx––––––xHx

––––––

example

= not anyone is H

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SL Rules that are Often Useful SL Rules that are Often Useful in Connection with in Connection with O and O and O O

( & )–––––––––

( )–––––––––

&

&OO

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(6)

(5)

(4)

(3)

(2)

(1)

Example 1Example 1

not every F is H / some F is un-H

5, x(Fx & Hx)

4, Fa & Ha

3, (Fa Ha)

1, x(Fx Hx)

DD: x(Fx & Hx)

Prx(Fx Hx)

IOO

O

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Example 2Example 2every F is G ; no G is H / no F is H

(12)

(13)

(14)

(10)

(11)

(15)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

8,10, Ga

9, Ga Ha

12,13, Ha

7, Fa

Ha

11,14,

6, (Ga & Ha)

1, Fa Ga

4, Fa & Ha

2, x(Gx & Hx)

DD : As x(Fx & Hx)

D: x(Fx & Hx)

Prx(Gx & Hx)

Prx(Fx Gx)

O&O

O

&O

I

O

O

O

O

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New Strategy/Rule; New Strategy/Rule; DD

:

:

°

°

D

DD

is any (official) formula

is any variable

AsD is a species of ID

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Example 1 Example 1 ReRe -- donedone Using ID Using ID

(13)

(12)

(10)

(11)

(14)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

10,11, Ha

Ha

8, Fa Ha

9, Fa

12,13,

7, Fa & Ha

6, (Fa & Ha)

5, (Fa Ha)

3, x(Fx & Hx)

1, x(Fx Hx)

DD : As x(Fx & Hx)D (ID): x(Fx & Hx)

Prx(Fx Hx)

O

&O

&O

I

OO

O

O

O

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Example 3a (using ID) Example 3a (using ID)

(12)(13)(14)

(10)(11)

(15)

(9)(8)(7)(6)(5)(4)(3)(2)(1)

8, (Ga & Ha)12, Ga Ha11,13, Ha

4,

Ha7,9, Ga

10,14,

Fa6,

x(Gx & Hx)1, Fa Ga2, Fa & HaDD : As x(Gx & Hx)D (ID): x(Gx & Hx)Prx(Fx & Hx)Prx(Fx Gx)

O&OO

O

O

I

&O

OO

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(5)

(10)

(9)

(8)

(7)

(6)

(4)

(3)

(2)

(1)

Example 3b (using DD) Example 3b (using DD)

1, Fa Ga

9, x(Gx & Hx)

7,8, Ga & Ha

5,6, Ga

Ha4,

Fa

2, Fa & Ha

DD (…I): x(Gx & Hx)

Prx(Fx & Hx)

Prx(Fx Gx)

O

I

&I

O

&O

O

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(8)

(10)

(11)

(12)

(9)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 4Example 4if anyone is F, then everyone is unH/ if someone is F, then no one is H

1, Fa yHy

7,8, yHy

10, Hb

9,11,

5, Hb

3, Fa

DD : As yHy

D : yHy

As xFx

CD: xFx yHy

Prx(Fx yHy)

O

OO

I

O

O

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(10)

(11)

(12)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 5 Example 5 if someone is F, then someone is unH/ if anyone is F, then not-everyone is H

9, Hb

6, Hb

10,11

1,8, yHy

4, xFx

DD : As yHy

ID : yHy

As Fa

CD : Fa yHy

UD: x(Fx yHy)

PrxFx yHy

O

O

O

OI

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THE ENDTHE END