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INTRO LOGICINTRO LOGICDAY 25DAY 25

Derivations in PLDerivations in PL44

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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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Rules Introduced – Day 1Rules Introduced – Day 1

OLD name

–––––

OLD name

–––––

a name counts as OLD precisely if it occurs somewhere

unboxed and uncancelled

O I

4

Rules Introduced – Day 2Rules Introduced – Day 2

NEW name

–––––

NEW name

:

:

a name counts as NEW precisely if it occurs nowhere

unboxed or uncancelled

O UD

5

Rules Introduced – Day 3Rules Introduced – Day 3

–––––

–––––

is any formula

is any variable

O O

7

StrategiesStrategies

DD or D

UD

SL strategy, , &,

show-strategymain operator

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Show-Show- Strategy (UD) Strategy (UD)

:

:

°

°

°

UD

??

must be a NEW name

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Show-Show- Strategy ( Strategy (D)D)

:

:

°

°

D

DD

As

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Example 1 (repeated)Example 1 (repeated)every 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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(10)(11)(12)

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

Example 2 (repeated)Example 2 (repeated)if someone is F, then someone is unH/ if anyone is F, then not-everyone is H

9, Hb6, Hb10,11

1,8, yHy4, xFxDD : As yHyID : yHyAs FaCD : Fa yHyUD: x(Fx yHy)PrxFx yHy

OOO

OI

12

(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 3Example 3there is someone whom everyone R’s

/ everyone R’s someone or other

8,9, 7, Rab

6, Rab

4, yRay

1, yRyb

DD : As yRay

D (ID) : yRay

UD: xyRxy

PrxyRyx

IO

O

O

O

13

(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 4Example 4there is someone who R’s no-one

/ everyone is dis-R’ed by someone or other

8,

Rba

6, yRby

4, yRya

1, yRby

DD : As yRya

D (ID) : yRya

UD: xyRyx

PrxyRxy

Rba

(11) 9,10,

7,

O

O

O

O

I

O

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(10)

(8)

(12)

(13)

(11)

(9)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 5Example 5there is someone who R’s every F/ every F is R’ed by someone or other

8, Fa Rba

1, y(Fy Rby)

4,10, Rba

11,12,

9, Rba

6, yRya

DD : As yRya

D (ID) : yRya

As Fa

CD : Fa yRya

UD: x(Fx yRyx)

Prxy(Fy Rxy)

O

O

OI

O

O

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(10)

(6)

(12)

(13)

(14)

(11)

(15)

(9)

(8)

(7)

(5)

(4)

(3)

(2)

(1)

Example 6Example 6there is some F who R’s no-one/ everyone is dis-R’ed by some F or other

7, (Fb & Rba)

1, Fb & yRby

10, Fb Rba

11, Rba

8,12 Rba

9, yRby

13,14,

yRby6,

Fb

4, y(Fy & Rya)

DD : As y(Fy & Rya)

D (ID) : y(Fy & Rya)

UD: xy(Fy & Ryx)

Prx(Fx & yRxy)

O

O

&O

OO

O

I

&O

O

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THE ENDTHE END