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INTRO LOGICINTRO LOGICDAY 23DAY 23

Derivations in PLDerivations in PL22

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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

+

+

+

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Rule SheetRule Sheet

available on courseweb page(textbook)

provided on exams

keep this in front of you when doing homework

don’t make

upyourownrules

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Sentential Logic RulesSentential Logic Rules

DD ID CD D &D etc.

DD ID CD D &D etc.

&I &O vO O O etc.

&I &O vO O O etc.

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Predicate Logic RulesPredicate Logic Rules

OTilde-Universal-Out

OUniversal-Out

UDUniversal Derivation

OTilde-Existential-Out

OExistential-Out

IExistential-In

day 3

day 1

day 2

day 3

day 2

day 1

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

–––––

–––––

is an OLD name

(more about this later)

O I

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Rules to be Introduced TodayRules to be Introduced Today

UDUniversal Derivation

OExistential-Out

9…

(?)

(c)

(?)

(b)

(?)

(a)

(3)

(2)

(1)

Example 1Example 1every F is H ; everyone is F / everyone is H

… …

?? ??

?? : Hc

?? ??

?? : Hb

?? ??

?? : Ha

??: xHx

PrxFx

Prx(Fx Hx)

(3) &.&.&.D: Ha & Hb & Hc & ………what

is

ultimately

involved

in

showing

a

universal

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(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1a Example 1a

5,6, Ha

2, Fa

1, Fa Ha

DD : Ha

??: xHx

PrxFx

Prx(Fx Hx)

one down,

a zillion to go!

OO

O

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(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1b Example 1b

5,6, Hb

2, Fb

1, Fb Hb

DD : Hb

??: xHx

PrxFx

Prx(Fx Hx)

two down,

a zillion to go!

OO

O

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(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1c Example 1c

5,6, Hc

2, Fc

1, Fc Hc

DD : Hc

??: xHx

PrxFx

Prx(Fx Hx)

three down,

a zillion to go!

OO

O

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But wait – the derivations all look alike!But wait – the derivations all look alike!

DD : Hc(4)

1,O Fc Hc(5)

2,O Fc(6)

5,6,O Hc(7)

DD : Hb(4)

1,O Fb Hb(5)

2,O Fb(6)

5,6,O Hb(7)

5,6,O Ha(7)

2,O Fa(6)

1,O Fa Ha(5)

DD : Ha(4)

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The Universal-Derivation StrategyThe Universal-Derivation StrategySo, all we need to do is

do one derivation with one name (say, ‘a’)

and then argue that

all the other derivations will look the same.

To ensure this,

we must ensure that the name is general,

which we can do by making sure

the name we select is NEW.

a name counts as NEW precisely if it occurs nowhere in the derivation

unboxed or uncancelled

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The Universal-Derivation Rule (UD)The Universal-Derivation Rule (UD)

:

:

°

°

°

°

UD

??

i.e., one that is occurs nowhere in the derivationunboxed or uncancelled

must be a NEW name,

replaces

is any (official) formula

is any variable

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Comparison with Universal-OutComparison with Universal-Out

OLD name

–––––

NEW name

:

:

a name counts as OLD precisely if it occurs

somewhere in the derivationunboxed and uncancelled

a name counts as NEW precisely if it occurs

nowhere in the derivationunboxed or uncancelled

O UD

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(8)

(7)

(6)

Ha

Fa

Fa Ha

6,7,

3,

1,

(5)

(4)

(3)

(2)

(1)

: Ha

: xHx

xFx

: xFx xHx

x(Fx Hx)

Example 2Example 2

DD

UD

As

CD

Pr

every F is H / if everyone is F, then everyone is H

a new

a old

a old

OO

O

18(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 3Example 3every F is G ; every G is H / every F is H

Ha

Ga

Ga Ha

Fa Ga

: Ha

Fa

: Fa Ha

: x(Fx Hx)

x(Gx Hx)

x(Fx Gx)

8,9,

5,7,

2,

As

CD

UD

Pr

Pr

a new

a old

a old

1,

DD

OOO

O

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(6)

(5)

(4)

(3)

(2)

(1)

Example 4Example 4everyone R’s everyone / everyone is R’ed by everyone

5, Rba

1, yRby

DD : Rba

UD : yRya

UD: xyRyx

PrxyRxy

b new

a new

b old

a oldO

O

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Existential-Out (Existential-Out (O) O)

any variable (z, y, x, w …)

any NEW name (a, b, c, d, …)

any formula replaces

–––––

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Comparison with Universal-OutComparison with Universal-Out

OLD name

––––––

NEW name

––––––

a name counts as OLD precisely if it occurs

somewhereunboxed and uncancelled

a name counts as NEW precisely if it occurs

nowhere unboxed or uncancelled

O O

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(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 5Example 5every F is un-H / no F is H

Ha

Ha

Fa

Fa Ha

Fa & Ha

DD : As x(Fx & Hx)

D: x(Fx & Hx)

Prx(Fx Hx)

new

old

8,9,

6,7,

5,

1,

3,

IO

&O

O

O

23(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 6Example 6some F is not H / not every F is H

Ha

Ha

Fa

Fa Ha

Fa & Ha

DD : As x(Fx Hx)

ID: x(Fx Hx)

Prx(Fx & Hx)

8,9,

6,7,

5,

3,

1,

IO

&O

O

O new

old

24(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 7Example 7every F is G ; some F is H / some G is H

9, x(Gx & Hx)

7,8, Ga & Ha

5,6, Ga

Ha4,

Fa

1, Fa Ga

2, Fa & Ha

DD: x(Gx & Hx)

Prx(Fx & Hx)

Prx(Fx Gx)

I

&I

O

&O

O

O new

old

25(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 8Example 8if anyone is F then everyone is H

/ if someone is F, then everyone is H

8, Ha

6,7, xHx

1, Fb xHx

3, Fb

DD : Ha

UD : xHx

As xFx

CD: xFx xHx

Prx(Fx xHx)

O

OO

O new

old

new

old

26(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 9Example 9if someone is F, then everyone is H/ if anyone is F then everyone is H

8, Hb

1,7, xHx

4, xFx

DD : Hb

UD : yHy

As Fa

CD : Fa yHy

UD: x(Fx yHy)

PrxFx xHx

O

OI

new

old

new

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(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 10 (a fragment) Example 10 (a fragment) someone R’s someone??missing premises??

/ everyone R’s everyone

6, Rcd

1, yRcy

?? : Rab

UD : yRay

UD: xyRxy

Pr???

PrxyRxy

(8) ?? ??

O

O

new

new

new

new

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THE ENDTHE END