INTRO LOGIC
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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
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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