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An overview of logic rules and derivations in both sentential and predicate logic. It includes information on various rules such as &i, &o, ~&o, &d, ∨i, ∨o, ∼∨o, ∨d, ∀i, ∀o, ∼∀o, ∃i, ∃o, and ∼∃o. The document also includes examples of derivations and explanations of how to use new names in derivations.
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Derivations in PL
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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
9 9 9 + + +
Exams 5 and 6 will be given on Friday, Dec 19 1:30-3:30 (4:00) Mahar Auditorium
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available on course web page (textbook)
provided on exams
keep this in front of you when doing homework
don’t
make
up
your
own
rules
Intro Logic &I (^) A A &O (^) A & B SENTENTIAL LOGIC A & B ∼&O (^) ∼ (^) (A & B) &DRules of Derivation: A&B B –––––– A (^) & B B––––––B (^) & A ––––––A ––––––B –––––––––A → ∼B |||Å|ÅAB ∨I ∨O ∼∨O ∨D ||(ID) A –––––– A (^) ∨ B A––––––B ∨ A A∼A–––––– ∨ B A∼B–––––– ∨ B ∼–––––––––∼A(A ∨ B) (^) ||∼ÅÎ(A: (^) ∨AB∨)B ↔I ↔O B^ A^ ∼↔O ∼B^ ↔D|| A B ––––––– ––––––– →→ AB AB →→ AB A–––––––A ↔→ BB A–––––––B ↔→ BA ∼––––––––––∼A(A ↔↔ BB ) (^) ||Å|: AA↔→BB A ↔ B B ↔ A | ||ÅB→A →I see CD →O (^) A A → C A∼C → C ∼→O (^) ∼––––––––––(A → C) CD|A: A→C ––––––– C –––––––∼A A & ∼C ||| : C DN (^) A ––––– DN (^) ∼∼A––––– Rep (^) A–– ∼D|A: ∼A ∼∼A A A | || : Î ÈI (^) A ∼A ÈO (^) Ζ–- DD (^) | : A ID|∼A: A –––– È A || (^) A ||| : Î In the following, [ o ] results by substituting ν is any variable; o for every free occurrence of [ν^ PREDICATE LOGIC ] is any formula in which ν, where ν occurs free; o is any old name. A name counts asotherwise it counts as[ n ] results by substituting old (^) if it occurs in a line that is neither boxed nor cancelled; new. n for every free occurrence of ν, where n is any new name. ∀I see UD ∀O (^) ∀ν–––––– [ν] ∼∀O (^) ∼––––––∀νΦ UD new |: ∀ν: [ n []ν] [ o ] old ∃ν∼Φ || || ∃I (^) ––––––[ o ] old ∃O (^) ∀ν–––––– [ν] ∼∃O (^) ∼––––––∀ν [ν ] ∃D (ID)|∼∃ν: ∃ν[ν] [ν] ∃ν [ν] [ n ] new ∃ν∼ [ν] | || : Î
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DD ID CD ~D &D etc.
&I &O vO →O ~∨O etc.
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…
(?)
(c)
(?)
(b)
(?)
(a)
(3)
(2)
(1)
every F is H ; everyone is F / everyone is H
… …
?? ??
: Hc ??
?? ??
: Hb ??
?? ??
: Ha ??
: ∀xHx ??
∀xFx Pr
∀x(Fx → Hx) Pr
(3) : Ha & Hb & Hc & ……… &.&.&.D
what is ultimately involved in showing a universal
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(7)
(6)
(5)
(4)
(3)
(2)
(1)
Ha 5,6,
Fa 2,
Fa → Ha 1,
: Ha DD
: ∀xHx ??
∀xFx Pr
∀x(Fx → Hx) Pr
one down, a zillion to go!
→O
∀O
∀O
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(7)
(6)
(5)
(4)
(3)
(2)
(1)
Hb 5,6,
Fb 2,
Fb → Hb 1,
: Hb DD
: ∀xHx ??
∀xFx Pr
∀x(Fx → Hx) Pr
two down, a zillion to go!
→O
∀O
∀O
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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)
: ∀v [v] : [n] ° ° ° °
UD ??
i.e., one that is occurs nowhere in the derivation unboxed or uncancelled
nn^ nn^ must be a^ NEW^ name,
n replaces v
[v] is any (official) formula
v is any variable
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OLD name
NEW name
a name counts as OLD precisely if it occurs somewhere in the derivation unboxed and uncancelled
a name counts as NEW precisely if it occurs nowhere in the derivation unboxed or uncancelled
∀O UD
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(8)
(7)
(6)
Ha
Fa
Fa → Ha
6,7,
3,
1,
(5)
(4)
(3)
(2)
(1)
}: H a
}: ∀xHx
∀xFx
}: ∀xFx → ∀xHx
∀x(Fx → Hx)
DD
UD
As
CD
Pr
every F is H / if everyone is F, then everyone is H
a new
a old a old →O
∀O
∀O
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any variable (z, y, x, w …)
any NEW name (a, b, c, d, …)
any formula (^) n replaces v
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OLD name
––––––
NEW name
––––––
a name counts as OLD precisely if it occurs somewhere unboxed and uncancelled
a name counts as NEW precisely if it occurs nowhere unboxed or uncancelled
(10) 21
(9)
(8)
(7)
(6)
(5)
(4)
(3)
(2)
(1)
every F is un- H / no F is H
È
∼Ha
Ha
Fa
F a → ∼H a
F a & H a
}: È DD
∃x(Fx & Hx) As
}: ∼∃x(Fx & Hx) ~D
∀x(Fx → ∼Hx) Pr
new old
8,9,
6,7,
5,
1,
3,
ÈI
→O
&O
∀O
∃O
(10) 22
(9)
(8)
(7)
(6)
(5)
(4)
(3)
(2)
(1)
some F is not H / not every F is H
È
Ha
∼Ha
Fa
F a → H a
F a & ∼H a
}: È DD
∀x(Fx → Hx) As
}: ∼∀x(Fx → Hx) ~D
∃x(Fx & ∼Hx) Pr
8,9,
6,7,
5,
3,
1,
ÈI
→O
&O
∀O
∃O new old
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(9)
(8)
(7)
(6)
(5)
(4)
(3)
(2)
(1)
if someone is F, then everyone is H / if anyone is F then everyone is H
H b 8,
∀xHx 1,7,
∃xFx 4,
}: H b DD
}: ∀yHy UD
Fa As
}: F a → ∀yHy CD
}: ∀x(Fx → ∀yHy) UD
∃xFx → ∀xHx Pr
∀O
→O
∃I
new
old
new
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(7)
(6)
(5)
(4)
(3)
(2)
(1)
someone R’s someone ??missing premises?? / everyone R’s everyone
Rc d 6,
∃yR c y 1,
}: Ra b ??
}: ∀yR a y UD
}: ∀x∀yRxy UD
??? Pr
∃x∃yRxy Pr
(8) ?? ??
∃O
∃O
new new
new new
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