Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Logic Rules and Derivations in Sentential and Predicate Logic - Prof. Gary Hardegree, Study notes of Reasoning

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.

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

koofers-user-u60
koofers-user-u60 🇺🇸

2

(1)

10 documents

1 / 14

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
1
1
INTRO LOGIC
DAY 23
Derivations in PL
2
2
Overview
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
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe

Partial preview of the text

Download Logic Rules and Derivations in Sentential and Predicate Logic - Prof. Gary Hardegree and more Study notes Reasoning in PDF only on Docsity!

1

INTRO LOGIC

DAY 23

Derivations in PL

2

2

Overview

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

3

Rule Sheet

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

4

Sentential Logic Rules

ƒ DD ƒ ID ƒ CD ƒ ~D ƒ &D ƒ etc.

ƒ &I ƒ &O ƒ vO ƒ →O ƒ ~∨O ƒ etc.

7

Rules to be Introduced Today

Universal Derivation UD
Existential-Out ∃O

8

(?)

(c)

(?)

(b)

(?)

(a)

(3)

(2)

(1)

Example 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

9

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1a

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

10

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1b

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

13

The Universal-Derivation Strategy

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

14

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

15

Comparison with Universal-Out

OLD name

∀v › [v]
› [o]

NEW name

}: ∀v › [v]
}: › [n]

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

16

(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)

Example 2

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

19

Existential-Out ( ∃∃∃∃ O)

any variable (z, y, x, w …)

any NEW name (a, b, c, d, …)

any formula (^) n replaces v

∃v › [v]
› [n]

20

Comparison with Universal-Out

OLD name

∀v › [v]

––––––

› [o]

NEW name

∃v › [v]

––––––

› [n]

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

∀O ∃O

(10) 21

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 5

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)

Example 6

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

25

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 9

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

26

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 10 (a fragment)

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

27

THE END