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INTRO LOGICINTRO LOGICDAY 12DAY 12

Derivations in SLDerivations in SL44

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Day 14

Day 13

Day 12

Day 11

Day 10

Day 09

ScheduleSchedule

Introductory Material

EXAM #2

show: conjunction

Indirect Derivationshow: atomicshow: disjunction

Conditional Derivation (CD)Negation Derivation (D)

Direct Derivation (DD)

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Exam 2 FormatExam 2 Format

6 argument forms, 15 points each, plus 10 free points

Symbolic argument forms (no translations) For each one, you will be asked to construct a

derivation of the conclusion from the premises. The rule sheetrule sheet will be provided.

1 problem from Set D2 problem from Set E2 problems from Set F1 problem from Set G (91-96)

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Inference Rules (so far)Inference Rules (so far)

––––––

––––––

DN –––––––

–––––––

O

––––––

––––––

I ––––––

––––––

O

–––––– &

–––––– &

&I & –––––––

& –––––––

&O

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Rules (so far)Rules (so far)

D: ID As :

CD: CD As:

DD: DD

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Affiliated RulesAffiliated RulesAssumption Rule (CD)

If one has a line of the form

: then one is entitled to write down the formula

on the very next line, as an assumption.

Assumption Rule (D)

If one has a line of the form

: then one is entitled to write down the formula

on the very next line, as an assumption.

Contradiction-In (I)if you have a formula

and you have its negation

then you are entitled to infer

––––a contradiction (absurdity)

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Direct-Derivation StrategyDirect-Derivation Strategy

: °

°

°

DD

In Direct Derivation (DD),

one directly arrives at

the very formula one is trying to show.

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Show-Conditional StrategyShow-Conditional Strategy

: As

: °

°

°

CD

??

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Show-Negation StrategyShow-Negation Strategy

: As

: °

°

D

DD

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Can we show the following?Can we show the following?

(1) P QPr

(2) P QPr

(3) : Q ??

We are stuck!!

we havePQ

so to apply OO

we must findP

or findQ

we also havePQ

so to apply OO

we must findP

or find Q

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Indirect DerivationIndirect Derivation

: As

: °

°

ID

DD

This is exactly parallel to DD, and is another version of

the traditional mode of reasoning known as REDUCTIO AD ABSURDUMREDUCTIO AD ABSURDUM

: As

: °

°

D

DD

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Using IDUsing ID

Although ID can, in principle, be used onanyany formula,

it is best used on two types of formulas.

1. atomic formulas P, Q, R, etc.

2. disjunctions

The difference between IDID and D D is thatDD applies only to negationsnegations,

whereas IDID applies (in principle) to all formulasall formulas;it is a generic rule, like direct-derivation.

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Show-Atomic StrategyShow-Atomic Strategy

: As

: °

°

ID

DD

is atomic (P,Q,R, etc.)

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(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 1Example 1

4,7, 2,6, Q

1,4, PDD : As QID: QPrP QPrP Q

IOO

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(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 2Example 2

1,7, 3,5, P & QDD : As QID : QAs P

CD: P QPr(P & Q)

I&I

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Show-Disjunction StrategyShow-Disjunction Strategy

: [] As

: °

°

ID

DD

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Affiliated Inference-Rule Affiliated Inference-Rule Tilde-Wedge-Out Tilde-Wedge-Out

((O)O)

––––––––– –––––––––

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(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 3Example 3

6,7, 1,5, Q

Q 3,

PDD : As (P Q)

ID: P QPrP Q

O

IO

19(11)

(10)

(9)

(8)

(7)

(6)

(5)

(4)

(3)

(2)

(1)

Example 4Example 4

3,10, 8,9, Q1,7, Q R

R 5,

PDD : As (P R)

ID : P RAs Q

CD: Q (P R)PrP (Q R)

O

IOO

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(9)

(8)

(6)

(5)

7,

(11)

(10)

(7)

(4)

(3)

(2)

(1)

Example 5Example 5

6,10, 8,9, P & Q

Q P

1,5, (P Q)

(P & Q)

O

(P & Q)

DD : As [(P & Q) (P & Q)]

ID: (P & Q) (P & Q)

Pr(P Q) (P & Q)

3, O

I&I

O

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THE ENDTHE END