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INTRO LOGICINTRO LOGICDAY 10DAY 10

Derivations in SLDerivations in SL22

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ReviewReview

We demonstrate (show) that an argument is valid

by derivingderiving (deducing) its conclusion

from its premises

using a few fundamental modes of reasoning.

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Initial Modes of ReasoningInitial Modes of ReasoningModus (Tollendo) Tollens(denying by denying)

––––––

Modus (Ponendo) Ponens(affirming by affirming)

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Modus Tollendo Ponens (1)(affirming by denying)

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Modus Tollendo Ponens (2)(affirming by denying)

––––––

aka disjunctive syllogism

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Example 1 (review)Example 1 (review)

5,9, Q4,8, P3,7, T1,2, RDD: QPrP QPrP TPrR TPrR SPrS

S ; RS ; RT ; PT ; PQ / Q

MP

MTP2

MTP1

MT

1.write all premises

3.apply rules to available lines

4.box and cancel(high-five!)

2.write ‘show’ conclusion

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other high-fivesother high-fives

Tek gives Manny a high-five Tek gives A-Rod a high-five

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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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Rules discussed TodayRules discussed Today

&I &O & & –––––– –––––– –––––– –––––– & & I O –––––– –––––– –––––– –––––– I see CD O ––––––– ––––––– DN DN ––––– –––––

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Rules of Inference – Basic IdeaRules of Inference – Basic Idea

Every connective has

(1) a take-out rule

also called an:

elimination-rule

OUTOUT-rule-rule

(2) a put-in rule

also called an:

introduction-rule

ININ-rule-rule

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Ampersand-Out (&O)Ampersand-Out (&O)

if you have a conjunctionconjunction && then you are entitled to infer –––––– its first conjunctfirst conjunct

if you have a conjunctionconjunction && then you are entitled to infer –––––– its second conjunctsecond conjunct

‘have’ means ‘have as a whole line’

rules apply only to whole linesnot pieces of lines

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Ampersand-IN (&Ampersand-IN (&II))

if you have a formulaformula and you have a formulaformula then you are entitled to infer ––––––their (first) conjunctionconjunction &&

if you have a formulaformula and you have a formulaformula then you are entitled to infer ––––––their (second) conjunctionconjunction &&

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Wedge-Out (Wedge-Out (O)O)

if you have a disjunctiondisjunction and you have the negationnegation of its 11stst disjunct disjunct then you are entitled to infer ––––––its second disjunctsecond disjunct

if you have a disjunctiondisjunction and you have the negationnegation of its 22ndnd disjunct disjunct then you are entitled to infer ––––––its first disjunctfirst disjunct

what we earlier called ‘modus tollendo ponens’

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Wedge-IN (Wedge-IN (II))

if you have a formulaformula then you are entitled to infer ––––––its disjunctiondisjunction with anyany formula to its right

if you have a formulaformula then you are entitled to infer ––––––its disjunctiondisjunction with anyany formula to its left

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Arrow-Out (Arrow-Out (O)O)

if you have a conditionalconditional and you have its antecedentantecedent then you are entitled to infer ––––––its consequentconsequent

if you have a conditionalconditional and you have the negationnegation of its consequentconsequent then you are entitled to infer ––––––the negationnegation of its antecedentantecedent

What we earlier called ‘modus ponens’ and ‘modus tollens’

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Arrow-IntroductionArrow-Introduction

THERE IS NO SUCH THING ASTHERE IS NO SUCH THING ASARROW-IN ARROW-IN ((I)I)

what we have instead is

CONDITIONAL DERIVATION CONDITIONAL DERIVATION (CD)(CD)

which we examine later

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Double-Negation (DN) Double-Negation (DN)

if you have a formula then you are entitled to infer –––––

its double-negative

if you have a double-negative then you are entitled to infer –––––

the formula

rules apply only to whole linesnot pieces of lines

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: °

°

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Direct DerivationDirect Derivation

In Direct Derivation (DD),one directly arrives at

DD

The Original and Fundamental -Rule

the very formula one is trying to show.

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Example 2Example 2

5,9, Q4,8, P3,7, T1,2, RDD: QPrP QPrP TPrR TPrR SPrS

S ; R S ; R T ; P T ; P Q / Q

O

O

O

O

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Arrow-Out StrategyArrow-Out Strategy

If you have a line of the form , then try to apply arrow-outarrow-out (OO),

which requires a second formula as input,

in particular, either or .

deduce

find

have

or

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Example 3Example 3

5,9, Q4,8, P3,7, T1,2, RDD: QPrP QPrP TPrR TPrR SPrS

S ; R S ; R T ; P T ; P Q / Q

O

OOO

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Wedge-Out StrategyWedge-Out Strategy

If you have a line of the form , then try to apply wedge-outwedge-out (OO),

which requires a second formula as input,

in particular, either or .

deduce

find

have

or

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Example 4Example 4

5,7, R

3,6, Q R

1,5, P Q

1,2, RDD: QPr(P Q) (Q R)

Pr[(P Q) R] RPr(P Q) R

(P Q) R ; [(P Q) R] R ; (P Q) (Q R) / Q

O

O

OO

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Example 5Example 5

2,7, T

6, R S1,5, S

3, P QDD: TPrP

Pr(R S) TPr(P Q) S

(P Q) S ; (R S) T) ; P / T

O

IOI

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Wedge-In StrategyWedge-In Strategy

If you need

then look for either disjunct:

find OR

find then

apply vvII to get

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Example 6Example 6

1,7, Q

3,6, R2,9, Q S

8,10, S

8,11, Q & S

P 4,&O

TDDQ & S

PrT & P

PrR TPrR (Q S)

PrP Q

P Q ; R (Q S) ; R T ; T & P / Q & S

O

O

OO

&I

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Ampersand-Out StrategyAmpersand-Out Strategy

If you have a line of the form

&&, then apply ampersand-outampersand-out (&O),

which can be applied immediatelyto produce

andand

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Ampersand-In StrategyAmpersand-In Strategy

If you are trying to find or show

&&then look for both conjuncts:

find AND

find then

apply &&II to get &&

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THE ENDTHE END