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INTRO LOGICINTRO LOGICDAY 09DAY 09

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UNIT 2UNIT 2DERIVATIONS INDERIVATIONS INSENTENTIAL LOGICSENTENTIAL LOGIC

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Basic IdeaBasic Idea

We start with a few argument forms,

which we presume are valid,

and we use these

to demonstrate that other argument forms are valid.

We demonstrate (show) that a given argument form is valid by

derivingderiving (deducing) its conclusion

from its premises

using a few fundamental modes of reasoning.

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Example 1 – Modus Ponens (MP)Example 1 – Modus Ponens (MP)

––––––

we can employ modus ponens (MP) to derive the conclusion from the premises.

ifif thenthen ––––––––––

PP QQ RR S––––––S

Example Argument Form

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Example 1 (continued) Example 1 (continued)

P ; PQ ; QR ; RS / SMP

QMP

RMP

S

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Example 2 – Modus Tollens (MT)Example 2 – Modus Tollens (MT)

we can employ modus tollens (MT) to derive the conclusion from the premises.

Example Argument Form

ifif thenthen not not ––––––––––not not

SR SQ RP Q––––––P

––––––

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Example 2 (continued) Example 2 (continued)

S ; RS ; QR ; PQ / PMT

RMT

QMT

P

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Example 3 – using both MP and MT Example 3 – using both MP and MT

we can employ a combination of MP and MT to derive the conclusion from the premises.

–––––––

–––––––

SR SR TP TP Q–––––––––Q

Example Argument Form

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Example 3 (continued) Example 3 (continued)

S ; RS ; R T ; P T ; P Q QMT

RMP

TMT

PMP

Q

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Derivations – How to StartDerivations – How to Start

argument : P ; PQ ; QR ; RS ; / S

1. write down premises

2. write down “:” conclusion

(1) P Pr (premise)

(2) P Q Pr

(3) Q R Pr

(4) R S Pr

(5) : S what one is trying to derive

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Derivations – How to ContinueDerivations – How to Continue

(1) P Pr

(premise)

(2) P Q Pr

(premise)

(3) Q R Pr

(premise)

(4) R S Pr

(premise)

(5) : S(goal)

(6) Q

(7) R

(8) S

1,2,MP

3,6,MP

4,7,MP

3. Apply rules, as applicable, to available lines until goal is reached.

follows from

lines 1 and 2

by modus ponens

follows from

lines 3 and 6

by modus ponens

follows from

lines 4 and 7

by modus ponens

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Derivations – How to FinishDerivations – How to Finish

4. Box and Cancel

(1) P Pr

(2) P Q Pr

(3) Q R Pr

(4) R S Pr

(5) : S

(6) Q 1,2,MP

(7) R 3,6,MP

(8) S 4,7,MP

DD Direct Derivation

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Derivation – Example 2Derivation – Example 2

argument : S ; RS ; QR ; PR ; / P

(1) S Pr

(2) R S Pr

(3) Q R Pr

(4) P Q Pr

(5) : P(6) R(7) Q(8) P

DD

1,2,MT

3,6,MT

4,7,MT

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Derivation – Example 3Derivation – Example 3

(1) S Pr

(2) R S Pr

(3) R T Pr

(4) P T Pr

(5) P Q Pr

(6) : Q(7) R(8) T(9) P(10) Q

DD

argument : S ; RS ; RT ; PT ; PQ / Q

1,2,MT

3,7,MP

4,8,MT

5,9,MP

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Initial Inference RulesInitial Inference Rules

Modus Tollens

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Modus Ponens

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Modus Tollendo Ponens (1)

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Modus Tollendo Ponens (2)

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Examples of Modus PonensExamples of Modus Ponens

(P&Q)(P&Q) (R(RS)S)P&QP&Q–––––––––––––RRSS

PP QQPP–––––––––QQ

––––––

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Examples of Modus TollensExamples of Modus Tollens

(P&Q)(P&Q) (R(RS)S)(R(RS)S)–––––––––––––(P&Q)(P&Q)

PP QQQQ–––––––––PP

––––––

PP QQQQ–––––––––PP

valid argument

BUTNOT an example of MT

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Form versus Content/ValueForm versus Content/Value

these ten pennies are worth

are the dime and ten pennies the same?

are they the same in value?

NO

YES

actually, in a very important sense, they are not the same in value!

$2,000,000

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Examples of MTP(1)Examples of MTP(1)

(P&Q)(P&Q) (R(RS)S)(P&Q)(P&Q)–––––––––––––RRSS

PP QQPP–––––––––QQ

––––––

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Examples of MTP(2)Examples of MTP(2)

(P&Q)(P&Q) (R(RS)S)(R(RS)S)–––––––––––––P&QP&Q

PP QQQQ–––––––––PP

––––––

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THE ENDTHE END