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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
––––––
Modus Ponens
––––––
Modus Tollendo Ponens (1)
––––––
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