1. 2 day 14 day 13 day 12 day 11 day 10 day 09 introductory material exam #2 show: conjunction...
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2
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)
3
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)
4
Inference Rules (so far)Inference Rules (so far)
––––––
––––––
DN –––––––
–––––––
O
––––––
––––––
I ––––––
––––––
O
–––––– &
–––––– &
&I & –––––––
& –––––––
&O
6
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)
7
Direct-Derivation StrategyDirect-Derivation Strategy
: °
°
°
DD
In Direct Derivation (DD),
one directly arrives at
the very formula one is trying to show.
10
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
11
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
12
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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(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
15
(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
17
Affiliated Inference-Rule Affiliated Inference-Rule Tilde-Wedge-Out Tilde-Wedge-Out
((O)O)
––––––––– –––––––––
18
(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
20
(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