LOGIC
DAY INTRO 06
TRANSLATIONS IN SENTENTIAL LOGIC 2
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Schedule for Unit 1
! Day 1 ! Day 2 ! Day 3 ! Day 4 ! Day 5 Day 6 Day 7 Day 8 Intro Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 4 Chapter 4 EXAM #1
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40% of Exam 60% of Exam
8 arguments 5 pts each 12 translations 5 pts each
1
Standard Connectives (have special symbols)
and or not if then &
3
Non-Standard Connectives (must be paraphrased)
xor neither-nor if only if if and only if unless
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if-otherwise unless-in-which-case necessary sufficient
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Non-Standard Connectives 1
exclusive or A A or B (AB) & xor but not B A and B (A&B)
(A B) & (A & B)
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Non-Standard Connectives 1b
exclusive or
alternative
A A but not B ( A & B )
xor or
B B but not A ( B & A )
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3
Non-Standard Connectives 2
neither-nor neither not A A A and & nor not B B B
A & B
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Non-Standard Connectives 2b
neither-nor
alternative
neither
A A
nor or or B B
B B )
not
either
A not ( A
(A B)
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4
Non-Standard Connectives 2c
recall truth-tables
~ ( R S ) //
F F F T
T T F F T T T F T F T F
~
F F T T
R
T T F F
&
F F F T
~
F T F T
S
T F T F
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Non-Standard Connectives 3
if A if B if B then A BA
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5
Non-Standard Connectives 3
if
A if B B
if
B
then A A
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Non-Standard Connectives 4
only if
A only if B not A if not B if not B B then not A A
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6
Non-Standard Connectives 5
if and only if A A if B (BA) if and only if and & B
A only if B (BA)
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Non-Standard Connectives 6
unless A A A if not B ~B unless if not if then B B not B A A
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7
ifotherwise
I will play tennis if it is sunny; otherwise, I will play racketball T if S ; otherwise, R this answers two questions: what will I do IF it is sunny? what will I do OTHERWISE (i.e., IF it is NOT sunny)? play tennis
play racketball
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if otherwise (cont.)
if its sunny, then Ill play tennis if S then T (ST) if its not furthermore sunny, then Ill play racketball and & if not S then R ( S R )
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8
unlessin which case
I will play tennis unless it rains, in which case I will play squash T unless R, in which case S this answers two questions: what will I do unless it rains (i.e., if it does not rain)? what will I do in case it rains (i.e., if it does rain)? play tennis play squash
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unless-in-which-case (cont.)
if it does not rain, then Ill play tennis if not-R then T ( R T ) if it does rain, then Ill play squash if R then S (RS)
furthermore
and &
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9
Necessary Conditions
in that order it is necessary that in order for it is necessary for in order it is necessary I get an A I take 4 exams me to get an A me to take 4 exams to get an A to take 4 exams taking 4 exams getting an A
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is necessary for
Necessary Conditions (cont)
simplest paraphrase: A is necessary for B this amounts to saying if A does not happen, then neither does B
if not A then not B
A B
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10
Example
taking 4 exams is necessary for getting an A if E does not happen, then neither does A E A
if not E then not A
E A
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Sufficient Conditions
in order that it is sufficient that in order for it is sufficient for in order it is sufficient I get an A I get a hundred me to get an A me to get a hundred to get an A to get a hundred getting a hundred getting an A
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is sufficient for
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Sufficient Conditions (cont)
simplest paraphrase: A is sufficient for B this amounts to saying if A does happen, then so does B
if A then B
AB
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Example
getting a hundred is sufficient for getting an A if H does happen, then so does A H A
if H then A
HA
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12
Negation of Necessity
getting a hundred is NOT necessary for getting an A not if H A
H is nec for A not H then not A H A
(
)
(H A )
it is not true that
if
you dont get a H
then
you wont get an A
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Negation of Sufficiency
taking all the exams is NOT sufficient for getting an A not if E E E
you (merely) take the exams
E A A A A
is suf for then
(
)
(E A )
then
you will get an A
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it is not true that
if
13
B...