INTRO LOGIC

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INTRO LOGIC

• DAY 06

TRANSLATIONS IN SENTENTIAL LOGIC2
2
Schedule for Unit 1
? Day 1 Intro
? Day 2 Chapter 1
? Day 3 Chapter 2
? Day 4 Chapter 3
? Day 5 Chapter 4
Day 6 Chapter 4
Day 7 Chapter 4
Day 8 EXAM 1
40 of Exam
10 arguments 4 pts each
60 of Exam
12 translations5 pts each
3
Standard Connectives(have special symbols)

and
?
or
?
not
?
if then
4
Non-Standard Connectives(must be paraphrased)
xor
if-otherwise
neither-nor
if
unless-in-which-case
necessary
only if
sufficient
if and only if
unless
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Non-Standard Connectives 1
exclusive or

• ? xor ?
• ? or ?, but not both ? and ?
• ( ? ? ? ) ?( ? ? )
• ?
• ( ? ?? ) ? ( ? ?? )

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Non-Standard Connectives 2
neither-nor
neither ? nor ? not ? and not ? ?? ?? ? ? ( ?
? ? )
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Non-Standard Connectives 3
if

• ? if ?
• if ? then ?
• ? ? ?

8
Non-Standard Connectives 4
only if
? only if ?
not ? if not ?
if not ?
then
not ?
? ?
?
? ?
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Non-Standard Connectives 5
if and only if
?
if and only if
?
? only if ?
and
? if ?
( ? ? ? ? ? )

( ? ? ? )
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Non-Standard Connectives 6
unless
?
unless
?
?
if not
?
not ?
if
?
?
then
if not ?
?
?
? ?
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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's 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 not sunny, then Ill play racketball
furthermore
if its sunny, then Ill play tennis
if not S then R
and
if S then T
( ?S ? R )

( S ? T )
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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 rain, then Ill play squash
furthermore
if it does not rain, then Ill play tennis
if R then S
and
if ?R then T
( R ? S )

( ?R ? T )
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Necessary Conditions
I get an A
in order that
I take 4 exams
it is necessary that
me to get an A
in order for
me to take 4 exams
it is necessary for
to get an A
in order
to take 4 exams
it is necessary
taking 4 exams
getting an A
is necessary for
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Necessary Conditions (cont)

• simplest paraphrase
• ? is necessary for ?

This amounts to saying if ? does not happen,
then neither does ?
if not ? then not ?
?? ? ??
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Example

• taking 4 exams E is necessary for getting an A A

if E does not happen, then neither does A
if not E then not A
?E ? ?A
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Sufficient Conditions
I get an A
in order that
I get a hundred
it is sufficient that
me to get an A
in order for
me to get a hundred
it is sufficient for
to get an A
in order
to get a hundred
it is sufficient
getting a hundred
getting an A
is sufficient for
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Sufficient Conditions (cont)

• simplest paraphrase
• ? is sufficient for ?

This amounts to saying if ? does happen, then
so does ?
if ? then ?
? ? ?
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Example

• getting a hundred H is sufficient forgetting an
  A A

if H does happen, then so does A
if H then A
H ? A
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Negations Of Necessity
getting a hundred His not necessary for getting
an A A
H
A
)
(
is nec for
not
not A
then
if not H
?A
)
(
?
?H
?
you wont get an A
then
if you dont get a hundred
it is not true that
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Negations of Sufficiency
taking all the exams Eis not sufficient
forgetting an A A
E
A
)
(
is suf for
not
A
then
if E
A
)
(
?
E
?
you will get an A
then
if you (merely) take all the exams
it is not true that
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Basic Statements

• ? is necessary for ?
• ? is sufficient for ?
• ? is not necessary for ?
• ? is not sufficient for ?

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Combinations

• ? is both necessary and sufficient for ?
• ? is necessary, but not sufficient, for ?
• ? is sufficient, but not necessary, for ?
• ? is neither necessary nor sufficient for ?

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Example 1

• averaging (at least) fifty A
• is both necessary and sufficient for
• passing P

( ?A ? ?P )
A is necessary for P

and
( A ? P )
A is sufficient for P
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Example 2

• taking four exams E
• is necessary but not sufficient for
• getting an A A

( ?E ? ?A )
E is necessary for A

but
? ( E ? A )
E is not sufficient for A
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Example 3

• getting a hundred H
• is sufficient but not necessary for
• getting an A A

( H ? A )
H is sufficient for A

but
? ( ?H ? ?A )
E is not necessary for A
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Example 4

• attending class A
• is neither necessary nor sufficient for
• passing P

? ( ?A ? ?P )
A is not necessary for P

and
? ( A ? P )
A is not sufficient for P
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THE END