logical connectives and or if…..then if and only if not
TRANSCRIPT
s : John studies.p : John passes.h : John is happy
This symbol replacesthe word “AND”
John passes AND John is happy
p h
s : John studies.p : John passes.h : John is happy
John does NOT study:
s
This symbol negates thestatement it precedes
~
~
s : John studies.p : John passes.h : John is happy
This symbol replacesthe word “OR”
John studies OR John does NOT pass
s ~ p
s : John studies.p : John passes.h : John is happy
This symbol replaces
the connective “if … then”
IF John studies THEN John passes.
s p
s : John studies.p : John passes.h : John is happy
This symbol replaces
the words “if and only if”
John is happy IF AND ONLY IF John studies.
h s
If Herman lives in Cleveland then Herman lives in Ohio.
c o
Herman lives in Ohio if Herman lives in Cleveland.
o cHerman lives in Cleveland only if Herman lives in Ohio.
John is happy ONLY IF he studies.
John is happy IF he studies.
John is happy IF AND ONLY IF he studies.
:~
:
:...
:
:
not
fifandonlyi
thenif
or
and
m: Herman takes math
h: Herman is happya: Herman gets an Ab: Herman bribes the teacher
Herman takes math and he is not happy. m ~h
If Herman gets an A then he is happy. a h
Herman takes math if and only if he bribes the teacher. m
b
( 3 + 5 ) x 2 is different from 3 + ( 5 x 2 )
similarly
aglagl is different from
When a statement has two or more connectives, one of the connectives is said to be dominant. This determines the grouping (placement of parentheses)
A SIMPLE STATEMENT has one verb. The following four statements are simple:
Today is Friday
f
It is hot
h
We will swim
s
We will picnic
p
Today is Friday
f
It is hot
h
We will swim
s
We will picnic
p
or
and
Now we have two COMPOUND STATEMENTS
Finally , we can connect these two compoundstatements:
Today is Friday
f
It is hot
h
We will swim
s
We will picnic
p
or
and
If
then
Is the dominantconnective. It is last one placed.
:~
:
:...
:
:
not
fifandonlyi
thenif
or
and
l: The teacher likes Herman
g: Herman does good worka: Herman gets an A
The teacher likes Herman and if Herman doesgood work then Herman gets an A.
If the teacher likes Herman and Herman doesgood work then Herman gets an A.
l g a
l g a
( 3 + 5 ) x 2 is different from 3 + ( 5 x 2 )
Without parentheseswe have a conventionto interpret:
3 + 5 x 2
to mean (3) + ( 5 x 2 )
When an arithmetic sentence contains + and signsand NO PARENTHESES to indicate otherwisewe assume that + is dominant
There is a similar convention in logic:
If a statement contains two or more connectivesand there are NO PARENTHESES to show you howto group symbols, then we will agree that theDOMINANT CONNECTIVE is whatever is higheston this chart:
~
~
If it rains then we go to a movie and if it
doesn’t rain then we swim.
( r m ) (~r s )
In this statement, the parentheses arenecessary because “and” is dominant
~
If Herman does not study then he willfail math and his life will be ruined.
~s f r
Because is the dominantconnective, the parenthesesare correctly placed…BUT …they are unnecessary
~
If Herman does not study then he willfail math and his life will be ruined.
~s f r
~s f r
Because this means
~
Interpret the following statement:
~p q r ~s r
Now we have two simpler statementsto deal with.
~p q r
In the pink statement , is highest on the chart
~
Interpret the following statement:
~p q r ~s r
Now we have two simpler statementsto deal with.
In the blue statement , is highest on the chart
~s r