LOGICAL CONNECTIVESandor

if…..thenif and only if

not

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

When using the connective

The direction of the arrow

is important.

cause effect

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

statement statementDOMINANTCONNECTIVE

( 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

This is higheston the chart.This is dominant

~

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

~

Interpret the following statement:

~p q r ~s r