Statement Form/Examples

Examples of Statement Forms

Napoleon

Napoleon is dead and the world is rejoicing

has the statement form:

$A \land B$

where:

$A$ stands for Napoleon is dead

$B$ stands for The world is rejoicing

Shape of Eggs

If all eggs are not square then all eggs are round

has the statement form

$A \implies B$

where:

$A$ stands for All eggs are not square

$B$ stands for All eggs are round

Barometer

If the barometer falls then either it will rain or it will snow

has the statement form

$A \implies \paren {B \lor C}$

where:

$A$ stands for The barometer falls

$B$ stands for It will rain

$C$ stands for It will snow

Arbitrary Example 1

If demand has remained constant and prices have been increased, then turnover must have decreased

has the statement form

$\paren {A \land B} \implies C$

where:

$A$ stands for Demand has remained constant

$B$ stands for Prices have been increased

$C$ stands for Turnover must have decreased.

Arbitrary Example 2

We shall win the election, provided that Jones is elected leader of the party

has the statement form

$A \implies B$

where:

$A$ stands for Jones is elected leader of the party

$B$ stands for We shall win the election.

Arbitrary Example 3

If Jones is not elected leader of the party, then either Smith or Robinson will leave the cabinet, and we shall lose the election

has the statement form

$\neg A \implies \paren {\paren {B \lor C} \land D}$

where:

$A$ stands for Jones is elected leader of the party

$B$ stands for Smith will leave the cabinet

$C$ stands for Robinson will leave the cabinet

$D$ stands for We shall lose the election.

Arbitrary Example 4

If $x$ is a rational number and $y$ is an integer, then $z$ is not real

has the statement form

$\paren {A \land B} \implies \neg C$

where:

$A$ stands for $x$ is a rational number

$B$ stands for $y$ is an integer

$C$ stands for $z$ is real.

Arbitrary Example 5

Either the murderer has left the country or somebody is harbouring him

has the statement form

$A \lor B$

where:

$A$ stands for The murderer has left the country

$B$ stands for Somebody is harbouring him.

Arbitrary Example 6

If the murderer has not left the country, then somebody is harbouring him

has the statement form

$\neg A \implies B$

where:

$A$ stands for The murderer has left the country

$B$ stands for Somebody is harbouring him.

Arbitrary Example 7

The sum of two numbers is even if and only if either both numbers are even or both numbers are odd

has the statement form

$A \iff \paren {B \lor C}$

where:

$A$ stands for The sum of two numbers is even

$B$ stands for Both numbers are even.

$C$ stands for Both numbers are odd.

Arbitrary Example 8

If $y$ is an integer then $z$ is not real, provided that $x$ is a rational number

has the statement form

$A \implies \paren {B \implies \neg C}$

where:

$A$ stands for $x$ is a rational number

$B$ stands for $y$ is an integer.

$C$ stands for $z$ is real.

Of the above statements:

Arbitrary Example 1 and Arbitrary Example 4 have the same statement form

Shape of Eggs and Arbitrary Example 2 have the same statement form.

Of the above statements:

Arbitrary Example 5 and Arbitrary Example 6 have the same meaning

and:

Arbitrary Example 4 and Arbitrary Example 8 have the same meaning.

Sources

• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives: Exercises $2 \ \text {(a)}, \text {(b)}$