# Statement Form/Examples

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## 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 $2 \ \text {(a)}, \text {(b)}$