Logical statements

Is it cold outside and is it raining? Only when both hold do you take a scarf and umbrella. Without noticing, you constantly combine statements into new statements. Propositional logic makes these combinations precise and checkable.

Statements are true or false

A statement is a sentence that is either true or false, but never both. The sentence $3 > 2$ is true, the claim that 3 is an even number is false. Questions and requests, in contrast, are not statements: neither asking for the time nor asking someone to close the window can be labelled true or false. We name statements briefly with letters like $A$ and $B$.

And, or, not

From statements you build new ones with three basic blocks. The and, written $A \land B$, is only true when both parts are true. The or, written $A \lor B$, is true as soon as at least one part is true. It deliberately includes the case where both are true, unlike the everyday either or. The not, written $\lnot A$, simply flips true and false.

The truth table

To fully understand a connective, you list all combinations of true and false. For one statement there are two rows, for two statements four, for three statements eight. With each further statement the number of rows doubles. The table then shows for each combination whether the compound statement is true or false. It also lets you check whether two differently written expressions have the same value in every row. If they do, they mean the same thing, and you may put an equals sign between them.

The if-then statement

Very common is the form if $A$, then $B$. It claims only one thing: if the first part is true, the second must be true as well. Such a statement is false only when $A$ is true but $B$ is false. If $A$ is false from the start, the statement counts as true, no matter what happens with $B$. An example: if it rains, the street is wet. You can refute this sentence only on a rainy day with a dry street, whereas on a sunny day it promises nothing at all.