# Propositional logic/Logical connectives/Applied sciences/Introduction/Section

Starting from several propositions, one can build new propositions. From the proposition

I eat my hat

one can build the

negative proposition

I do not eat my hat[1],

from the two propositions

Martians are green

and

I eat my hat

one can produce the following new propositions.

Martians are green and I eat my hat
Martians are green or I eat my hat
If Martians are green, then I eat my hat
If it is not true that Martians are green, then I eat my hat
If Martians are green, then I do not eat my hat
If it is not true that Martians are green, then I do not eat my hat
Martians are green if and only if I eat my hat

Here, the two propositions involved are not changed (maybe some slight changes due to grammar), they are just brought together into a new logical relationship. Such a logical combination (operation) is characterized by the fact that its logical value is determined by the logical values of the propositions involved and the meaning of the logical combinations (in propositional logic, these are called logical connectives). The proposition

Martians are green and I do not eat my hat

is true if and only if both propositions are true. This is the logical meaning of the and-connective. A further connection between the two statements is not necessary.

To contrast, let us consider a proposition like

The green Martians eat hats

Here a completely new proposition arises, where just some words or predicates of the two propositions occur, but its logical value can not be deduced from the given propositions.

We have a logical connective of propositions if the logical value of the whole proposition can be deduced from the logical values of the propositions involved. The logical connectives determine how the logical value of the compound proposition has to be computed from the logical values of the propositions involved.

1. An easy way to get the negation of a proposition is to use a construction like "it is not the case that ...“.