Propositional logic/Propositions/Applied sciences/Introduction/Section

A proposition is a structure of language (a sentence) that can be either true or false. It might be the case that it is difficult to decide whether the proposition is true or false. The important thing is that the predicates true and false are reasonable predicates for the structure due to its syntactic and semantic form.

The condition that the meaning of a sentence is completely clear is usually not fulfilled by statements of the natural language. Let's have a look at the sentence

This horse is fast.

On one hand, we do not have any information about what horse we are talking, and the validity of the statement depends probably on the horse. On the other hand, the meaning of "fast“ is not so clear that even if we knew about which horse we are talking, it is difficult to agree whether we want to consider it as rather fast or not. Further examples are

Martians are green.

I eat my hat.

Heinz Ngolo and Mustafa Müller are friends.

In a natural language, we have the possibility to produce a situation (by adding information, considering the context, making agreements) where one can clarify the meaning of the statement. In logic and in mathematics however this kind of solution is not allowed. Instead, the meaning of the proposition has to be clear from the terms used in the proposition alone, and these terms have to be defined in advance. Some mathematical statements are (maybe true, maybe false)

${}5>3$.

${}5<3$.

5 is a natural number.

We have ${}7+5=13$.

Prime numbers are odd.

If we understand these statements and in particular the concepts and symbols involved, then we recognize that we are dealing with a proposition that may be true or false, and this holds independently whether we know its validity. Whether a structure of language is a proposition does not depend from our knowledge whether it is true or false, nor on the effort necessary to decide whether it is true or false. In the following we present some mathematical objects which are not propositions:

5

5+11

The set of prime numbers

${}A\cap B$

A sum of five squares

${}\int _{a}^{b}f(t)dt$.

Instead of dealing with concrete propositions, in the following we take up the position that a proposition is just a propositional variable ${}p$ which may have the logical value (truth value) true or false. We are then interested on how the assignment of logical values behaves when we construct new propositions from old propositions.