Proposition

Sentences considered in propositional logic are not arbitrary sentences but are the ones that are either true or false, but not both. This kind of sentences are called propositions.

If a proposition is true, then we say it has a truth value of "true"; if a proposition is false, its truth value is "false". For example, "Grass is green", and "2 + 5 = 5" are propositions.

The first proposition has the truth value of "true" and the second "false".



But "Close the door", and "Is it hot outside ?"are not propositions.

Also "x is greater than 2", where x is a variable representing a number, is not a proposition, because unless a specific value is given to x we can not say whether it is true or false, nor do we know what x represents.

Similarly "x = x" is not a proposition because we don't know what "x" represents hence what "=" means. For example, while we understand what "3 = 3" means, what does "Air is equal to air" or "Water is equal to water" mean ? Does it mean a mass of air is equal to another mass or the concept of air is equal to the concept of air ? We don't quite know what "x = x" mean. Thus we can not say whether it is true or not. Hence it is not a proposition.