Counterexamples Here are some examples of statements that are not propositions in a mathematical sense.

• $\blue{ \text{Red is a nice colour.} }$
• $\blue{ \text{Clowns are creepy.} }$
• $\blue{ \text{Do you know the way to the supermarket?} }$
• $\blue{ \text{Come home!} }$

Opinions, questions, and orders are never propositions. By the way, a statement that involves an opinion but is not an opinion, can be a proposition. For example, "$\blue{\textit{Jesse thinks} \text{ clowns are creepy.}}$" is a valid proposition.

True propositions The following are examples of propositions that are always true.

• $\blue{ \text{Paris is the capital of France.} }$
• $\blue{ \text{Red is a colour.} }$
• $\blue{ \text{Blue flowers exist.} }$

False propositions The following are examples of propositions that are always false.

• $\blue{ \text{Apples grow under the ground.} }$
• $\blue{ \text{The sky is always green.} }$
• $\blue{ \text{Water tastes spicy.} }$

Excluded middle
By stating that we can assign the value true or false to a proposition we are being quite optimistic. For instance, we do not currently know which of the two values to assign to the statements

• $\blue{ \text{ There is life on another planet than Earth. }}$
• $\blue{ \text{ There are parallel universes. }}$
• $\blue{ \text{ The digit }9\text{ occurs nine consecutive times in the decimal representation of }\pi \text{.} }$ (it is known that the digit $9$ occurs eight consecutive times, for instance starting at position $66,\!780,\!105$)

Nevertheless, classical logic assumes that every proposition is either true or false, and that there is no other option (like partially true, undefined or unknown) in the middle of these two extremes. This principle is called the law of excluded middle.

Context Some propositions depend on context. A proposition such as "$\blue{\text{It is raining.}}$" could be either true or false, depending on time and place, and the amount of water pouring down per minute. However, once we know the time and place and give a proper measuring criterion, there is no discussion about whether or not it is raining. Examples of such propositions are

• $\blue{ \text{Jane is wearing jeans.}}$
• $\blue{ \text{Pete likes to read.}}$
• $\blue{ \text{It is sunny.}}$

Similarly, statements that express opinions including the word "I" can be a bit ambiguous. Is "I" the writer, the reader, or some other unknown person? Depending on the meaning of "I", a statement with "I" can be either a proposition or not. If "I" is the writer, then the statement "I like apples" is a proposition, because we can objectively decide whether or not the author likes apples. If "I" is the reader, then we cannot objectively decide this anymore, since it can differ who the reader is. In this course, we consider opinions with "I" to be propositions. We will try not to use this in cases which can be confusing.

For the sake of simplicity, we will refrain from referring to context in the rest of this chapter.

Complex proposition Up until this point it seems as if all propositions are simple sentences. In general this is not true. There are much more complicated propositions. In fact, in Negation, conjunction, and disjunction we will discuss some ways to construct new propositions from simple ones. In most instances, these new propositions will be more complicated.