Often, numbers are thought to be the core of mathematics. But mathematics is about reasoning, and logic guides us in that process. Mathematical reasoning concerns the making of statements and determining whether they are provable in a given context. We discuss the classical approach to working with propositions, known as propositional logic.
Propositions are statements to which we can objectively assign either the value true or false.
In doing so, we often refer implicitly to a context in which the proposition can be evaluated.
Examples
- #\blue{ \text{It is raining.}}#
- #\blue{ \textrm{James likes candy.}}#
- #\blue{5 \text{ is a prime number.}}#
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.
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.} }#
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.} } #
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.
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.
It may not be immediately clear why #\blue{ \text{"Red is a nice colour."}}# is not called a proposition. In good mathematical tradition, we could have given a precise meaning to the adjective "nice" for a colour and then verify whether this definition applies to "red" (or does not). Thus, the notion of proposition is quite informal at this point. But in mathematics we can be very precise if needed, and s#{}#et up a system of axioms and deduction rules that will enable us to define precisely what a proposition is and what it means to establish its truth (or falsity). Such an approach is best left for a second course in logic since it is very formal. For now we content ourselves with gaining some hands-on experience.
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.
Which of the statements listed below are propositions?
\[\begin{array}{l} \bullet\quad\text{The waves were crashing on the shore.} \\\bullet\quad\text{She did a happy dance because all of the socks from the dryer matched.}\\\bullet\quad\text{It is much more difficult to play tennis with a bowling ball than it is to bowl with a tennis ball.}\\\bullet\quad\text{There are few things better in life than a slice of pie.}\\ \end{array}\]
The following three sentences are propositions.
- The waves were crashing on the shore.
- She did a happy dance because all of the socks from the dryer matched.
- It is much more difficult to play tennis with a bowling ball than it is to bowl with a tennis ball.
"There are few things better in life than a slice of pie." is not a proposition.
The statement "There are few things better in life than a slice of pie." expresses an opinion and so is not a proposition. The truth or falsity of the other statements can be verified objectively.
Propositions and truth
