Propositions

What is a proposition?

In simple words, a proposition is a statement with specific properties. These properties of the statement include that it must assert or declare something, and it should be either true or false but not both at the same time. That means a proposition is a declarative sentence, either true or false (but not both). We’ll also use “statement” as a synonym for propositions. Let’s look at a few examples to determine what kind of sentences qualify as a proposition and which do not.

Examples of propositions

Let’s start by considering the following proposition labeled $I.$

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• $I$: Islamabad is the capital of Pakistan.

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If we examine the above statement $I$, it declares a special status of Islamabad, and this declaration will be either true or false.

We typically label our propositions with an upper case letter. This allows us to refer to various propositions in the subsequent discussion.

Let’s consider,

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• $S$: The sun revolves around the earth.

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In the proposition $S$, the declarative part is about the sun orbiting around the earth. Yes, $S$ is a false statement, but that is not the point here — we should instead notice that this statement can not be true and false at the same time. Therefore, $S$ is a bona fide proposition.

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• $P$: Patric has longer hair compared to Alex.

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In the proposition $P$, although we do not know who Patric and Alex are, we can see that $P$ will be either true or false. Therefore, $P$ is an example of a proposition. If Patric indeed has longer hair compared to Alex, then $P$ is true, and if it is not the case, then $P$ is false.

Examples that are non-propositions

Let’s now look at a few sentences which are not propositions.

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• $J$: Is John from Alaska?

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$J$ is an interrogative sentence with a yes or no answer, but can we say that $J$ is either true or false? No. The problem with this sentence is that it is an interrogative, not a declarative sentence. It does not assert anything. Instead, it is inquiring about something. Therefore, $J$ is not a proposition.

Key features

A proposition must have the following two features in it:

1. It must proclaim or declare something.

2. It must have a definite truth value; that is, either true or false but not both.

Any sentence missing any of the two stated features will not be considered a proposition. The attribute of a proposition being true or false is called its truth value.

In the following examples, the first feature is missing, and due to this, these sentences are non-propositions.

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• $T$: What time is it?
• $O$: Open the second drawer.
• $G$: Give me a break.

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Let’s look at an example in which the second feature is missing.

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• $\mathcal{E}$: This statement is false.

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$\mathcal{E}$ does declare something and therefore has the first property. However, a close examination reveals that it is missing the second property. It does not have a definite truth value.

If we say that $\mathcal{E}$ is true, it claims that it is a false statement. Therefore, it appears both true and false simultaneously, which is not possible. On the other hand, if we say that $\mathcal{E}$ is false, then the claim made in this statement is not valid, which means $\mathcal{E}$ is a true statement. We observe the same problem again and cannot determine the truth value of $\mathcal{E}$. So, $\mathcal{E}$ is not a proposition. However, to realize this, we have to use very subtle reasoning.

Note: A sentence expressing a question, exclamation, wish, request, or command is not a proposition.

Some other examples

Let’s look at the following table for a few more examples of propositions and non-propositions:

Propositions	Non-propositions
$Q$: $7+8 = 15$.	$X$: $x+2=8$.
$W$: Today is Wednesday.	$D$: What is the day today?
$L$: $2021$ is a leap year.	$\hat{L}$: Let us do it.
$M$: Mars is bigger than Earth.	$R$: Please do what is required.

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Consider another proposition.

• $H$: David is happy.

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The states of being happy and not happy cannot co-occur, and this observation is enough to say that $H$ is a proposition. To determine whether David is really happy or not is about finding the truth value of the proposition $H$. There can be many factors involved in finding out whether $H$ is true or false. For example, we might want to find out the dopamine levels of David to figure out if David is pretending or is actually happy. It can be very tedious to find out the truth value of a proposition. However, it is usually a more straightforward task to determine whether a given statement is a proposition or not.

The following illustration classifies the various examples that we have discussed:

We can see that the propositions $W$, $H$, and $P$ can either be true or false— their verdict remains uncertain. Our inability to determine the truth value of a proposition is irrelevant. For a statement to qualify as a proposition, the criteria do not demand that we should be able to find its truth value.

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Consider the proposition:

• $N$: Neil Armstrong got a haircut on April 1, 1954.

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$N$ is a proposition. It might be impossible to find the truth value of $N$, but that is irrelevant. It is enough to convince ourselves that $N$ is true or false—but not both. Therefore, $N$, is a bona fide proposition.

Quiz