Introduction to Logic

Logic is, basically, the study of valid reasoning. When searching the internet, we use logic – terms like “and” and “or” – to help us find specific web pages that fit the sets we are interested in.  We will look at logical arguments and how we can determine the validity of a claim.

Statements

A statement is a declarative sentence or assertion that is either true or false.  Statements are the foundation of mathematical logic, and we shall see some examples of statements below.

Examples of Statements

  1.   The Eiffel tower is in Paris.
  2.   A rat is a fish.
  3.   4 < 6
  4.   The integer 17 is odd.

Let's look a little deeper as to why the above examples are all statements.

Examples of Statements (Revisited)

  1.  Notice that the Eiffel tower is located in Paris, France, so the sentence is actually true.  Therefore, the declarative sentence is a statement.
  2.  Although a rat is not a type of fish, our false declarative sentence provides us with another statement.
  3.  Upon first glance, observing the mathematical inequality 4 < 6, it does not seem that we have a sentence on our hands.  However, this is just a mathematical representation of the sentence "The integer 4 is less than the integer 6." which happens to be a true declarative sentence.  This implies that we have another statement on our hands.
  4.  This declarative sentence focuses on our set of integers.  Recall that an odd integer is just an integer (which 17 is) that is not a multiple of 2.  Therefore this sentence is actually true, and we find ourselves with another statement.  Also take note, that even if switched the word "odd" to "even" we would still have a statement on our hands.

After seeing examples of what makes a statement, it is also beneficial to see examples of things that are not statements.  We shall take a look at such sentences and questions here.

Examples of Things that are Not Statements

  1. Who is Rey's father?
  2. Dance with me.
  3. The Royal Tenenbaums is the best movie ever!
  4. This statement is false.

As we have seen before, an explanation is in order to verify why each of these are not statements.  We shall revisit the above examples below.

Examples of Things that are Not Statements (Revisited)

  1. Notice that this is an example of a question.  Since it is neither a declarative sentence nor an assertion, it can not be a statement.
  2. For this example, we have a command.  Just like we saw in the previous example, a command is not a declarative sentence.  Therefore, the phrase can not have an associated truth value implying it is not a statement.
  3. Once again, we find ourselves with something that is neither a declarative sentence nor an assertion.  Since this is exclamatory, it is not a statement.
  4. This example is a bit different since we begin with a declarative sentence.  However, the possible truth values create a paradox (this is one of a small number of examples of such an occurrence).  Notice that if the sentence "This statement is false." were assumed to be true, then the sentence itself is false.  Hence the statement could not be true.  However, if we assume the sentence "This statement is false." is false, then the sentence itself would be true.  As a result, the sentence could not be false either.  Since the sentence is neither true nor false, it is not a statement.

Similar to our introduction of variables in a secondary school algebra course, we shall use variables to denote statements.  In particular, we often resort to P, Q, and R to notate a statement.

Learning Activities

In order to further comprehend the material, please attempt the following exercises.

  1. Determine which of the following sentences are statements.
  2. Determine the truth value of the following statements.
  3. Provide an example of a sentence that is not a statement.  Justify your answer.
  4. Does there exist a question that is a statement.  Justify your answer.