Short Little Lessons in Logic: Operators




Lesson 9: Operators

What you’ll learn in this lesson:

  • How operators work on simple sentences
  • Determining truth value of compound statements
  • Challenges of compound statements with more than one operator

We learned in a previous lesson how simple statements can be combined to form compound statements. Simple statements are joined using statement operators. Operators are grammatical in the sense that they function as a part of the syntax or structure of sentences. In a language like English, they also function symantically to help a reader or listener understand the meaning of what is written or spoken. In logic, operators also have an important role to play in helping us determine the truth value of a logical statement. In this lesson, we’ll explore more on how operators work to build compound statements and how these statements function in logic.

We’ve looked at the operators "and" and "or" and we’ll learn more about these two operators in the next lesson. Let’s look at part of a statement we saw before: I was at the store AND I saw Aidan. This is a compound statement composed of two simple statements joined by the word ‘and.’ All simple statements have a truth value in logic and the truth value of the compound statement is a property of the truth value of the simple statements along with the function of the statement operator on those simple statements. This should make sense since that’s how things work grammatically.

For example, let’s stipulate that the statement "I was at the store" is true and the truth value of the statement "I saw Aidan" is false. What would be the truth value of the compound statement, "I was at the store and I saw Aidan"? Intuitively you may be inclined to say that the whole statement is false because the second simple statement is false. The role the operator plays is known as ‘truth function’ which we’ll study more in an upcoming lesson. The important point here is that the statement as a whole has a truth value that is a function of both simple sentences along with the sentence operator. Put another way, the ‘and’ operates on the truth value of the simple statements giving us a truth value for the compound statement as a whole.

Using Tables to Track Truth Values

As arguments get more complex, it can become tricky to remember all the truth values of each of the simple sentences and the operators that join them. Using a simple table is a helpful device for keeping track of everything. As we learned in the previous lesson, we can symbolize our simple sentences as well to make things easier. So, in our statement above we have the following two simple statements:

  1. I was at the store
  2. I saw Aidan

We haven’t learned how to symbolize operators yet so for now, let’s symbolize this statement as "S and A" where S stands for statement 1 and A stands for statement 2.

A common practice in logic is to use the first letter of the prominent noun in a sentence for the variable for that statement. If two (or more) statements have nouns with the same letter, move to the second noun (or second letter of the main noun) etc. This makes symbolic statements easier to read and remember later.

Now in order to keep track of the truth values, we can put the symbolic statement in a table and earmark the truth values for easier reference. This practice will set us up for the lesson on truth tables which takes this concept even further. A table for the statement above would look like this.

S and A

You’ll notice that both statements have a letter T or F above them to indicate whether they’re true or false. The statement operator also has an F above it. This tells us that we think the compound statement as a whole is false. As we’ll learn later, in a compound statement, we really care about the truth value of the statement as a whole and use the individual simple statements to help us determine what the truth value of the statement as a whole should be.

Let’s add a third simple statement to the mix. We’ll symbolize this statement as ‘P’ and assume it’s true:

  1. I saw Philippa

Assuming statement 2 is still false, what is the truth value of the compound statement, I saw Aidan or I saw Philippa? Notice that because we used the "or" in this sentence, your intuition might be that the compound statement is true because at least one of the simple statements is true. Our table would look like this:

A and P

More Than One Operator

What if we have a compound statement with more than a single operator? For example, suppose we added statement 3 to our compound statement above so the compound statement becomes the following: I was at the store and I saw Aidan or I saw Philippa. Our table would now look like this:

T ? F ? T
S and A or P

We now have all our symbols represented but notice that I left out the truth values for the operators. The reason is that there is some ambiguity in how we should group these statements. Think about the compound statement above with all three simple statements. What is the truth value of the whole statement? It’s hard to tell, right? We can understand this statement in two ways and I’ll use commas to show how they can be grouped:

A. I was at the store and, I saw Aidan or I saw Philippa

B. I was at the store and I saw Aidan, or I saw Philippa

If you look carefully at each sentence and think about whether they are true or false, your answer might be different for each sentence based on how the operators are grouped. There is a method for determining the truth value of compound statements like these and we’ll need more tools to be able to apply that method.

In the next two lessons we’ll look at four different operators we can use to create compound statements. Then we’ll look at the idea of truth function a little bit more deeply and those lessons will give us the tools we’ll need to figure out how to figure out the truth value of compound statements regardless of how many operators are involved.

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