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Short Little Lessons in Logic: Operator of the Largest Scope

Lesson 13: Operator of the Largest Scope

What you’ll learn in this lesson:

  • How to work with multiple operators in a compound statement
  • How to determine the operator of the largest scope
  • Further explore the relationship between operators and truth value

In earlier lessons, we’ve been looking at how to construct compound statements using four operators and how the truth value of the compound statement is a function of those operators. In the previous lesson, we saw that an operator can operate on simple statements and compound statements and briefly explored how to read the truth value of a compound statement that contains another compound statement as a part of it. In this lesson, we’ll look more closely at these types of compound statements and how to figure out the truth value of the statement as a whole.

Nested Compound Statements

As we saw previously, if we’re using an operator to construct a compound statement and one side of the operator is another compound statement, we need a method to determine how to figure out the truth value of the compound statement as a whole. Consider this example.

  • If Megan goes skiing then Joe gets cocoa and Jasmine rides the gondola or the dog sits by the fire

We’ll stipulate that the simple statements have the following symbols and truth values:

  1. M = Megan goes skiing, true
  2. J = Joe gets cocoa, false
  3. G = Jasmine rides the gondola, false
  4. D = the dog sits by the fire, true

You should see that this compound statement has three different operators–a conditional, a conjunction, and a disjunction. We know that we can solve for the truth value of the individual compound statements but how do we group the statements and in which order do we solve for the truth values of those statements?

For the statement above, we have a few options. We’ll use punctuation to show these options more clearly.

  1. If Megan goes skiing then Joe gets cocoa and Jasmine rides the gondola, or the dog sits by the fire
  2. If Megan goes skiing then Joe gets cocoa, and Jasmine rides the gondola or the dog sits by the fire
  3. If Megan goes skiing then, Joe gets cocoa and Jasmine rides the gondola, or the dog sits by the fire

But it still may not be entirely clear how exactly we should group the compound statements so let’s use brackets and parentheses to make it clearer.

  1. [If Megan goes skiing then (Joe gets cocoa and Jasmine rides the gondola)] or the dog sits by the fire
  2. (If Megan goes skiing then Joe gets cocoa) and (Jasmine rides the gondola or the dog sits by the fire)
  3. If Megan goes skiing then [(Joe gets cocoa and Jasmine rides the gondola) or the dog sits by the fire]

We can also use symbols to simplify this even further.

  1. [M > (J & G)] v D
  2. (M > J) & (G v D)
  3. M > [(J & G) v D]

So the proper grouping of the compound statements largely is a matter of how the sentence is written and it’s entirely up to the person making the statement to determine what he or she means. As we’ll see below, it’s important to determine the proper grouping as the way statements are grouped has an impact on how we evaluate the truth value of compound statements.

Once you determine the proper grouping, you can use brackets and parentheses to make it easier to determine which operators should go with which statements. You can also see more clearly which operators are working on simple statements or compound statements. For example, the conjunction in statement 2 operates on two compound statements: the conditional as the first conjunct and the disjunction as the second conjunct.

The Operator of the Largest Scope

Now we’re ready to figure out the truth value of the compound statement as a whole and there’s a method for doing so. You may remember when you first learned mathematics that you solve for equations involving parentheses by working from the inside out. That is, you look for the parentheses or brackets that are ‘nested’ inside other parentheses or brackets. The grouping that is nested the deepest is where you would start1. For example, consider the following equation.

x = 1 + [23 - (4 x 5)]

In this equation, you would solve the multiplication operation first and get 20. Then you’d work on the subtraction operation taking 20 from 23 leaving 3. Finally, you’d add 1 and 3 for a total of 4 as your final answer and the value of ‘x’. Logic works the same way. You solve for the innermost operations first and work out. When you get to the final operator, that’s called the operator of the largest scope and its truth value is the truth value of the entire compound statement.

Let’s see how this works with our three statements above using tables. We’ll walk through the first statement step by step then you can try to solve for the next two on your own and see what you come up with.

Statement 1:

Step 1: First we solve for the conjunction since that is the inner-most grouping. Our rule says that if one or more conjunct is false, then then the conjunction as a whole is false so we’ll write F above the column for the conjunction operator to show that the compound statement is false.

T F F F T
[M > (J & G)] v D

Step 2: Next we’ll solve for the conditional since that is the next grouping working outward. Notice we use the truth value of the conjunction and the truth value of the M statement to solve for the conditional. The M statement is the antecedent and the conjunction is the consequent. Our rule states that if the antecedent is true and the consequent is false, the conditional as a whole is false so we’ll write ‘F’ above the conditional operator.

T F F F F T
[M > (J & G)] v D

Step 3: Finally, we solve for the disjunction. The conditional is the first disjunct and the D statement is the second disjunct. Our rule states that if one disjunct is true and the other is false, the disjunction as a whole is true so we put a ‘T’ above the disjunction symbol. We’ll aso use the arrow to indicate that the disjunction operator is the operator of the largest scope.

T F F F F T T
[M > (J & G)] v D

So in this compound statement, given the groupings we’re using, we can conclude that the statement as whole is true!

Now try the next two on your own and see what you come up with. If you’re unsure of your answers, use the discussion board to comment on your answer and see what others have to say about them. Hint: In statement 2, the order in which you solve for the conditional and disjunction doesn’t matter since they’re both at the same “level.”

Statement 2:

T F F T
(M > J) & (G v D)

Statement 3:

T F F T
M > [(J & G) v D]

Assuming that you’ve done the operations correctly, you should notice that statements 2 and 3 have different truth values for the operator of the largest scope. This is because we grouped the compound statements differently and the order in which we solved for the operators has an impact on truth value. The way you understand a compound statement can make a difference in the truth value of the statement as a whole. So its important to make sure you understand the meaning of the person making the statement so you group your operators properly.

The ideas in this lesson will be used in later lessons to help us form and analyze arguments. We also use these concepts in everyday speech, you just may not have recognized them. But if you think about it, when people state facts, their statements have a truth value that follow the patterns we talked about in this lesson. Pretty cool.

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  1. As a matter of convention, logicians will alternate between brackets and parentheses to keep things clearer. While you can use just parentheses or just brackets, it can get confusing to determine groupings in complex statements so alternating keeps things clearer.

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