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Short Little Lessons in Logic: Truth Function

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## Lesson 12: Truth Function

What you’ll learn in this lesson:

• The definition of truth function
• How understanding truth function can help in determining the truth value of compound statements
• How to think about truth function for conditional statements

We’ve learned that simple statements have a truth value and in the study of operators, we’ve seen that compound statements have a truth value as well. We said that operators “operate” on the value of simple or compound statements to form a new compound statement that has its own truth value. This should make intuitive sense since this is how things work in general conversation. We know that if someone says, “I went to the store.” that this statement is either true or false depending on whether they actually went to the store or not. But if they say, “I went to the store and I bought milk.” that statement as a whole is true or false only if they do both things and false if they fail to do one of the two things. 1

### Operators and Compound Statements

The ideas above hopefully are clear when it comes to simple statements. Let’s take a look at how this operation works with compound statements. Let’s modify the statement above and add a compound statement as one of the conjuncts:

1. I went to the store and, I bought milk or I bought bread

What is the truth value of the entire statement? In normal conversation, we might think that the statement is true if the person who is making the claim went to the store AND bought either milk or bread or both. Again, our conjunction rule and our disjunction rule supports this intuition. But we need a way to think about how this works formally. So we could rewrite statement 1 as follows to make this clearer:

1. I went to the store and, (I bought milk or I bought bread)

The comma in the English sentence helps us to know where to put the parentheses. We group the disjunction and determine the truth value of that statement and use the value of the disjunction and the first simple statement to determine the value of the conjunction. And if we put the sentence into a table we can see how we’d solve for the truth value of this compound statement:

T T T T F
I went to the store and (I bought milk or I bought bread)

The arrow indicates the truth value of the compound statement as a whole. This is what it means to say that an operator operates on simple or compound statements.

The truth value of the entire compound statement, then, is a product of the truth value of the statements that make it up (simple or compound) along with the function of the operator on those statements. Once you understand how the operators function, you can determine the truth value of the entire compound statement.

### Truth Function

We’ll learn more about how to determine truth value for compound statements like the one above in the next lesson, “Operator of the Largest Scope”. But this gives us enough of a foundation to talk about why operators work like this and the reason is that the logic we’ve been studying is “truth functional” Put simply:

A Truth Function is operation that takes place when we evaluate the truth value of all the statements (simple or compound) along with all their operators.

Dr. Paul Herrick in his excellent text Introduction to Logic states it this way:

A function is a rule that relates one set of values to another set of values. A function is a truth-function if the values it relates are truth values. A sentence operator is truth-functional if the compound sentence that it forms is a truth-functional compound sentence. A compound sentence is a truth-functional compound if its truth-value is a function of the truth-value of its component ot components. (Paul Herrick, Introduction to Logic, 2013 Oxford University Press, p. 289)

This is a lot more technical (it comes from a text book after all) but it does describe how each of the parts of a compound statement relate to one another to create a single truth value. Solving for the overall truth value of the statement above is an example of a truth-functional operation. We used the value of the component parts to come up with a single truth value.

### Truth Function and Conditional Statements

Truth function can help us better understand the seemingly puzzling way a conditional works as well. You’ll recall that we said in the lesson on conditionals that a conditional only is false if the antecedent is true and the consequent is false, otherwise it’s true. Suppose we made the statement, “if I went to the store, then I bought milk.” If it’s false that I went to the store but true that I bought milk, the conditional statement as a whole is true. This seems counter-intuitive. But we can use the concept of “truth function” to help us sort it out.

One big difference between conditionals in logic and the conditionals we use in common speech is that conditionals in logic do not imply a causal relationship. For example, the conditional in English, “If the car is going 20MPH and hits a solid object, then the airbag will deploy” implies that the car going 20 and hitting the object causes the airbag to deploy. In formal logic, the conditional is sometimes referred to as material implication which involves the idea that the antecedent doesn’t cause the consequent but only that the two are related.

Remember that in a conditional, the antecedent is the sufficient condition–what could be true. But we also said that if the antecedent is true, the consequent must be true and that’s the necessary condition. So this tells us what the truth function should produce. If the sufficient condition is true and the necessary condition doesn’t happen, the conditional “fails” to follow the rule so it’s overall value is false. Let’s look at this more closely.

#### An Example

Suppose you’re in charge of determining when a certain substance is present in the air. We’ll call this substance “Chemical X”. You’ve noticed that every time Chemical X is present, another substance “Element R” glows red. You run 100 tests and each time Chemical X is present, Element R glows red and you create a hypothesis: when Chemical X is present, Element R will glow red. You can state your hypothesis as a conditional:

Hypothesis 1: If Chemical X is present, then Element R glows red.

Notice that this conditional doesn’t mean that Chemical X causes Element R to glow red or that Chemical X is the only chemical by which Element R will glow red. It does mean though that if Chemical X is present in the air, Element R will glow red. We’ll say that your hypothesis is false only if Element R doesn’t glow red when Chemical X is present. The following table describes all the possible truth values for the conditional under all possible situations:

Case Number Conditions Antecedent / Consequent Outcome Truth Value of the Hypothesis
1 Chemical X is present and Element R glows red True/True The case is in line with what the hypothesis predicts True
2 Chemical X is present but Element R doesn’t glow red True/False This is the “failure” case so the hypothesis doesn’t hold False
3 Chemical X is not present and Element R glows red False/True This conditional doesn’t claim that only Chemical X causes Element R to glow red so the hypothesis holds True 2
4 Chemical X is not present and Element R doesn’t glow red False/False This case is consistent with the claim in the conditional so the hypothesis holds True

In each of these cases, you’re testing to see if the conditions of the test support what you describe in the conditional: does Element R glow red when Chemical X is present? If it doesn’t, then the hypothesis fails and the conditional is false. In any other case, we say the hypothesis is true. Think of this as an “innocent until proven guilty” approach.

### An Analogy

The following example might help make it clearer. It’s an analogy and all analogies break down eventually. But this example does, I think, illustrate the truth-functional nature of compound operators particularly as it applies to the conditional operator.

Let’s suppose you build a box with slots on the left and right sides of the box. You build the box so that when you put a special token in the left side of the box, a peanut is dispensed from the right side. You design the box to do other things but you know the box is working when you insert the token on the left and a peanut is dispensed on the right. If that doesn’t happen, the box is not working as you designed it.

You can state your device’s design as a conditional: `If a token goes in the left, then a peanut comes out of the right`. We can examine the function of the box by looking at the cases for the conditional. This conditional states your only rule for failure: if a token goes in the left and a peanut does not come out of the right, the box is not working. In all other cases you don’t care and will say that the box is working as it should.

Case 1 (antecedent is true, consequent is true): You put the token in the left and a peanut is dispensed from the right. This is what you said the box must do so it’s working as you designed it.

Case 2 (antecedent is true, consequent is false): You put the token in the left and a walnut is dispensed from the right. The box is not working as you designed so the conditional is false.

Case 3 (antecedent is false, consequent is true): You put an almond in the left and a peanut is dispensed from the right. Since you’ve only said the box fails to work if a peanut is not dispensed when the token is inserted in the left, this case doesn’t violate that rule so the box is working as far as you’re concerned.

Case 4 (antecedent is false, consequent is false): You put an almond in the left and a walnut is dispensed from the right. Since you made no claims about what should happen when almonds are inserted, it doesn’t fall under your rule for failure so the box is working.

In this analogy, the box itself is the “function”: it takes an input, does some work, and produces some output. Your rule for what the box must do, is your test for the claim, “the box is working.” That statement is false if it doesn’t do what you said it should and true otherwise.

We’ve focused mainly on the conditional operator in this lesson but remember that all the operators we’ve studied are truth functional in the way we’ve learned in this lesson. As we study deductive arguments in Module 3, this idea of truth function will be essential for understanding how deductive arguments work. Before we get to deductive and inductive arguments, we need to look more deeply at how to get to the overall truth value of a compound statement. We’ll do that in the next lesson.

1. If you haven’t studied the lessons on operators, you should take some time and review those before getting into this lesson. You’ll need that background in order to better understand what is going on in this lesson. Browse to the outline to find the lessons on operators.

2. Case 3 might seem a bit odd at first but notice that it follows the conditional rule perfectly and also illustrates the power of necessary and sufficient conditions. The conditional says that if Chemical X is present, then Element R glows red. But it doesn’t specific under what other conditions Element R might glow red. Perhaps there are a number of other chemicals that could cause Element R to glow red. Your conditional doesn’t specify what those might be. All it states is that in the presence of Chemical X, Element R glows red.

Put another way, if we know that Element R is glowing red, the conditional tells us nothing about what else is true. The conditional tells us what must be true (the consequent) if something else is true (the antecedent) but not the reverse.

Notice too that the Hypothesis 1 is not the same as this:

Hypothesis 2: If Element R glows red, then Chemical X is present

This conditional states that Element R glowing red is one condition under which Chemical X is present and that if Chemical X is not present then Element R doesn’t glow red. It does not tell us that if Chemical X is present what Element R must do do. It could glow red or glow some other color. For example, Element R could glow green in 10% of the cases Chemical X is present. But we do know that every time it glows red, Chemical X must be present.

If you actually were running this experiment, you might need to consider which conditional gives you the right conditions for your test. Since you’re only hypothesizing (and not ready to say that Chemical X causes Element R to glow red), which conditional better expresses what you want to hypothesize: hypothesis 1 or hypothesis 2?

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