Truth Tables

The aim of this tutorial is to help you learn to construct truth tables and use them to test the validity of

arguments.

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Truth tables often seem complex and difficult. This initial perception, however, is misleading. By following a consistent, step-by-step process, constructing truth tables will soon become second nature. The practice it does take to gain this skill is worth the effort for at least two reasons.

1. Truth tables are powerful and let you test the validity of arguments with certainty.

2. The skills involved in constructing truth tables are foundational for more sophisticated sorts of symbolic logic.

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Before we begin it is important to take a minute to think about what truth tables are and what their purpose is. Like Venn Diagrams, truth tables are a way of representing an argument symbolically for the purpose of determining the argument’s validity or invalidity.

Validity, we recall, refers to a deductive argument whose true premises guarantee the truth of its conclusion. On the contrary, an invalid argument is one where the truth of the premises do not guarantee the truth of the conclusion.

Truth tables, then, offer a systematic way to investigate these relationships and determine validity or invalidity.

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First things first. As with Venn Diagrams, truth tables require an argument to be translated into a form using claim variables for the claims.

Each claim variable stands for a complete sentence.

Each claim variable has a truth value; that is, it is either true or false.

PTF

This is a truth table. As you can see it shows the possible truth values of the claim “P.”

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Next, realize that whatever truth value a claim has, its negation (contradictory claim) has the opposite truth value.

P ~PT FF T

We use the “~” to represent the negation and pronounce this truth function symbol as “not.”

This truth table is the definition of negation. So, if “P” is true then “not P” is false and if “P” is false then “not P” is true.

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Let’s look at the remaining truth function symbols. They each represent the relationship between two claims.

A conjunction is a compound claim asserting both of the simpler claims contained in it. A conjunction is true if and only if both of the simpler claims are true.

P & Q This is a conjunction using the ”&” symbol. It is pronounced “P and Q.”

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So, what would a truth table for a conjunction look like? Remember it must show all the possible combinations of truth values of its claims.

P Q P&QT T ?T F ?F T ?F F ?

This table must contain 4 lines because it needs to show all possible combinations of truth values of “P” and “Q.”

“P” True and “Q” True “P” True and “Q” False

“P” False and “Q” True

“P” False and “Q” False

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Now, look at the third column. The lines below “P and Q” show the truth values of this conjunction based on the truth values of its parts. Since a conjunction is only true when both its parts are true we can see how the lines are assigned values.

P Q P&QT T T F F T F F

T

FF

F

Study this table carefully. Make sure you understand what the purpose of each part is and what it tells you. It is essential that you understand these basics in order to use truth tables to test validity.

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A disjunction is a compound claim asserting either or both of the simpler claims contained in it. A disjunction is false if and only if both of the simpler claims are false.

P v Q This is a disjunction using the ”v” symbol. It is pronounced “P or Q.”

P Q P v QT T T F F T F F

T

TT

F

This truth table represents the rule of disjunction. As you can see, the only way “P or Q” is false is the case where both “P” is false and “Q” is false.

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A conditional is a compound claim asserting the second simpler claim on the condition that the first is true. A conditional is false if and only if the first claim is true and the second false.

P Q This is a conditional using the ”” symbol. It is pronounced “if P then Q.”

P Q P QT T T F F T F F

T

TF

T

This truth table represents the conditional. As you can see, the only way “ if P then Q” is false is the case where “P” is true and “Q” is false.

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We are now ready to look at truth table themselves. When constructing truth tables, keep the following three rules in mind:

1. Use parentheses, as in algebra, to represent where the truth function operation is doing its work.

2. The table must capture all possible combinations of truth values for individual sentences contained in the complex expression.

3. The table must contain columns for the parts of the final complex expression, if any of those parts is not a single claim variable.

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Now, use a truth table to determine the validity or invalidity of this argument:

First, translate this argument into standard form

“If building the bookshelf requires a screw driver then I will not be able to build it. After reading the directions I see that a screw driver is needed. So, I can’t build it.”

If S then not BS _Not B

Now into symbolsS ~BS _~B

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S ~BS _~B

Now, build a truth table. We have two claim variables, “S” and “~B” which will each need a column.

S ~BT TT FF TF F

Next, we need a column for each premise and the conclusion. The second premise is already represented, so we only need to add the first premise to our table.

S ~B

~B

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S ~BT TT FF TF F

S ~B

S ~BS _~B

Now, fill in the truth values for the first premise based on the rule of the conditional.

T F T T

For convenience we can add columns for the second premise and the conclusion, though they are already in the table.

~B T F T F

STTFF

We’re done. Our truth table now tells us whether or not the argument is valid. What do you think?

S ~BT TT FF TF F

S ~B T F T T

~B T F T F

S ~BS _~B

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To use the table to test the validity of our argument, we need to concentrate on the definition of validity. An argument is valid if it is impossible for the premises to be true and the conclusion to be false. Since our truth table represents all possible combinations of the truth values of the parts of the argument, we simply need

STTFF

to inspect it to see if any line shows true premises and a false conclusion. As you can see there are no such lines, so this is a valid argument. When the premises are true so is the conclusion.

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Now, use a truth table to determine the validity or invalidity of this argument:

First, translate this argument into standard form

“Martin is not buying a new car since he said he would buy a new car or take a Hawaiian vacation and I just heard him talking about his trip to Maui.

C or HH _Not C

Now into symbolsC v HH _~ C

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C v HH _~ C

Now, build a truth table. We have two claim variables, “C” and “H” which will each need a column.

C HT TT FF TF F

Next, we need a column for each premise and the conclusion. The second premise is already represented, so we only need to add the first premise to our table.

C v H

~C

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C HT TT FF TF F

C v H

C v HH _~ C

Now, fill in the truth values for the first premise based on the rule of disjunction.

T T T F

For convenience we can add columns for the second premise, though it is already in the table. We merely recopy the “H” column and determine the truth values of the conclusion “~ C” column by negating the “C.”

~C F F T T

HTFTF

We’re done. Our truth table now tells us whether or not the argument is valid. What do you think?

C HT TT FF TF F

C v H T T T F

~C F F T T

HTFTF

C v HH _~ C

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When you inspect the truth table you want to see if it is possible for the premises to be true and the conclusion false.

Note the red shaded line. It is possible for the premises to be true and the conclusion false. This is an invalid argument.

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Now, use a truth table to determine the validity or invalidity of this argument:

First, translate this argument into standard form

“If you want to over-clock your processor you must make both hardware and software changes. Unfortunately, you are either ignorant of hardware or software. So, you won’t be over-clocking your processor

If O then H and SNot H or Not S _Not O

Now into symbolsO (H & S)~H v ~S _~ O

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O (H & S)~H v ~S _~ O

Now, build a truth table. We have three claim variables: “O,” “H,” and “S,” which will each need a column. With

three variables we will need 8 lines to show all possible truth value combinations. Note the the “S” column alternates one true and one false all the way down, the “H” line alternates pairs of trues and falses and the “O” line alternates four trues with four falses. If we had another column it would alternate eight trues with eight falses.

O H ST T TT T FT F TT F FF T TF T FF F TF F F

Now add columns for the premises.

O H ST T TT T FT F TT F FF T TF T FF F TF F F

O (H & S)~H v ~S _~ O

We need columns for “~H” and “~S”.We get these truth values by negating “H” and “S”.

~H ~S F F F T T F T T F F F T T F T T

H & S T F F F T F F F

Next we add a column for “H & S” by applying the rule of conjunction to the “H” and “S” truth values we have already drawn.

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O H ST T TT T FT F TT F FF T TF T FF F TF F F

O (H & S)~H v ~S _~ O

~H ~S F F F T T F T T F F F T T F T T

H & S T F F F T F F F

Now add a column for the first premise by applying the rule of the conditional to the “O” and “H&S” values in the table.

O (H&S)TFFFTTTT

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O H ST T TT T FT F TT F FF T TF T FF F TF F F

O (H & S)~H v ~S _~ O

~H ~S F F F T T F T T F F F T T F T T

H & S T F F F T F F F

Now add a column for the second premise by applying the rule of disjunction to the “~H” and “~S” values in the table.

O (H&S)TFFFTTTT

~H v ~SFTTTFTTT

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O H ST T TT T FT F TT F FF T TF T FF F TF F F

O (H & S)~H v ~S _~ O

~H ~S F F F T T F T T F F F T T F T T

H & S T F F F T F F F

Now add a column for the conclusion by negating the “O” column and determine the validity or invalidity.

O (H&S)TFFFTTTT

~H v ~SFTTTFTTT

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~ O F F F F T T T T

O H ST T TT T FT F TT F FF T TF T FF F TF F F

O (H & S)~H v ~S _~ O

~H ~S F F F T T F T T F F F T T F T T

H & S T F F F T F F F

There are no cases where the premises are true and the conclusion is false. Thus, this is a valid argument.

O (H&S)TFFFTTTT

~H v ~SFTTTFTTT

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~ O F F F F T T T T

This is the end of this tutorial

Obviously there are more complex arguments than the ones addressed in this tutorial. However, if you are systematic in applying the rules, the complexity is irrelevant.

If you would like more practice with building truth tables click here.