chapter 9 deductive arguments ii truth-functional...

Chapter 9

Deductive Arguments IITruth-Functional Logic

Tutorial: Truth Tables

Truth Tables

The aim of this tutorial is to helpyou learn to construct truth tablesand use them to test the validity of

arguments.

2© 2009 McGraw-Hill Higher Education. All rights reserved.

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 secondnature. The practice it does take to gain this skill isworth the effort for at least two reasons:


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

2. The skills involved in constructing truth tables arefoundational for more sophisticated sorts ofsymbolic logic.

Before we begin, it is important to take a minute to thinkabout what truth tables are and what their purpose is. LikeVenn diagrams, truth tables are a way of representing anargument symbolically for the purpose of determining theargument’s validity or invalidity.

Validity, we recall, refers to a deductive argument whosetrue premises guarantee the truth of its conclusion. Incontrast, an invalid argument is one where the truth of thepremises do not guarantee the truth of the conclusion.

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


PTF

This is a truth table. As you can seeit shows the possible truth values ofthe claim “P.”

First things first. As with Venn diagrams, truth tablesrequire an argument to be translated into a form usingclaim variables for the claims.

Each claim variable stands for a completesentence.

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

© 2009 McGraw-Hill Higher Education. All rights reserved. 5

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

P ~PT FF T

We use the “~” to represent the negation andpronounce 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.


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

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

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


So, what would a truth table for a conjunction looklike? Remember it must show all the possiblecombinations 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 showall 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


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

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

T

FF

F

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


A disjunction is a compound claim asserting either or bothof the simpler claims contained in it. A disjunction is falseif 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 ofdisjunction. As you can see, the onlyway “P or Q” is false is the case whereboth “P” is false and “Q” is false.


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

P _ QThis 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 theconditional. As you can see, the onlyway “ if P then Q” is false is the casewhere “P” is true and “Q” is false.


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

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

2. The table must capture all possible combinations oftruth values for individual sentences contained in thecomplex expression.

3. The table must contain columns for the parts of thefinal complex expression, if any of those parts is nota single claim variable.


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

First, translate this argument intostandard form

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

If S then not BS _Not B

Now into symbolsS _ ~BS _~B


S _ ~BS _~B

Now, build a truth table. We have two claimvariables, “S” and “~B” which will eachneed a column.

Next, we need a column for eachpremise and the conclusion.The second premise is alreadyrepresented, so we only need toadd the first premise to our table.

S ~BT TT FF TF F

S _ ~B ~B


S _ ~BS _~B

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

For convenience we can add columns for the secondpremise and the conclusion, though they are already inthe table. S ~B

T TT FF TF F

S _ ~B T F T T

~B T F T F

STTFF

We’re done. Our truth table nowtells us whether or not the argumentis valid. What do you think?


To use the table to test the validity of ourargument, we need to concentrate on thedefinition of validity. An argument is valid if itis impossible for the premises to be true and theconclusion to be false. Since our truth tablerepresents all possible combinations of the truthvalues of the parts of the argument, we simplyneed to inspectit to see if any line shows truepremises and a false conclusion. Asyou can see, there are no such lines;this is a valid argument. When thepremises are true so is the conclusion.

S _ ~BS _~B


S ~BT TT FF TF F

T F T T

~B T F T F

STTFF

S _ ~B


First, translate this argumentinto standard form

“Martin is not buying a new car, since he said he wouldbuy a new car or take a Hawaiian vacation and I just heardhim talking about his trip to Maui.”

C or HH _Not C

Now into symbolsC v HH _~ C


C v HH _~ C

Now, build a truth table. We have two claimvariables, “C” and “H” which will each needa column.

Next, we need a column for eachpremise and the conclusion.The second premise is alreadyrepresented, so we only need to addthe first premise to our table.

C HT TT FF TF F

C v H

~C


C v HH _~ C

Now, fill in the truth values for the first premisebased on the rule of disjunction.For convenience we add columns for the second

premise, though it is already in the table. We merely recopythe “H” column and determine the truth values of theconclusion “~ C” column by negating the “C.”

C HT TT FF TF F

C v H T T T F

~C F F T T

HTFTF

We’re done. Our truth table nowtells us whether or not the argumentis 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

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

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



First, translate this argumentinto standard form

“If you want to over-clock your processor you must make bothhardware and software changes. Unfortunately, you are eitherignorant of hardware or software. So, you won’t be over-clockingyour processor.”

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

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


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

Now, build a truth table. We have three claimvariables, “O,” “H,” and “S,” which will eachneed a column. With three variables we willneed 8 lines to show all possible true valuecombinations. Note the “S” column alternatesone true and one false all the way down, the“H” line alternates pairs of trues and falsesand the “O” line alternates four trues with fourfalses. If we had another column it wouldalternate 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 & S)~H v ~S _~ O

We need columns for “~H” and “~S”; we getthese 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 therule of conjunction to the“H” and “S” truth valueswe have already drawn.


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

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




O _ (H&S)TFFFTTTT


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

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




O _ (H&S)TFFFTTTT

~H v ~SFTTTFTTT


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

Now add a column for the conclusion bynegating the “O” column, and determinethe validity or invalidity.




O _ (H&S)TFFFTTTT

~H v ~SFTTTFTTT

~ O F F F F T T T T


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

There are no cases where the premises aretrue and the conclusion false; this is avalid argument.




O _ (H&S)TFFFTTTT

~H v ~SFTTTFTTT

~ O F F F F T T T T


Obviously there are more complex arguments thanthe ones addressed in this tutorial. However, if you are

systematic in applying the rules, the complexityis irrelevant.


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