LOG UNIT 6

UNIT6

Further Applications

of the Truth Table Method

A. INTRODUCTION

Now that you know the basic process of constructing truth tables, you can apply

this method in a variety of other circumstances. You can use it to show whether

someone is contradicting himself, whether a statement has any real significance,

or whether two statements have the same meaning. If someone claims, for in-

stance, that he has been offered a job as a petroleum engineer, which he will take

provided it pays $100,000 a year, and that if it doesn't pay that much, he will in-

stead take a job as vice-president of a small coal company, but that he won't take

either job, you will be able to show by the truth table method that what he says

cannot be true, since he has just contradicted himself. Or, if some self-appointed

economic expert "predicts" that either there will be inflation and unemployment,

or no inflation and no unemployment, or inflation but no unemployment, or unem-

ployment but no inflation, you will be able to recognize, using the truth table

method, that such a claim really says nothing at all, has absolutely no content. The

truth table method can also be used to show when two statements, although they

seem to be making different claims, really mean the same. "The rate of inflation

will be reduced provided there is a steep increase in interest rates and high unem-

ployment" and "If there is no reduction in the inflation rate, then either unemploy-

ment will not be high or there will be no steep increase in interest rates," for

instance, really say exactly the same thing.

In this unit you will learn all these things and more: how to test a set of state-

ments or statement forms for consistency, how to determine whether two state-

ments or forms say the same thing, and whether a given statement or form is

9596 Unit 6 Further Applications of the Truth Table Method

significant or is simply a tautology, giving no real information. You will need to

learn to distinguish clearly the kinds of problems that can be solved using truth ta-

bles, and you will learn some interesting relationships between the various truth

table concepts, such as the fact that if the premises of an argument form are incon-

sistent, or contradictory, then the argument form is valid rather than invalid. The

things you will be expected to master are listed below in the "Objectives" section.

B. UNIT 6 OBJECTIVES

• Learn the definitions of tautology, contradiction, and contingency, given

at the end of the unit.

• Be able to use the truth table method to determine the logical status of sin-

gle statement forms, that is, whether they are tautologous, contradictory,

or contingent.

• Learn the definitions of logical implication and logical equivalence and

the relation between them.

• Be able to use the truth table method to determine whether two or more

statement forms are logically equivalent or whether one logically implies

another.

• Learn the definition of consistency.

• Be able to use the truth table method to determine whether a set of state-

ment forms is consistent.

• Be able to state clearly the four different kinds of truth table problems we

have encountered and the concepts applicable to each. (It will make no

sense, for instance, to say that a single statement form is valid.)

• Be able to state several relationships between the various truth table con-

cepts.

c. UNIT 6 TOPICS

1. Tautologies, Contradictions, and Contingencies

So far you have been using the truth table method just to test argument forms

for validity, which means making up a joint truth table for the premises and con-

clusion and then inspecting the results for all the formulas taken together. It is also

possible to make up a truth table for just a single, individual statement form, and

we sometimes want to do this to determine the logical status of the form. Certain

forms, for instance, have an interesting property: they can never, under any cir-

cumstances, tum out to be false, and this is something that can be determined by

the truth table. Other forms have just the opposite property: they can never, in anyUnit 6 Further Applications of the Truth Table Method 97

case, turn out to be true. The former sort of statement forms, which can never be

false, are called tautologies, and the latter, which can never be true, are called, sen-

sibly enough, contradictions. The intermediate cases, forms that are true in some

instances and false in others, we will call contingencies. (In ordinary parlance, a

contingency is something that might or might not occur; we are using the term in a

different, but closely related, sense.)

Testing single statement forms to determine their logical status, that is,

whether they are tautologous, contradictory, or contingent, is a simple matter once

you know how to construct truth tables. You simply take all the variables that

occur in the form, construct your base columns for those variables, and then com-

pute the result for each instance. An example should make this process clear:

This form turns out to be a tautology

because there is no instance in which

the value under the major operator is

false.

We will compute the results here in the same way as we have done before,

starting with the smallest components first, placing the truth tables for the subfor-

mulas underneath the operators for those formulas, and gradually working our

way up to the major operator. It is the truth values under the major operator that

determine the logical status. Notice that in this example, for instance, we have

many F's in various places in the truth table, but under the major operator the re-

sult is always T, so the form is a tautology. We may formally define tautology as

follows: a tautology is a statement form that is true (under its major operator) for

every substitution instance. Another way we could put this would be to say that for

every row in the truth table, the result of the computation under the major operator

is "true."

We can consider a statement, a specific instance, to be a tautology just in case

its form is a tautology. To determine whether a statement is a tautology, then, we

can use the same kind of procedure we used in Unit 5 to determine whether a par-

ticular argument was valid: symbolize, extract the form, and apply the truth table

method. Here, we will symbolize the English statement, systematically replace the

capital letters with variables to obtain the statement form, and then test the form to

see whether it ever comes out false. Using this procedure, we find that the eco-

nomic "prediction" mentioned in the introduction to this unit is, in fact, a tautology.

The statement "Either there will be inflation and unemployment or no inflation and no

unemployment or inflation but no unemployment or unemployment but no inflation"

98 Unit 6 Further Applications of the Truth Table Method

would be symbolized as (U· U) v (~ / . ~ U» v (U· ~ U) v (U • ~ I»~. The cor-

responding form would be ((p. q) v (~p. ~ q» v ((p. ~ q) v (q. ~ p», and the

truth table below shows that it is indeed a tautology.

Tautologies play little role in ordinary language because they give us no in-

formation. Since they are always true under any circumstances, they don't make

any definite claim about the way things actually are. If the weather forecasters tell

you, for instance (as they are wont to do), that it will either rain tomorrow or not

rain, that is not much of a forecast. It certainly doesn't help you decide whether to

go on a picnic. A tautology is an "empty" claim; it really says nothing about the

world. Tautologies do play a very important role in logic, however. They are the

axioms and theorems of formal logical systems, the "truths" of the system, just as

"x + y = y + x" is a truth of most of our mathematical systems. They are useful

in logic precisely because they do not make any claim about the empirical world,

but are, we might say, true no matter what the worldly facts. They are formulas

whose truth we can absolutely depend on, and whatever its limited usefulness in

ordinary language, this is a highly desirable property in logic. In doing the prob-

lems, you should pay particular attention to formulas that tum out to be tautologies

(and also those that tum out to be contradictions); some of them will be important

later, and, in any case, it will help to develop your logical intuitions.

There are other statement forms that can never turn out to be true, no mat-

ter what the values of the component parts. Such forms, as already noted, are

called contradictions. An example of a contradiction, with its truth table, is the

following:

p q (p -

~ q) - (p - q)

T T T F F

F T This is a contradiction

T F F T T T F because, as the truth

F T F T F T F table shows, there is no

instance in which it

F F T F T F T turns out to be true.

~

(3) (2)(1) (4) (2) (1)

A contradiction is a statement form that is false (under its major operator)

for every substitution instance or, equivalently, one that turns out false for every

Unit 6 Further Applications of the Truth Table Method 99

row in the truth table. Notice that the negation of a tautology will be a contradic-

tion, since a negation changes the truth values, so that if we start with all trues-a

tautology-then, if we negate it, we will end up with all falses-a contradiction.

Similarly, the negation of a contradiction will always be a tautology.

A contingency is a form that is neither a tautology nor a contradiction; that is,

it does not have either all T's or all F's in its truth table under the major operator,

which means simply that it has some of each. We can define contingency as fol-

lows: A contingency is a statement form that is true for some substitution instances

and false for others, under its major operator. An example of a contingency is

given below, with its truth table.

F FF This form is contingent

F FT because it has some T's

T FF and some F's under its

T TT major operator.

Notice that the negation of a contingency will be another contingency, because all

the T's will change to F's and all the F's will change to T's, so there will still be

some of each.

As with tautologies, to determine whether an English statement, a specific

instance, is a contradiction or a contingency, we first symbolize the statement, then

obtain its form by systematically replacing capital letters with variables, and then

test the form using truth tables. The following statement is a contradiction, since

the truth table for its form has no T's under the major operator. "I'll get a job if!

run out of money, and I'll run out of money if and only if I don't get a job, but I

won't get a job." This could be symbolized as ((M :) J) • (M == ~ J» • ~ J. The

corresponding form would be ((p :) q) • (p == ~ q» • ~ q. The truth table is below.

The following statement is a contingency, since its form has both T's and F's under

the major operator. "I'll get a job if I run out of money, but I won't run out of

money." The symbolization would be (M:) J)' ~ M, and the form is

(p :) q)' ~ p. The very simple truth table is below.

100 Unit6 Further Applications of the Truth Table Method

p q (p:J q) • ~ P

T T T

T F F F F

F T T TT

~F

F F T TT

(1) (2)(1)

If you thoroughly understand the concepts of tautology, contradiction, and

contingency, you should be able to answer the following sort of question: what

would be the result of disjoining a tautology and a contradiction? Answer: it would

be a tautology, since a disjunction is true provided at least one disjunct is true;

since one side of the disjunction would be a tautology, we would have at least one

true disjunct in every row. What would be the result of conjoining a tautology and

a contradiction? A contingency? No. It would be a contradiction, since in every

row there would be one false conjunct, which would make the conjunction false

for every row. What would be the result of conjoining two contingent forms? Here

we don't know for sure; it might be contingent (if, for instance, the two forms were

p and ~ q), or it might be a contradiction (for instance, if the two forms were p and

~ p). The only thing we can be sure of is that it won't be a tautology, since there

will be at least one false row in the truth table. To take one more example, what

would be the result of placing a triple bar between two contradictions? Another

contradiction? No, here we will have a tautology, since we will have "F == F" in

every row, which will turn out to be true. Exercises at the end of the unit will give

you more practice in working out such combinations; you might even try to think

up other combinations and work out their results.

2. Logical Implication and Logical Equivalence

In Unit 4, when learning to symbolize statements, you learned that certain

forms were interchangeable and meant exactly the same, such as ~(pvq) and

(~p. ~ q). We are now in a position to see why they mean the same and why one

can be used in place of the other. Such statement forms have the very important

property of being logically equivalent to each other, which means that they turn

out to be true or false in exactly the same circumstances. In other words, they have

identical truth tables under their major operators. This will be our primary defini-

tion of logical equivalence: two or more statement forms will be logically equiva-

lent if and only if the truth tables under their major operators are identical. You

test statement forms for logical equivalence by constructing a joint truth table for

them and computing the results for each of the formulas. If the truth tables have

the same values under the major operator, in every single row, then the formulas

are equivalent. If there is some row in which the values are different, then they are

not equivalent. The following two forms, then, are equivalent, since they are each

false in the second row and true in all the others.p q Unit 6 Further Applications of the Truth Table Method (~ (p • q) :J ~ p) (~q:J ~ (p v q))

101

T T F T

T F T F F F F T T F T T F F T F T T wF

(2) (1) (3) (1) T F F T

F W F

T

F T F T

T T T F

(1) (3) (2) (1)

We can also test more than two formulas at a time. In the following example

we have four. We can say which are equivalent to which just by comparing their

truth tables.

p q 1. ~ (p. q) 2. (~p • ~ q) 3. ~(pvq) 4. (p:J ~ q)

T T

F

mT wF T F T F F F T F T T T

F T T F T F F F T T F

F F T F T T T F T T

mT F

m

(2) (1) (1) (2)(1) (2) (1) (2) (1)

In this instance, we can say that formulas 2 and 3 are equivalent and that formulas

1 and 4 are equivalent, but neither 2 nor 3 is equivalent to either 1 or 4.

It is also important to realize that formulas may be logically equivalent even

though they have different numbers of variables! You can determine this, again, by

drawing up the joint truth table and working out the results for both formulas. In

this way we can see that (~p. ~q) and ((~p. ~q)·r)v((~p· ~q). ~r) are

equivalent. The truth table follows.

T T T FFF F F F F F F

T T F F F F F F F F F T

T F T F F T F F F F F F

T F F F F T F F F F FT

F T T T F F F F F F F F

F T F T F F F F F F F T

F F T T T T T T T T F F

F F F T l T T F c.L T TT

(1) (2) (1) (1) (2) (3) (1) (2)(1)

There is an interesting connection between the concepts of logical equiva-

lence and tautology: if two formulas that are logically equivalent are joined into a

biconditional, the result will be a tautology; and if the two formulas are not equiv-

alent, the result of joining them will not be a tautology. The reason for this is that102 Unit 6 Further Applications of the Truth Table Method

formulas that are logically equivalent, as we have defined it, have identical truth

tables; this means their truth values are the same for every row in the truth table.

Thus, given that a biconditional is true if and only if its components have the same

truth values, the biconditional with logically equivalent components must be true

for every row. Hence, the biconditional will be a tautology. On the other hand, if

the formulas are not equivalent, there will be some row in which their values dif-

fer; in that row the biconditional will tum out to be false and so will not be a tau-

tology. Thus we can give the following alternative definition of logical

equivalence: two statement forms are logically equivalent if and only if the result

of joining them with the biconditional is a tautology. This definition is, in fact,

used by some textbook authors, and it should be evident that the two definitions

mean the same.

In the Introduction, we gave an example of two sentences that we claimed said

the same thing. We can now show that these sentences, ''The rate of inflation will be re-

duced provided there is a steep increase in interest rates and high unemployment" and

"If there is no reduction in the inflation rate, then either unemployment will not be high

or there will be no steep increase in interest rates" are logically equivalent. These sen-

tences could be symbolized as follows: ((S • H) ::J R), (~ R ::J (~H v ~ S)). The cor-

responding forms would be ((p. q) ::J r), (~r ::J (~qv ~ p)); the truth table below

shows that they are equivalent.

A third definition of logical equivalence can be obtained from a closely re-

lated concept, that of logical implication, which is a relationship between two

statement forms. One statement form logically implies a second if and only if there

is no row in their joint truth table in which the first is true and the secondfalse. If

we make up the truth tables for ~(p v q) and ~ p, for instance, we see that

~(p v q) logically implies ~ p, because there is no row in the truth table in which

~ (p v q) is true while ~ p is false.Unit 6 Further Applications of the Truth Table Method p q ~(p V q) ~p

103

T T

T

T F T

F T T

F F F

~

Logical implication, unlike logical equivalence, is not a symmetric relation: that

one form logically implies another does not necessarily mean that the second will

imply the first (although it may). In the example above, for instance, although

~ (p v q) does logically imply ~ p. the converse relation does not hold; ~ p does

not logically imply ~(p v q) because in the third row ~p is true while ~(p v q)

is false.

When two formulas do both logically imply each other, they must be logi-

cally equivalent. To say that the first implies the second is to say that there is no

row where the first is true and the second false. To say the second implies the first

is to say that there is no row where the second is true while the first is false. If we

have this two-way logical implication, then there can be no row in which one is

true and the other is false, that is, no row in which the truth values are different.

Thus their truth tables must be identical, so they are logically equivalent. Our third

definition of logical equivalence, then, is the following: Two statement forms are

logically equivalent if and only if they logically imply each other.

3. Rules of Inference, Logical Implication,

and Logical Equivalence

The concepts of logical implication and logical equivalence are extremely

important and are closely related to the concept of rules of inference, which will be

discussed in Units 7-9, where we develop the proof method. The most critical fea-

ture of a rule of inference (a rule that, as the term suggests, tells you when you may

correctly infer one statement or form from others) is that it is truth-preserving. To

be truth-preserving means that it is never possible to go from statements (prem-

ises) that are true to a statement (the conclusion) that is false. But this is precisely

the idea of logical implication: there is no instance where the first form is true and

the second false. Thus we will be able to say that in a correct rule of inference, the

premises (or conjunction of premises) logically imply the conclusion. This will be

especially relevant to the basic rules of inference, such as "Given (p :) q) and ~ q,

you may infer ~ p," which are introduced in Unit 7.

The basic rules are what we might call "one-way" rules: they allow you to

infer the conclusion from the premises, but not the reverse. One very basic rule,

Simplification, says "Given (p • q), you may infer p." It would obviously not be

~104 Unit 6 Further Applications of the Truth Table Method

correct, however, to turn this around to say "Given p, you may infer (p. q)." The

rules in Unit 8, however, which we will call "replacement rules," are reversible, or

"two-way," rules. This is because the relationship between premise and conclusion

is the stronger one of logical equivalence, rather than just logical implication. For

each rule, the premise and conclusion are logically equivalent, and this means that

they have identical truth tables, so that there can be no instance in which one is

true and the other false. Thus, we can correctly infer the first formula from the sec-

ond as well as the second from the first, since in neither case will we be in danger

of going from a true statement to a false statement.

The concepts of logical implication and logical equivalence are among the

most important in logic. If you haven't done so already, you should be sure you

know the definitions that have been given in this section and can state the relation-

ships between them.

4. Consistency

Consistency may be the hobgoblin of little minds, but it is also a very important

property of sets of formulas. If a set of premises is inconsistent, for instance, it makes

the argument form virtually worthless, since absolutely anything will follow from it,

including both the conclusion and the negation of the conclusion. If you say some-

thing inconsistent, you might just as well have remained silent, because what you said

could not possibly be true. Thus we all, or at least most of us, want to be consistent.

But what does this mean? It will be a little easier to understand consistency if we talk

first about inconsistency, so we will give the following definition: a set of fonnulas is

inconsistent if there is no row in their joint truth table in which they all come out true

at once. The following set of three formulas, for instance, is inconsistent:

p q (~q -::J ~p), ((p • ~ q) v (~p • ~ q)), ( ~ P ::J q)

T T F

F

FF F

FF ~ F

T F T F T T T F FT F

F T F T F F F T FF T

F F T T F T T T TT T

~

(1) (2) (1) (2)(1) (3) (1) (2)(1) (1) (2)

In the first row of this truth table the middle formula is false; in the second

row the left-hand formula is false; in the third row the middle formula is again

false; and in the last row the right-hand formula is false. Thus there is no row in

which all the formulas come out true at once, and so the set is inconsistent.

Our first example in the Introduction to this unit was a set of sentences con-

cerning job offers, which we claimed were contradictory, or mutually inconsistent.

We are now in a position to show that the set of statements made there was indeed

inconsistent. Those statements could be symbolized as follows: (J. (P -::J T)),

~Unit 6 Further Applications of the Truth Table Method 105

(~P :J V), ~ (T v V). The corresponding forms would be (p. (q :J r)),

( ~ q :J s), ~ (r v s), and the 16-row truth table below indicates that the sen-

tences are indeed inconsistent, since there is no row in which the forms are all

true at once.

There is obviously a very close relationship between inconsistency and contradic-

tion; the difference is that a contradiction is a single formula, whereas consistency is a

property of sets of formulas. We can say, however, that a set of formulas is inconsistent if

and only if the conjunction of all the formulas is a contradiction. If we conjoined all the

formulas above, for instance, we would have a contradiction, since a conjunction is false if

at least one of the conjuncts is false. Here there would be a false conjunct in every row, so

the conjunction as a whole would be a contradiction.

Given the definition of inconsistency, we can now define consistency: we will say

that a set of formulas is consistent if and only if there is at least one row in their joint truth

table in which they all come out true. The following set of three formulas is consistent:

p q (p :J (p. q)), (~p :J ~ (p v q)), (~ (p • q) == (~p • ~ q))

T T

T F F F T F T T F F F FT

T F rnF T F T

FF

F T F T F F T T F F T FF

~F

F F F T T T F T F T T TT

(2) (1) (1) (3)(2) (1) (2) (1) (3) (1) (2)(1)

~106 Unit 6 Further Applications of the Truth Table Method

These formulas all come out true in both the top row and the bottom row; thus,

there is at least one row where they all come out true, so they form a consistent

set.

It is important not to confuse the concept of tautology with that of consis-

tency. A single statement form is a tautology, remember, if it comes out true in

every row under the major operator. This means that the vertical column underneath

the major operator consists entirely of "trues." With a consistent set of formulas,

however, we are saying that there is a horizontal row in which each of several for-

mulas comes out true.

S. Four Kinds of Truth Table Problems

and the Relations Between Them

In this unit and Unit 5, you have learned four or five different applications of

the truth table method-different kinds of problems that can be solved using this

method. Some are used on a single form, some on pairs of forms, and some on sets

of forms. It is extremely important that you keep these sorts of problems straight

and be clear about the various concepts, so let us summarize:

The concept of validity is applied to argument forms, that is, sets of statement

forms consisting of premises and a conclusion. When you test an argument form

for validity, you are checking to see whether it has any counterexamples.

The concepts of tautology, contradiction, and contingency are applied to sin-

gle statement forms and concern the kind of truth tables the forms have. A form

with all T's in its final table, for instance, is a tautology.

The concepts of logical implication and logical equivalence apply to pairs of

statement forms. We say that one form logically implies another or is logically

equivalent to another. (We can also apply the concept of equivalence to more than

two formulas.)

Finally, the concept of consistency applies to any set offormulas.

You should never say, then, that a single statement form is valid, since valid-

ity is a property of argument forms (or arguments) only; it doesn't even make

sense to apply it to single statement forms. Nor would it make sense to say that an

argument form is a tautology, since argument forms have several formulas in

them. There are, however, some very interesting relationships between these con-

cepts, and to thoroughly understand the concepts, you need to understand these re-

lationships. We will first discuss the way in which validity is related to the other

truth table concepts.

A very interesting fact about argument forms is that if the premises are in-

consistent the argument form is valid (not invalid)! A closely related property is

that if the conclusion of an argument form is a tautology, then the argument form is

also valid. Why should this be so? Well, in both cases the reason is that there can-

not possibly be a counterexample. In the first case, if the premises are inconsistent,Unit 6 Further Applications of the Truth Table Method 107

this means by definition that there can be no row in which they are all true at once.

But if there can be no row in which they are all true, there can certainly be no row

in which they are all true and the conclusion is false; thus there is no counterex-

ample, and so the argument form is valid. The same sort of reasoning, turned

around a little, goes for the second case as well. If the conclusion is a tautology,

there is no row in which it is false; but if there is no row in which the conclusion is

false, then there is no row in which the conclusion is false and the premises are all

true. Again, there is no counterexample, and thus the argument form is valid.

There is a slightly more complicated relationship between the concepts of

logical implication and validity: if the conjunction of the premises logically im-

plies the conclusion, then the argument form is valid. The reason for this is found

simply in the definition of validity. If the conjunction of the premises logically im-

plies the conclusion, then there is no row in which the conjunction of the premises

is true, but the conclusion false. Since the conjunction of the premises can only be

true if all the premises are true in that row, this means that there is no row in which

all the premises are true, but the conclusion false.

We have already seen that two forms are logically equivalent if and only if

the biconditional they form is a tautology. Similarly, formula 1 logically implies

formula 2 if and only if the conditional with formula 1 as antecedent and for-

mula 2 as consequent is a tautology. This is because in both cases there is no row

in the truth table in which the first formula is true and the second false.

Keep in mind that one of your definitions of logical equivalence is in terms

of a dual logical implication. Other interesting facts about logical equivalence are

the following: any two contradictions are logically equivalent, and any two tau-

tologies are logically equivalent. This is evident, since all contradictions have

identical truth tables (all false), and the same holds for tautologies (all true). You

may be able to think of even more ways in which these concepts are related. If so,

and if you understand these, you can be reasonably sure you have mastered the

material of these last two units.

1. 2. DEFINITIONS

A tautology is a single statement form that is true for every substitution in-

stance; that is, it comes out true under the major operator for every row in the

truth table. I

A contradiction is a single statement form that is false for every substitution in-

stance; that is, it comes out false under the major operator for every row in the

truth table.

IBy extension, we can say that a statement (instance) is a tautology if its form is a tautology. The

other definitions here can be extended to statements in the same way.108 3. 4. 5. 6. 7. 8. 9. Unit 6 Further Applications of the Truth Table Method

A contingency is a single statement form that is false for some substitution in-

stances and true for others; that is, it has both T's and F's in its truth table under

the major operator.

Two (or more) statement forms are logically equivalent if and only if their truth

tables are identical under their major operators.

One statement form logically implies another if and only if there is no row in

their joint truth table in which the first comes out true and the second comes out

false.

Two statement forms are logically equivalent if and only if they logically imply

each other.

Two statement forms are logically eqnivalent if and only if the result of joining

them with a biconditional is a tautology.

A set of statement forms is inconsistent if and only if there is no row in their

joint truth table in which they all come out true at once.

A set of statement forms is consistent if and only if there is a row in their joint

truth table in which they all come out true at once.

1. 2. 3. 4. 5. 6. STUDY QUESTIONS

Why is an argument form with inconsistent premises valid rather than invalid?

What is the relationship between validity and logical implication?

Why do all three definitions of logical equivalence amount to the same thing?

Can one say that a statement form is valid? Why or why not?

What are the four sorts of truth table problems you have learned?

What are some of the relationships between the various truth table concepts you

have learned in the last two units?