UNIT 3 - Textbook
UNIT 3
Computing Truth Values
A. INTRODUCTION
In Unit 1 it was emphasized that an argument form is valid if and only if it has no
instances with true premises and a false conclusion. If we are ever to be able to de-
termine whether or not a form is valid, then, we obviously must be able to tell
whether the premises and conclusions of the instances are true or false, which
means we must be able to determine the truth value of the compound sentence
once we are given the truth values of its component parts. For reasons that need
not concern us now, we will not have to worry about determining the truth values
of the simple sentences; this will be done for us.
In Unit 2 we saw that there are an infinite number of possible sentential op-
erators, out of which we have chosen just five. We will now see that one very good
reason for choosing these five, aside from the fact that they are very common, is
that they have a very special property, which sets them off from a great many other
ways of combining sentences in the English language and which will be of great
importance for our logical purposes. That is, these five operators are all truthfunc-
tional, which means that the truth or falsity of the compound sentences that they
form can always be determined just by knowing the truth or falsity of their compo-
nent parts. Another way of putting this is to say that for these five operators the
truth value of the compound sentence is completely determined by, is afunction of,
the truth values of the component sentences.
Of course, to determine the truth value of a compound sentence given the
truth values of its components, you will have to know the rules of computation for
each of our five sentential operators. These rules will be given by means of
schematic truth tables for the operators, which will indicate, for each possible
3334 Unit 3 Computing Truth Values
combination of truth values for the components of the formula, what truth value
must be assigned to the compound. Once you learn the truth tables for each opera-
tor, you will be able, given the truth values of the elementary components, to com-
pute the truth value of compound formulas by working your way up from smaller
subformulas to larger ones.
In this unit, then, you will be learning the meaning of the term truth functional
(and what it means for an operator not to be truth functional), the truth tables for the
five operators, and how to compute the truth value of a compound formula given the
truth values of the components. What you will need to know is stated more explicitly
in the "Objectives" section below.
B. UNIT 3 OBJECTIVES
• Memorize the truth tables for the five sentential operators and be able to
state informally the computation rules for each operator.
• Be able to compute the truth value of compound sentences of any degree
of complexity, given the truth values of the simple sentences they contain.
• Learn the definition of truth functional.
• Be able to show that an operator is not truth functional.
C. UNIT 3 TOPICS
1. Truth Tables for the Operators
As noted above, a very important property of the operators we will be using
is that they are all truth functional, that is, the truth value of the compound that
they form can be determined solely by the truth values of their components. This
means that there will be a rule telling us exactly what the value of the compound
must be for each combination of values for the components. The rule for conjunc-
tion, for instance, is that a conjunction will be true only if both conjuncts are true,
and thus, if one or both of the conjuncts is false, the entire conjunction must be
counted false. Such rules, as we shall see, can be given more formally by means of
a little truth table for each operator, which will list systematically all possible
combinations of truth values for the components and the result of the computation
for each of these possibilities.
One of the very important presuppositions of this procedure is that a sen-
tence must be either true or false. We noted in the Introduction to Unit 2 that in el-
ementary logic we will be dealing only with declarative sentences; we now go a
little further to say that (1) our sentences must have a truth value (they cannot be
indeterminate) and (2) the truth value must be either true or false (we will have no
such value as "nearly true"). What we have, then, is a two-valued logic, whichUnit 3 Computing Truth Values 35
simply means that whatever sentences we use in our logical operations must have
one of the truth values true or false.
It should be mentioned that there are logical systems (many-valued logics)
that investigate the logical properties of sentences that have three possible values
(for instance, true, false, and undetermined), four possible values (for instance,
necessarily true, contingently true, contingently false, and necessarily false), or
even more. But these are specialized disciplines, and standard logic assumes, as
noted, that the sentences it uses are simply true or simply false.
Given that we have a two-valued logic, that each sentence is either true or
false, we can draw up a list of all possible combinations of truth values for the sim-
ple components of a compound formula. l In the formula «A :::J B) • (A v B)), for ex-
ample, we know that A must be true or false, and that B may be either true or false
when A is true, and may also be true or false when A is false. (Note that another very
important assumption we will be making is that the truth values of the simple sen-
tences are independent of each other.) We may indicate these four possibilities in a
little schematic table; each of the four possibilities will be referred to as a row in the
truth table.
A B
1.
T
T
2.
T
F
3.
F
T
4.
F
F
The fact that we can systematically list all these combinations is one of the things
that makes it possible to give rules for the computation of our compound formulas.
We can say, for each combination, what the value of the compound must be, and,
since these are the only possible combinations given that we have a two-valued
logic, we will have stated a complete rule. The rules of computation for the five
operators are given below.
a. "and." The rule for computing the truth value of conjunctions is just
what you would expect given the meaning of "and." When we say "p and q" we
mean to assert that both p and q are true, so the first row is the only row in which
the conjunction will be considered true. Going through each of the possibilities, if
lIt is also possible in a three-valued, four-valued, or any finite-valued logic to list all possible com-
binations. However, it is more cumbersome for the higher-valued logics, since the number of possi-
bilities gets very large very fast. In a four-valued logic, for example, there are 16 possibilities
(instead of 4) for 2 different sentence letters, and 64 different combinations (instead of 8) for 3.36 Unit 3 Computing Truth Values
p and q are both true, then (p • q) is true; if p is true and q is false, (p • q) is false; if
p is false and q is true, (p • q) is false; and if p and q are both false, then (p • q) is
false. We can give the rule for computing the value of conjunctions by means of
the following truth table for "and":
p q (p. q)
T T T We can summarize this table by saying that a con-
T F F junction is to be counted true in only one case:
where both conjuncts are true. If one or the other,
F T F
or both, of the conjuncts is false, the conjunction
F F F will be false.
Note that we use the variables p and q here. This indicates that the conjuncts
may be any formulas, simple or complex. One instance of (p' q) is the more com-
plex formula «(A v B) :::J C) . ~(D v E)); this will be true only if «A v B) :::J C) is
true and also ~ (D v E) is true.
b. "or." The truth table for disjunction will be very nearly what we would
expect from the meaning of the English "or"; the only question is what happens in
the top row, where p and q are both true. It is easy to say what must happen in the
last three rows; if one disjunct is true and one is false, the disjunction as a whole
will be true, and if both are false, the disjunction will be false.
What about the top row, however? What happens if both p and q are true?
Here we need to distinguish between inclusive disjunction and exclusive disjunc-
tion. Inclusive disjunction allows the possibility that both disjuncts are true, while
exclusive disjunction rules this out and declares that only one or the other, but not
both, of the disjuncts is true. Thus, in inclusive disjunction the top row in the truth
table comes out true, whereas in exclusive disjunction it comes out false. These
are two different operators, and we must make a decision as to which one we will
use here.
Both senses of "or" occur in English. In many cases it is clear that the inclusive
sense is intended, since we count the sentence true when both of the disjuncts are
true. One example would be "Ronald Reagan was either a movie actor or a politi-
cian." Even though he was both, we would still count the sentence true. Or if we see
Mary at a university function we might say "Mary is either a student or the wife of a
student," and of course she might very well be both, in which case we would still
count the disjunction as true. Instances of exclusive disjunction would be "Either
coffee or tea is included in the price of the meal" (where it is clear that you would
pay extra if you wanted both), and "John is married to either Josephine or Carolyn."
It is the inclusive sense of disjunction, in which the top row of the truth table
comes out true, that we will be using in this book. Its computation rule is given by
means of the following truth tablefor "or":Unit 3 Computing Truth Values 37
p q (p V q)
We can summarize this table by saying that an in-
T T T
clusive "or" is to be counted false in only one case:
T F T where both disjuncts are false. If one or the other,
F T T or both, of the disjuncts is true, the disjunction will
F F F be considered true.
The exclusive sense of "or" is also a truth-functional operator, and its truth
table is the same as that for the inclusive sense, except that the top row comes out
false instead of true. It is sometimes symbolized with a bar above the wedge: v.
However, we will not be including this as one of our operators because we can al-
ways say the same thing by combining some of our other operators. If we want to
say, for instance, that either John or Bob will be promoted, but not both, we can
simply conjoin the "not both" phrase to the inclusive disjunction. This would give
us the following symbolization: ((l v B)· ~(l· B». In general, the exclusive
sense of disjunction, where p and q are the two disjuncts, can always be symbol-
ized as ((p v q) • ~ (p • q», and so a separate operator is not needed. In this book,
then, we will be using (exclusively) the inclusive sense of "or," and this means that
whenever you are computing the value of a disjunction, the top row of the truth
table, where both of the disjuncts are true, will always turn out to be true.
c. ~~if and only if." A biconditional statement says that one thing happens if
and only if (in just the same circumstances in which) another takes place. An in-
stance of this would be "Mary will be elected student body president if and only if
Fred is elected treasurer," which could be symbolized as (M == F). (Note that the
biconditional, as the name indicates, is a two-way conditional; the sentence above
asserts both that if Mary is elected student body president, then Fred will be elected
treasurer, and that if Fred is elected, then Mary will be elected.) Now, what should be
the truth table for this operator? Suppose both Fred and Mary are elected; then both
M and F are true, which would correspond to the top row in our truth table. Surely
we would then consider the biconditional (M == F) to be true. What if they are both
defeated? This would correspond to the last row of the table, in which both M and F
are false. Since the biconditional says that one will be elected if and only if the other
is elected, or, roughly, that they will stand or fall together, it will turn out to be true
when both components are false. (In this case, they have fallen together.) Thus the
top and bottom rows of the truth table, where the truth values of the components are
the same, will be considered true. What about the middle two rows, in which the
truth values of the components are different? These would correspond to the cases in
which one of the candidates was elected and the other defeated. But in these cases
we would surely say that the biconditional, which asserted that one would be elected
if and only if the other was elected, was false, and this is how we complete the truth
table. The truth table for the biconditional, then, will be as follows:38 Unit3 Computing Truth Values
p q (p == q) We can summarize this truth table by saying that if
T T T the truth values of the components are the same, as
T F F in the first and fourth rows, the biconditional will
F T F be true, and if they are different, as in the second
and third rows, the biconditional will be false.
F F T
d. "not." The truth table for negation will contain only two rows instead of
four, because the negation sign is placed in front of a single formula (which may
be complex as well as simple) instead of joining two formulas together. A single
formula can be only true or false, so we need consider only two cases: we must say
what happens to ~ p when p is true and what happens to ~ p when p is false. The
truth table for negation is just what you would expect: if p is true, then ~p is false,
and if p is false, ~p turns out to be true. This is schematized in the following truth
table for negation:
p ~p
T F
F T
This can be summarized by saying that the nega-
tion of p will have the truth value opposite that
ofp.
A negation, in other words, simply reverses the truth value of the formula
being negated. This is why double negations, such as ~ ~ p, cancel each other out.
The inner negation reverses the truth value once, and the outer negation reverses it
again, right back to where it started. (This is why your elementary school teachers
always warned you not to say things like "I ain't got no bad habits." The two nega-
tions cancel each other out, so that the sentence literally, although probably not
colloquially, means "I do have some bad habits.")
e. "if-then. " We have left the conditional until last because its truth table
is the least intuitive and the most difficult to justify of any of our five operators.
We can begin by noting, however, that if the antecedent of a conditional is true
and its consequent is false, the conditional will always be false. If we had pre-
dicted in 2000, for instance, that if unemployment rates remained low, then the
Democrats would be reelected in a landslide, the prediction would now be seen
to be false, since unemployment rates did remain low (the antecedent is true),
but the Democrats were not reelected in a landslide (the consequent is false.)
Again, the conditional (p ::J q) will be false whenever p is true and q is false.
What happens if p and q are both true? Suppose we had said in 1984, "If Reagan
is elected, the stock market will climb." Since Reagan was elected and the stock
market did climb, we would count this sentence as true. This example corre-
sponds to the top row in the truth table, and the previous example corresponds to
the second row.Unit 3 Computing Truth Values 39
What happens when p is false, however? Suppose we let B stand for the false
sentence "Bush was reelected in 1992," M stand for "There was an invasion from
Mars in 1993," and C stand for "Christmas comes but once a year." What would be
the truth values of (B =:J C) ("If Bush was reelected in 1992, then Christmas comes
but once a year"), which would correspond to the third row of our truth table, and
(B =:J M) ("If Bush was reelected in 1992, then there was a Martian invasion in
1993"), which would correspond to the fourth row? How do we decide cases like
this, in which we can never observe the antecedent condition (in this case, Bush
being reelected in 1992)? There seems to be no way of judging. Classicallogi-
cians, whose lead we will be following in this book, dispose of the matter rather
neatly simply by declaring that any conditional with a false antecedent will be
counted true. If the antecedent is true, as we have seen earlier, then the condition-
al will be true or false according to the truth value of the consequent. The truth
table for the conditional, then, will look like this:
p q (p =:J q) This table can be summarized by saying that the
T T T only time a conditional is to be considered false is
T F F when the antecedent is true and the consequent is
F T T false. Whenever the antecedent is false, the condi-
F F T tional is true, and whenever the consequent is true,
the conditional is true.
This table has some very odd consequences. For instance, since the condi-
tional will be true whenever the antecedent is false, we will have to say, if the
"if-then" is taken as the horseshoe, that all of the following are true:
If cats speak French, they produce great novels.
If Lincoln was president in 1993, then dinosaurs were discovered alive and well in
South Dakota.
If a woman landed on the moon in 1963, she discovered highly intelligent moon men and
produced one of their offspring, who is now masquerading as the Queen of England.
If no one landed on the moon in 1969, then California experienced a severe earth-
quake in 1990 and has dropped into the Pacific Ocean.
If we have found intelligent beings on Mars, then 7T is greater than 3.9.
It also follows from the truth table that whenever the consequent of a condi-
tional is true (the first and third rows) the conditional as a whole is true, whether or
not there is any connection between antecedent and consequent. This means that
all the following sentences must also be counted as true:
If my cat sleeps a lot, then 5 + 7 = 12.
If cats speak Farsi, then men landed on the moon in 1969.
If Jupiter is a planet, then cats sleep a lot.40 Unit 3 Computing Truth Values
What is perhaps most disconcerting about this truth table is that conditionals
that seem to contradict each other both have to be considered true, provided the
antecedents are false. For instance, all of the following are true:
If an atomic bomb was dropped on New York City in May 2001, then millions of
people were killed.
If an atomic bomb was dropped on New York City in May 2001, then no one was
hurt and most people benefited greatly from the strong dose of radiation.
If cats speak all Western European languages, then they speak French.
If cats speak all Western European languages, then they do not speak French.
At this point, you may very well wonder what gives logicians the right to
make up such rules as "whenever the antecedent of a conditional is false, the con-
ditional as a whole is true," especially when they lead to such odd results. Isn't it
obviously false rather than true that if a bomb was dropped on New York, then no
one was hurt and everyone benefited? A satisfactory reply to this sort of perplexity
would take us far afield into the philosophy of logic, so just a few observations
will have to suffice here.
The most important thing to keep in mind is that the logical operator, the
horseshoe, is not the same as the ordinary English "if-then." The logical operator,
which is sometimes called the material conditional, is a kind of "minimal"
if-then, which captures only the common logical content ofthe various uses of the
English "if-then." This common logical content, which is reflected in the truth
table for the horseshoe, is simply that it is not the case that the antecedent is true
and the consequent false. The horseshoe is a very weak operator, which says noth-
ing more than is indicated in its truth table. Thus, the appearance of paradox in the
previous examples vanishes; when we keep in mind that the "if-then" is intended
to be simply the material conditional, we may properly claim both that if a bomb
was dropped, then millions were killed and that if a bomb was dropped, then no
one was hurt and everyone benefited. Both conditionals are true simply because in
neither case do we have a true antecedent with a false consequent, since in both
cases the antecedent is false. One way to think of it might be the following: since
the only time the conditional is false is when the antecedent is true and the conse-
quent is false, if we have a false antecedent, then we have not said anything false.
But if a sentence is not false, then we must count it as true, since this is a two-
valued logic. Thus, a sentence with a false antecedent must be true.
One possibility that may have occurred to you is that we might simply leave
the third and fourth rows blank in the truth table for the horseshoe. There is one
very good reason, however, for filling in those blanks by making a decision one
way or the other. We want to be able to determine the truth values of compound
sentences in all cases, so that we can always tell whether an argument is valid or
P q l. (p ~ q) 2. (p ~ q) 3. (p ~ q) 4. (p ~ q) | ||||
T | T T | T | T | |
T | F F | F | F | |
F T [i] [:J m [i] F F |
Unit 3 Computing Truth Values 41
not. If we left blanks, many of our sentences would have unknown truth values,
and so we could never determine whether the argument forms containing them had
instances with true premises and a false conclusion. We want a complete decision
procedure for sentential logic, and we can have this only if all our truth tables are
complete. We want all our operators, then, to have rules determining in every case
what the truth value of the compound will be.
Given that we want complete truth tables for our operators, it turns out that
the rule we have given for the horseshoe is the only acceptable candidate. It is fairly
clear that the top two rows must be T and F, respectively. For the bottom two rows,
there are only four possibilities: they may both be true, as in the truth table we
have; they may both be false; the third row could be true with the fourth false; or
the fourth row could be true with the third false. We can list these possibilities, in-
cluding the top two rows as well and outlining the bottom two rows, as follows:
OUR VERSION OTHER POSSIBLE VERSIONS
T
F
Notice that we could not use the second version for the horseshoe because that is
the truth table for conjunction, and we certainly want "if p then q" to mean some-
thing different from-be true in different circumstances than-"p and q." Similar-
ly, we could not use the fourth version because that is the truth table for the
biconditional. Finally, if we compare the third version with the list of possibilities
on the far left, we see that it simply represents q, and we certainly want "if p then
q" to mean something different from just q. Thus, ifwe are going to have a truth-
functional, two-valued logic in which all operators have complete rules for com-
putation, the only possible truth table for the horseshoe is the one we have given.
One final comment on the use of the horseshoe. There are alternative logics
that use a stronger "if-then" operator. Many logicians claim that these stronger op-
erators are closer to our English "if-then" and are thus better candidates for an ac-
curate system of logic. We will make no judgment on this point, except to say that
it may well be true and that these stronger systems of logic deserve close study.
Not, however, in an elementary textbook. The logical systems with stronger
"if-then" operators are considerably more complex than our truth-functional ver-
sion, and even if you eventually decide that one of them is more nearly correct,
you will probably not be able to understand it unless you have first thoroughly
mastered the simpler system. If nothing else, then, we could justify the use of the42 Unit 3 Computing Truth Values
horseshoe in elementary logic on the grounds that it is the most easily understood
version of "if-then" and is thus the most suitable for an introduction to logic.
2. Computing Truth Values
By now you should have memorized the truth tables for the five operators
and should be able to state informally their computation rules (for instance, that a
disjunction is false only if both disjuncts are false, and is otherwise true). You are
now in a position to see how it is possible to compute the truth value of any com-
pound formula given the truth values of the simple component sentences, and that
will be the topic of this section. As in the exercises at the end of the unit, we will
here assume that A, B, andC are true, while X, Y, and Z are false. We will also
adopt the convention of occasionally dropping the outside parentheses on our for-
mulas if this makes them easier to read, since no ambiguity results as long as we
do not use such a formula as a part of another. In longer formulas, there will be
enough parentheses as it is, without including the outermost pair.
We will begin our computations with a fairly simple example. How, for in-
stance, would we compute the truth value of ((~ A v ~ B) ::J ~C)? Since we are
counting A, B, and C as true, ~ A, ~ B, and ~ C will all be false. The disjunction
(~A v ~B) will then be false, since both the disjuncts are false. Both antecedent
and consequent, then, are false, and by consulting the truth table for" ::J ," we see
that F ::J F turns out to be true. The truth value for the whole, then, is true. We
could represent the computation for the formula above in the following way:
T T T
( ~ A v ~ B) ::J ~ C
F F F
~I I
F~J
T
It will help to adopt the following conventions: We will place the truth values
of the simple sentences immediately above the sentence letters. We will place the
truth values of the various subformulas (formulas that occur as a part of a larger for-
mula) immediately below the major operator for that subformula. (These may be re-
peated, connected by dotted lines, in order to clarify what formulas are being used in
the later computations.) The arrows indicate how the truth values of the subformulas
"feed into" the computation of the value of the next-largest formula. Our computation
procedure will be to fIrst determine the truth values of the smallest subformulas, thenUnit 3 Computing Truth Values 43
use these to compute the values of the next-largest subformulas, and so on, until we
reach the value of the sentence as a whole. Another example will illustrate this pro-
cedure for a slightly more complex formula: ~ ((A Z) ::J (~A ~ Z)).
T F T F
~((A Z) ::J (~A ~ Z))
I i J J
T F F T
\1 ~/
F F
~/
/
T
Since A is true, ~ A is false, and since Z
is false, ~ Z is true.
A Z and ~A ~ Z are both false,
since in both cases, one side is true and
the other false.
The conditional is true, since both an-
tecedent and consequent are false.
The formula as a whole, the negation of
the conditional, is false, since the condi-
tional is true.
The following formula is even more complex:
T T F T F T
~ ((A v B) ::J Z) v ((B v Z) ::J ~ A)
\/ i \J I
T F T F
~ "V
~/ /F
Since A and B are true, (AvB) and (BvZ)
are true, and ~ A is false.
Both conditionals are false, since in
both, antecedent is true and consequent
is false.
The disjunction is false since both dis-
juncts are false.
The formula as a whole, that is, the
negated disjunction, is true, since the
disjunction is false.
Negations can be confusing, and it is essential that you understand what
formula is affected by, that is, what is the scope of, the negation operator. The tilde
will operate on, or negate (and thus reverse the truth value of), the first complete
formula following it; this will be indicated by the use of parentheses. If there are44 Unit 3 Computing Truth Values
no parentheses immediately following the tilde, then it negates only the sentence
letter immediately following. In
(~A v B), for instance, we would be negating only
A. The sentence would be read "Either not A or else B" and would be true since B
is true. If the tilde is followed by a parenthesis, as in ~(A v B), then it negates the
fonnula contained between the left parenthesis immediately following it and the
right parenthesis that is paired with it. In
~(A v B), for instance, the tilde negates
the whole fonnula (A v B), so the sentence would be read "It is not the case that ei-
ther A or B" or, in other words, "Neither A nor B," and would be false since (A v B)
is true. In the more complex fonnula ~[(~ A == (B v C)), ~(X ::::> (2 v Y))], we
have three tildes, all with different scopes. Reading from left to right, the first tilde
negates the entire fonnula (and hence will be the last operator to be computed), the
second negates only the A, and the third negates the conditional (X::::> (2 v Y)).
The computation would be as follows:
T T T F F F
~ [(~A=(BvC)) '~(X::::>(2vY))]
I \1 i \ I S· A B d C A ·11 b
F T F F mce" an are true, ~ WI e
false and (B v C) will be true. Since X,
Y, and 2 are false, (2 v Y) will be false.
\ / \/
F T The biconditional will be false and the
~ / conditional will be true.
~ The conjunction will be false, since one
T~F conjunct is false.
The major operator of this fonnula is
the first negation. This will be true,
since the conjunction, which it negates, is
false.
If you do not understand the results of the computations above, go back and review
the truth tables for the operators. You will need to know them very thoroughly so that
you can compute the results for the subfonnulas with a minimum of effort. Remem-
ber that the procedure must always be to work from the inside out, to start with the
smallest subfonnulas and work your way up step by step to the larger ones. Exercise
1, at the end of the unit, will give you practice in this sort of computation.
There are some shortcuts you can use if you know your truth table rules well.
For instance, since it takes only one true disjunct to make a disjunction true, if you
find that one side is true, you need make no other calculations, but can conclude
without further ado that the entire disjunction is true. Similarly, if you find oneUnit 3 Computing Truth Values 4S
conjunct false, you may conclude that the conjunction as a whole is false. And if
either the antecedent is false or the consequent is true in a conditional, you may
conclude that the conditional itself is true. Given that A and B are true, and X,
Y, and Z are false, for instance, we know almost immediately that
(~(A v B) ::J (~Z ::J ~Y» v ((Z' A) v B) is true, since B is true and this makes
the second disjunct true. Similarly, we know that ~A ::J [((Z v y). (A v ~ B» ==
((A v ~Z) v (X v B»] is true as soon as we see that ~ A is false, since a false an-
tecedent always yields a true conditional. We need not bother at all with the very
complex consequent! A summary of these shortcut rules is as follows:
If one disjunct is true, the entire disjunction is true.
If one conjunct is false, the entire conjunction is false.
If the antecedent is false, the conditional is true.
If the consequent is true, the conditional is true.
Knowing these shortcuts will considerably facilitate your work when it comes to
truth tables in Units 5 and 6. Exercise 2 at the end of the unit will give you practice
in using these shortcuts.
3. Truth-functional Operators
The concept of truth functionality was discussed briefly in the Introduction
to this unit. Now that you know the truth tables for the five operators and are able
to do the computations, you are in a position to understand this concept in a little
more depth. As we noted in the introduction, what it means for an operator to be
truth functional is that the truth value of the compound that it forms is completely
determined by the truth values of the component parts. There are other ways in
which this could be stated. We might say, for instance, that there are rules that tell
us what the value of the compound must be, given the value of the components,
or we could say that the truth value of the compound is a function solely of the
truth values of the components. This also means that the truth value of the com-
pounds will be the same whenever the truth values of their respective components
are the same. That is, identical truth-value input yields identical truth-value out-
put. However we put it, the important thing to remember is that it is the truth val-
ues only that determine the value of the compound and not, for instance, the
meanings of or the relations between the sentences. All we need to know is the
truth values of the components in order to determine the value of the compound.
As we shall see in the next section, there are many operators, perhaps most, for
which this is not the case.
In the system of logic we will be using, all our operators are truth functional,
which means that we have a truth-functional logic. A system of logic is truth func-
tional if and only if each of its operators is truth functional. There are many systems of46 Unit 3 Computing Truth Values
logic in which some operators are truth functional and others are not, for instance,
systems of modal logic, which explore the concepts of possibility and necessity.
Such systems are not considered to be truth functional even though some of the op-
erators have this property; to be truth functional, every operator in the logical system
must have a complete rule for determining the truth value of the compounds.
4. Non-truth-functional Operators
You will have a better grasp of what it means for an operator to be truth func-
tional if you understand, by contrast, what it means for an operator not to have this
property. A non-truth-functional operator is by definition, of course, simply an op-
erator for which we cannot determine the truth value of a compound given the
truth values of the components. Some examples of non-truth-functional operators
are "John believes that ," "it is possible that ," "it is necessary that __ ,"
" because ," and " , after ." In fact, most of the operators in
general use in English are non-truth-functional. With none of these operators is it
possible, given just the truth values of the components, to determine the truth
value of the whole. Something else is needed as well, some outside information; in
the case of "after," for instance, we would need the times at which the stated ac-
tions took place.
A typical example of a non-truth-functional operator involves the concept of
belief. We cannot determine the truth value of "John believes that __ " just by
knowing the truth value of the sentence that goes into the blank. That a sentence is
true does not guarantee that John (or any of us, unfortunately) believes it; that is, it
does not determine the value of the compound sentence. Nor does the fact that a
sentence is false guarantee that John, or we, won't believe it. We all believe all
sorts of false things (although we may not believe that we do). There is, in short,
no rule that determines the truth value of sentences of the form "x believes that
__ ," given only the truth values of the component sentences. Of course, we
may determine the truth values of belief sentences by other means; we might, for
instance, simply ask people what they believe. What we cannot do is to determine
whether they believe something simply on the basis of whether or not it is true.
Another example, and one that raises some very interesting philosophical
questions, is "because." It is particularly important not to confuse this strong non-
truth-functional operator with our weak truth-functional "if-then." With "be-
cause," unlike our "if-then," the truth value of the components does not
determine the truth value of the compound. That two sentences are true, for in-
stance, does not guarantee that the corresponding "because" statement is true (al-
though it would guarantee the truth of the material conditional). It is true, for
instance, that it rained in Moorhead, Minnesota, in October 2000, and it is alsoUnit 3 Computing Truth Values 47
true that the Yankees won the World Series in 2000. But it would be absurd, or
worse, to suppose that the Yankees won the World Series because it rained in
Moorhead in October.2
We need to be more precise, however, in what it means to say that an opera-
tor such as "because" is not truth functional, and in particular we need to show
how we can demonstrate that it is not. To show that an operator is not truth func-
tional, we need to be able to show that the truth values of the components do not
determine the truth value of the compound. But how do we do this? We could
begin by noting that to sayan operator is truth functional-to say that the value of
the compound is determined solely by the values of the components-is to say,
among other things, that, given the same truth values for two different sets of com-
ponents, you will always get the same end result. Identical input implies identical
output. "Gore was president in 2001," for instance, is false, and so is "Toyotas are
made in Liechtenstein." If we negate the two sentences, using a truth-functional
operator, we get the same results in both cases; both the negated sentences "Gore
was not president in 2001" and "Toyotas are not made in Liechtenstein" are true.
This suggests that if we could come up with pairs of component sentences with
identical truth values and show that the results of compounding them were
different, this would be a demonstration of non-truth-functionality, since it would
show that identical truth-value input does not imply the same truth-value output,
and thus that the value of the compound is not a function solely of the truth values
of the components. This is exactly what we do to show that an operator is not truth
functional. We will use this method to demonstrate the non-truth-functionality of
"because."
It is true that much ofthe state of Washington was covered with ash on May 20,
1980. It is also true that Mt. St. Helens erupted in the middle of May 1980. And it
is obviously true that the state was covered with ash because the volcano erupted.
In this case, then, we have "T because T," resulting in T. Another pair of true sen-
tences is "Toyotas are made in Japan" and "Lincoln was president during the Civil
War." It is safe to say, however, that the sentence "Toyotas are made in Japan
because Lincoln was president during the Civil War" is false (and even safer to say
that "Lincoln was president during the Civil War because Toyotas are made in
2In less silly cases, however, people do sometimes make the mistake of thinking that because two
statements are true, especially if one event happened after another, that one thing caused another. The
error is so common, in fact, that it has been given a special name: post hoc ergo propter hoc (after the
thing, therefore because of the thing). This kind of error is probably the source of all sorts of super-
stitions, such as the belief in the efficacy of the rain dance. No doubt in some cases the dance was
done and rain followed; thus it was believed, fallaciously, that the rain occurred because of the rain
dance. The question of when (or even if) causal statements are justified is one of the most interesting
questions in the philosophy of science.48 Unit 3 Computing Truth Values
Japan" is false). In this case we have "T because T" coming out false. What these
four sentences show is that the same truth-value input does not yield the same
truth-value output for the "because" operator, since in one case two true compo-
nents yield a true compound, while in the other case two true components yield
a false compound. This shows that the truth value of the output is not a func-
tion solely of the truth values of the input; therefore, "because" is not a truth-
functional operator.
It is even easier to show that the operator " before " is not truth
functional. The following sentences are all true: "Washington was a U.S. presi-
dent," "Lincoln was a U.S. president," and "Truman was a U.S. president." If we
put the first two sentences, in that order, into the blanks, we get the true com-
pound sentence "Washington was a U.S. president before Lincoln was a U.S.
president." If we put the sentence about Truman in the first blank, however, fol-
lowed by the one about Lincoln, we get the false compound sentence "Truman
was a U.S. president before Lincoln was a U.S. president." Thus, identical truth-
value input yields different truth-value outputs, so the "before" operator is not
truth functional. Now try, just for the fun of it, to answer the following question:
could you ever yourself show that "I believe that __ " is not truth functional?
Why or why not?
In summary, in our system of logic we will be using five operators, all of
which are truth functional. The advantage of a truth-functional sentential logic is
that it makes it possible to determine the validity of any given argument form,
since we can always find out for any of its instances whether or not it has true
premises with a false conclusion. There are a great many operators in English,
however, that are not truth functional, including causal operators, temporal opera-
tors, possibility and necessity operators, and operators using terms such as "be-
lieves" and "hopes." To show that an operator is truth functional, we need to come
up with a rule for computing the truth value of the compound given the truth val-
ues of the components. This we have done for all our operators by using the little
truth tables. To show that an operator is not truth functional, we need to come up
with examples that show that the same truth values for the components may result
in different truth values for the compound.
1. 2. DEFINITIONS
An operator is truth functional if and only if the truth value of the compound
that it forms is completely determined by the truth values of the component
parts.
A system of logic is truth functional if and only if each operator of that system
is truth functional.1. 2. 3. 4. 5. 6. 7. Unit 3 Computing Truth Values 49
STUDY QUESTIONS
Write down the truth tables for the five operators we will be using.
State informally the rules for computing the values of our five operators.
What does it mean for a logic to be two valued?
What is the advantage of using a two-valued (or at least a finite-valued) logic?
Give an example of a truth-functional operator other than the ones we will be
using, and write down the truth table for it. (You may make one up if you wish.)
What does it mean for an operator not to be truth functional? Give two or three
examples of your own.
What are the advantages and disadvantages of using material implication, the
horseshoe, as our "if-then" operator?
EXERCISES
1. Compute the truth values of the following, given that A, B, and C are true and X, Y, and
Z are false.
*a. ~A v ~B
b. ~A::J ~C
*c. (A . X) v (B· Y)
d. (X v A) ::J~B
*e. ~B::J ~(X v Y)
f. ~ (A v B) ::J (X v Y)
*g. (A B) (Z == X)
h. (A X) (B == Y)
*i. (X::J (Y ::J Z» ::J ((Z ::J X) ::J Y)
j. (~X· ~Y) == (A ::J~(X· Y»
*k. ~((A v B) ::J C) ::J ((X v Y) ::J Z)
l. ~((A v B) v C) v (C ::J ~(X v ~ A»
*m. ~[A::J (~A ::J (~B v X))]
n. ~[((A::J ~B) ::J~C) ::J~X]
*0. ~(~A v ~(B v ~(C v ~ X»)
p. ~~(A· ~(B ::J ~(C ::J ~(X v Y»»
*q. ((X· A) v ~(X • B» ::J ((A v X)· Y)
r. ((X· Y) v (A· ~B» ::J ((X v A)· (Y v~B»
*s. ~ [~(A v ~B) v ~(~A v X)] ::J ~(X v A)
t. [(((A::J B) ::J X) ::J Z) ::J Y] ::J [((X ::J Z) ::J A) ::J X]50 Unit 3 Computing Truth Values
2. Compute the truth values of the following, using the values given in Exercise 1, but
without being given the values of G, H, or f.
*a. (G v H) ·~A
b. (B v G)· (A v H)
*c. ~~Av(X·(G:J~A))
d. (A v B) == «G v H) :J A)
*e. ~(A v B) :J ~(G v H)
f. (X v Y) :J «G v A) :J ~(X v Z))
*g. ~(X v ~Y) :J (G == (~H ·f))
h. (G == (H· A)):J «~H v I) :J (X:J Z))
*i. ~ [A v (G == [(H· I) v (X· Y)])l :J (A :J X)
j. ~[(G v (H == ~A))·~Al
3. Given the values from Exercise 2, can you figure out the values of the following? Why
or why not?
*a. (A· G) v (B· H)
b. ~(X v G)
*c. H:J (G :J H)
d. (A v G) • (B v H)
*e. ~ (A v G)
f. H:J (G ·~G)
*g. (H G) :J (A B)
h. (H A) :J (G B)
*. 1. ~(A· G)
j. ~(X· G)
4. Show that the following operators are not truth functional.
*a. after
b. *c. e. It is logically possible that __
It is logically necessary that __