Symbolic Logic
9. It is not the case that Egypt’s food shortage worsens, and Jordan re-
quests more U.S. aid.
*10. It is not the case that either Egypt’s food shortage worsens or Jordan re-
quests more U.S. aid.
11. Either it is not the case that Egypt’s food shortage worsens or Jordan re-
quests more U.S. aid.
12. It is not the case that both Egypt’s food shortage worsens and Jordan re-
quests more U.S. aid.
13. Jordan requests more U.S. aid unless Saudi Arabia buys five hundred
more warplanes.
14. Unless Egypt’s food shortage worsens, Libya raises the price of oil.
*15. Iran won’t raise the price of oil unless Libya does so.
16. Unless both Iran and Libya raise the price of oil neither of them does.
17. Libya raises the price of oil and Egypt’s food shortage worsens.
18. It is not the case that neither Iran nor Libya raises the price of oil.
19. Egypt’s food shortage worsens and Jordan requests more U.S. aid, un-
less both Iran and Libya do not raise the price of oil.
*20. Either Iran raises the price of oil and Egypt’s food shortage worsens, or
it is not the case both that Jordan requests more U.S. aid and that Saudi
Arabia buys five hundred more warplanes.
21. Either Egypt’s food shortage worsens and Saudi Arabia buys five hun-
dred more warplanes, or either Jordan requests more U.S. aid or Libya
raises the price of oil.
22. Saudi Arabia buys five hundred more warplanes, and either Jordan re-
quests more U.S. aid or both Libya and Iran raise the price of oil.
Conditionalstatement
Ahypothetical 23. Either Egypt’s food shortage worsens or Jordan requests more U.S. aid,
statement; a compound but neither Libya nor Iran raises the price of oil.
proposition or statement
of the form “If pthen q.” 24. Egypt’s food shortage worsens, but Saudi Arabia buys five hundred
more warplanes and Libya raises the price of oil.
Antecedent
In a conditional *25. Libya raises the price of oil and Egypt’s food shortage worsens; howev-
statement
er, Saudi Arabia buys five hundred more warplanes and Jordan requests
(“If... then...”), the
more U.S. aid.
component that
immediately follows the
“if.” Sometimes called
the implicansor the 3 Conditional Statements
protasis.
and Material Implication
Consequent
In a conditional
Where two statements are combined by placing the word “if” before the first and
statement
(“If... then...”), the inserting the word “then” between them, the resulting compound statement is a
component that conditional statement (also called a hypothetical, an implication, or an implicative
immediately follows the
statement). In a conditional statement the component statement that follows the
“then.” Sometimes
“if” is called the antecedent (or the implicans or—rarely—the protasis), and the
called the implicate, or
the apodosis. component statement that follows the “then” is the consequent(or the implicate
(cid:22)(cid:20)(cid:27)
Symbolic Logic
or—rarely—the apodosis). For example, “If Mr. Jones is the brakeman’s next-door
neighbor, then Mr. Jones earns exactly three times as much as the brakeman” is a
conditional statement in which “Mr. Jones is the brakeman’s next-door neigh-
bor” is the antecedent and “Mr. Jones earns exactly three times as much as the
brakeman” is the consequent.
A conditional statement asserts that in any case in which its antecedent is
true, its consequent is also true. It does not assert that its antecedent is true, but
only that if its antecedent is true, then its consequent is also true. It does not as-
sert that its consequent is true, but only that its consequent is true if its an-
tecedent is true. The essential meaning of a conditional statement is the
relationship asserted to hold between the antecedent and the consequent, in that
order. To understand the meaning of a conditional statement, then, we must un-
derstand what the relationship of implication is.
Implicationplausibly appears to have more than one meaning. We found it
useful to distinguish different senses of the word “or” before introducing a spe-
cial logical symbol to correspond exactly to a single one of the meanings of the
English word. Had we not done so, the ambiguity of the English would have in-
fected our logical symbolism and prevented it from achieving the clarity and
precision aimed at. It will be equally useful to distinguish the different senses of
“implies” or “if–then” before we introduce a special logical symbol in this con-
nection.
Consider the following four conditional statements, each of which seems to
assert a different type of implication, and to each of which corresponds a differ-
ent sense of “if–then”:
A. If all humans are mortal and Socrates is a human, then Socrates is mortal.
B. If Leslie is a bachelor, then Leslie is unmarried.
C. If this piece of blue litmus paper is placed in acid, then this piece of blue litmus
paper will turn red.
D. If State loses the homecoming game, then I’ll eat my hat.
Even a casual inspection of these four conditional statements reveals that they
are of quite different types. The consequent of A follows logically from its an-
tecedent, whereas the consequent of B follows from its antecedent by the very
definition of the term bachelor, which means “unmarried man.” The consequent
of Cdoes not follow from its antecedent either by logic alone or by the definition
of its terms; the connection must be discovered empirically, because the implica-
tion stated here is causal. Finally, the consequent of D does not follow from its
antecedent either by logic or by definition, nor is there any causal law involved.
Statement D reports a decision of the speaker to behave in the specified way
under the specified circumstances. Implication
These four conditional statements are different in that each asserts a different Therelation that holds
between the antecedent
type of implication between its antecedent and its consequent. But they are not
and the consequent of a
completely different; all assert types of implication. Is there any identifiable com-
true conditional or
mon meaning, any partial meaning that is common to these admittedly different hypothetical statement.
(cid:22)(cid:20)(cid:28)
Symbolic Logic
types of implication, although perhaps not the whole or complete meaning of
any one of them?
The search for a common partial meaning takes on added significance when
we recall our procedure in working out a symbolic representation for the English
word “or.” In that case, we proceeded as follows: First, we emphasized the differ-
ence between the two senses of the word, contrasting inclusive with exclusive
disjunction. The inclusive disjunction of two statements was observed to mean
that at least one of the statements is true, and the exclusive disjunction of two
statements was observed to mean that at least one of the statements is true but
not both are true. Second, we noted that these two types of disjunction had a
common partialmeaning. This partial common meaning—that at least one of the
disjuncts is true—was seen to be the wholemeaning of the weak, inclusive “or,”
and a partof the meaning of the strong, exclusive “or.” We then introduced the
special symbol “ ” to represent this common partial meaning (which is the en-
tire meaning of “or” in its inclusive sense). Third, we noted that the symbol rep-
¡
resenting the common partial meaning is an adequate translation of either sense
of the word “or” for the purpose of retaining the disjunctive syllogism as a valid
form of argument. It was admitted that translating an exclusive “or” into the
symbol “ ” ignores and loses part of the word’s meaning. The part of its mean-
ing that is preserved by this translation is all that is needed for the disjunctive
¡
syllogism to remain a valid form of argument. Because the disjunctive syllogism
is typical of arguments involving disjunction, with which we are concerned here,
this partial translation of the word “or,” which may abstract from its “full” or
“complete” meaning in some cases, is wholly adequate for our present purposes.
Now we wish to proceed in the same way, this time in connection with the
English phrase “if–then.” The first part is already accomplished: We have al-
ready emphasized the differences among four senses of the “if–then” phrase
corresponding to four different types of implication. We are now ready for the
second step, which is to discover a sense that is at least a part of the meaning of
all four types of implication.
We approach this problem by asking: What circumstances suffice to establish
the falsehood of a given conditional statement? Under what circumstances
should we agree that the conditional statement
If this piece of blue litmus paper is placed in that acid solution, then this piece of blue
litmus paper will turn red.
is false? It is important to realize that this conditional does not assert that any
blue litmus paper is actually placed in the solution, or that any litmus paper ac-
tually turns red. It asserts merely that ifthis piece of blue litmus paper is placed
in the solution, thenthis piece of blue litmus paper will turn red. It is proved false
if this piece of blue litmus paper is actually placed in the solution and does not
turn red. The acid test, so to speak, of the falsehood of a conditional statement is
available when its antecedent is true, because if its consequent is false while its
antecedent is true, the conditional itself is thereby proved false.
Any conditional statement, “If pthen q,” is known to be false if the conjunc-
#
tion p 'qis known to be true—that is, if its antecedent is true and its consequent
(cid:22)(cid:21)(cid:19)
Symbolic Logic
is false. For a conditional to be true, then, the indicated conjunction must be false;
#
that is, its negation '(p 'q) must be true. In other words, for any conditional,
#
“If pthen q,” to be true, the statement '(p 'q),which is the negation of the con-
junction of its antecedent with the negation of its consequent, must also be true.
#
We may then regard '(p 'q)as a part of the meaning of “If pthen q.”
Every conditional statement means to deny that its antecedent is true and its
consequent false, but this need not be the whole of its meaning. Aconditional
such as Aon earlier page also asserts a logical connection between its antecedent
and consequent, as B asserts a definitional connection, C a causal connection,
and Da decisional connection. No matter what type of implication is asserted by
a conditional statement, part of its meaning is the negation of the conjunction of
its antecedent with the negation of its consequent.
We now introduce a special symbol to represent this common partial mean-
ing of the “if–then” phrase. We define the new symbol “ )”, called a horseshoe
(other systems employ the symbol “:” to express this relation), by taking p)q
#
as an abbreviation of '(p 'q).The exact significance of the ) symbol can be in-
dicated by means of a truth table:
# #
p q !q p !q !(p !q) p!q
T T F F T T
T F T T F F
F T F F T T
F F T F T T
Here the first two columns are the guide columns; they simply lay out all
possible combinations of truth and falsehood for p and q. The third column is
filled in by reference to the second, the fourth by reference to the first and third,
and the fifth by reference to the fourth; the sixth is identical to the fifth by
definition.
The symbol ) is not to be regarded as denoting themeaning of “if–then,” or
standing for the relation of implication. That would be impossible, for there is no
single meaning of “if–then”; there are several meanings. There is no unique rela-
tion of implication to be thus represented; there are several different implication
relations. Nor is the symbol ) to be regarded as somehow standing for all the Horseshoe
meanings of “if–then.” These are all different, and any attempt to abbreviate all The symbol for material
implication, ".
of them by a single logical symbol would render that symbol ambiguous—as
ambiguous as the English phrase “if–then” or the English word “implication.” Material implication
#
The symbol ) is completely unambiguous. What p)qabbreviates is '(p 'q), Atruth-functional
relation (symbolized by
whose meaning is included in the meanings of each of the various kinds of impli-
the horseshoe, ") that
cations considered but does not constitute the entire meaning of any of them. may connect two
We can regard the symbol ) as representing another kind of implication, statements. The
statement “pmaterially
and it will be expedient to do so, because a convenient way to read p)qis “If p,
implies q” is true when
then q.” But it is not the same kind of implication as any of those mentioned ear-
either pis false, or qis
lier. It is called material implicationby logicians. In giving it a special name, we true.
(cid:22)(cid:21)(cid:20)
Symbolic Logic
admit that it is a special notion, not to be confused with other, more usual, types
of implication.
Not all conditional statements in English need assert one of the four types of
implication previously considered. Material implication constitutes a fifth type
that may be asserted in ordinary discourse. Consider the remark, “If Hitler was a
military genius, then I’m a monkey’s uncle.” It is quite clear that it does not as-
sert logical, definitional, or causal implication. It cannot represent a decisional
implication, because it scarcely lies in the speaker’s power to make the conse-
quent true. No “real connection,” whether logical, definitional, or causal, obtains
between antecedent and consequent here. Aconditional of this sort is often used
as an emphatic or humorous method of denying its antecedent. The consequent
of such a conditional is usually a statement that is obviously or ludicrously false.
And because no true conditional can have both its antecedent true and its conse-
quent false, to affirm such a conditional amounts to denying that its antecedent is
true. The full meaning of the present conditional seems to be the denial that
“Hitler was a military genius” is true when “I’m a monkey’s uncle” is false. Be-
cause the latter is so obviously false, the conditional must be understood to deny
the former.
The point here is that no “real connection” between antecedent and conse-
quent is suggested by a material implication. All it asserts is that it is not the case
that the antecedent is true when the consequent is false. Note that the material
implication symbol is a truth-functional connective, like the symbols for conjunc-
tion and disjunction. As such, it is defined by the following truth table:
p q p!q
T T T
T F F
F T T
F F T
As thus defined by the truth table, the symbol ) has some features that may
at first appear odd: The assertion that a false antecedent materially implies a true
consequent is true; and the assertion that a false antecedent materially implies a
false consequent is also true. This apparent strangeness can be dissipated in part
by the following considerations. Because the number 2 is smaller than the num-
ber 4 (a fact notated symbolically as 2 6 4), it follows that any number smaller
than 2 is smaller than 4. The conditional formula
If x 6 2, then x 6 4.
is true for any number xwhatsoever. If we focus on the numbers 1, 3, and 4, and
replace the number variable x in the preceding conditional formula by each of
them in turn, we can make the following observations. In
If 1 6 2, then 1 6 4.
both antecedent and consequent are true, and of course the conditional is true. In
If 3 6 2, then 3 6 4.
(cid:22)(cid:21)(cid:21)
Symbolic Logic
Visual Logic
Material Implication
Photodisc/Getty Images
© Bettmann/CORBIS All Rights Reserved
Source:Photodisc/Getty Images
“If the moon is made of green cheese, then the Earth is flat.”
This proposition, in the form G)F,is a material implication. Amaterial im-
plication is true when the antecedent (the “if” clause) is false. Therefore a ma-
terial implication is true when the antecedent is false and the consequent is
also false, as in this illustrative proposition.
Source:Photodisc/Getty Images Source:Photodisc/Getty Images
“If the moon is made of green cheese, then the Earth is round.”
This proposition, in the similar form G)R,is also a material implication. A
material implication is true when the antecedent (the “if” clause) is false.
Therefore a material implication is true when the antecedent is false and the
consequent is true, as in this illustrative proposition.
Amaterial implication is false only if the antecedent is true and the consequent is
false.Therefore a material implication is true whenever the antecedent is false,
whether the consequent is false or true.
segamI
ytteG/csidotohP
segamI
ytteG/csidotohP
Source:Photodisc/Getty Images
(cid:22)(cid:21)(cid:22)
Symbolic Logic
the antecedent is false and the consequent is true, and of course the conditional is
again true. In
If 4 6 2, then 4 6 4.
both antecedent and consequent are false, but the conditional remains true.
These last two cases correspond to the third and fourth rows of the table defining
the symbol ).So it is not particularly remarkable or surprising that a condition-
al should be true when the antecedent is false and the consequent is true, or
when antecedent and consequent are both false. Of course, there is no number
that is smaller than 2 but not smaller than 4; that is, there is no true conditional
statement with a true antecedent and a false consequent. This is exactly what the
defining truth table for ) lays down.
Now we propose to translate anyoccurrence of the “if–then” phrase into our
logical symbol ).This proposal means that in translating conditional statements
into our symbolism, we treat them all as merely material implications. Of course,
most conditional statements assert more than that a merely material implication
holds between their antecedents and consequents. So our proposal amounts to
suggesting that we ignore, or put aside, or “abstract from,” part of the meaning
of a conditional statement when we translate it into our symbolic language. How
can this proposal be justified?
The previous proposal to translate both inclusive and exclusive disjunctions
by means of the symbol was justified on the grounds that the validity of the
disjunctive syllogism was preserved even if the additional meaning that attaches
¡
to the exclusive “or” was ignored. Our present proposal to translate all condi-
tional statements into the merely material implication symbolized by ) may be
justified in exactly the same way. Many arguments contain conditional state-
ments of various kinds, but the validity of all valid arguments of the general type
with which we will be concerned is preserved even if the additional meanings of
their conditional statements are ignored. This remains to be proved, of course,
and will occupy our attention in the next section.
Conditional statements can be formulated in a variety of ways. The
statement
If he has a good lawyer, then he will be acquitted.
can equally well be stated without the use of the word “then” as
If he has a good lawyer, he will be acquitted.
The order of the antecedent and consequent can be reversed, provided that the
“if” still directly precedes the antecedent, as
He will be acquitted if he has a good lawyer.
It should be clear that, in any of the examples just given, the word “if” can be re-
placed by such phrases as “in case,” “provided that,” “given that,” or “on condi-
tion that,” without any change in meaning. Minor adjustments in the phrasings
(cid:22)(cid:21)(cid:23)
Symbolic Logic
of antecedent and consequent permit such alternative phrasings of the same con-
ditional as
That he has a good lawyer implies that he will be acquitted.
or
His having a good lawyer entails his acquittal.
Ashift from active to passive voice may accompany a reversal of order of an-
tecedent and consequent, yielding the logically equivalent
His being acquitted is implied (or entailed) by his having a good lawyer.
Other variations are possible:
There is no way he won’t be acquitted if he has a good lawyer.
Any of these is symbolized as L)A.
The notions of necessary and sufficient conditions provide other formula-
tions of conditional statements. For any specified event, many circumstances are
necessary for it to occur. Thus, for a normal car to run, it is necessary that there be
fuel in its tank, that its spark plugs be properly adjusted, that its oil pump be
working, and so on. So if the event occurs, every one of the conditions necessary
for its occurrence must have been fulfilled. Hence to say
That there is fuel in its tank is a necessary condition for the car to run.
can equally well be stated as
The car runs only if there is fuel in its tank.
which is another way of saying that
If the car runs then there is fuel in its tank.
Any of these is symbolized as R)F.Usually “qis a necessary conditionfor p” is
symbolized as p)q.Likewise, “p only if q” is also symbolized as p)q.
For a specified situation there may be many alternative circumstances, any
one of which is sufficient to produce that situation. For a purse to contain more
than a dollar, for example, it is sufficient for it to contain five quarters, or eleven
dimes, or twenty-one nickels, and so on. If any one of these circumstances ob-
tains, the specified situation will be realized. Hence, to say “That the purse con-
tains five quarters is a sufficient condition for it to contain more than a dollar” is
to say “If the purse contains five quarters then it contains more than a dollar.” In
general, “pis a sufficient conditionfor q” is symbolized as p)q.
To illustrate: Recruiters for the Wall Street investment firm Goldman Sachs
(where annual bonuses are commonly in the millions) grill potential employees
repeatedly. Those who survive the grilling are invited to the firm’s offices for a
full day of interviews, culminating in a dinner with senior Goldman Sachs exec-
utives. As reported recently, “Agile brains and near-perfect grades are necessary
but not sufficient conditions for being hired. Just as important is fitting in.”2
(cid:22)(cid:21)(cid:24)
Symbolic Logic
If pis a sufficient condition for q, we have p)q,and qmust be a necessary
condition for p. If pis a necessary condition for q, we have q)p,and qmust be a
sufficient condition for p. Hence, if p is necessary and sufficient for q, then q is
sufficient and necessary for p.
Not every statement containing the word “if” is a conditional. None of the
following statements is a conditional: “There is food in the refrigerator if you
want some,” “Your table is ready, if you please,” “There is a message for you if
you’re interested,” “The meeting will be held even if no permit is obtained.” The
presence or absence of particular words is never decisive. In every case, one must
understand what a given sentence means, and then restate that meaning in a
symbolic formula.
EXERCISES
A. If A, B, and Care true statements and X, Y, and Zare false statements, deter-
mine which of the following are true, using the truth tables for the horseshoe,
the dot, the wedge, and the curl.
*1. A)B 2. A)X
3. B)Y 4. Y)Z
*5. (A)B))Z 6. (X)Y))Z
7. (A)B))C 8. (X)Y))C
9. A)(B)Z) *10. X)(Y)Z)
11. [(A)B))C])Z 12. [(A)X))Y])Z
13. [A)(X)Y)])C 14. [A)(B)Y)])X
*15. [(X)Z))C])Y 16. [(Y)B))Y])Y
17. [(A)Y))B])Z
#
18. [(A X))C])[(A)C))X]
#
19. [(A X))C])[(A)X))C]
#
*20. [(A X))Y])[(X)A))(A)Y)]
# # #
21. [(A X) ('A 'X)])[(A)X) (X)A)]
#
22. {[A)(B)C)])[(A B))C]})[(Y)B))(C)Z)]
¡
23. {[(X)Y))Z])[Z)(X)Y)]})[(X)Z))Y]
# #
24. [(A X))Y])[(A)X) (A)Y)]
#
*25. [A)(X Y)])[(A)X) (A)Y)]
¡
B. Symbolize the following, using capital letters to abbreviate the simple state-
ments involved.
*1. If Argentina mobilizes, then if Brazil protests to the UN, then Chile will
call for a meeting of all the Latin American states.
2. If Argentina mobilizes, then either Brazil will protest to the UN or Chile
will call for a meeting of all the Latin American states.
(cid:22)(cid:21)(cid:25)
