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Symbolic Logic from this text. It may

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1 Modern Logic and Its Symbolic Language (Belvedere by

M.C. Escher)

2 The Symbols for Conjunction, Negation, and Disjunction

3 Conditional Statements and Material Implication

4 Argument Forms and Refutation by Logical Analogy

5 The Precise Meaning of “Invalid” and “Valid”

6 Testing Argument Validity Using Truth Tables

7 Some Common Argument Forms

8 Statement Forms and Material Equivalence

9 Logical Equivalence

10 The Three “Laws of Thought”

1 Modern Logic and Its Symbolic Language

To have a full understanding of deductive reasoning we need a general theory of

deduction. Ageneral theory of deduction will have two objectives: (1) to explain

the relations between premises and conclusions in deductive arguments, and

(2) to provide techniques for discriminating between valid and invalid deduc-

tions. Two great bodies of logical theory have sought to achieve these ends. The

first is called classical (or Aristotelian) logic. The second, called modern, symbolic,

or mathematical logic, is this chapter.

Although these two great bodies of theory have similar aims, they proceed in

very different ways. Modern logic does not build on the system of syllogisms. It

does not begin with the analysis of categorical propositions. It does seek to dis-

criminate valid from invalid arguments, although it does so using very different

concepts and techniques. Therefore we must now begin afresh, developing a

modern logical system that deals with some of the very same issues dealt with by

traditional logic—and does so even more effectively.

Modern logic begins by first identifying the fundamental logical connectives

on which deductive arguments depend. Using these connectives, a general ac-

count of such arguments is given, and methods for testing the validity of argu-

ments are developed.

This analysis of deduction requires an artificial symbolic language. In a

natural language—English or any other—there are peculiarities that make

exact logical analysis difficult: Words may be vague or equivocal, the

construction of arguments may be ambiguous, metaphors and idioms may

confuse or mislead, emotional appeals may distract. These difficulties can be

largely overcome with an artificial language in which logical relations can be

From Chapter 8 of Introduction to Logic, Fourteenth Edition. Irving M. Copi, Carl Cohen, Kenneth McMahon.

Copyright © 2011 by Pearson Education, Inc. Published by Pearson Prentice Hall. All rights reserved.

Symbolic Logic

formulated with precision. The most fundamental elements of this modern

symbolic language will be introduced in this chapter.

Symbols greatly facilitate our thinking about arguments. They enable us to

get to the heart of an argument, exhibiting its essential nature and putting

aside what is not essential. Moreover, with symbols we can perform, almost

mechanically, with the eye, some logical operations which might otherwise

demand great effort. It may seem paradoxical, but a symbolic language there-

fore helps us to accomplish some intellectual tasks without having to think too

much. The Indo-Arabic numerals we use today (1, 2, 3, . . .) illustrate the ad-

vantages of an improved symbolic language. They replaced cumbersome

Roman numerals (I, II, III, . . .), which are very difficult to manipulate. To mul-

tiply 113 by 9 is easy; to multiply CXIII by IX is not so easy. Even the Romans,

some scholars contend, were obliged to find ways to symbolize numbers more

efficiently.

Classical logicians did understand the enormous value of symbols in analy-

sis. Aristotle used symbols as variables in his own analyses, and the refined sys-

tem of Aristotelian syllogistics uses symbols in very sophisticated ways.

However, much real progress has been made, mainly during the twentieth century,

in devising and using logical symbols more effectively.

The modern symbolism with which deduction is analyzed differs greatly

from the classical. The relations of classes of things are not central for modern lo-

gicians as they were for Aristotle and his followers. Instead, logicians look now

to the internal structure of propositions and arguments, and to the logical links—

very few in number—that are critical in all deductive argument. Modern sym-

bolic logic is therefore not encumbered, as Aristotelian logic was, by the need to

transform deductive arguments into syllogistic form.

The system of modern logic we now begin to explore is in some ways less el-

egant than analytical syllogistics, but it is more powerful. There are forms of de-

ductive argument that syllogistics cannot adequately address. Using the

approach taken by modern logic, with its more versatile symbolic language, we

can pursue the aims of deductive analysis directly and we can penetrate more

deeply. The logical symbols we shall now explore permit more complete and

more efficient achievement of the central aim of deductive logic: discriminating

between valid and invalid arguments.

2 The Symbols for Conjunction, Negation,

and Disjunction

In this chapter we shall be concerned with relatively simple arguments such as:

The blind prisoner has a red hat or the blind prisoner has a white hat.

The blind prisoner does not have a red hat.

Therefore the blind prisoner has a white hat.

and

(cid:22)(cid:19)(cid:25)

Symbolic Logic

If Mr. Robinson is the brakeman’s next-door neighbor, then Mr. Robinson lives

halfway between Detroit and Chicago.

Mr. Robinson does not live halfway between Detroit and Chicago.

Therefore Mr. Robinson is not the brakeman’s next-door neighbor.

Every argument of this general type contains at least one compound state-

ment. In studying such arguments we divide all statements into two general cat-

egories: simple and compound. Asimple statementdoes not contain any other

statement as a component. For example, “Charlie is neat” is a simple statement.

Acompound statementdoes contain another statement as a component. For ex-

ample, “Charlie is neat and Charlie is sweet” is a compound statement, because

it contains two simple statements as components. Of course, the components of a

compound statement may themselves be compound. In formulating definitions

and principles in logic, one must be very precise. What appears simple often

proves more complicated than had been supposed. The notion of a “component

of a statement” is a good illustration of this need for caution.

One might suppose that a component of a statement is simply a part of a

statement that is itself a statement. But this account does not define the term with

enough precision, because one statement may be a partof a larger statement and

Simple statement

yet not be a component of it in the strict sense. For example, consider the state-

Astatement that does

ment: “The man who shot Lincoln was an actor.” Plainly the last four words of not contain any other

this statement are a part of it, and could indeed be regarded as a statement; it is statement as a

component.

either true or it is false that Lincoln was an actor. But the statement that “Lincoln

was an actor,” although undoubtedly a part of the larger statement, is not a Compound statement

Astatement that

componentof that larger statement.

contains two or more

We can explain this by noting that, for part of a statement to be a componentof

statements as

that statement, two conditions must be satisfied: (1) The part must be a statement components.

in its own right; and(2) if the part is replaced in the larger statement by any other

Component

statement, the result of that replacement must be meaningful—it must make A part of a compound

sense. statement that is itself a

statement, and is of

The first of these conditions is satisfied in the Lincoln example, but the sec-

such a nature that, if

ond is not. Suppose the part “Lincoln was an actor” is replaced by “there are

replaced in the larger

lions in Africa.” The result of this replacement is nonsense: “The man who shot statement by any other

there are lions in Africa.” The term component is not a difficult one to under- statement, the result will

be meaningful.

stand, but—like all logical terms—it must be defined accurately and applied

carefully. Conjunction

A truth-functional

connective meaning

A. Conjunction “and,” symbolized by

the dot, •. A statement

There are several types of compound statements, each requiring its own logical of the form p• qis true

notation. The first type of compound statement we consider is the conjunction. if and only if pis true

and qis true.

We can form the conjunctionof two statements by placing the word “and” be-

tween them; the two statements so combined are called conjuncts. Thus the com- Conjunct

Each one of

pound statement, “Charlie is neat and Charlie is sweet,” is a conjunction whose

thecomponent

first conjunct is “Charlie is neat” and whose second conjunct is “Charlie is

statements connected in

sweet.” a conjunctive statement

(cid:22)(cid:19)(cid:26)

Symbolic Logic

The word “and” is a short and convenient word, but it has other uses besides

connecting statements. For example, the statement, “Lincoln and Grant were

contemporaries,” is not a conjunction, but a simple statement expressing a rela-

tionship. To have a unique symbol whose only function is to connect statements

#

conjunctively, we introduce the dot“ ” as our symbol for conjunction. Thus the

#

previous conjunction can be written as “Charlie is neat Charlie is sweet.” More

generally, where p and q are any two statements whatever, their conjunction is

#

written p q.In some books, other symbols are used to express conjunction, such

as “ ” or “&”.

We know that every statement is either true or false. Therefore we say that

¿

every statement has a truth value, where the truth value of a true statement is

true, and the truth value of a false statement is false. Using this concept, we can

divide compound statements into two distinct categories, according to whether

the truth value of the compound statement is determined wholly by the truth

Dot The symbol for values of its components, or is determined by anything other than the truth val-

conjunction, •, meaning ues of its components.

“and.”

We apply this distinction to conjunctions. The truth value of the conjunction

Truth value The status of two statements is determined wholly and entirely by the truth values of its

of any statement as true two conjuncts. If both its conjuncts are true, the conjunction is true; otherwise it

or false (Tor F).

is false. For this reason a conjunction is said to be a truth-functional compound

statement, and its conjuncts are said to be truth-functional componentsof it.

Truth-functional

component Not every compound statement is truth-functional. For example, the truth

Anycomponent of a value of the compound statement, “Othello believes that Desdemona loves Cas-

compound statement

sio,” is not in any way determined by the truth value of its component simple

whose replacement

there by any other statement, “Desdemona loves Cassio,” because it could be true that Othello be-

statement having the lieves that Desdemona loves Cassio, regardless of whether she does or not. So

same truth value would the component, “Desdemona loves Cassio,” is not a truth-functional component

leave the truth value of

of the statement, “Othello believes that Desdemona loves Cassio,” and the state-

the compound

statement unchanged. ment itself is not a truth-functional compound statement.

For our present purposes we define a component of a compound statement

Truth-functional

as being a truth-functional componentif, when the component is replaced in the

compound statement

compound by any different statements having the same truth value as each

A compound statement

whose truth value is other, the different compound statements produced by those replacements also

determined wholly by have the same truth values as each other. Now a compound statement is defined

the truth values of its

as being a truth-functional compound statement if all of its components are

components.

truth-functional components of it.1

Truth-functional We shall be concerned only with those compound statements that are truth-

connective Any logical functionally compound. In the remainder of this text, therefore, we shall use the

connective (e.g.,

term simple statement to refer to any statement that is not truth-functionally

conjunction, disjunction,

material implication and compound.

material equivalence) Aconjunction is a truth-functional compound statement, so our dot symbol

between the

is a truth-functional connective. Given any two statements, p and q, there are

components of a truth-

only four possible sets of truth values they can have. These four possible cases,

functionally compound

statement. and the truth value of the conjunction in each, can be displayed as follows:

(cid:22)(cid:19)(cid:27)

Symbolic Logic

#

Where pis true and qis true, p qis true.

#

Where pis true and qis false, p qis false.

#

Where pis false and qis true, p qis false.

#

Where pis false and qis false, p qis false.

If we represent the truth values “true” and “false” by the capital letters Tand

F, the determination of the truth value of a conjunction by the truth values of its

conjuncts can be represented more compactly and more clearly by means of a

truth table:

#

p q p q

T T T

T F F

F T F

F F F

This truth table can be taken as defining the dot symbol, because it explains

#

what truth values are assumed by p qin every possible case.

We abbreviate simple statements by capital letters, generally using for this

purpose a letter that will help us remember which statement it abbreviates. Thus

#

we may abbreviate “Charlie is neat and Charlie is sweet” as N S.Some conjunc-

tions, both of whose conjuncts have the same subject term—for example, “Byron

was a great poet and Byron was a great adventurer”—are more briefly and per-

haps more naturally stated in English by placing the “and” between the predi-

cate terms and not repeating the subject term, as in “Byron was a great poet and

a great adventurer.” For our purposes, we regard the latter as formulating the

#

same statement as the former and symbolize either one as P A.If both conjuncts

of a conjunction have the same predicate term, as in “Lewis was a famous explor-

er and Clark was a famous explorer,” the conjunction is usually abbreviated in

English by placing the “and” between the subject terms and not repeating the

predicate, as in “Lewis and Clark were famous explorers.” Either formulation is Truth table An array on

#

symbolized as L C. which all possible truth

values of compound

As shown by the truth table defining the dot symbol, a conjunction is true if

statements are

and only if both of its conjuncts are true. The word “and” has another use in displayed, through the

which it does not merely signify (truth-functional) conjunction, but has the sense display of all possible

of “and subsequently,” meaning temporal succession. Thus the statement, “Jones combinations of the

truth values of their

entered the country at New York and went straight to Chicago,” is significant

simple components. A

and might be true, whereas “Jones went straight to Chicago and entered the truth table may be used

country at New York” is hardly intelligible. There is quite a difference between to define truth-functional

connectives; it may also

“He took off his shoes and got into bed” and “He got into bed and took off his

be used to test the

shoes.”*Such examples show the desirability of having a special symbol with an

validity of many

exclusively truth-functional conjunctive use. deductive arguments.

*In The Victoria Advocate, Victoria, Texas, 27 October 1990, appeared the following report: “Ramiro Ramirez

Garza, of the 2700 block of Leary Lane, was arrested by police as he was threatening to commit suicide and

flee to Mexico.”

(cid:22)(cid:19)(cid:28)

Symbolic Logic

Note that the English words “but,” “yet,” “also,” “still,” “although,” “howev-

er,” “moreover,” “nevertheless,” and so on, and even the comma and the semicolon,

can also be used to conjoin two statements into a single compound statement, and

in their conjunctive sense they can all be represented by the dot symbol.

B. Negation

The negation (or contradictory or denial) of a statement in English is often

formed by the insertion of a “not” in the original statement. Alternatively, one

can express the negation of a statement in English by prefixing to it the phrase “it

is false that” or “it is not the case that.” It is customary to use the symbol “'”,

called a curlor a tilde, to form the negation of a statement. (Again, some books

use the symbol “-” for negation.) Thus, where Msymbolizes the statement “All

humans are mortal,” the various statements “Not all humans are mortal,” “Some

humans are not mortal,” “It is false that all humans are mortal,” and “It is not the

case that all humans are mortal” are all symbolized as 'M. More generally,

where p is any statement whatever, its negation is written 'p. Some logicians

Negation treat the curl as another connective, but since it does not actually connect two or

Denial; symbolized by more units, it is sufficient to note that it performs an operation—reversing truth

the tilde or curl. !p

value—on a single unit, and thus may be referred to as an operator. It is a truth-

simply means “it is not

functionaloperator, of course. The negation of any true statement is false, and the

the case that p,” and

may be read as “not-p.” negation of any false statement is true. This fact can be presented very simply

Curl or tilde and clearly by means of a truth table:

Thesymbol for negation, p !p

!. It appears

immediately before (to T F

the left of) what is

F T

negated or denied.

Disjunction This truth table may be regarded as the definition of the negation “'” symbol.

A truth-functional

connective meaning

C. Disjunction

“or”; components so

connected are called

The disjunction(or alternation) of two statements is formed in English by insert-

disjuncts. There are two

types of disjunction: ing the word “or” between them. The two component statements so combined

inclusive and exclusive. are called disjuncts(or alternatives).

Inclusive disjunction The English word “or” is ambiguous, having two related but distinguishable

A truth-functional meanings. One of them is exemplified in the statement, “Premiums will be

connective between two

waived in the event of sickness or unemployment.” The intention here is obvi-

components called

disjuncts. A compound ously that premiums are waived not only for sick persons and for unemployed

statement asserting persons, but also for persons who are both sick and unemployed. This sense of

inclusive disjunction is the word “or” is called weak or inclusive. An inclusive disjunctionis true if one

true when at least one of

or the other or both disjuncts are true; only if both disjuncts are false is their in-

the disjuncts (that is,

one or both) is true. clusive disjunction false. The inclusive “or” has the sense of “either, possibly

Normally called simply both.” Where precision is at a premium, as in contracts and other legal docu-

“disjunction,” it is also

ments, this sense is often made explicit by the use of the phrase “and/or.”

called “weak disjunction”

The word “or” is also used in a strong or exclusive sense, in which the mean-

and is symbolized by the

wedge, . ing is not “at least one” but “at least one and at most one.” Where a restaurant

¡

(cid:22)(cid:20)(cid:19)

Symbolic Logic

lists “salad or dessert” on its dinner menu, it is clearly meant that, for the stated

price of the meal, the diner may have one or the other but not both. Where preci-

sion is at a premium and the exclusive sense of “or” is intended, the phrase “but

not both” is often added.

We interpret the inclusive disjunction of two statements as an assertion that

at least one of the statements is true, and we interpret their exclusive disjunc-

tion as an assertion that at least one of the statements is true but not both are

true. Note that the two kinds of disjunction have a part of their meanings in

common. This partial common meaning, that at least one of the disjuncts is

true, is the whole meaning of the inclusive “or” and a part of the meaning of

the exclusive “or.”

Although disjunctions are stated ambiguously in English, they are unam-

biguous in Latin. Latin has two different words corresponding to the two differ-

ent senses of the English word “or.” The Latin word vel signifies weak or

inclusive disjunction, and the Latin word autcorresponds to the word “or” in its

strong or exclusive sense. It is customary to use the initial letter of the word velto

stand for “or” in its weak, inclusive sense. Where pand qare any two statements

whatever, their weak or inclusive disjunction is written p q.Our symbol for in-

clusive disjunction, called a wedge(or, less frequently, a vee) is also a truth-func-

¡

tional connective. Aweak disjunction is false only if both of its disjuncts are false.

We may regard the wedge as being defined by the following truth table:

p q p q

T T T¡

Exclusive disjunction

T F T

or strong disjunction

F T T

A logical relation

F F F meaning “or” that may

connect two component

The first specimen argument presented in this section was a disjunctive syllo- statements. A

gism. (Asyllogism is a deductive argument consisting of two premises and a con- compound statement

asserting exclusive

clusion.)

disjunction says that at

The blind prisoner has a red hat or the blind prisoner has a white hat. least one of the

The blind prisoner does not have a red hat. disjuncts is true andthat

at least one of the

Therefore the blind prisoner has a white hat.

disjuncts is false. It is

Its form is characterized by saying that its first premise is a disjunction; its contrasted with an

“inclusive” (or “weak”)

second premise is the negation of the first disjunct of the first premise; and its

disjunction, which says

conclusion is the same as the second disjunct of the first premise. It is evident

that at least one of the

that the disjunctive syllogism, so defined, is valid on either interpretation of the disjuncts is true and that

word “or”—that is, regardless of whether an inclusive or exclusive disjunction is they may both be true.

intended. The typical valid argument that has a disjunction for a premise is, like

Wedge

the disjunctive syllogism, valid on either interpretation of the word “or,” so a

The symbol for weak

simplification may be effected by translating the English word “or” into our log- (inclusive) disjunction, .

ical symbol “ ”—regardless of which meaning of the English word “or” is intended. Any statement of the

form p qis true if pi¡s

Only a close examination of the context, or an explicit questioning of the speaker

¡ true, or if qis true, or if

or writer, can reveal which sense of “or” is intended. This problem, often impos- both pa¡nd qare true.

(cid:22)(cid:20)(cid:20)

Symbolic Logic

sible to resolve, can be avoided if we agree to treat any occurrence of the word

“or” as inclusive. On the other hand, if it is stated explicitly that the disjunction is

intended to be exclusive—by means of the added phrase “but not both,” for ex-

ample—we have the symbolic machinery to formulate that additional sense, as

will be shown directly.

Where both disjuncts have either the same subject term or the same predicate

term, it is often natural to compress the formulation of their disjunction in Eng-

lish by placing the “or” so that there is no need to repeat the common part of the

two disjuncts. Thus, “Either Smith is the owner or Smith is the manager” might

equally well be stated as “Smith is either the owner or the manager,” and either

one is properly symbolized as O M. And “Either Red is guilty or Butch is

guilty” may be stated as “Either Red or Butch is guilty”; either one may be sym-

¡

bolized as R B.

The word “unless” is often used to form the disjunction of two statements.

¡

Thus, “You will do poorly on the exam unless you study” is correctly symbolized

as P S, because that disjunction asserts that one of the disjuncts is true, and

hence that if one of them is false, the other must be true. Of course, you may

¡

study and do poorly on the exam.

The word “unless” is sometimes used to convey more information; it may

mean (depending on context) that one or the other proposition is true but that

not both are true. That is, “unless” may be intended as an exclusive disjunction.

Thus it was noted by Ted Turner that global warming will put New York under

water in one hundred years and “will be the biggest catastrophe the world has

ever seen—unless we have nuclear war.” Here the speaker did mean that at

least one of the two disjuncts is true, but of course they cannot both be true.

Other uses of “unless” are ambiguous. When we say, “The picnic will be held

unless it rains,” we surely do mean that the picnic will be held if it does not

rain. Do we mean that it will not be held if it does rain? That may be uncertain.

It is wise policy to treat every disjunction as weak or inclusive unless it is cer-

tain that an exclusive disjunction is meant. “Unless” is best symbolized simply

with the wedge ( ).

¡

D. Punctuation

In English, punctuation is absolutely required if complicated statements are to

be clear. Many different punctuation marks are used, without which many sen-

tences would be highly ambiguous. For example, quite different meanings attach

to “The teacher says John is a fool” when it is given different punctuations: “The

teacher,” says John, “is a fool”; or “The teacher says ‘John is a fool.’” Punctuation

is equally necessary in mathematics. In the absence of a special convention, no

Punctuation number is uniquely denoted by 2 * 3 + 5,although when it is made clear how

Theparentheses, its constituents are to be grouped, it denotes either 11 or 16: the first when punc-

brackets, and braces

tuated (2 * 3) + 5,the second when punctuated 2 * (3 + 5).To avoid ambigui-

used in mathematics

and logic to eliminate ty, and to make meaning clear, punctuation marks in mathematics appear in the

ambiguity. form of parentheses, ( ), which are used to group individual symbols; brackets,

(cid:22)(cid:20)(cid:21)

Symbolic Logic

[], which are used to group expressions that include parentheses; and braces, { },

which are used to group expressions that include brackets.

In the language of symbolic logic those same punctuation marks—parenthe-

ses, brackets, and braces—are equally essential, because in logic compound

statements are themselves often compounded together into more complicated

#

ones. Thus p q r is ambiguous: it might mean the conjunction of p with the

disjunction of qwith r, or it might mean the disjunction whose first disjunct is the

¡

conjunction of p and q and whose second disjunct is r. We distinguish between

#

these two different senses by punctuating the given formula as p (q r)or else

#

as (p q) r. That the different ways of punctuating the original formula do

¡

make a difference can be seen by considering the case in which p is false and q

¡

and rare both true. In this case the second punctuated formula is true (because

its second disjunct is true), whereas the first one is false (because its first conjunct

is false). Here the difference in punctuation makes all the difference between

truth and falsehood, for different punctuations can assign different truth values

#

to the ambiguous p q r.

The word “either” has a variety of different meanings and uses in English. It

¡

has conjunctive force in the sentence, “There is danger on either side.” More

often it is used to introduce the first disjunct in a disjunction, as in “Either the

blind prisoner has a red hat or the blind prisoner has a white hat.” There it con-

tributes to the rhetorical balance of the sentence, but it does not affect its mean-

ing. Perhaps the most important use of the word “either” is to punctuate a

compound statement. Thus the sentence

The organization will meet on Thursday and Anand will be elected or the election will

be postponed.

is ambiguous. This ambiguity can be resolved in one direction by placing the

word “either” at its beginning, or in the other direction by inserting the word “ei-

ther” before the name “Anand.” Such punctuation is effected in our symbolic

#

language by parentheses. The ambiguous formula p q rdiscussed in the pre-

ceding paragraph corresponds to the ambiguous sentence just examined. The

¡

two different punctuations of the formula correspond to the two different punc-

tuations of the sentence effected by the two different insertions of the word

“either.”

The negation of a disjunction is often formed by use of the phrase

“neither–nor.” Thus the statement, “Either Fillmore or Harding was the greatest

U.S. president,” can be contradicted by the statement, “Neither Fillmore nor

Harding was the greatest U.S. president.” The disjunction would be symbolized

#

as F H, and its negation as either '(F H) or as ('F) ('H). (The logical

equivalence of these two symbolic formulas will be discussed in Section 9.) It

¡ ¡

should be clear that to deny a disjunction , which states that one or another state-

ment is true, requires that both statements be stated to be false.

The word “both” in English has a very important role in logical punctuation,

and it deserves the most careful attention. When we say “Both Jamal and Derek are

not . . .” we are saying, as noted just above, that “Neither Jamal nor Derek is . . .”;

(cid:22)(cid:20)(cid:22)

Symbolic Logic

we are applying the negation to each of them. But when we say “Jamal and Derek

are not both . . .” we are saying something very different; we are applying the

negation to the pair of them taken together, saying that “it is not the case that

they are both . . . .” This difference is very substantial. Entirely different mean-

ings arise when the word “both” is placed differently in the English sentence.

Consider the great difference between the meanings of

Jamal and Derek will not both be elected.

and

Jamal and Derek will both not be elected.

# #

The first denies the conjunction J Dand may be symbolized as '(J D).The

second says that each one of the two will not be elected, and is symbolized as

#

'(J) '(D).Merely changing the position of the two words “both” and “not” al-

ters the logical force of what is asserted.

Of course, the word “both” does not always have this role; sometimes we use

it only to add emphasis. When we say that “Both Lewis and Clark were great ex-

plorers,” we use the word only to state more emphatically what is said by “Lewis

and Clark were great explorers.” When the task is logical analysis, the punctua-

tional role of “both” must be very carefully determined.

In the interest of brevity—that is, to decrease the number of parentheses

required—it is convenient to establish the convention that, in any formula, the

negation symbol will be understood to apply to the smallest statement that the

punctuation permits. Without this convention, the formula 'p q is ambigu-

ous, meaning either ('p) q,or '(p q).By our convention we take it to mean

¡

the first of these alternatives, for the curl can (and therefore by our convention

¡ ¡

does) apply to the first component, p, rather than to the larger formula, p q.

Given a set of punctuation marks for our symbolic language, it is possible to

¡

write not just conjunctions, negations, and weak disjunctions in that language,

but exclusive disjunctions as well. The exclusive disjunction of p and q asserts

that at least one of them is true but not both are true, which is written as

# #

(p q) '(p q).Another way of expressing the exclusive disjunction is “ ”.

The truth value of any compound statement constructed from simple state-

¡ ¡

ments using only the curl and the truth-functional connectives—dot and

wedge—is completely determined by the truth or falsehood of its component

simple statements. If we know the truth values of simple statements, the truth

value of any truth-functional compound of them is easily calculated. In working

with such compound statements we always begin with their inmost components

and work outward. For example, if Aand Bare true statements and Xand Yare

false statements, we calculate the truth value of the compound statement

# # #

'['(A X) (Y 'B)] as follows: Because X is false, the conjunction A X is

#

false, and so its negation '(A X)is true. Bis true, so its negation 'Bis false, and

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because Yis also false, the disjunction of Ywith 'B,Y 'B,is false. The brack-

# #

eted formula ['(A X) (Y 'B)]is the conjunction of a true with a false state-

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ment and is therefore false. Hence its negation, which is the entire statement, is

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Symbolic Logic

true. Such a stepwise procedure always enables us to determine the truth value

of a compound statement from the truth values of its components.

In some circumstances we may be able to determine the truth value of a

truth-functional compound statement even if we cannot determine the truth or

falsehood of one of its component simple statements. We may do this by first cal-

culating the truth value of the compound statement on the assumption that a

given simple component is true, and then by calculating the truth value of the

compound statement on the assumption that the same simple component is

false. If both calculations yield the same truth value for the compound statement

in question, we have determined the truth value of the compound statement

without having to determine the truth value of its unknown component, because

we know that the truth value of any component cannot be other than true or

false. Truth tables allow us to expand this method to cases with more than one

undetermined component.

overview

Punctuation in Symbolic Notation

The statement

I will study hard and pass the exam or fail

is ambiguous. It could mean “I will study hard and pass the exam or I will fail

the exam” or “I will study hard and I will either pass the exam or fail it.”

The symbolic notation

#

S P F

is similarly ambiguous. Parentheses resol¡ve the ambiguity. In place of “I will

study hard and pass the exam or I will fail the exam,” we get

#

(S P) F

and in place of “I will study hard and I wil¡l either pass the exam or fail it,”

we get

#

S (P F)

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EXERCISES

A. Using the truth-table definitions of the dot, the wedge, and the curl, deter-

mine which of the following statements are true:

*1. Rome is the capital of Italy Rome is the capital of Spain.

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2. '(London is the capital of England Stockholm is the capital of

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Norway).

#

3. 'London is the capital of England 'Stockholm is the capital of Norway.

4. '(Rome is the capital of Spain Paris is the capital of France).

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