Symbolic Logic

B. Use truth tables to characterize the following statement forms as tautolo-

gous, self-contradictory, or contingent.

*1. [p)(p)q)])q 2. p)[(p)q))q]

# #

3. (p q) (p) 'q) 4. p)['p)(q 'q)]

# #

*5. p)[p)(q 'q)] 6. (p)p))(q 'q)

¡

7. [p)(q)r)])[(p)q))(p)r)]

8. [p)(q)p)])[(q)q)) '(r)r)]

# #

9. {[(p)q) (r)s)] (p r)})(q s)

# #

*10. {[(p)q) (r)s)] (q s)})(p r)

¡ ¡

C. Use truth tables to decide ¡which of th¡e following biconditionals are

tautologies.

*1. (p)q) K ('q) 'p) 2. (p)q) K ('p) 'q)

3. [(p)q))r] K [(q)p))r] 4. [p)(q)r)] K [q)(p)r)]

# #

*5. p K [p (p q)] 6. p K [p (p q)]

# #

7. p K [p (p)q)] 8. p K [p (q)p)]

¡ ¡

9. p K [p (p)q)] *10. (p)q) K [(p q) K q]

# # #

11. p K [p (q 'q)] 12. p K [p (q 'q)]

¡ ¡

#

13. p K [p (q 'q)] 14. p K [p (q 'q)]

¡

# # # #

*15. [p (q r)] K [(p q) (p r)]

¡ ¡ ¡

# #

16. [p (q r)] K [(p q) (p r)]

¡

# # #

17. [p (q r)] K [(p q) (p r)]

¡ ¡ ¡

# #

18. [p (q r)] K [(p q) (p r)]

¡ ¡

#

19. [(p q))r] K [p)(q)r)]

¡ ¡ ¡

# # #

*20. [(p)q) (q)p)] K [(p q) ('p 'q)]

¡

9 Logical Equivalence

At this point we introduce a new relation, important and very useful, but not a

connective, and somewhat more complicated than any of the truth-functional

connectives just discussed.

Statements are materially equivalent when they have the same truth value.

Because two materially equivalent statements are either both true, or both false,

we can readily see that they must (materially) imply one another, because a false

antecedent (materially) implies any statement, and a true consequent is (materi-

ally) implied by any statement. We may therefore read the three-bar sign, K,as

“if and only if.”

However, statements that are merely materially equivalent most certainly

cannot be substituted for one another. Knowing that they are materially equiva-

lent, we know only that their truth values are the same. The statements, “Jupiter

(cid:22)(cid:23)(cid:26)

Symbolic Logic

is larger than the Earth” and “Tokyo is the capital of Japan,” are materially equiv-

alent because they are both true, but we obviously cannot replace one with the

other. Similarly, the statements, “All spiders are poisonous” and “No spiders are

poisonous,” are materially equivalent simply because they are both false, but

they certainly cannot replace one another!

There are many circumstances, however, in which we must express the rela-

tionship that does permit mutual replacement. Two statements can be equivalent

in a sense much stronger than that of material equivalence. They may be equiva-

lent in the sense that any proposition that incorporates one of them could just as

well incorporate the other. If there is no possible case in which one of these state-

ments is true while the other is false, those statements are logically equivalent.

Of course, any two statements that are logically equivalent are materially

equivalent as well, for they obviously have the same truth value. Indeed, if two

statements are logically equivalent, they are materially equivalent under all cir-

cumstances—and this explains the short but powerful definition of logical

equivalence: Two statements are logically equivalent if the statement of their material

equivalence is a tautology. That is, the statement that they have the same truth

value is itself necessarily true. This is why, to express this very strong logical re-

T

lationship, we use the three-bar symbol with a small Timmediately above it, K,

indicating that the logical relationship is of such a nature that the material equiv-

alence of the two statements is a tautology. Because material equivalence is a bi-

conditional (the two statements implying one another), we may think of this

T

symbol of logical equivalence, K,as expressing a tautological biconditional.

Some simple logical equivalences that are very commonly used will make

this relation, and its great power, very clear. It is a commonplace that pand ''p

Logical equivalence mean the same thing; “he is aware of that difficulty” and “he is not unaware of

When referring to truth-

that difficulty” are two statements with the same content. In substance, either of

functional compound

propositions, the these expressions may be replaced by the other because they both say the same

relationship that holds thing. This principle of double negation, whose truth is obvious to all, may be

between two

exhibited in a truth table, where the material equivalence of two statement forms

propositions when the

is shown to be a tautology:

statement of their

material equivalence is a

T

tautology. A very strong p 'p ''p p K ''p

relation; statements that

T F T T

are logically equivalent

must have the same F T F T

meaning, and may

therefore replace one This truth table proves that p and ''p are logically equivalent. This very useful

another wherever they logical equivalence, double negation, is symbolized as

occur.

T

p K ''p

Double negation

Anexpression of the The difference between material equivalence on the one hand and logical equiva-

logical equivalence of lenceon the other hand is very great and very important. The former is a truth-

any symbol and the

functional connective, K, which may be true or false depending only on the

negation of the negation T

of that symbol.

truth or falsity of the elements it connects. But the latter, logical equivalence, K,

Symbolized as is not a mere connective, and it expresses a relation between two statements that

p KT !!p. is not truth-functional. Two statements are logically equivalent only when it is

(cid:22)(cid:23)(cid:27)

Symbolic Logic

absolutely impossible for them to have different truth values. However, if they

alwayshave the same truth value, logically equivalent statements may be substi-

tuted for one another in any truth-functional context without changing the truth

value of that context. By contrast, two statements are materially equivalent if

they merely happento have the same truth value, even if there are no factual con-

nections between them. Statements that are merely materially equivalent certain-

ly may not be substituted for one another!

There are two well-known logical equivalences (that is, logically true bicon-

ditionals) of great importance because they express the interrelations among

conjunction and disjunction, and their negations. Let us examine these two logi-

cal equivalences more closely.

First, what will serve to deny that a disjunction is true? Any disjunction

p qasserts no more than that at least one of its two disjuncts is true. One can-

not contradict it by asserting that at least one is false; one must (to deny it) assert

¡

that both disjuncts are false. Therefore, asserting the negation of the disjunction

(p q)is logically equivalent to asserting the conjunction of the negations of p and

of q. To show this in a truth table, we may formulate the biconditional,

¡ #

'(p q) K ('p 'q),place it at the top of its own column, and examine its truth

value under all circumstances, that is, in each row.

¡

# T #

p q p q '(p q) 'p 'q 'p 'q '(p q) K ('p 'q)

T T T¡ F¡ F F F ¡ T

T F T F F T F T

F T T F T F F T

F F F T T T T T

Of course we see that, whatever the truth values of pand of q, this bicondi-

tional must always be true. It is a tautology. Because the statement of that mate-

rial equivalence is a tautology, we conclude that its two component statements

are logically equivalent. We have proved that

!(p q) KT (!p# !q)

De Morgan’s

theorems

Simila¡rly, asserting the conjunction of pand qasserts that both are true, so to

Twoexpressions of

contradict this assertion we need merely assert that at least one is false. Thus, as-

# logical equivalence. The

serting the negation of the conjunction (p q) is logically equivalent to asserting first states that the

the disjunction of the negations of p and of q. In symbols, the biconditional, negation of a disjunction

'(p# q) K ('p 'q) may be shown, in a truth table, to be a tautology. Such a is logically equivalent to

the conjunction of the

table proves that

negations of its

!(p# q) KT (¡ !p !q) disjuncts: !(p q) KT

(!p• !q).

These two tautologo¡us biconditionals, or logical equivalences, are known as De The second sta¡tes that

the negation of a

Morgan’s theorems, because they were formally stated by the mathematician

conjunction is logically

and logician Augustus De Morgan (1806–1871). De Morgan’s theorems can be

equivalent to the

formulated in English thus: disjunction of the

negations of its

a. The negation of the disjunction of two statements is logically equivalent to the T

conjuncts: !(p• q) K

conjunction of the negations of the two statements; (!p !q).

¡

(cid:22)(cid:23)(cid:28)

Symbolic Logic

and

b. The negation of the conjunction of two statements is logically equivalent to the

disjunction of the negations of the two statements.

These theorems of De Morgan are exceedingly useful.

Another important logical equivalence is very helpful when we seek to ma-

nipulate truth-functional connectives. Material implication, ) , was defined (in

#

Section 3) as an abbreviated way of saying '(p 'q). That is, “p materially im-

plies q” simply means, by definition, that it is not the case that pis true while qis

#

false. In this definition we see that the definiens, '(p 'q),is the denial of a con-

junction. And by De Morgan’s theorem we know that any such denial is logical-

ly equivalent to the disjunction of the denials of the conjuncts; that is, we know

#

that '(p 'q)is logically equivalent to ('p ''q);and this expression in turn,

applying the principle of double negation, is logically equivalent to 'p q.Log-

¡

ically equivalent expressions mean the same thing, and therefore the original

# ¡

definiensof the horseshoe, '(p 'q),may be replaced with no change of meaning

by the simpler expression 'p q.This gives us a very useful definition of mate-

rial implication: p)qis logically equivalent to 'p q.In symbols we write:

¡

(p !q) KT (!p q) ¡

This definition of material implication is widely relied on in the formulation

¡

of logical statements and the analysis of arguments. Manipulation is often essen-

tial, and manipulation is more efficient when the statements we are working

with have the same central connective. With the simple definition of the horse-

shoe we have just established, (p)q)"T ('p q),statements in which the horse-

shoe is the connective can be conveniently replaced by statements in which the

¡

wedge is the connective; and likewise, statements in disjunctive form may be

readily replaced by statements in implicative form. When we seek to present a

formal proof of the validity of deductive arguments, replacements of this kind

are very useful indeed.

Before going on to the methods of testing for validity and invalidity in the

next section, it is worthwhile to pause for a more thorough consideration of the

meaning of material implication. Implication is central in argument but, as we

noted earlier, the word “implies” is highly ambiguous. Material implication, on

which we rely in this analysis, is only one sense of that word, although it is a very

important sense, of course. The definition of material implication explained just

above makes it clear that when we say, in this important sense, that “pimplies q,”

we are saying no more than that “either qis true or pis false.”

Asserting the “if–then” relation in this sense has consequences that may

seem paradoxical. For in this sense we can say, correctly, “If a statement is true,

then it is implied by any statement whatever.” Because it is true that the earth is

round, it follows that “The moon is made of green cheese implies that the earth is

round.” This appears to be very curious, especially because it also follows that

“The moon is not made of green cheese implies that the earth is round.” Our pre-

cise understanding of material implication also entitles us to say, correctly, “If a

statement is false, then it implies any statement whatever.” Because it is false that

(cid:22)(cid:24)(cid:19)

Symbolic Logic

the moon is made of green cheese, it follows that “The moon is made of green

cheese implies that the earth is round,” and this is the more curious when we re-

alize that it also follows that “The moon is made of green cheese implies that the

earth is notround.”

Why do these true statements seem so curious? It is because we recognize that

the shape of the earth and the cheesiness of the moon are utterly irrelevant to each

other. As we normally use the word “implies,” a statement cannot imply some other

statement, false or true, to which it is utterly irrelevant. That is the case when “im-

plies” is used in most of its everyday senses. And yet those “paradoxical” statements

in the preceding paragraph are indeed true, and not really problematic at all, because

they use the word “implies” in the logical sense of “material implication.” The pre-

cise meaning of material implication we have made very clear; we understand that

to say pmaterially implies qis only to say that either pis false or qis true.

What needs to be borne in mind is this: Meaning—subject matter—is strictly

irrelevant to material implication. Material implication is a truth function. Only the

truth and falsity of the antecedent and the consequent, not their content, are rel-

evant here. There is nothing paradoxical in stating that any disjunction is true

that contains one true disjunct. Well, when we say that “The moon is made of

green cheese (materially) implies that the earth is round,” we know that to be

logically equivalent to saying “Either the moon is not made of green cheese or

the earth is round”—a disjunction that is most certainly true. And any disjunc-

tion we may confront in which “The moon is not made of green cheese” is the

first disjunct will certainly be true, no matter what the second disjunct asserts.

So, yes, “The moon is made of green cheese (materially) implies that the earth is

square” because that is logically equivalent to “The moon is not made of green

cheese or the earth is square.” Afalse statement materially implies any statement

whatever. Atrue statement is materially implied by any statement whatever.

Every occurrence of “if–then” should be treated, we have said, as a material

implication, and represented with the horseshoe, ) .The justification of this prac-

tice, its logical expediency, is the fact that doing so preserves the validity of all

valid arguments of the type with which we are concerned in this part of our log-

ical studies. Other symbolizations have been proposed, adequate to other types

of implication, but they belong to more advanced parts of logic, beyond the

scope of this text.

10 The Three “Laws of Thought”

Some early thinkers, after having defined logic as “the science of the laws of

thought,” went on to assert that there are exactly three basiclaws of thought, laws

so fundamental that obedience to them is both the necessary and the sufficient

condition of correct thinking. These three have traditionally been called:

! The principle of identity. This principle asserts that if any statement is true, Principle of identity

The principle that

then it is true. Using our notation we may rephrase it by saying that the prin-

asserts that if any

ciple of identity asserts that every statement of the form p)pmust be true, statement is true then it

that every such statement is a tautology. is true.

(cid:22)(cid:24)(cid:20)

Symbolic Logic

! The principle of noncontradiction. This principle asserts that no statement

can be both true and false. Using our notation we may rephrase it by saying

that the principle of noncontradiction asserts that every statement of the

#

form p 'pmust be false, that every such statement is self-contradictory.

! The principle of excluded middle. This principle asserts that every statement

is either true or false. Using our notation we may rephrase it by saying that the

principle of excluded middle asserts that every statement of the form p 'p

must be true, that every such statement is a tautology.

¡

It is obvious that these three principles are indeed true—logically true—but

the claim that they deserve privileged status as the most fundamental laws of

thought is doubtful. The first (identity) and the third (excluded middle) are tau-

tologies, but there are many other tautologous forms whose truth is equally cer-

tain. The second (noncontradiction) is by no means the only self-contradictory

form of statement.

We do use these principles in completing truth tables. In the initial columns

of each row of a table we place either a Tor an F, being guided by the principle

of excluded middle. Nowhere do we put both Tand F, being guided by the prin-

ciple of noncontradiction. Once having put a Tunder a symbol in a given row,

being guided by the principle of identity, when we encounter that symbol in

other columns of that row, we regard it as still being assigned a T. So we could re-

gard the three laws of thought as principles governing the construction of truth

tables.

Nevertheless, in regarding the entire system of deductive logic, these three

principles are no more important or fruitful than many others. Indeed, there are

tautologies that are more fruitful than they for purposes of deduction, and in that

sense more important than these three, such as De Morgan’s theorems, which are

more applicable in a system of natural deduction than these more abstract prin-

ciples. Nonetheless, these principles are useful in guiding informal argumenta-

tion, in which axiomatic deductive systems seldom obtain. A more extended

treatment of this point lies beyond the scope of this text.6

Some thinkers, believing themselves to have devised a new and different

logic, have claimed that these three principles are in fact not true, and that obedi-

ence to them has been needlessly confining. But these criticisms have been based

on misunderstandings.

The principle of identity has been attacked on the ground that things change,

Principle of and are always changing. Thus, for example, statements that were true of the

noncontradiction United States when it consisted of the thirteen original states are no longer true

Theprinciple that

of the United States today, which has fifty states. But this does not undermine the

asserts that no

statement can be both principle of identity. The sentence, “There are only thirteen states in the United

true and false. States,” is incomplete, an elliptical formulation of the statement that “There were

Principle of excluded only thirteen states in the United States in 1790”—and that statement is as true

middle today as it was in 1790. When we confine our attention to complete, nonelliptical

The principle that

formulations of propositions, we see that their truth (or falsity) does not change

asserts that any

statement is either true over time. The principle of identity is true, and it does not interfere with our

or false. recognition of continuing change.

(cid:22)(cid:24)(cid:21)

Symbolic Logic

The principle of noncontradiction has been attacked by Hegelians and Marxists

on the grounds that genuine contradiction is everywhere pervasive, that the world

is replete with the inevitable conflict of contradictory forces. That there are conflict-

ing forces in the real word is true, of course—but to call these conflicting forces “con-

tradictory” is a loose and misleading use of that term. Labor unions and the private

owners of industrial plants may indeed find themselves in conflict—but neither the

owner nor the union is the “negation” or the “denial” or the “contradictory” of the

other. The principle of noncontradiction, understood in the straightforward sense in

which it is intended by logicians, is unobjectionable and perfectly true.

The principle of excluded middle has been the object of much criticism, be-

cause it leads to a “two-valued orientation,” which implies that things in the

world must be either “white or black,” and which thereby hinders the realization

of compromise and less-than-absolute gradations. This objection also arises from

misunderstanding. Of course the statement “This is black” cannot be jointly true

with the statement “This is white”—where “this” refers to exactly the same

thing. However, although these two statements cannot both be true, they can

both be false. “This” may be neither black nor white; the two statements are

contraries, not contradictories. The contradictory of the statement “This is white”

is the statement “It is not the case that this is white” and (if “white” is used in

precisely the same sense in both of these statements) one of them must be true

and the other false. The principle of excluded middle is inescapable.

All three of these “laws of thought” are unobjectionable—so long as they are

applied to statements containing unambiguous, nonelliptical, and precise terms.

Plato appealed explicitly to the principle of noncontradiction in Book IV of his

Republic (at numbers 436 and 439); Aristotle discussed all three of these princi-

ples in Books IV and XI of his Metaphysics. Of the principle of noncontradiction,

Aristotle wrote: “That the same attribute cannot at the same time belong and not

belong to the same subject and in the same respect” is a principle “which every-

one must have who understands anything that is,” and which “everyone must

already have when he comes to a special study.” It is, he concluded, “the most

certain of all principles.” The “laws of thought” may not deserve the honorific

status assigned to them by some philosophers, but they are indubitably true.

chapter Summary

This chapter has presented the fundamental concepts of modern symbolic logic.

In Section 1, we explained the general approach of modern symbolic logic

and its need for an artificial symbolic language.

In Section 2, we introduced and defined the symbols for negation (the curl: ');

#

and for the truth-functional connectives of conjunction (the dot: ) and disjunc-

tion (the wedge: ). We also explained logical punctuation.

In Section 3, we discussed the different senses of implication and defined the

¡

truth-functional connective material implication (the horseshoe: )).

In Section 4, we explained the formal structure of arguments, defined argument

forms, and explained other concepts essential in analyzing deductive arguments.

(cid:22)(cid:24)(cid:22)