F F F

Since two of the connectives in the target sentence apply to atomic sen-

tences whose values are speci¬ed in the reference column, we can ¬ll in these

columns using the truth tables for § and ¬ given earlier.

A B C (A § B) ∨ ¬C

T F

t t t

T T

t t f

F F

t f t

F T

t f f

F F

f t t

F T

f t f

F F

f f t

F T

f f f

This leaves only one connective, the main connective of the sentence. We ¬ll

in the column under it by referring to the two columns just completed, using

the truth table for ∨.

Section 4.1

98 / The Logic of Boolean Connectives

A B C (A § B) ∨ ¬C

T

t t t t f

T

t t f t t

F

t f t f f

T

t f f f t

F

f t t f f

T

f t f f t

F

f f t f f

T

f f f f t

When we inspect the ¬nal column of this table, the one beneath the con-

nective ∨, we see that the sentence will be false in any circumstance where

Cube(c) is true and one of Cube(a) or Cube(b) is false. This table shows that

our sentence is not a tautology. Furthermore, since there clearly are blocks

worlds in which c is a cube and either a or b is not, the claim made by our

original sentence is not logically necessary.

Let™s look at one more example, this time for a sentence of the form

¬(A § (¬A ∨ (B § C))) ∨ B

This sentence, though it has the same number of atomic constituents, is con-

siderably more complex than our previous example. We begin the truth table

by ¬lling in the columns under the two connectives that apply directly to

atomic sentences.

A B C ¬(A § (¬A ∨ (B § C))) ∨ B

F T

t t t

F F

t t f

F F

t f t

F F

t f f

T T

f t t

T F

f t f

T F

f f t

T F

f f f

We can now ¬ll in the column under the ∨ that connects ¬A and B § C by

referring to the columns just ¬lled in. This column will have an F in it if and

only if both of the constituents are false.

Chapter 4

Tautologies and logical truth / 99

A B C ¬(A § (¬A ∨ (B § C))) ∨ B

T

t t t f t

F

t t f f f

F

t f t f f

F

t f f f f

T

f t t t t

T

f t f t f

T

f f t t f

T

f f f t f

We now ¬ll in the column under the remaining §. To do this, we need to

refer to the reference column under A, and to the just completed column. The

best way to do this is to run two ¬ngers down the relevant columns and enter

a T in only those rows where both your ¬ngers are pointing to T™s.

A B C ¬(A § (¬A ∨ (B § C))) ∨ B

Tf

t t t t t

Ff

t t f f f

Ff

t f t f f

Ff

t f f f f

Ft

f t t t t

Ft

f t f t f

Ft

f f t t f

Ft

f f f t f

We can now ¬ll in the column for the remaining ¬ by referring to the previously

completed column. The ¬ simply reverses T™s and F™s.

A B C ¬(A § (¬A ∨ (B § C))) ∨ B

F

t t t tft t

T

t t f fff f

T

t f t fff f

T

t f f fff f

T

f t t ftt t

T

f t f ftt f

T

f f t ftt f

T

f f f ftt f

Finally, we can ¬ll in the column under the main connective of our sentence.

We do this with the two-¬nger method: running our ¬ngers down the reference

column for B and the just completed column, entering T whenever at least

one ¬nger points to a T.

Section 4.1

100 / The Logic of Boolean Connectives

A B C ¬(A § (¬A ∨ (B § C))) ∨B

T

t t t f tft t

T

t t f t fff f

T

t f t t fff f

T

t f f t fff f

T

f t t t ftt t

T

f t f t ftt f

T

f f t t ftt f

T

f f f t ftt f

We will say that a tautology is any sentence whose truth table has only T™s

tautology

in the column under its main connective. Thus, we see from the ¬nal column

of the above table that any sentence of the form

¬(A § (¬A ∨ (B § C))) ∨ B

is a tautology.

You try it

................................................................

1. Open the program Boole from the software that came with the book. We

will use Boole to reconstruct the truth table just discussed. The ¬rst thing

to do is enter the sentence ¬(A § (¬A ∨ (B § C))) ∨ B at the top, right of

the table. To do this, use the toolbar to enter the logical symbols and

the keyboard to type the letters A, B, and C. (You can also enter the

logical symbols from the keyboard by typing &, |, and ∼ for §, ∨, and

¬, respectively. If you enter the logical symbols from the keyboard, make

sure you add spaces before and after the binary connectives so that the

columns under them will be reasonably spaced out.) If your sentence is

well formed, the small “(1)” above the sentence will turn green.

2. To build the reference columns, click in the top left portion of the table to

move your insertion point to the top of the ¬rst reference column. Enter C

in this column. Then choose Add Column Before from the Table menu

and enter B. Repeat this procedure and add a column headed by A. To ¬ll

in the reference columns, click under each of them in turn, and type the

desired pattern of T™s and F™s.

3. Click under the various connectives in the target sentence, and notice

that turquoise squares appear in the columns whose values the connective

depends upon. Select a column so that the highlighted columns are already

Chapter 4

Tautologies and logical truth / 101

¬lled in, and ¬ll in that column with the appropriate truth values. Continue

this process until your table is complete. When you are done, click on the

button Verify Table to see if all the values are correct and your table

complete.

4. Once you have a correct and complete truth table, click on the Assess-

ment button under the toolbar. This will allow you to say whether you

think the sentence is a tautology. Say that it is (since it is), and check your

assessment by clicking on the button Verify Assess. Save your table as

Table Tautology 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations

There is a slight problem with our de¬nition of a tautology, in that it

assumes that every sentence has a main connective. This is almost always the

case, but not in sentences like: main connectives

P§Q§R

For purposes of constructing truth tables, we will assume that the main con-

nective in conjunctions with more than two conjuncts is always the rightmost

§. That is to say, we will construct a truth table for P § Q § R the same way

we would construct a truth table for:

(P § Q) § R

More generally, we construct the truth table for:

P1 § P2 § P3 § . . . § Pn

as if it were “punctuated” like this:

(((P1 § P2 ) § P3) § . . .) § Pn

We treat long disjunctions similarly.

Any tautology is logically necessary. After all, its truth is guaranteed sim- tautologies and logical

necessity

ply by its structure and the meanings of the truth-functional connectives.

Tautologies are logical necessities in a very strong sense. Their truth is inde-

pendent of both the way the world happens to be and even the meanings of

the atomic sentences out of which they are composed.

It should be clear, however, that not all logically necessary claims are

tautologies. The simplest example of a logically necessary claim that is not

a tautology is the fol sentence a = a. Since this is an atomic sentence, its

truth table would contain one T and one F. The truth table method is too