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F F T
F F F

Since two of the connectives in the target sentence apply to atomic sen-
tences whose values are speciп¬Ѓed in the reference column, we can п¬Ѓll in these
columns using the truth tables for в€§ and В¬ given earlier.

A B C (A в€§ B) в€Ё В¬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 п¬Ѓll
in the column under it by referring to the two columns just completed, using
the truth table for в€Ё.

Section 4.1
98 / The Logic of Boolean Connectives

A B C (A в€§ B) в€Ё В¬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 п¬Ѓnal column of this table, the one beneath the con-
nective в€Ё, 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.
LetвЂ™s look at one more example, this time for a sentence of the form

В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓlling in the columns under the two connectives that apply directly to
atomic sentences.
A B C В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓll in the column under the в€Ё that connects В¬A and B в€§ C by
referring to the columns just п¬Ѓlled 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 В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓll in the column under the remaining в€§. 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 п¬Ѓngers down the relevant columns and enter
a T in only those rows where both your п¬Ѓngers are pointing to TвЂ™s.

A B C В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓll in the column for the remaining В¬ by referring to the previously
completed column. The В¬ simply reverses TвЂ™s and FвЂ™s.

A B C В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓll in the column under the main connective of our sentence.
We do this with the two-п¬Ѓnger method: running our п¬Ѓngers down the reference
column for B and the just completed column, entering T whenever at least
one п¬Ѓnger points to a T.

Section 4.1
100 / The Logic of Boolean Connectives

A B C В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€ЁB
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 TвЂ™s
tautology
in the column under its main connective. Thus, we see from the п¬Ѓnal column
of the above table that any sentence of the form

В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 п¬Ѓrst thing
to do is enter the sentence В¬(A в€§ (В¬A в€Ё (B в€§ C))) в€Ё 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 в€ј for в€§, в€Ё, and
В¬, 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 вЂњ(1)вЂќ 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 п¬Ѓrst 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 п¬Ѓll
in the reference columns, click under each of them in turn, and type the
desired pattern of TвЂ™s and FвЂ™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

п¬Ѓlled in, and п¬Ѓll 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 deп¬Ѓnition 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
Pв€§Qв€§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
в€§. That is to say, we will construct a truth table for P в€§ Q в€§ R the same way
we would construct a truth table for:

(P в€§ Q) в€§ R

More generally, we construct the truth table for:

P1 в€§ P2 в€§ P3 в€§ . . . в€§ Pn

as if it were вЂњpunctuatedвЂќ like this:

(((P1 в€§ P2 ) в€§ P3) в€§ . . .) в€§ 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
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