| truth value |
Definition. Truth and Falsity (abbreviated T and F) are TRUTH VALUES. |
| truth table |
Comment. When an argument is valid, its con-
clusion cannot be false when its premises are all
true. One way to discover whether an argument is
valid is to consider explicitly all the possible
combinations of truth values among the premises
and the conclusion. In this chap-ter we show how to
do this. The idea is to assign truth values variously
to the sentence letters of the argu-ment and see how
the premises and the conclusion turn out. The following
rules, codified in TRUTH TABLES
(TTs), enable us to do this.
|
|
Comment. For this method to work, it has to be the
case that the truth values of compound sentences are
determined by the truth values of the sentence letters
that appear in them. |
| truth-functional connectives |
Comment. All the sentential connectives introduced in chapter 1
have the property described in the previous comment. Since the
truth values of com-pound sentences containing these connect-
ives are functions of the truth values of the component wffs, they
are known as TRUTH-FUNCTIONAL CON-NECTIVES
(Not all English connectives are truth-functional.) |
| TT for negation |
In order for a negation ~f to be true, f must be false.
|
|
Table 2.1 Truth function for negation. |
| TT for conjunction |
In order for a conjunction (f & y) to be true, both
conjuncts f and y must be true. |
| TT for disjunction |
In order for a disjunction (f v y) to be false, both
disjuncts f and y must be false. |
| TT for conditional |
In order for a conditional (f -> y) to be false, the
antecedent f must be true while the consequent y is
false. |
| TT for biconditional |
In order for a biconditional (f <-> y) to be true, f
and y must have the same truth value.
|
|
f y | f & y | f v y | f -> y | f <-> y |
| T T |
T |
T |
T |
T |
| T F |
F |
T |
F |
F |
| F T |
F |
T |
T |
F |
| F F |
F |
F |
T |
T |
Table 2.2 Truth functions for the binary connectives. |
|
Comment Observe that if a conditional's antecedent
is false, then the conditional is true no matter
what the truth value of its con-sequent. Also, if its
consequent is true, then it is true, regardless of the
truth value of its antecedent. These are the truth
table analogues of the derived rules False Antecedent
and True Consequent. |
| TTs for sentences |
By means of these rules we can construct TTs for
compound wffs, exhibiting how their truth values are
determined by the truth values of their sentence
letters. |
|
Example.
| P Q R |
(P -> Q) v (~Q & R) |
| T T T |
T T F F |
| T T F |
T T F F |
| T F T |
F T T T |
| T F F |
F F T F |
| F T T |
T T F F |
| F T F |
T T F F |
| F F T |
T T T T |
| F F F |
T T T F |
Table 2.3 TT for the wff (P -> Q)
v (~Q & R). |
|
Comment. By referring to the columns for P and
Q, we construct column (a), for (P -> Q), using the
TT for conditionals (see table 2.2). Next,
we construct column (b), for ~Q, (see table 2.1)
. Column (c), for (~Q & R) is constructed by referring to the
columns for its conjuncts, ~Q and R and using the
TT for conjunction (see table 2.2).
Finally, we construct column (d), for (P -> Q) v (~Q & R), by
referring to those for its disjuncts, (P -> Q) and
(~Q & R) (see table 2.2). |
|
Comment. The column for a given component of a
sentence (other than the sentence letters) is placed
under that component's connective. For example,
the column for (P -> Q) in table 2.3
falls under its arrow. |
| Exercise 2.1 |
Construct TTs for the following sentences. |
| i* |
P v (~P v Q) |
| ii* |
~(P & Q) v P |
| iii* |
~(P -> Q) -> P |
| iv* |
(P v Q) v (~P & Q) |
| v* |
P v Q -> R v ~P |
| vi* |
R <-> ~P v (R & Q) |
| vii* |
(P & Q <-> Q) -> (Q -> P) |
| viii* |
(P <-> ~Q) <-> (~P <-> ~Q) |
| ix* |
(P <-> Q) <-> (P v R -> (~Q -> R)) |
| x* |
(P & Q) v (R & S) -> (P & R) v (Q & S) |
|
For additional practice, construct TTs for wffs in
chapter 1. |