Bridge to Abstract Mathematics: Mathematical Proof and Structures

3.5 ANALYSIS OF ARGUMENTS FOR LOGICAL VALIDITY, PART II (OPTIONAL) 111

3. No p's are 9's.


4. Some p's are not q's.

As seen earlier in this chapter, statements of each of these four types may
be recast, in terms of either formal logical symbolism or statements about
the truth sets involved. We may summarize the situation as follows:

Logical symbolism Corresponding statement (s) about truth sets

Let us consider some examples.

EXAMPLE 1 Analyze for logical validity the argument, "all dinosaurs are

cold-blooded animals. All cold-blooded animals are vegetarians. There-

fore all dinosaurs are vegetarians."

Solution Denote by d(x), c(x), and v(x), respectively, the predicates, "x is

a dinosaur," "x is a cold-blooded animal," and "x is a vegetarian," with

D, C, and V representing the respective truth sets. Our argument can

then be symbolized:

(Vx)(d(x) + c(x)) (or D G C)

(W(c(x) + o((4) (or c c V)

therefore (Vx)(d(x) + v(x)) (OJ D G V)

The validity or nonvalidity of the argument in this case "boils down" to

the truth or falsehood of the theorem from set theory, "if D is a subset

of C and C is a subset of V, then D is a subset of V, (for any three sets

D, C, and V)" [recall (6) of Fact 1, Article 1.41. Assuming the truth of

this statement, the argument is seen to be valid.

The situation in Example 1 can also be represented on a Venn diagram,

as shown in Figure 3.2. With three or fewer predicates involved in an

argument, Venn diagrams can be a useful tool in deciding validity.

EXAMPLE 2 Analyze the argument, "all violence is crime. Some violence

is necessary. Therefore some crime is necessary."

Solution Denote the three predicates involved in this argument by u(x), c(x),

and n(x), with truth sets V, C, and N (refer to the solution to Example 1

for guidance in writing out these predicates in detail). The argument