112 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3
Figure 3.2 Venn diagram representation
of Example I. We symbolize the fact that
a portion of a circle is necessarily empty,
due to the given premises, by the dotted
design within the region. Portions I and 4
are empty since D is a subset of C. Since C
is a subset of V, then 2 pnd 3 are empty.
Since I and 2 are empty, we may conclude
D is a subset of V, so that the argument
is valid.
may then be represented:A fairly convincing Venn diagram presentation may be given, as shown
in Figure 3.3. Taking a more rigorous approach, we may argue first that
there exists x such that x E V and x E N. Since this x is an element of V,
and since V is a subset of C, then x E C. Since x E C and x E N, then
x E C n N, so that C n N # @, the conclusion of the argument. Since
the conclusion is thereby seen to follow from the premise, the argument
may be deemed valid.EXAMPLE 3 Analyze the argument, "all professors are logical. Some men
are logical. Some professors are overweight. Therefore either some men
are overweight or some professors are men."Solution Proceeding as in Examples 1 and 2, we may symbolize this argu-
ment in terms of truth sets: