3.5 ANALMS OF ARGUMENTS FOR LOGICAL VALIDllY, PART I1 (OPTIONAL)^113Figure 3.3 The argument in Exmnpk 2,
represented by the Venn diagram Let us
agree that a rqwn contained inside a
darkened circular arc is necessarily non-
empty. The union of regions 2 and 3 is
nonempty, whereas 2 is empty. Hence 3
is nonempty, so that the union of 3 and
4 is necessarily nonempty. This is the
conclusion of the argument, which is
thereby did.Recalling the tautology [p -* (q v r)] cr [(p A -- q) 4 r] [Theorem I@),
Article 2.31, we note first that we may add the equation M n 0 = 0 to
our list of hypotheses and try to deduce the conclusion P n M # 0 from
the expanded list. Several approaches similar to that taken in Example 2
may be tried in an attempt to draw this conclusion. After noting that
all these approaches fail, we will not be surprised to find that an example
such as P = (l,2), L = (1,z 3,4), M = (4,5,6), and 0 = (2,3) can be
found to show that the conclusion does not need to follow from the
premise, that is, the argument is invalid. OExercises
Analyze these arguments for logical validity:- All men are mortal. Socrates is a man. Therefore Socrates is mortal.
2. Some students are athletes. Some athletes fail courses. Therefore some students
fail courses.
*3. All good citizens register to vote. Some registered voters do community service.
No lazy people do community service. Therefore some good citizens are not lazy.
- All statesmen are politicians. Some statesmen are wise. Some politicians are
dishonest. Therefore either no politicians are wise or tlo wise people are dishonest.
5. All pessimists are unhappy. Some happy people are healthy. Therefore some
healthy people are not pessimists.
6. All bigots are intolerant. Some fanatics are bigots. All fanatics hate the truth
Therefore every lover of truth is tolerant
q. All fields are rings. Some rings are integral domains. Some integral domains
are not fields. Therefore some rings are not fields.