Using Venn Diagrams with Deductive Reasoning

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Using Venn Diagrams with Deductive Reasoning
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In Deductive Reasoning a set of statements are
assumed to be true. The rules of logic are then
applied to deduce (come up with) more valid
statements.
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Below, Statements A and B are the hypotheses,
that is the statements that are assumed to be
true. Note that Statement C begins with the word
therefore. That identifies it as the
conclusion. Together the statements (hypotheses
plus conclusion) are called an argument. An
argument may be valid or invalid. You will need
to determine which it is based on the Laws of
Deductive Reasoning found in your text. The
argument below is valid.
A. All students love to read. B. Mona is a
student. C. Therefore, Mona loves to read.
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All students love to read. Mona is a
student. Therefore, Mona loves to read.
Argument
This argument is valid. Why is this? To be
valid the conclusion must follow from the
hypotheses. If we assume that the first two
statements are true, then it follows that the
third statement is also true.
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We can use a Venn diagram to show that the
argument is valid.
Argument All students love to read. Mona is a
student. Therefore, Mona loves to read.
Mona
The Venn diagram models the hypothesis
Students love to read. Note that this can be
rewritten as if one is a student, then one loves
to read.
Since Mona is a student, the only spot she can be
placed is inside the student circle.
This places Mona inside the circle of those who
love to read. Therefore, it follows, that Mona
must love to read.
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You may protest that this is poor reasoning
because all students do NOT love to read.
Remember though that you are determining
whether the ARGUMENT is VALID, not whether the
statements used in the argument are true or
false.
A source of confusion Notice that we used the
word hypothesis when discussing if, then
statements. There the hypothesis was the first
part of the statement and the conclusion was the
part of the statement which followed the word
then. Now we are discussing arguments. The
hypotheses of an argument are the statements that
precede the word therefore. The conclusion is
the statement that follows the word therefore.
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Determine whether the argument is valid. Use a
Venn diagram.
Argument All multiples of 10 are divisible by
2. 6 is divisible by 2. Therefore, 6 is a
multiple of 10.
6
Note that the first statement is a conditional
statement. It may be written in if, then
format as follows If a number is a multiple of
10, then it is divisible by 2.
6
The argument is INVALID. Where did the reasoning
go wrong? Look at the Venn diagram. Multiples
of 10 are a subset of Numbers divisible by 2.
This comes from the first hypothesis. Where can
6 be placed? According to the second hypothesis
it is divisible by 2. Therefore it must go
somewhere in the outer circle.
There are two places within the outer circle that
the 6 can be placed. The hypotheses do not
state whether or not 6 is a multiple of 10. It
may be and it may not be. Since there is a
possibility that it is not a multiple of 10, the
argument is invalid. In reading your text, you
will find that this error involves equating a
statement and its converse.
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It is important to note that nothing changes if
one substitutes a 60 for 6.
Argument All multiples of 10 are divisible by
2. 60 is divisible by 2. Therefore, 60 is a
multiple of 10.
60
Note that the first statement is a conditional
statement. It may be written in if, then
format as follows If a number is a multiple of
10, then it is divisible by 2.
60
The argument is INVALID, even though, the
conclusion is true! The hypotheses still do not
state whether or not 60 is a multiple of 10. And
we have to base our deductive argument solely on
our the hypotheses. We cannot bring in
extraneous information. Since there is a
possibility that it is not a multiple of 10, the
argument it invalid.
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Lets try one more. Determine whether the
argument is valid. Use a Venn diagram.
Argument If a figure is a square, then it is a
rectangle. This figure is not a
square. Therefore, it is not a rectangle.
figure
figure
The argument is INVALID. Where did the reasoning
go wrong? Looking at the Venn diagram, where can
the figure go? According to the second
hypothesis it is not a square. So it cannot go
in the inner circle. However, there are still
two places that it can go. It can go inside the
outer circle (rectangle) as long as it is not
inside the inner (square) circle. It can also go
outside both circles.
The hypotheses do not state that the figure is a
rectangle. Since there is a possibility that it
is not a rectangle, the argument it invalid. In
reading your text, you will find that this error
involves equating a statement and its inverse.
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• Remember that in determining whether an argument
  is valid, you must
• assume that the hypotheses are true, and
• determine whether the conclusion follows from
  the hypotheses.
• What matters is not whether the hypotheses and
  conclusion are actually true, but whether the
  conclusion follows deductively from the
  hypotheses.

Your text discusses deductive arguments in terms
of inverse, converse and contrapostive
statements. I will not use those terms on any
quizzes or tests. Instead, I will expect you to
determine whether an argument is true or false
and explain why by constructing a Venn Diagram.
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If you have found this topic challenging, do not
be alarmed. Logic is not something that you can
expect to catch on to right away. After reading
Section 1.5 in your text you should go through
this lecture again. The exercises have a link to
another lesson that you might want to explore. In
mathematics perseverance accompanied by
reflection eventually chisel away at the mode and
reveal whats underneath. Please feel free to
contact me with your questions.
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