CHAPTER 1
Mathematical Logic and Reasoning
Valid Arguments
Section 1.6 - Arguments
Argument: A sequence of statements called
premises leading to a conclusion
Example: If p then q
p
therefore q Edna
If Edna studies, then Edna will get good grades.
Edna studies.
Therefore, Edna will get good grades.
Section 1.6 - Arguments
Argument: A sequence of statements called premises
leading to a conclusion
Example: Example:
x , x 0 x x
for all x, if p(x) then q(x)
3 0
p(c), for a particular c
3 (3)
therefore q(c)
Section 1.6 - Arguments
An argument is valid if and only if the form is
true regardless of the truth or falsity of the
statements in it.
Law of Detachment (or Modus Ponens)
Simple Form: Universal Form:
If p then q for all x, if p(x) then q(x)
p p(c), for a particular c
therefore q therefore q(c)
Section 1.6 - Arguments
Use truth tables to prove the Simple
Form of the Law of Detachment
Law of Detachment (or Modus Ponens)
Simple Form: Universal Form:
If p then q for all x, if p(x) then q(x)
p p(c), for a particular c
therefore q therefore q(c)
Section 1.6 - Arguments
The conclusion of a valid argument
is called a valid conclusion.
Rex
Rex is an English Sheepdog
Example:
If someone loves linguini, then they are Italian.
Rex loves linguini.
Therefore, Rex is Italian.
Note: A conclusion does not have to be true for the
argument to be valid
Section 1.6 - Arguments
Which of the following are valid argument forms?
If p then q If p then q
~p If q then r
Therefore q Therefore, if p then r
Law of Transitivity
Section 1.6 - Arguments
Law of Transitivity
Simple Form: Universal Form:
If p then q For all x, if p(x) then q(x)
If q then r For all x, if q(x) then r(x)
Therefore, if p then r Therefore, for all x,
if p(x) then r(x)
Give an example using the universal form!
Section 1.6 - Arguments
Law of Indirect Reasoning (or Modus Tollens)
Simple Form: Universal Form:
If p then q For all x, if p(x) then q(x)
~q ~ q(c) for a particular c
Therefore, ~ p Therefore, ~ p(c) for that c
1. Law of Detachment
Simple Form: Universal Form:
If p then q For all x, if p(x) then q(x)
Create p p(c), for a particular c
Therefore q Therefore q(c)
examples of
the three 2. Law of Transitivity
Simple Form: Universal Form:
types of If p then q For all x, if p(x) then q(x)
arguments: If q then r For all x, if q(x) then r(x)
Therefore, if p then r Therefore, if p(x) then r(x)
3. Indirect Reasoning
Simple Form: Universal Form:
If p then q For all x, if p(x) then q(x)
~q ~q(c) for a particular c
Therefore, ~p Therefore, ~p(c) for that c
Section 1.6 - Homework
Homework: Read Section 1.6 and do 1-15
1.6 - Valid Arguments
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