Thinking Mathematically

1
Thinking Mathematically

• Arguments and Truth Tables

2
Definition of a Valid Argument
An argument is valid if the conclusion is true
whenever the premises are assumed to be true. An
argument that is not valid is said to be an
invalid argument, also called a fallacy.
3
An Example of an Argument
p ? q If I get an A on the final I will pass
the course. p I got an A on the final. ?q I
will pass the course The argument is If I get
an A on the final I will pass the course and I
got an A on the final therefore I will pass the
course. (p ? q)/\p ? q
4
Valid Arguments
Valid arguments are tautologies. That is they
are always true.

• p ? q If I get an A on the final I will pass
  the course.
• p I got an A on the final.
• ?q I will pass the course

5
Testing the Validity of an Argument with a Truth
Table

• Use a letter to represent each simple statement
  in the argument.
• Express the premises and the conclusion
  symbolically.
• If the argument contains n premises, write the
  symbolic conditional statement of the form
  (premise 1)/\(premise 2)/\.../\(premise
  n)?conclusion.

6
Testing the Validity of an Argument with a Truth
Table

• Construct a truth table for the conditional
  statement in step 3.
• If the final column of the truth table has all
  trues, the conditional statement is a tautology,
  and the argument is valid. If the final column
  does not have all trues, the conditional
  statement is not a tautology, and the argument is
  invalid.

7

• Discuss the Standard Forms of Arguments and some
  Fallacies (page 144)
• Show Transitive Reasoning is valid. This uses a
  larger truth table.