This document defines and explains logical concepts such as simple and compound statements, truth tables, logical operators like negation and conjunction, and argument validity. It discusses translating statements into symbolic logic using variables, determining statement truth values from truth tables, and classifying statements as tautologies, contradictions, or contingencies. Common valid argument forms like modus ponens and modus tollens are also defined.

1.  Simple Statements:
◦ James Joyce wrote Ulysses.
 Compound Statements:
◦ Contain at least one simple statement.
◦ It is not the case that Al Qaeda is a humanitarian
organization translated to: It is not the case that A.

2.  Translating and Symbolizing Negation,
Conjunction, Disjunction, Implication and
Equivalence.

3.  Example of negation
◦ It is not the case that A: ~A.
 Example of conjunction
◦ Tiffany sells jewelry and Gucci sells cologne: T• G

4.  Example of disjunction:
◦ Aspen sells snowboards or Teluride does: A ˅ T.
 Examples of conditionals (implication):
◦ If Purdue raises tuition, then so does Notre Dame: P
⊃ N.

5.  Examples of Biconditionals (equivalence):
◦ JFK tightens security if and only if O’Hare does: J ≡ O.
 Common Confusions:

6.  A Truth Function is any compound proposition
whose truth value is completely determined by the
truth values of its components.
 The definitions of the Logical Operators are
presented in the forms of Statement Variables,
which are lowercase letters that can stand for any
statement.

7.  A Truth Table is an arrangement of truth values
that shows in every possible case how the truth
value of a compound proposition is determined by
the value of its simple components.

8.  Computing the truth value of Longer
Propositions:
If A, B, and C are true and D, E and F are false, then :
(A v D) ⊃ E with these truth values below:
(T v F) ⊃ F. Next:
T ⊃ F. Finally:

9.  Further Comparisons with Ordinary Language:
◦ She got married and had a baby: M• B.
◦ She had a baby and got married: B • M.

10.  Constructing Truth Tables: the relationship
between the columns of numbers is L = 22
Number of
different simple
propositions
Number of lines in
truth table
1 2
2 4
3 8
4 16
5 32
6 64

11.  An Alternate Method:
◦ [(a v b) • (b ⊃ a)] ⊃ b
a b [(a v b) • (b ⊃ a)] ⊃ b
T T T T T T
T F T T T F
F T T F F T
F F F F T T

12.  Classifying Statements:
◦ A Tautologous or Logically True Statement is true
regardless of the truth values of its components.
◦ A Self-contradictory or Logically False Statement is false
regardless of the truth values of its components.
◦ A Contingent Statement’s truth value varies according to
the truth values of its components.

13.  Comparing Statements:
◦ Logically Equivalent statements have the same truth
value on each line under their main operators.
◦ Contradictory Statements have the opposite truth value
on each line under their main operators. Consistent
Statements have at least one line on which both (or all)
the truth values are true.
◦ Inconsistent Statements have no line on which both (or
all) the truth values are true.
◦ Compare main operator columns; logically equivalent
and contradictory statements take precedence over
consistent and inconsistent ones

14.  Constructing a Truth Table for an Argument:
◦ Symbolize the argument using letters for simple
propositions.
◦ Write out the argument using a single slash between
premises and a double slash between the last premises
and the conclusion.
◦ Draw a truth table for the symbolized argument as if it
were a proposition broken into parts.
◦ Look for a line where all the premises are true and the
conclusion false. If there is no such line, the argument is
valid. If there is not, it is invalid.

15.  They are shorter and faster than ordinary truth
tables and work best for arguments with a large
number of simple propositions.
◦ However, you must be able to work backward from the
truth value of the main operator of a compound
statement to the truth values of its simple components.

16.  Testing Arguments for Validity
1. Assume the argument is invalid.
2. Work backward to derive the truth values of the
separate components.
3. Can you derive a contradiction? If you can, it is not
possible for the premises to be true and the
conclusion false, so the argument is valid.

17.  Testing Statements for Consistency is similar to
the method for testing arguments.
1. Assume the statements are consistent.
2. Can you derive a contradiction?

18.  An Argument Form is an arrangement of
statement variables and operators so that
uniformly substituting statements in place of
variables results in arguments. Common forms
are as follows:
◦ Disjunctive Syllogism (DS)
p v q
~p
q

19. ◦ Pure Hypothetical Syllogism (HS):
p ⊃ q
q ⊃ r
p ⊃ r
◦ Modus Ponens (MP):
p ⊃ q
p
q

20. ◦ Modus Tollens (MT):
p ⊃ q
~q
~p
◦ Affirming the Consequent (AC):
p ⊃ q
q
p

21. ◦ Denying the Antecedent (DA):
p ⊃ q
~p
~q
◦ Constructive Dilemma (CD):
(p ⊃ q) • (r ⊃ s)
p v r
q v s
◦ Destructive Dilemma (DD):
(p ⊃ q) • (r ⊃ s)
~q v ~s
~p v ~r

22.  Refuting Constructive and Destructive Dilemmas.

23.  Note on Invalid Forms: any substitution instance
of a valid argument form is a valid argument.
However, this result does not apply to invalid
forms.