Truth Tables
Keith Burgess-Jackson
12 October 2016
This handout has two parts. In Part I, I distinguish and discuss four uses
of truth tables. In Part II, I explain how to construct both long and short
truth tables.
I. The Four Main Uses of Truth Tables
Truth tables, like squares of opposition and Venn diagrams, are tools;
they help us perform certain tasks. The traditional (Aristotelian) and
modern (Boolean) squares of opposition are used to display the logical
relations between standard-form categorical propositions (All S are P; No
S are P; Some S are P; Some S are not P). Venn diagrams are used to test
the validity of standard-form categorical syllogisms (e.g., EAO-3). Truth
tables have at least four uses. (This should not be surprising. A claw
hammer can be used to pound a nail, pull a nail, stop a door, open a paint
can, keep paper from blowing away, and kill someone.) Here are the four
main uses of truth tables:
1. To define truth-functional connectives. A truth table shows
how the truth value of a truth-functional compound proposition
(such as ‘p ‫ ﬤ‬q’) depends on (i) the truth values of its component
(simple) propositions (‘p’ and ‘q’) and (ii) the logical connectives
involved (‘‫)’ﬤ‬. We can think of the truth table as defining the
various truth-functional connectives:
a.
b.
c.
d.
e.
tilde (‘’);
dot (‘•’);
wedge (‘’);
horseshoe (‘‫ ;)’ﬤ‬and
tribar (‘’).
2. To classify propositional forms. There are four logical properties of propositional forms:
a. tautologousness;
b. self-contradictoriness;
c. contingency; and
1
d. self-consistency.1
3. To compare propositional forms. There are nine logical relations between (among) propositional forms:
a.
b.
c.
d.
e.
f.
g.
h.
i.
logical implication;
logical equivalence;
contradictoriness;
contrariety;
subcontrariety;
subalternation;
independence;
consistency; and
inconsistency.2
4. To classify argument forms. There are two logical properties
of (deductive) argument forms:
a. validity; and
b. invalidity.
If there is any row of the truth table for the argument in which
the premises are true and the conclusion false, then the argument is invalid. If there is no such row, then the argument is
valid.
You will be tested on each of these concepts, as well as on your ability to
use truth tables in the ways just described.
II. How to Construct Truth Tables
The purpose of a truth table is to show how the truth value of a truthfunctional compound proposition depends on the truth values of its components. Truth tables may be long or short, depending on whether there
are separate columns for the simple propositions. I will illustrate this by
using both long and short truth tables to define the five truth-functional
connectives:
1
2
See the handout entitled “Logical Properties of Propositional Forms.”
See the handout entitled “Logical Relations Between (Among) Propositional
Forms.”
2
Truth-Functional
Connective

•

‫ﬤ‬

Long Truth Table
Short Truth Table
p | p
--|--T | F
F | T
p | q | p • q
--|---|-----T | T |
T
T | F |
F
F | T |
F
F | F |
F
p | q | p  q
--|---|-----T | T |
T
T | F |
T
F | T |
T
F | F |
F
p | q | p ‫ ﬤ‬q
--|---|-----T | T |
T
T | F |
F
F | T |
T
F | F |
T
p | q | p  q
--|---|-----T | T |
T
T | F |
F
F | T |
F
F | F |
T
p
-FT
TF
p • q
----T T T
T F F
F F T
F F F
p  q
----T T T
T T F
F T T
F F F
p ‫ ﬤ‬q
----T T T
T F F
F T T
F T F
p  q
----T T T
T F F
F F T
F T F
3