Propositional Logic
– The Basics (2)
Truth-tables for Propositions
Assigning Truth
True or false? –
“This is a class in introductory-level logic.”
“This is a class in introductory-level logic,
which does not include a study of informal
fallacies.”
L●~F
How about this one?
“This is a class in introductory logic,
logic (T),
which
which
includes
includes
a study
a study
of informal
of informal
fallacies.”
fallacies
(F).”
L ● F
T F F
Propositional Logic and Truth
The truth of a compound proposition is a
function of:
a. The truth value of it’s component, simple
propositions, plus
b. the way its operator(s) defines the relation
between those simple propositions.
p●q
pvq
T F F
T T F
Truth Table Principles and Rules
Truth tables enable you to determine the
conditions under which you can accept a
particular statement as true or false.
Truth tables thus define operators; that is,
they set out how each operator affects or
changes the value of a statement.
Truth and the Actual World
Some statements describe the actual world - the existing
state of the world at “time x”; the way the world in fact is.
“This is a logic class and I am seated in SOCS 203.”
- Actually and currently true on a class
day.
- Possibly true, but not “currently” true on
Monday, Wednesday or Friday.
Truth and Possible Worlds
Some statements describe possible worlds - particular
states of the world at “time y”; a way the world could be..
“This is a history class and I am seated in SOCS 203.”
Possibly true, but not currently true.
Actually true, if you have a history class here
and it is a history class day/time.
A truth table describes all possible combinations of truth
values for a statement. It will, in fact, even tell you if a
statement could not possibly be true in any world.
Constructing Truth Tables
1. Write your statement in symbolic form.
2. Determine the number of truth-value lines you must
have to express all possible conditions under which
your compound statement might or might not be true.
Method: your table will represent 2n power, where n = the
number of propositions symbolized in the statement.
3. Distribute your truth-values across all required lines
for each of the symbols (operators will come later).
Method: Divide by halves as you move from left
to right in assigning values.
Constructing Truth Tables - # of Lines
For statement forms, there are only two symbols. Thus,
these require lines numbering 22 power, or 4 lines.
p
q
p
1.
1.
2.
2.
3.
3.
4.
4.
q
Constructing Truth Tables – Distribution
across all Symbols
Under “p,” divide the 4 lines by 2. In rows 1 & 2 (1/2 of 4 lines), enter “T.” In
rows 3 & 4, (the other ½ of 4 lines), enter “F.”
p
●
q
p
1.
T
1.
T
2.
T
2.
T
3.
F
3.
F
4.
F
4.
F
≡
q
Constructing Truth Tables – Distribution across all
Symbols
Under “q,” divide the 2 “true” lines by 2. In row 1 (1/2 of 2 lines), enter “T.” In
row 2, (the other ½ of 2 lines), enter “F.”
Repeat for lines 3 & 4, inserting “T” and “F” respectively.
p
●
≡
q
p
q
1.
T
T
1.
T
T
2.
T
F
2.
T
F
3.
F
T
3.
F
T
4.
F
F
4.
F
F
Constructing Truth Tables – Operator Definitions
Thinking about the corresponding English expressions for each of
the operators, determine which truth value should be assigned for
each row in the table.
p
●
q
p
≡
q
1.
T
T
T
1.
T
T
T
2.
T
F
F
2.
T
F
F
3.
F
F
T
3.
F
F
T
4.
F
F
F
4.
F
T
F
Constructing Truth Tables - # of Lines
Remember that you are counting each symbol, not how
many times symbols appear.
(p
≡
q)
●
q
1.
2.
3.
4.
2 symbols: 1 appearance of “p” and 2 appearances of “q”
Exercises - 1
Using the tables which define the operators, determine the values of this
statement.
(M
>
P )
v
(P
>
M )
T
T
T
T
T
T
T
T
F
F
T
F
T
T
F
T
T
T
T
F
F
F
T
F
T
F
T
F
1.
2.
3.
4.
Exercises – 2
Using the tables which define the operators, determine the values of this statement.
[(Q >
P)
●
(~
Q
>
R)] ●
~
(P
v
R)
1.
T
T
T
T
F
T
T
T
F
F
T
T
T
2.
T
T
T
T
F
T
T
F
F
F
T
T
F
3.
T
F
F
F
F
T
T
T
F
F
F
T
T
4.
T
F
F
F
F
T
T
F
F
T
F
F
F
5.
F
T
T
T
T
F
T
T
F
F
T
T
T
6.
F
T
T
F
T
F
F
F
F
F
T
T
F
7.
F
T
F
T
T
F
T
T
F
F
F
T
T
8.
F
T
F
F
T
F
F
F
F
T
F
F
F