Propositional Logic

1
Propositional Logic The Basics (2)

• Truth-tables for Propositions

2
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.
This is a class in introductory-level logic,
which does not include a study of informal
fallacies.
L ? F
3
How about this one?

• This is a class in introductory logic, which
  includes a study of informal fallacies.

This is a class in introductory logic (T), which
includes a study of informal fallacies (F).
L ? F T F
F
4
Propositional Logic and Truth

• The truth of a compound proposition is a function
  of

1. The truth value of its component, simple
   propositions, plus
2. the way its operator(s) defines the relation
   between those simple propositions.

p ? q p v q
T F T F
F
T
5
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.

6
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.
7
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.
8
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.
9
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
1.
2.
3.
4.
p q
1.
2.
3.
4.
10
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
1.
2.
3.
4.
p q
1.
2.
3.
4.
TTFF
TTFF
11
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
1.
2.
3.
4.
p q
1.
2.
3.
4.
TTFF
TTFF
TF
TF
TF
TF
12
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
1.
2.
3.
4.
p q
1.
2.
3.
4.
TTFF
TTFF
TF
TF
T
T
FF
FFF
TF
TF
T
13
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
14
Exercises - 1
Using the tables which define the operators,
determine the values of this statement.

1.
2.
3.
4.
( M gt P ) v ( P gt M )
TTFF
TFTF
TTFF
TFTF
TFTT
TTFT
TTTT
15
Exercises 2 Using the tables which define the
operators, determine the values of this statement.
(Q gt P) ? ( Q gt R) ? (P v R)
1.
2.
3.
4.
5.
6.
7.
8.
TTTTFFFF
TTTTFFFF
TTFFTTFF
TTFFTTFF
TFTFTFTF
TTFFTTFF
FFFFTTTT
TFTFTFTF
TTFFTTTT
TTTTTFTF
TTTFTTTF
FFFTFFFT
TTFFTFTF
FFFFFFFF