1. Propositional Logic – The Basics (2)
Truth-tables for Propositions

2. Assigning Truth L ● ~ F 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? L ● F T F F
“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:
The truth value of it’s component, simple propositions, plus
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.
T T F F
T T F F

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.
T T F F
T T F F
T F
T F
T F
T F

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.
T T F F
T T F F T
T F
T F T
F F
F F F
T F
T F 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 1. 2. 3. 4. ( M > P ) v ( P > M ) T T F F T F T T
Using the tables which define the operators, determine the values of this statement. 1. 2. 3. 4.
( M > P ) v ( P > M )
T T F F
T F T T
T F T F
T T T T
T F T F
T T F T
T T F F

15. Exercises – 2
Using the tables which define the operators, determine the values of this statement.
[(Q
> P) ●
( ~ Q
R)] ~ (P v R) 1. 2. 3. 4. 5. 6. 7. 8.
T T T T F F F F
T T F F T T T T
T T F F T T F F
T T F F T F T F
F F F F T T T T
T T T T F F F F
T T T T T F T F
T F T F T F T F
F F F F F F F F
F F F T F F F T
T T F F T T F F
T T T F T T T F
T F T F T F T F
T T F F T T F F