At the end of the class, students should be able to:
- Define primitive statements
- Form a compound statements
- define negation, conjunction, disjunction, conditional, biconditional, tautology, contradiction and contingency statements.
- construct truth table.
Main components of symbolic logic are:
- Proposition/Statements
- Connectives
- A proposition is a declarative sentence that is either true or false.
- The truth or falsify of a statement is called truth value.
- Usually denoted by letters p, q, r, s and so on.
Example:
✔ You will be late to school if you miss the bus;
✔ Ms. Nina will have a broader audience next month;
✔ I did not join the competition;
✔ Today is Monday;
✔ I love reading thriller books;
✔ The number 3 is an odd integer;
| Logical Connective | Compound statement | Symbolic compound statement |
|---|---|---|
| ∧ | p and q (conjunction) | p ∧ q |
| ∨ | p or q (disjunction) | p ∨ q |
| → | if p then q (conditional/implication) | p → q |
| ↔ | p if and only if q (biconditional) | p ↔ q |
- If the compound statement has two variables (p and q) the truth table must be constructed with 4 rows.
- If 3 variables (p, q and r), then 8 rows.
Two primitive statements are defined as follows,
s: Your handbag is stylish
c: I like its colour.
Connectives join primitives statements into more complex statement:
Your handbag is stylish and I like its colour.
In symbolic: s ∧ c
Connective “but” has an identical role as “and”, thus, use same symbol ∧
Your handbag is stylish but I like its colour
In symbolic: s ∧ c
Answer :
- Boston is the capital of Massachusetts. Truth value: True.
- Miami is the capital of Florida. Truth value: False. (The capital of Florida is Tallahassee)
- 2 + 3 = 5. Truth value: True.
- 5 + 7 = 10. Truth value: False.
- x + 2 = 11
x + 2 = 11" not a proposition because it contains a variable (x) and is not a statement that can be determined as true or false without additional information about the value of x.
| p | ¬q |
|---|---|
| True | False |
| False | True |
- The operation “not” or ¬ turns single statement into negation and it is not a connective.
For example, the negation of I like you is I don’t like you.
p: I like you.
¬q: I don't like you.
Combine primary statements by the word “and”denoted by p ∧ q (if 2 statements)
If p is true and q is true, then p ∧ q is true. Otherwise, p ∧ q is false.
| p | q | p ∧ q |
|---|---|---|
| False | False | False |
| False | True | False |
| True | False | False |
| True | True | True |
p: I play the piano (false)
q: I study logic (true)
Thus,
p ∧ q: I play the piano and study logic is a false statement
Combine primary statements by the word “or”. Denoted by p ∨ q (if 2 statements)
If p is true or q is true or both p and q are true, then p ∨ q is true. Otherwise, p ∨ q is false.
| p | q | p ∨ q |
|---|---|---|
| False | False | False |
| False | True | True |
| True | False | True |
| True | True | True |
p: 2+3=6 (false)
q: 3>2 (true)
Thus,
p ∨ q: (2+3=6) or (3>2) is a true statement
A compound statement of the form “If p then q”, p → q In p → q, p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence).
If p is true and q is false, then the conditional p → q (p implies q) is false. Otherwise, p → q is true.
| p | q | p → q |
|---|---|---|
| False | False | True |
| False | True | True |
| True | False | False |
| True | True | True |
If Amy is a human being, then Amy has feeling. Defined primitive statement:
p : Amy is a human being (is the antecedent/hypothesis).
q : Amy has feeling (is the consequent).
Thus, the only way the statement is false , that is:
p → q: If Amy is a human being and she doesn’t have feeling.
If p and q have the same truth value, then p ↔ q is true. If p and q have opposite truth value, then p ↔ q is false.
| p | q | p ↔ q |
|---|---|---|
| False | False | True |
| False | True | False |
| True | False | False |
| True | True | True |