Propositional Logic
January 2026
Propositions
A proposition is a statement which is either
trueorfalse.
Propositions can have truth values, for example here are propositions with a truth value of true:
- 1 is greater than 0
- Python is an interpreted programming language
- A week is longer than a day
And here are propositions with a false truth value:
- Green looks the same as Red
- C++ is an interpreted programming language
- A year is longer than a decade
Logic
Absolute truth values are not as important in logic, we focus more on propositions.
We aren’t philosophers, after all!
Axioms are propositions which are assumed to be true.
Logic is used to determine the consequences of these axioms.
We do not prove that axioms are true.
However, if the axiom is true, then so are its consequences - for example:
This shows that getting wet will be true if I stand in the rain.
However, if I don't stand in the rain, I may still get wet, we can’t tell.
Logical Operations
That’s right - It’s time to learn MORE NOTATION!!!P will refer to a proposition.
NOT ¬ (negation)
| P | ¬P |
|---|---|
T | F |
F | T |
Examples:
¬(2 + 2 = 5)istrue¬(6 + 6 = 12)isfalse
AND $\wedge$ (conjunction)
| P | Q | P $\wedge$ Q |
|---|---|---|
F | F | F |
F | T | F |
T | F | F |
T | T | T |
OR $\vee$ (disjunction)
| P | Q | P $\vee$ Q |
|---|---|---|
F | F | F |
F | T | T |
T | F | T |
T | T | T |
AND $\wedge$ (conjunction)
| P | Q | P $\wedge$ Q |
|---|---|---|
F | F | F |
F | T | F |
T | F | F |
T | T | T |
IMPLIES $\Rightarrow$ (implication)
If P is true, then Q must be true
| P | Q | P $\Rightarrow$ Q |
|---|---|---|
F | F | T |
F | T | T |
T | F | F |
T | T | T |
It does
NOT imply causation.LOGICALLY EQUIVALENT $\Leftrightarrow$ (equal)
| P | Q | P $\Leftrightarrow$ Q |
|---|---|---|
F | F | T |
F | T | F |
T | F | F |
T | T | T |
Precedence
Logical operations have the following prcedence:
- Not/negation
- And
- Or
- Implication
- Equivalence