Propositions and connectives
A proposition is a statement that is either true or false. Connectives join propositions: not (¬), and (∧), or (∨), implies (→) and if and only if (↔).
| p | q | ¬p | p ∧ q | p ∨ q | p → q | p ↔ q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
A tautology is always true (p ∨ ¬p), and a contradiction is always false (p ∧ ¬p). The converse of p → q is q → p, the inverse is ¬p → ¬q and the contrapositive is ¬q → ¬p, which is equivalent to the original.
Set p and q to 1 (true) or 0 (false). The connectives are computed.
Arguments
An argument has premises and a conclusion. It is valid if the conclusion must be true whenever the premises are true. Common valid forms are modus ponens (p → q, p, therefore q) and modus tollens (p → q, ¬q, therefore ¬p). A valid argument can still have a false conclusion if a premise is false.
Electrical networks
A switch network is a logic circuit. Two switches in series act as AND (current flows only if both are closed). Two switches in parallel act as OR (current flows if either is closed). The statement p ∧ (q ∨ r) is a switch p in series with the parallel pair q and r.
Key terms
- Proposition
- A statement that is true or false.
- Tautology
- A statement that is always true.
- Modus ponens
- From p → q and p, conclude q.
- Contrapositive
- The statement ¬q → ¬p.
Summary
- Connectives: ¬, ∧, ∨, →, ↔ have truth tables.
- An argument is valid if the conclusion follows from the premises.
- Series switches are AND; parallel switches are OR.
- The contrapositive is equivalent to the original statement.