→ Biconditional(↔):
Given two propositions p and q, the biconditional p↔q is a proposition that is true when p and q have same truth value, meaning they are either both true or both false and false otherwise.
•It is also known as “if and only if statement”.
Example:
p=You can enter the restricted area .
q= You have a valid access card.
p↔q=”You can enter the restricted area if and only if you have a valid access card.”
Truth Table
| p | q | p↔q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
• “p” and “q” represent the truth values of the propositions.
• “p ↔ q” represents the truth value of the biconditional statement.
→ Tautology:
A tautology is a compound proposition that is always true, no matter what the truth values of the atomic propositions.
Truth Table
| p | ⁓p | p⋁⁓p |
| T | F | T |
| F | T | T |
• “p” represents the truth value of a proposition.
• “⁓p” represents the negation (NOT) of proposition p.
• “p⋁⁓p” represents the tautology.
→ Contradiction:
A contradiction is a compound proposition that is always false, regardless of the truth values of its component propositions.
Truth Table
| p | ⁓p | pΛ⁓p |
| T | F | F |
| F | T | F |
• “p” represents the truth value of a proposition.
• “⁓p” represents the negation (NOT) of proposition p.
• “pΛ⁓p” represents the contradiction.
→ Contingency:
A contingency is a compound proposition that is neither a tautology (always true) nor a contradiction (always false).
Note: To clarify, “contingency” is not a formal term in logic like “tautology” or “contradiction.” Rather, a contingency is a term used to describe a logical statement that can be true in some cases and false in others, depending on the specific truth values of its component propositions.
A contingency doesn’t have a specific truth table associated with it.
# Show that pΛq is contingency.
| p | q | pΛq |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
