→ 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.
Thank you for reading this post, don't forget to subscribe!•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 |