Laws of Propositional Logic
Laws of Propositional Logic | ||
|---|---|---|
Name | Sequent | Description |
Modus Ponens | If p then q; p; therefore q | |
Modus Tollens | If p then q; not q; therefore not p | |
Hypothetical Syllogism | If p then q; if q then r; therefore, if p then r | |
Disjunctive Syllogism | Either p or q, or both; not p; therefore, q | |
Constructive Dilemma | If p then q; and if r then s; but p or r; therefore q or s | |
Destructive Dilemma | If p then q; and if r then s; but not q or not s; therefore not p or not r | |
Bidirectional Dilemma | If p then q; and if r then s; but p or not s; therefore q or not r | |
Simplification | p and q are true; therefore p is true | |
Conjunction | p and q are true separately; therefore they are true conjointly | |
Addition | p is true; therefore the disjunction (p or q) is true | |
Composition | If p then q; and if p then r; therefore if p is true then q and r are true | |
De Morgan’s Theorem (1) | The negation of (p and q) is equivalent to (not p or not q) | |
De Morgan’s Theorem (2) | The negation of (p or q) is equivalent to (not p and not q) | |
Commutation (1) | (p or q) is equivalent to (q or p) | |
Commutation (2) | (p and q) is equivalent to (q and p) | |
Commutation (3) | (p is equivalent to q) is equivalent to (q is equivalent to p) | |
Association (1) | p or (q or r) is equivalent to (p or q) or r | |
Association (2) | p and (q and r) is equivalent to (p and q) and r | |
Distribution (1) | p and (q or r) is equivalent to (p and q) or (p and r) | |
Distribution (2) | p or (q and r) is equivalent to (p or q) and (p or r) | |
Double Negation | p is equivalent to the negation of not p | |
Transposition | If p then q is equivalent to if not q then not p | |
Material Implication | If p then q is equivalent to not p or q | |
Material Equivalence (1) | (p is equivalent to q) means (if p is true then q is true) and (if q is true then p is true) | |
Material Equivalence (2) | (p is equivalent to q) means either (p and q are true) or (both p and q are false) | |
Material Equivalence (3) | (p is equivalent to q) means, both (p or not q is true) and (not p or q is true) | |
Exportation | From (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true) | |
Importation | ||
Tautology (1) | p is true is equivalent to p is true or p is true | |
Tautology (2) | p is true is equivalent to p is true and p is true | |
Tertium non datur (Law of Excluded Middle) | p or not p is true | |
Law of Non-Contradiction | p and not p is false, is a true statement | |