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SYMBOLIC LOGIC
From unpublished symbolic logic handouts by Harlan B. Miller.
| Negation | ~p |
| Conjunction | p & q |
| Disjunction | p ∨ q |
| Conditional | p → q |
| Biconditional | p ↔ q |
Truth Tables
| p | q | p & q | p ∨ q | p → q | p ↔ q |
|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
Inference Rules
| Name of Rule | Abbreviation | Argument Form |
|---|
| modus ponens | MP | p → q | p | ∴ q |
| modus tollens | MT | p → q | ~q | ∴ ~p |
| conjunction | Conj | p | q | ∴ p & q |
| simplification | Simp | p & q | ∴ p |
| | p & q | ∴ q |
| disjunctive syllogism | DS | p ∨ q | ~p | ∴ q |
| | p ∨ q | ~q | ∴ p |
| addition | Add | p | ∴ p ∨ q |
| | p | ∴ q ∨ p |
| biconditional modus ponens | BMP | p ↔ q | p | ∴ q |
| | p ↔ q | q | ∴ p |
| hypothetical syllogism | HS | p → q | q → r | ∴ p → r |
| dilemma | Dilm | (p → q) & (r → s) | p ∨ r | ∴ q ∨ s |
Equivalence Rules
| Name of Rule | Abbreviation | Form |
|---|
| double negation equivalence | DNE | p | equiv | ~~p |
| DeMorgan's equivalences | DeME | ~(p & q) | equiv | ~p ∨ ~q |
| | ~(p ∨ q) | equiv | ~p & ~q |
| commutation equivalences | ComE | p & q | equiv | q & p |
| | p ∨ q | equiv | q ∨ p |
| | p ↔ q | equiv | q ↔ p |
| transposition equivalence | TrnE | p → q | equiv | ~q → ~p |
| tautology equivalences | TauE | p | equiv | p ∨ p |
| | p | equiv | p & p |
| distribution equivalences | DstE | p ∨ (q & r) | equiv | (p ∨ q) & (p ∨ r) |
| | p & (q ∨ r) | equiv | (p & q) ∨ (p & r) |
| regrouping equivalences | RgrE | p & (q & r) | equiv | (p & q) & r |
| | p ∨ (q ∨ r) | equiv | (p ∨ q) ∨ r |
| | p ↔ (q ↔ r) | equiv | (p ↔ q) ↔ r |
| biconditional equivalence | BicE | p ↔ q | equiv | (p → q) & (q → p) |
| conditional equivalence | ConE | p → q | equiv | ~p ∨ q |
| exportation equivalence | ExpE | (p & q) → r | equiv | p → (q → r) |
| negated biconditional equivalence | NBE | ~(p ↔ q) | equiv | p ↔ ~q |
Contact Information
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