Symbolic Logic: Propositional Calculus
What this domain tests
Translating ordinary arguments into symbols and testing them by truth-functional analysis. You must use the standard symbols, build well-formed formulas, and classify statement forms as tautology, contradiction or contingent.
Symbols
- Negation: ~p (not p)
- Conjunction: p · q (and)
- Disjunction: p ∨ q (or, inclusive)
- Conditional: p ⊃ q (if p then q)
- Biconditional: p ≡ q (if and only if)
Key notions
- A statement form is a schema; a substitution instance fills it with actual statements.
- Tautology: true on every assignment (p ∨ ~p).
- Contradiction: false on every assignment (p · ~p).
- Contingent: true on some assignments, false on others.
- Forms with identical truth columns are logically equivalent, e.g. De Morgan: ~(p · q) ≡ ~p ∨ ~q.
Worked example
Test (p · q) ⊃ p. Rows for (p, q): (T,T) gives T ⊃ T = T; (T,F) gives F ⊃ T = T; (F,T) gives F ⊃ T = T; (F,F) gives F ⊃ T = T. Every row is true, so the form is a tautology - a valid argument schema. Note the main operator is ⊃, and that alone determines the final column.
Traps that cost marks
Reading ⊃ as a causal or temporal "implies"; assuming p ⊃ q is equivalent to q ⊃ p, when the valid converse is contraposition, ~q ⊃ ~p; misidentifying the main operator; forgetting that inclusive "or" is true when both disjuncts are true.