Problem Solving Guide

Mathematical Logic · Axiom Academy

Master problem-solving strategies for all major topics in Mathematical Logic. Setup: List all atomic propositions first, then count rows needed (2 n where n is the number of atoms) Systematic evaluation: Build from innermost operations outward, one column per connective Check tautology: All true in final column = tautology; all false = contradiction; mixed = contingent Common mistake: Don't skip intermediate columns - they help catch errors and show your work Strategy: Identify the main connective first, then translate each component piece by piece Predicates and domains: Clearly define what each predicate means and specify the domain of quantification "Only": Translates to ∀x(P(x) → Q(x)) , not ∀x(Q(x) → P(x)) "Unless": Means "if not", so "P unless Q" becomes ¬Q → P or equivalently Q ∨ P "Except": Often requires conjunction with negation, e.g., "all except x" means ∀y(y ≠ x → ...) Proof Problems: Step-by-Step Strategy Choose your approach: Use direct proof when the implication is straightforward; use indirect proof (contradiction or contrapositive) when the negation gives you more to work with Work forwards and backwards: Start from premises, work toward conclusion. Also start from the conclusion and work backwards to see what you need Common patterns to recognize: Universal instantiation: From ∀x P(x) , derive P(c) for any constant c Existential generalization: From P(c) , derive ∃x P(x) Modus ponens: From P and P → Q , derive Q

This is the written version of the interactive lesson above. See the full Mathematical Logic course.