Logic Puzzles
Formal Logic Proofs
Engage with the precise machinery of propositional logic: conditionals, contrapositives, De Morgan's Laws, and valid inference forms. These exercises train you to distinguish between what an argument's structure guarantees and what it merely suggests — the foundation of mathematical reasoning, legal argumentation, computer programming, and any domain where precision of thought is non-negotiable.
Context
Why this exercise
Formal logic is the discipline that turns reasoning into something you can verify the way you verify arithmetic. Once you have the machinery — conditionals, contrapositives, De Morgan's Laws, valid inference forms — an argument either follows or it does not, and you can show the result step by step. This precision is what makes formal logic foundational for mathematics, computer science, law, philosophy, and any domain where a single broken step in a chain can produce confident conclusions that are entirely wrong. This exercise drills the core formal moves on examples drawn from realistic reasoning, not abstract symbol-pushing.
Before you start
Propositional logic, the level this exercise operates at, was systematized by George Boole in the 1850s and put on rigorous foundations by Gottlob Frege in the 1880s. The key insight is that statements have truth-functional structure: complex statements are built from simpler ones using connectives (and, or, not, if-then, if-and-only-if), and the truth of the whole is determined by the truth of the parts. This is what makes the truth-table method possible: enumerate every combination of truth values for the basic statements, compute the truth value of the complex statement in each case, and you have completely characterized when it is true and when it is false. Modus ponens (P → Q; P; ∴ Q) and modus tollens (P → Q; ¬Q; ∴ ¬P) are the two valid forms involving conditionals; affirming the consequent (P → Q; Q; ∴ P) and denying the antecedent (P → Q; ¬P; ∴ ¬Q) are the corresponding fallacies that fool listeners because they look symmetric to the valid forms.
Several specific relationships repay deliberate study. The contrapositive of P → Q is ¬Q → ¬P, and the two are logically equivalent — they say exactly the same thing in different words, which is why a deductive proof can always be replaced by the contrapositive proof of the same conditional. The converse of P → Q is Q → P, and the two are NOT equivalent — confusing them is the affirming-the-consequent fallacy. De Morgan's Laws give the rules for negating conjunctions and disjunctions: ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q, and ¬(P ∨ Q) is equivalent to ¬P ∧ ¬Q. These laws come up constantly in legal drafting, contract interpretation, software conditionals, and any setting where you need to negate a complex condition without making an error.
The practical payoff of formal logic is that it lets you check a conclusion mechanically rather than judging it by feel. When a chain of legal conditionals, a software conditional with multiple branches, or a contract clause with nested negations is making your head spin, the formal apparatus gives you a procedure: write out the propositional structure, apply the equivalences, and the answer either follows or it does not. Computer programs that verify mathematical proofs (Coq, Lean, Isabelle) and that check software correctness operate by exactly this kind of formal manipulation, scaled up. As you work the scenarios, practice translating English statements into propositional form before evaluating them, applying contrapositive and De Morgan transformations when they simplify the structure, and distinguishing valid forms from their fallacious near-twins. For the broader landscape of inference patterns, see Types of Reasoning.