Logic
Mathematics argues only about sentences that are true or false. How are they built, and when does one force another?
Where you are 9 inside
What this area is about
Logic is the grammar underneath every proof. It starts from statements, sentences that are true or false, asks how the truth of a sentence built from them depends on the truth of its parts, and ends with the question a proof has to answer: when do some statements force another to be true?
How these fit together
The starting point. Statements and negation: what counts as a statement, and \neg P, its opposite.
Building statements. A logical connective, such as “and” or “or”, joins statements into one whose truth depends only on the truth of its parts. Each connective is a definition. Conjunction and disjunction: P \land Q is true only when both parts are true, and P \lor Q is false only when both are false (“or” is inclusive). Conditionals cover “if ... then” and its turn-arounds. The biconditional, P \leftrightarrow Q, is true when the parts agree: two conditionals at once, (P \to Q) \land (Q \to P). Quantifiers, not connectives, add “for all” and “there exists”.
Deciding statements. Truth tables list every case. Logical equivalence holds results about statements whose tables agree, such as De Morgan’s laws.
Drawing conclusions. What an argument is sorts valid forms from fallacies.
A path through it
Negation, then and, or, the conditional, if and only if. Then truth tables and equivalence; quantifiers and arguments in either order.
Finish here and you can say what a claim asserts, what would break it, and whether a step follows, which is what direct proof, proof by contraposition and proof by contradiction are built on.