Foundations Summary

Intro to Proofs · Axiom Academy

SUMMARY Foundations of Logic and Proof How precise logical language and systematic reasoning combine to build mathematical proofs that establish truth with absolute certainty. Logic is the language; proofs are the method. Together they turn mathematics from computation into rigorous reasoning. A proof flows from hypothesis through justified logical steps to a conclusion — every step backed by a definition, axiom, or earlier theorem, with no gaps allowed. Connectives, quantifiers, and their negation rules give you the precise grammar to state exactly what you mean. Knowing the four core techniques — direct, contrapositive, contradiction, and induction — lets you choose the right strategy for the claim in front of you. Core Concept Propositional Logic A proposition is a declarative statement that is definitively true or false, with no ambiguity — exactly the precision mathematics requires. Logical connectives combine propositions: (and), (or), (not), (implies), (if and only if). Truth tables determine the truth value of a compound statement under every possible combination of inputs. Logical equivalence: two statements are equivalent when they share identical truth values in all cases — the basis for techniques like the contrapositive. Core Concept Predicate Logic & Quantifiers A predicate is a statement with variables that becomes a proposition once the variables are specified — e.g. P(x): " x is even."

This is the written version of the interactive lesson above. See the full Intro to Proofs course.