Logical proof

A logical proof is a proof that shows an argument is valid by using logic laws and rules of inference to go from hypotheses to conclusion while preserving truth. The format of a logical proof differs from that of all other proofs, as you'll see below.

Logical proofs take the form of a table with numbered rows, a proposition column, and a justification column. First, understand what the hypotheses and conclusion of the argument are. Then, introduce the hypotheses into the first few rows, each with the justification "Hypothesis". From there, you can freely use logic laws and rules of inference to obtain new propositions that bring you closer to the conclusion. Make sure to justify each step, mentioning the name of which law or rule brought you to that proposition, as well as the row(s) it was applied to.

1. Proposition Hypothesis
2. Proposition Hypothesis
3. Proposition Hypothesis
4. Proposition Modus ponens (1, 2)
5. Proposition Hypothetical syllogism (3, 4)
⚠ Once a step has been justified, the justified proposition is known to be true. This allows you to use it as a hypothesis for a rule of inference! To speak more broadly, every justified proposition in a logical proof is true. So, never hesitate to use a proposition from a previous row because it's supposedly too far away... it's always in your toolkit.
Logic & Proofs
Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form
Set Theory
Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof
Functions
Function • Floor function • Ceiling function • Inverse function
Algorithms
Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm
Relations
Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class
Number Theory
Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm
Induction
Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition
Combinatorics
Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle
Graph Theory
Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm
Recursion
Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort