Argument

An argument is a group of propositions in which one (the conclusion) is claimed to follow logically from the others (the hypotheses). In other words, an argument claims that a conclusion can be reached through a series of inferences that maintain truth from the hypotheses to the conclusion.

$h_1$
$...$
$\underline{h_n\,\,}$
$\therefore c$

Above is the typical format of an argument, where $h_1 ... h_n$ are the hypotheses and $c$ is the conclusion. The corresponding conditional of this argument is:

$$(h_1 \land ... \land h_n) \rightarrow c$$

⚠ The order in which you write the hypotheses of an argument does not matter. It will not change the meaning of the argument. This is because conjunction is commutative.

Contents

Validity

When the corresponding conditional of an argument is a tautology, the argument is said to be valid. This means that the conjunction of the hypotheses implies the conclusion. In a valid argument, for every possible combination of truth values that make all the hypotheses true, the conclusion is also true.

A truth table can help you determine the validity of an argument. Create columns for the hypotheses, the conclusion, and any individual propositions that make them up. If the argument is valid, the conclusion must be true in every row where all the hypotheses are true. If there is even a single exception to this rule, the argument is rendered invalid.

It's important to be able to recognize invalid arguments to avoid making false conclusions.

⚠ However, just because an argument is valid does not mean its conclusion is true! Validity only means that the conclusion would be true if all the hypotheses are true, which they might not be.

Alternatively, if you want to show an argument is invalid without using a truth table, come up with some assignment of truth values to its individual propositions that make all the hypotheses true and the conclusion false. You need only one such assignment to prove an argument is invalid.

Form

The form of an argument in English can be examined by rewriting all of its propositions and logical connectives as letters and symbols. It is possible for different arguments to have the same form, when their subject matters differ but the logic behind them is the same.

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