Proof and Precise Reasoning: From Arguments to Theorems

A proof is an argument that leaves no logical gap between accepted starting points and a claim over a stated domain. It is a different thing from an example, a computation, a picture or a simulation, however convincing those feel. Learning the grammar of proof is the quickest way to stop being fooled, by others and by yourself.

What a proof is, and is not

• An example shows a claim holds once. "Every odd number is prime" survives 3, 5 and 7, then fails at 9.
• A computation checks particular cases. It can refute a universal claim but cannot establish one.
• A diagram can suggest a truth and can also hide an assumption. Geometry proofs have long depended on features that merely look right in a figure.
• A simulation gives evidence about a model, not a theorem about every case.
• A proof derives the claim from definitions and accepted statements by steps that each follow. Demonstrated in the strict mathematical sense when the proof is valid.

The same discipline sits behind the trivium of grammar, logic and rhetoric described in The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium. Applied to mathematical work:

Anatomy of an argument

Any argument has premises, a conclusion, and usually some hidden assumptions. To evaluate it, state the conclusion, list the premises, add the unstated assumptions, check the inference, and then name the strongest objection. Reconstruct the other person's position fairly before criticizing it. Peirce on the Fixation of Belief shows the same discipline applied to ways of settling belief.

Validity, soundness, consistency and truth

These are different properties, and confusing them causes most bad arguments.

• Valid: if the premises were true, the conclusion would have to be true. Validity concerns form, not whether the premises are actually true.
• Sound: valid, and the premises are in fact true.
• Consistent: the statements can all be true together.
• True: a property of a single statement.

A valid argument can have a false conclusion if a premise is false. An invalid argument can have a true conclusion by luck. A classic invalid form is affirming the consequent: "If it rained, the road is wet; the road is wet; so it rained." The road could be wet for other reasons. Established

Propositional logic and truth tables

Statements combine with not, and, or, if...then and if and only if. A truth table lists the result for every assignment of true and false. The one that surprises people is the conditional: "if P then Q" is false only when P is true and Q is false. It does not claim that P causes Q, and it is not the same as its converse "if Q then P". Tables let you test whether two statements are equivalent, whether an argument is valid, and whether a set of statements is consistent. The contrapositive "if not Q then not P" always has the same truth value as the original.

Quantifiers and scope

"For all" and "there exists" are the quantifiers. Their order matters: "for every person there is someone they trust" is not "there is someone every person trusts". To negate: not (for all x, P) means there exists x with not P; not (there exists x, P) means for all x, not P. Practise these until they are automatic, because most invalid proofs go wrong at a quantifier.

The main proof forms

The infinitude of the primes (Euclid, Elements IX.20) is a standard proof by contradiction, and Aristotle (Prior Analytics I.23) already refers to proof by reductio that the diagonal of a square is incommensurable with its side. Demonstrated These are the techniques to learn from a text such as Hammack's Book of Proof.

Domain matters

State the domain before manipulating. "If x squared equals 4, then x equals 2" is false over the real numbers, because x could be negative 2. The same statement is true if the domain is restricted to positive numbers. A familiar-looking manipulation can fail when the domain is forgotten. The same applies to dividing by an expression that might be zero, or to taking square roots.

Counterexample and falsification

A single valid counterexample refutes a universal claim. This asymmetry is why "what would refute this?" is the most useful question in reasoning. It is the same move as the elenchus in Socratic questioning, and it underlies the testing habit in Hypotheses, Predictions and Tests and the way Evidence-Based Troubleshooting: Separating Explanations chooses the next test that separates competing causes. Euclid's method, where everything rests on stated definitions and postulates, is the ancestor of this style; Geometry, Euclid and Trigonometry shows how to study it by reconstruction.

Building a proof notebook

Keep a notebook with one entry per proof: the claim, the domain, the definitions used, your attempt, a corrected version, and a note of the first step that failed if you got stuck. Revisit selected proofs after a week and rewrite them without looking. The standard is validity, not elegance.

A proof checker, whether a person, a textbook solution or software, validates steps. It does not check that your formal statement faithfully captures the claim you meant. Translation from plain words to symbols is its own skill and needs separate practice.

Beyond the first course

Sets, functions and relations lead into Discrete Mathematics and Graphs: Counting, Relations and Networks. Further on, logic studies itself: the difference between syntax (what can be derived by rules) and semantics (what is true in a model). Godel's first incompleteness theorem (1931, in the form later sharpened by Rosser) says that any consistent, effectively axiomatized theory strong enough to express basic arithmetic contains statements it can neither prove nor refute. Demonstrated for such theories. It does not say that "nothing can be proved" or that "truth is unknowable"; popular summaries often overreach. Proof habits also carry into writing for others, as in Technical Writing as Reasoning, and into reading claims about health, where Reading Nutrition Evidence: What a Study Can and Cannot Show applies the same distinctions to study design.

Try this

1. Write a proof that the sum of two odd integers is even, then a proof by contrapositive that if n squared is odd, n is odd. State the domain at the start of each.
2. Negate in words and in symbols: "Every student passed some exam." Then invent a situation where the original is true and its (wrong) quantifier-swapped version is false.
3. Find a flaw: "I tested the formula on n = 1, 2, 3 and 4, so it holds for all n." Write what a proof would need to add.

Further reading

• Richard Hammack, Book of Proof (3rd ed., free online).
• Daniel Velleman, How to Prove It (3rd ed., Cambridge University Press, 2019).
• P. D. Magnus and others, forall x: Calgary Remix (Open Logic Project, free online).
• Charles S. Peirce, "The Fixation of Belief" (1877).
• Kurt Godel, "Uber formal unentscheidbare Satze..." (1931); for a readable modern treatment, the Open Logic Project's Incompleteness and Computability.