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In formal logic, a proof of validity is a rule-governed derivation that starts with an argument’s premises and reaches its conclusion. Each step must be justified by an inference rule accepted in the chosen proof system. The phrase can mean different things in other fields, so this definition is specifically about logic.
What does proof of validity mean?
A proof of validity shows, within a formal system, how a conclusion follows from stated premises. It is not just a statement of the conclusion or an assertion that the argument seems convincing: it is a sequence of steps, with each derived line licensed by an accepted rule. The system’s rules and proof format determine which steps are permitted. Colorado Community Colleges Online’s introductory logic text presents proofs and derivations as ways to demonstrate validity by applying inference rules.
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For example, from the premises “If it rains, the ground gets wet” and “It rains,” the conclusion “The ground gets wet” follows by modus ponens. A formal proof would identify the premises, derive the conclusion, and cite modus ponens as the rule that licenses the final step. The example illustrates the structure; a course or proof system may require a particular notation or line format.
What does validity say about an argument?
Validity concerns the relationship between premises and conclusion, not whether the premises happen to be true in the real world. An argument is invalid if there is an interpretation under which every premise is true while the conclusion is false. If no such counterexample exists under the relevant semantics, the argument is semantically valid.
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A derivation provides a syntactic demonstration: it constructs the conclusion from the premises using the system’s permitted rules. A successful proof therefore shows that the conclusion follows from those premises under that system. It does not independently establish that the premises are factually true. A valid argument can have false premises; truth of premises is a separate question.
How is a formal proof different from a semantic test?
Both approaches address argument validity, but they establish it in different ways. A semantic test examines interpretations or truth assignments; a formal proof examines a derivation’s steps.
| Approach | What is evaluated? | What establishes success? | What assumptions matter? |
|---|---|---|---|
| Semantic test | Interpretations or truth assignments for the premises and conclusion | No interpretation makes all premises true and the conclusion false | The intended semantics and the interpretations considered |
| Formal derivation | The lines and inference steps in a proof | A derivation reaches the conclusion using permitted rules | The proof system’s axioms, rules, and proof format |
A counterexample can show that an argument is invalid. A derivation can make explicit how a conclusion follows and where each inference is licensed. One is not simply a different way of writing the other: the first reasons about interpretations, while the second reasons through a rule-governed proof.
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What does “valid proof” mean in proof-theoretic semantics?
In more advanced work, “validity” may refer not only to an argument’s semantic status but to whether a proof meets technical criteria in a specified proof-theoretic framework. The Stanford Encyclopedia of Philosophy’s Spring 2013 entry on proof-theoretic semantics describes proof validity relative to an underlying atomic system and accepted proof reductions. It distinguishes closed canonical proofs, closed noncanonical proofs that reduce appropriately, and open proofs assessed through closed substitutions.
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That account is system-relative: deciding whether a proof is valid requires knowing the chosen base system and what reductions it accepts. It should not be collapsed into the introductory definition of a derivation or treated as a universal test for any written proof. The current Stanford Encyclopedia entry likewise discusses validity relative to an atomic system.
How to read or write an introductory proof
- Identify the premises and target. Write down the statements assumed and the conclusion the proof must reach.
- Choose the stated proof system. Use its notation, axioms, and permitted inference rules rather than assuming every textbook uses the same format.
- Derive one line at a time. For each new statement, identify the earlier line or lines that support it and the rule that licenses the inference.
- Check the final line. The derivation must reach the specified conclusion; an unsupported jump or a merely similar statement does not complete the proof.
This line-by-line approach is useful because it exposes exactly where an inference is licensed. For introductory study, the Colorado Community Colleges Online text provides a treatment of premises, derivations, and inference rules in A Concise Introduction to Logic: CCA Edition.
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Proof of validity is not proof of factual truth
A proof establishes a consequence relative to premises and a formal system; it does not show on its own that the premises describe reality. To establish a conclusion as true in an everyday or empirical sense, the premises also need support. Keep these questions separate: “Does the conclusion follow from these premises?” asks about validity; “Are these premises true?” asks about their truth.
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