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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 licensed by the rules of the chosen proof system. This is different from checking whether the premises happen to be true in the real world, and it is not a general definition for legal, software, or statistical validity.
What does proof of validity mean?
An argument is valid when its conclusion follows from its premises: there is no interpretation in which all the premises are true and the conclusion is false. A formal proof demonstrates that relationship syntactically, by deriving the conclusion from the premises through permitted inference rules.
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For example, consider these premises:
- If the device is connected, its status light is on.
- The device is connected.
The conclusion “The status light is on” follows by modus ponens: from P → Q and P, infer Q. The conclusion is not merely guessed or recognized; the derivation identifies the rule that licenses it.
In an introductory setting, a proof is therefore a sequence of justified lines. The Colorado Community Colleges Online introduction to logic explains proofs and derivations as ways to demonstrate validity using established inference rules.
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How a formal proof differs from a semantic test
Both methods concern whether a conclusion follows from premises, but they establish success differently:
| Approach | What it examines | What counts as success |
|---|---|---|
| Semantic test | Interpretations or truth assignments | No interpretation makes every premise true and the conclusion false |
| Syntactic proof | A sequence of derivation steps | The conclusion is reached using rules permitted by the proof system |
A semantic counterexample shows invalidity: it supplies a case where the premises hold but the conclusion fails. A formal derivation instead makes explicit how the conclusion is obtained. Which approach is useful depends on the question; they are related methods, not the same procedure.
What makes each proof step valid?
A proof is not just a list of statements that seem plausible. Each derived line must follow from earlier lines or from an allowed premise or axiom, according to the rules of the system being used. The proof format and accepted rules matter: a line justified in one formal system may not be licensed in another.
- Premises are the starting statements the argument grants.
- Inference rules specify which conclusions may be drawn from which earlier statements.
- The conclusion is the target statement the derivation must reach.
- Justification identifies the rule or premise supporting each line.
To check a proposed proof, verify that it begins with the stated premises, that every step has a permitted justification, and that the final line is the intended conclusion. Identifying a conclusion without showing the licensed steps is not, by itself, a formal proof.
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Does a valid proof show that its premises are true?
No. A proof establishes that the conclusion follows if the premises are granted. It does not independently establish that those premises describe reality. Validity concerns the relation between premises and conclusion; truth concerns whether statements are in fact true. An argument can be valid while one or more premises are false.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Proof-theoretic validity in advanced logic
In some technical discussions, “validity” refers not simply to whether a written derivation follows rules, but to a proof-theoretic property defined relative to an underlying system. The Stanford Encyclopedia of Philosophy’s account of proof-theoretic semantics assesses proofs in relation to an 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 specialized use is system-relative: what qualifies depends on the chosen atomic base and the permitted reductions. The current Stanford Encyclopedia entry likewise discusses validity in relation to an atomic system. For an introductory logic question, the practical meaning is usually simpler: derive the conclusion from the premises, and justify every step under the specified rules.
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