Starting from true general statement or statements, logical steps prove conclusion true {deduction}. Deduction is true if premises are true.
Proposition proofs have finite numbers of steps {decision procedure}.
Proofs {existence proof} can try to show that something exists, preliminary to showing what it is like. Disproving non-existence or proving no non-existence cannot establish existence.
Logic {natural deduction} can have only inference rules, with no axioms. It reaches results but is not about truth. Natural deduction uses sequent calculus. Basic sequent statements are premises or conclusions. Statement sequence shows reasoning chain and conclusion. Introduction rules make more-complex formulas from simpler ones. Elimination rules change complex formulas to simpler formulas. Proofs and truth-trees eliminate formulas by reductio ad absurdum {cut elimination theorem, natural deduction}.
Proof methods {reductio ad impossibile} {reductio ad absurdum}| {indirect proof} {method of contradiction} {contradiction method} can assume that negative of theorem is true, and then prove that theorem or its premise is false, establishing contradiction. For any component-statement truth-values, contradictions are always false.
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Date Modified: 2022.0225