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Maths › 03. Logic and Foundations ›

3.4 Proof Transformations

Proofs are not only objects that certify the truth of formulas, but also mathematical structures that can be analyzed and transformed, and studying these transformations reveals important properties about the nature of reasoning and the efficiency of different proof systems. A proof transformation takes a derivation and produces another derivation of the same conclusion, often with improved structure, reduced complexity, or clearer logical form, and such transformations are central in...
logic proof systems proof theory normalization May 2, 2026
Maths › 03. Logic and Foundations ›

3.5 Cut Elimination

The cut rule allows a proof to introduce an intermediate formula and then eliminate it, and while this rule can shorten derivations by reusing previously proved statements, it is not essential for the power of the sequent calculus, and its elimination reveals a deeper structural property of proofs. The Cut Rule The cut rule has the form: $$ \frac{\Gamma \vdash \Delta, A \qquad \Gamma, A \vdash \Delta}{\Gamma \vdash \Delta} $$...
logic proof systems sequent calculus cut elimination May 2, 2026
Maths › 03. Logic and Foundations ›

3.2 Sequent Calculus

The sequent calculus is a proof system in which reasoning is expressed in terms of sequents, and it makes the structure of logical arguments explicit by separating assumptions from conclusions and by allowing multiple formulas to appear on both sides of a derivation. A sequent has the form: $$ \Gamma \vdash \Delta $$ where $\Gamma$ and $\Delta$ are finite collections of formulas, and it is understood that $\Gamma$ represents assumptions...
logic proof systems sequent calculus May 2, 2026
Maths › 03. Logic and Foundations ›

3.1 Natural Deduction

Natural deduction is a proof system designed to reflect the way reasoning is carried out in ordinary mathematical arguments, where conclusions are obtained from assumptions by applying rules that correspond directly to the meaning of logical connectives. In this system, proofs are constructed step by step, and each step is justified either by an assumption or by applying an inference rule to earlier steps, so that the entire derivation forms...
logic proof systems natural deduction May 2, 2026
Maths › 03. Logic and Foundations ›

3.3 Hilbert Systems

Hilbert systems provide a very compact and minimalistic approach to formal proofs, where almost all logical reasoning is encoded in a fixed collection of axiom schemes, and only a small number of inference rules are used to derive new formulas. In contrast with natural deduction and the sequent calculus, where many rules correspond directly to logical connectives, Hilbert systems concentrate the logical content in axioms and rely heavily on repeated...
logic proof systems hilbert systems May 2, 2026