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Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.

History, Proofs and type theory & Classical and modal logics

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History

HilbertFregeRussellWhiteheadPrincipia Mathematica
27 related topics

Proofs and type theory

TurnstileInductionType theoryProductArrow
18 related topics

Classical and modal logics

Excluded middleHeytingParigotΛμCallcc
16 related topics

First and higher-order extensions

TermsCountablePredicatesTerm algebraAtomic formula
11 related topics

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Overview

History

Notation

Propositional language syntax

Gentzen-style propositional logic

Consistency, completeness, and normal forms

First and higher-order extensions

Proofs and type theory

Classical and modal logics

Comparison with sequent calculus

Advanced semantic analysis

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Map overview Semantic statistics

Natural deduction

Nodes126
Edges125
Triples109
Avg. degree1.98
Density0.015873
Components1

How this topic connects Entity context

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Natural deduction

Top relations

related to history · 35
Natural deduction → An, Fitch, Frege, Gentzen, Gentzen's, Gerhard Gentzen, German, Göttingen, Hilbert, His, IEP, In, It, Jaśkowski, Kleene, Kleene's, Lemmon, Natural, Notation, Poland
related to Modal substitution theorem · 12
Natural deduction → Avron, Belnap's, However, Kripke, Labels, Pottinger's, S5, Simpson, Stouppa, The, This, To
related to Substitution theorem · 11
Natural deduction → For, If, In, Normalisability, Propositions, Recall, So, The, Thus, To, Type
related to Comparison with sequent calculus · 9
Natural deduction → Gentzen, In, Inference, Introduction, Kleene, Metamathematics, The, Thus, To
related to Cut (substitution) · 7
Natural deduction → For, However, In, Now, Proof, This, Thus
related to Suppes–Lemmon-style inference rules · 6
Natural deduction → Fitch, Gentzen-style, Lemmon, Lemmon-style, Suppes, The
related to Gentzen's tree notation · 5
Natural deduction → Gentzen, In Gentzen's, Let's, Representing, This
related to Proofs and type theory · 5
Natural deduction → Following, The, This, To, We
see also · 4
Natural deduction → Argument, GentzenSystem, Mathematical, Philosophy
is a · 3
Natural deduction → kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the, pair of soundness and completeness theorems, syntactic proof system

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Important terminology

natural deduction rules logic calculus displaystyle proof theory sequent type notation proofs introduction rule inference elimination lemmon one propositions logical

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Natural deductionis akind of proof calculus in which logical reasoning is expressed by inference rules closely related to the0.90text
Natural deductionis asyntactic proof system0.90text
Natural deductionis apair of soundness and completeness theorems0.90text
Fitch notation or Suppes' methodinstance ofHis proposals led to different notations0.80text
for which Lemmon gave a variant now known as Suppesinstance ofHis proposals led to different notations0.80text
Patrick Suppesinstance ofwhere assumptions could be opened within a subderivation and discharged later.Later logicians and educators0.80text
Einstance ofwhere assumptions could be opened within a subderivation and discharged later.Later logicians and educators0.80text
the calculus of constructionsinstance ofPopular modern logical frameworks0.80text
LF are based on higher-order dependent type theoryinstance ofPopular modern logical frameworks0.80text
with various trade-offs in terms of decidabilityinstance ofPopular modern logical frameworks0.80text
expressive powerinstance ofPopular modern logical frameworks0.80text
labelling or systems of deep inference.The addition of labels to formulae permits much finer control of the conditions under which rules applyinstance ofextensions0.80text

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