# 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 the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.

### Other articles related to "natural, natural deduction, deduction":

Metamath - A Generic Proof Checker
... low level would make it difficult) it could possibly even be a syntactic checker for a natural language (same remark) ... The original natural deduction system (due to Gerhard Gentzen), which uses an extra stack, is an example of a system that cannot be implemented with Metamath ... In the case of natural deduction however it is possible to append the stack to the formulas (transforming the natural deduction formulas into a sort of ...
Sequent Calculus - Introduction
... One way to classify different styles of deduction systems is to look at the form of judgments in the system, i.e ... simplest judgment form is used in Hilbert-style deduction systems, where a judgment has the form where is any formula of first-order-logic (or whatever logic the deduction ... Concrete arguments about proofs in such a system almost always appeal to the deduction theorem ...
Curry–Howard Correspondence - Correspondence Between Natural Deduction and Lambda Calculus
... emphasized the syntactic correspondence between Hilbert-style deduction and combinatory logic, Howard made explicit in 1969 a syntactic analogy ... formalizes intuitionistic implicational natural deduction as a calculus of sequents (the use of sequents is standard in discussions of the Curry–Howard isomorphism as it allows the deduction ... Intuitionistic implicational natural deduction Lambda calculus type assignment rules To paraphrase the correspondence, proving Γ ⊢ α means having a program that, given values with the types listed in ...
Natural Deduction - Comparison With Other Foundational Approaches - Sequent Calculus
... The sequent calculus is the chief alternative to natural deduction as a foundation of mathematical logic ... In natural deduction the flow of information is bi-directional elimination rules flow information downwards by deconstruction, and introduction rules flow information upwards by assembly ... Thus, a natural deduction proof does not have a purely bottom-up or top-down reading, making it unsuitable for automation in proof search ...
Proof-theoretic Semantics
... the meaning of logical connectives in their introduction rules within natural deduction ... Dag Prawitz extended Gentzen's notion of analytic proof to natural deduction, and suggested that the value of a proof in natural deduction may be understood as its normal form ... by means of cut-elimination theorems and for natural deduction by means of normalisation theorems ...

### Famous quotes containing the word natural:

In my state, on the basis of the separate but equal doctrine, we have made enormous strides over the years in the education of both races. Personally, I think it would have been sounder judgment to allow that progress to continue through the process of natural evolution. However, there is no point crying about spilt milk.
Lyndon Baines Johnson (1908–1973)