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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. [1] This contrasts with Hilbert-style systems , which instead use axioms as much as possible to express the logical laws of deductive reasoning .
The class of Hilbert systems, [2] of which the most famous example is the 1928 Hilbert–Ackermann system of first-order logic; Gerhard Gentzen's calculus of natural deduction, which is the first formalism of structural proof theory, and which is the cornerstone of the formulae-as-types correspondence relating logic to functional programming;
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi systems, LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although ...
First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.
There are numerous deductive systems for first-order logic, including systems of natural deduction and Hilbert-style systems. Common to all deductive systems is the notion of a formal deduction. This is a sequence (or, in some cases, a finite tree) of formulae with a specially designated conclusion.
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables.
The weakening rule may be expressed as a natural deduction sequent: Γ ⊢ C Γ , A ⊢ C {\displaystyle {\frac {\Gamma \vdash C}{\Gamma ,A\vdash C}}} This can be read as saying that if, on the basis of a set of assumptions Γ {\displaystyle \Gamma } , one can prove C, then by adding an assumption A, one can still prove C.