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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 .
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 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.
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.
The definition of a deduction is such that it is finite and that it is possible to verify algorithmically (by a computer, for example, or by hand) that a given sequence (or tree) of formulae is indeed a deduction. A first-order formula is called logically valid if it is true in every structure for the language of the formula (i.e. for any ...
It is an open problem whether this theory is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. [ 2 ] [ 3 ] In contrast, the extension of the theory of real closed fields with the sine function is undecidable since this allows encoding of the undecidable theory of integers (see Richardson's theorem ).
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 ...
In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic [1] (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.