A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics $I_{\langle \alpha,\beta \rangle}$

Abstract

In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics I ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, such that I ⟨0,0⟩ is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbertstyle calculus HI ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, for each logic in question, in order to construct examples of effects of Glivenko theorem's generalization. Here we, first, propose a subordinated natural deduction system N I ⟨α,β⟩ , α, β ∈ {0, 1, 2, 3, . . . ω}, for each logic in question, with a precise definition of a N I ⟨α,β⟩ -inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each α, β ∈ {0, 1, 2, 3, . . . ω}, a Hilbert-style calculus HI ⟨α,β⟩ and a natural deduction system N I ⟨α,β⟩ are equipollent, that is, a formula A is provable in HI ⟨α,β⟩ iff A is provable in N I ⟨α,β⟩ .

The Logic of Internal Rational Agent

“…Let us introduce the natural deduction system ND ¬∼ LIRA which has the following inference and proof construction rules: Definition 4.1. The notion of an inference of a formula C from a set of formulas Γ in ND ¬∼ LIRA is understood in the spirit of [11, p. 366] and [35] as a finite non-empty sequence of formulas which satisfies the following conditions:…”