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Classical and Intuitionistic Subexponential Logics are Equally Expressive
"... Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by doublenegation, while the other direction has no truthpreserving propositional ..."
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Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by doublenegation, while the other direction has no truthpreserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics. 1
“macrorule”. In
, 2011
"... Focused proof systems such as LKF allow us to change the size of inference rules with which we work. Let us call individual introduction rules “microrules”. An entire phase within a focused proof can be seen as a ..."
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Focused proof systems such as LKF allow us to change the size of inference rules with which we work. Let us call individual introduction rules “microrules”. An entire phase within a focused proof can be seen as a
The inference rules for the LKF focused proof system [1] for classical logic are given in Figure 1. Structural Rules
, 2011
"... F ocus ⊢ ¬P, Θ ⇓ P Id (literal P) Introduction of negative connectives ⊢ Θ ⇑ Γ, t − ⊢ Θ ⇑ Γ, A ⊢ Θ ⇑ Γ, B ..."
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F ocus ⊢ ¬P, Θ ⇓ P Id (literal P) Introduction of negative connectives ⊢ Θ ⇑ Γ, t − ⊢ Θ ⇑ Γ, A ⊢ Θ ⇑ Γ, B