Proof Terms for Generalized Natural Deduction

In previous work it has been shown how to generate natural deduction rules for propositional connectives from truth tables, both for classical and constructive logic. The present paper extends this for the constructive case with proof-terms, thereby extending the Curry-Howard isomorphism to these new connectives. A general notion of conversion of proofs is defined, both as a conversion of derivations and as a reduction of proof-terms. It is shown how the well-known rules for natural deduction (Gentzen, Prawitz) and general elimination rules (Schroeder-Heister, von Plato, and others), and their proof conversions can be found as instances. As an illustration of the power of the method, we give constructive rules for the nand logical operator (also called Sheffer stroke). As usual, conversions come in two flavours: either a detour conversion arising from a detour convertibility, where an introduction rule is immediately followed by an elimination rule, or a permutation conversion arising from an permutation convertibility, an elimination rule nested inside another elimination rule. In this paper, both are defined for the general setting, as conversions of derivations and as reductions of proof-terms. The properties of these are studied as proof-term reductions. As one of the main contributions it is proved that detour conversion is strongly normalizing and permutation conversion is strongly normalizing: no matter how one reduces, the process eventually terminates. Furthermore, the combination of the two conversions is shown to be weakly normalizing: one can always reduce away all convertibilities

Classical Natural Deduction from Truth Tables

In earlier articles we have introduced truth table natural deduction which allows one to extract natural deduction rules for a propositional logic connective from its truth table definition. This works for both intuitionistic logic and classical logic. We have studied the proof theory of the intuitionistic rules in detail, giving rise to a general Kripke semantics and general proof term calculus with reduction rules that are strongly normalizing. In the present paper we study the classical rules and give a term interpretation to classical deductions with reduction rules. As a variation we define a multi-conclusion variant of the natural deduction rules as it simplifies the study of proof term reduction. We show that the reduction is normalizing and gives rise to the sub-formula property. We also compare the logical strength of the classical rules with the intuitionistic ones and we show that if one non-monotone connective is classical, then all connectives become classical

Deriving natural deduction rules from truth tables

We develop a general method for deriving natural deduction rules from the truth table for a connective. The method applies to both constructive and classical logic. This implies we can derive “constructively valid” rules for any classical connective. We show this constructive validity by giving a general Kripke semantics, that is shown to be sound and complete for the constructive rules. For the well-known connectives (˅, ˄, →, ¬) the constructive rules we derive are equivalent to the natural deduction rules we know from Gentzen and Prawitz. However, they have a different shape, because we want all our rules to have a standard “format”, to make it easier to define the notions of cut and to study proof reductions. In style they are close to the “general elimination rules” studied by Von Plato [13] and others. The rules also shed some new light on the classical connectives: e.g. the classical rules we derive for → allow to prove Peirce’s law. Our method also allows to derive rules for connectives that are usually not treated in natural deduction textbooks, like the “if- then-else”, whose truth table is clear but whose constructive deduction rules are not. We prove that “if-then-else”, in combination with ┴ and ┬, is functionally complete (all other constructive connectives can be defined from it). We define the notion of cut, generally for any constructive connective and we describe the process of “cut-elimination”

Strong Normalization for Truth Table Natural Deduction

We present a proof of strong normalization of proof-reduction in a general system of natural deduction called truth table natural deduction. In previous work, we have defined truth table natural deduction, which is a method for deriving intuitionistic derivation rules for a connective from its truth table. This yields natural deduction rules for each connective separately. Moreover, these rules adhere to a standard format which gives rise to a general notions of detour and permutation conversion for natural deductions. The aim is to remove all convertibilities and obtain a deduction in normal form. In general, conversion of truth table natural deductions is non-deterministic, which makes it more challenging to study. It has already been shown that this conversion is weakly normalizing. To prove strong normalization, we construct a conversionpreserving translation from deductions to terms in an extension of simply typed lambda calculus which we call parallel simply typed lambda calculus and which we prove to be strongly normalizing. This makes it possible to get a grip on the non-deterministic character of conversion in the intuitionistic truth table natural deduction system

Strong Normalization for Truth Table Natural Deduction

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Strong Normalization for Truth Table Natural Deduction

We present a proof of strong normalization of proof-reduction in a general system of natural deduction called truth table natural deduction. In previous work, we have defined truth table natural deduction, which is a method for deriving intuitionistic derivation rules for a connective from its truth table. This yields natural deduction rules for each connective separately. Moreover, these rules adhere to a standard format which gives rise to a general notions of detour and permutation conversion for natural deductions. The aim is to remove all convertibilities and obtain a deduction in normal form. In general, conversion of truth table natural deductions is non-deterministic, which makes it more challenging to study. It has already been shown that this conversion is weakly normalizing. To prove strong normalization, we construct a conversionpreserving translation from deductions to terms in an extension of simply typed lambda calculus which we call parallel simply typed lambda calculus and which we prove to be strongly normalizing. This makes it possible to get a grip on the non-deterministic character of conversion in the intuitionistic truth table natural deduction system