Classical Natural Deduction Research Articles

Completeness theorems for first-order logic analysed in constructive type theory

AbstractWe study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak Kőnig’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

Are the open-ended rules for negation categorical?

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument.

Characterizing of Strong Normalization for Λμ-Calculus

λμ-calculus is introduced by Parigot as an extension isomorphic to an alternative presentation of classical natural deduction. Since then, many properties of it have been studied and, in particular, it does not enjoy the separation property shown by David and Py. Λμ-calculus is proposed by de Groote and developed by Saurin as an extension of λμ-calculus Saurin demonstrates that the separation property holds for the Λμ -calculus. Bakel gives a characterization of strong normalization of λμ-calculus in the view of intersection type. In this paper, we will extend the intersection type assignment system to the Λμ -calculus, and show that it characterizes those terms that are strongly normalizing. The system satisfies the subject reduction and subject expansion properties.

Normality, Non-contamination and Logical Depth in Classical Natural Deduction

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity.

Postponement of $$\mathsf {raa}$$ raa and Glivenko’s Theorem, Revisited

We study how to postpone the application of the reductio ad absurdum rule ( $$\mathsf {raa}$$ ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of $$\mathsf {raa}$$ , which induces a negative translation (a variant of Kuroda’s one) from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

A note on strong normalization in classical natural deduction

In the context of natural deduction for propositional classical logic, with classicality given by the inference rule reductio ad absurdum, we investigate the De Morgan translation of disjunction in terms of negation and conjunction. Once the translation is extended to proofs, it obtains a reduction of provability to provability in the disjunction-free subsystem. It is natural to ask whether a reduction is also obtained for, say, strong normalization; that is, whether strong normalization for the disjunction-free system implies the same property for the full system, and whether such lifting of the property can be done along the De Morgan translation. Although natural, these questions are neglected by the literature. We spell out the map of reduction steps induced by the De Morgan translation of proofs. But we need to "optimize" such a map in order to show that a reduction sequence in the full system from a proof determines, in a length-preserving way, a reduction sequence in the disjunction-free system from the De Morgan translation of the proof. In this sense, the above questions have a positive answer.

A verified abstract machine for functional coroutines

Functional coroutines are a restricted form of control mechanism, where each coroutine is represented with both a continuation and an environment. This restriction was originally obtained by considering a constructive version of Parigot's classical natural deduction which is sound and complete for the Constant Domain logic. In this article, we present a refinement of de Groote's abstract machine for functional coroutines and we prove its correctness. Therefore, this abstract machine also provides a direct computational interpretation of the Constant Domain logic.

Compositional Z: Confluence Proofs for Permutative Conversion

This paper gives new confluence proofs for several lambda calculi with permutation-like reduction, including lambda calculi corresponding to intuitionistic and classical natural deduction with disjunction and permutative conversions, and a lambda calculus with explicit substitutions. For lambda calculi with permutative conversion, naive parallel reduction technique does not work, and (if we consider untyped terms, and hence we do not use strong normalization) traditional notion of residuals is required as Ando pointed out. This paper shows that the difficulties can be avoided by extending the technique proposed by Dehornoy and van Oostrom, called the Z theorem: existence of a mapping on terms with the Z property concludes the confluence. Since it is still hard to directly define a mapping with the Z property for the lambda calculi with permutative conversions, this paper extends the Z theorem to compositional functions, called compositional Z, and shows that we can adopt it to the calculi.

On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

We present a new Curry–Howard correspondence for classical first-order natural deduction. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for raising and catching multiple exceptions, and from the viewpoint of logic, the excluded middle over arbitrary prenex formulas. The machinery will allow to extend the idea of learning – originally developed in Arithmetic – to pure logic. We prove that our typed calculus is strongly normalizing and show that proof terms for simply existential statements reduce to a list of individual terms forming an Herbrand disjunction. A by-product of our approach is a natural-deduction proof and a computational interpretation of Herbrand's Theorem.

Some properties of the -calculus

In this paper, we present the -calculus which at the typed level corresponds to the full classical propositional natural deduction system. The Church–Rosser property of this system is proved using the standardisation and the finiteness developments theorem. We also define the leftmost reduction and prove that it is a winning strategy.

Towards a canonical classical natural deduction system

This paper studies a new classical natural deduction system, presented as a typed calculus named λ̲μlet. It is designed to be isomorphic to Curien and Herbelinʼs λ¯μμ˜-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigotʼs λμ-calculus with the idea of “coercion calculus” due to Cervesato and Pfenning, accommodating let-expressions in a surprising way: they expand Parigotʼs syntactic class of named terms.This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of “read-back” of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the λ¯μμ˜-calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. λ̲μlet is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and Θ provides the “read-back” map and the formalized semantics, based on the precise notions of context and “hole-expression” provided by λ̲μlet.We use “read-back” to achieve a precise connection with Parigotʼs λμ, and to derive λ-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics Θ, when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in λ¯μμ˜ (“co-context”), with which one can give a new definition of η-reduction.

NORMAL DERIVABILITY IN CLASSICAL NATURAL DEDUCTION

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

Classical Natural Deduction for S4 Modal Logic

This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dualcontext to formulate S4 modal logic. The modal λμ-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λμ-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λμ-calculus are shown. Finally, the computational interpretation of the modal λμ-calculus, especially the computational meaning of the modal possibility operator, is discussed.

Proof-graphs: a Thorough Cycle Treatment, Normalization and Subformula Property

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure, where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof. Moreover, the subformula and separation properties are determined.

Strong normalization results by translation

We prove the strong normalization of full classical natural deduction (i.e. with conjunction, disjunction and permutative conversions) by using a translation into the simply typed λ μ -calculus. We also extend Mendler’s result on recursive equations to this system.

Data Compression for Proof Replay

We describe a compressing translation from SAT solver generated propositional resolution refutation proofs to classical natural deduction proofs. The resulting proof can usually be checked quicker than one that simply simulates the original resolution proof. We use this result in interactive theorem provers, to speed up reconstruction of SAT solver generated proofs. The translation is fast and scales up to large proofs with millions of inferences.

Strong normalization of classical natural deduction with disjunctions

This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.

Full classical S5 in natural deduction with weak normalization

Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on Mathematics, 2006] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic only for a language without ∨ , ∃ and with a restricted application of reduction ad absurdum. Reduction steps related to ∨ , ∃ and classical negation bring about many problems solved only rather recently. For classical S5 modal logic, Prawitz defined a normalizable system, but for a language without ∨ , ∃ , ◊ and, for a propositional language without ◊ , Medeiros [M.da P.N. Medeiros, A new S4 classical modal logic in natural deduction, Journal of Symbolic Logic 71 (3) (2006) 799–809] presented a normalizable system for classical S4. We can mention many cut-free Gentzen systems for S4, S5 and K45/K45D, some normalizable natural deduction systems for intuitionistic modal logics and one more for full classical S4, but not for full classical S5. Here our focus is on the definition of a classical and normalizable natural deduction system for S5, taking not only □ and ◊ as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure is based on the strategy proposed by Massi [C.D.B. Massi, Provas de normalizaçaõ para a lógica clássica, Ph.D. Thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990] and Pereira and Massi [L.C. Pereira, C.D.B. Massi, Normalização para a lógica clássica, in: O que nos faz pensar, Cadernos de Filosofia da PUC-RJ, vol. 2, 1990, pp. 49–53] for first-order classical logic to cope with the combined use of classical negation, disjunction and the existential quantifier. Here we extend such results to deal with □ and ◊ too. The elimination rule for ◊ uses the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of □ . Beyond weak normalization, we also prove the subformula property for full S5.

A proof-theoretic foundation of abortive continuations

We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a implementation of this logic is Parigot's classical natural deduction. We then move on to the computational side and emphasize that Parigot's ? μ corresponds to minimal classical logic. A continuation constant must be added to ? μ to get full classical logic. The extended calculus is isomorphic to a syntactical restriction of Felleisen's theory of control that offers a more expressive reduction semantics. This isomorphic calculus is in correspondence with a refined version of Prawitz's natural deduction.

Strong normalization proofs by CPS-translations

This paper proposes a new proof method for strong normalization of classical natural deduction and calculi with control operators. For this purpose, we introduce a new CPS-translation, continuation and garbage passing style ( CGPS ) translation. We show that this CGPS-translation method gives simple proofs of strong normalization of λ μ → ∧ ∨ ⊥ , which is introduced in [P. de Groote, Strong normalization of classical natural deduction with disjunction, in: S. Abramsky (Ed.), Typed Lambda Calculi and Applications, 5th International Conference, TLCA 2001, in: Lecture Notes in Comput. Sci., vol. 2044, Springer, Berlin, 2001, pp. 182–196] by de Groote and corresponds to the classical natural deduction with disjunctions and permutative conversions.