Comparing Sense and Denotation in Bilateralist Proof Systems for Proofs and Refutations

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

We prove in HOL that three proof systems for classical first-order predicate logic, the Hilbertian axiomatization, the system of natural deduction, and a variant of sequent calculus, are isomorphic. The isomorphism is in the sense that provability of a conclusion from hypotheses in one of these proof systems is equivalent to provability of this conclusion from the same hypotheses in the others. Proving isomorphism of these three proof systems allows us to guarantee that meta-logical provability properties about one of them would also hold in relation to the others. We prove the deduction, monotonicity, and compactness theorems for Hilbertian axiomatization, and the substitution theorem for the system of natural deduction. Then we show how these properties can be translated between the proof systems. Besides, by proving a theorem which states that provability in flattened sequent calculus implies provability in standard sequent calculus, we show how some meta-logical provability results about Hilbertian axiomatization and natural deduction can be translated to sequent calculus. We use higher-order logic as the metalogic for reasoning about first-order proof systems and formalize proofs in a theorem-proving environment, thereupon reducing susceptibility to errors and bringing up subtle issues which are usually overlooked when the reasoning is done in a natural language.KeywordsInference RuleProof SystemCompactness TheoremNatural DeductionSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.

Constructive logics Part I: A tutorial on proof systems and typed λ-calculi

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different focusingannotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics.

Towards a canonical classical natural deduction system

This is an introduction to proof theory of nonclassical logic, which is directed at people who have just started the study of nonclassical logics, using proof-theoretic methods. In our paper, we will discuss only its proof theory based on sequent calculi. So, we will discuss mainly cut elimination and its consequences. As this is not an introduction to sequent systems themselves, we will assume a certain familiarity with standard sequent systems LK for the classical logic and LJ for the intuitionistic logic. When necessary, readers may consult e.g. Chapter 1 of Proof Theory [43] by Takeuti, Chapters 3 and 4 of Basic Proof Theory [45] by Troelstra and Schwichtenberg, and Chapter 1 of the present Memoir by M. Takahashi [41] to supplement our paper. Also, our intention is not to give an introduction of nonclassical logic, but to show how the standard proof-theoretic methods will work well in the study of nonclassical logic, and why certain modifications will be necessary in some cases. We will take only some modal logics and substructural logics as examples, and will give remarks on further applications. Notes at the end of each section include some indications for further reading. An alternative approach to proof theory of nonclassical logic is given by using natural deduction systems. As it is well-known, natural deduction systems are closely related to sequent calculi. For instance, the normal form theorem in natural deduction systems corresponds to the cut elimination theorem in sequent calculi. So, it will be interesting to look for results and techniques on natural deduction systems which are counterparts of those on sequent calculi, given in the present paper.

Many important logical systems have proof-theoretic presentations of more than one type, say, an axiomatization, a natural deduction proof system, and a sequent calculus presentation. Usually, this is a rather fortunate situation. It may happen that certain axiom schemata are characterizable by algebraic or relational properties expressible in an interesting fragment of first-order logic, and that Gentzen-style proof systems lend themselves to automated deduction. As we have seen, display logic is an elegant and powerful refinement of Gentzen’s sequent calculus and meets quite a few methodological requirements of a more philosophical and a more technical nature. It would be nice to relate the modal display calculus to natural deduction proof systems for intensional logics, and in the present chapter we shall relate DL to generalized Fitch-style natural deduction systems for modal logics. Moreover, we shall show that DL may have repercussions on the axiomatic presentation of logical systems and define an apparently new axiomatization of Kt. Eventually, we shall consider a generalization of DL, namely four-place display sequents. We shall redisplay Nelson’s system N4 and obtain a display sequent calculus for N3 by the addition of a purely structural sequent rule.

A bilateralist take on proof-theoretic semantics can be understood as demanding of a proof system to display not only rules giving the connectives’ provability conditions but also their refutability conditions. On such a view, then, a system with two derivability relations is obtained, which can be quite naturally expressed in a proof system of natural deduction but which faces obstacles in a sequent calculus representation. Since in a sequent calculus there are two derivability relations inherent, one expressed by the sequent sign and one by the horizontal lines holding between sequents, in a truly bilateral calculus both need to be dualized. While dualizing the sequent sign is rather straightforwardly corresponding to dualizing the horizontal lines in natural deduction, dualizing the horizontal lines in sequent calculus, uncovers problems that, as will be argued in this paper, shed light on deeper conceptual issues concerning an imbalance between the notions of proof vs. refutation. The roots of this problem will be further analyzed and possible solutions on how to retain a bilaterally desired balance in our system are presented.

In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cut-elimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural deduction. The reduction relation is simulated soundly and completely by a cut-elimination procedure which consists of local proof transformations. It follows that the sequent calculus with our cut-elimination procedure is a proper extension that is conservative over natural deduction with normalization.

We investigate the correlation between empty antecedent and succedent of the intutionistic (respectively dual-intuitionistic) sequent calculus and discharge of assumptions and the constants absurdity (resp. discharge of conclusions and triviality) in natural deduction. In order to be able to express and manipulate the sequent calculus phenomena, we add two units to sequent calculus. Depending on the sequent calculus considered, the units can serve as discharge markers or as absurdity and triviality.

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the internal interpretation, which has to be developed simultaneously with, and is justified by, an equivalent, external interpretation, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the formal duality between control and co-control is studied, in the context of classical logic. The duality cannot be observed in the sequent calculus, but rather in a system unifying sequent calculus and natural deduction.

In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a Curry-Howard interpretation of sequent calculus into a variant of the λ-calculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambda-like calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The proof-theoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cut-elimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.KeywordsReduction RuleNatural DeductionSequent CalculusApplicative ContextTyping RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodi_ed from the usual Prawitz introduction and elimination rules — the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules. The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.