Supposition: no Problem for Bilateralism

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.

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.

Gentzen-style sequent calculi and Gentzen-style natural deduction systems are introduced for a family (C-family) of connexive logics over Wansing’s basic constructive connexive logic C. The C-family is derived from C by incorporating Peirce’s law, the law of excluded middle, and the generalized law of excluded middle. Natural deduction systems with general elimination rules are also introduced for the C-family. Theorems establishing the equivalence between the proposed sequent calculi and natural deduction systems are demonstrated. Cutelimination and normalization theorems are established for the proposed sequent calculi and natural deduction systems, respectively. Additionally, similar results are obtained for a family (N-family) of paraconsistent logics over Nelson’s constructive four-valued logic N4.

In this chapter we compare the hybrid-logical natural deduction system given in Section 2.2 to a labelled natural deduction system for modal logic. The chapter is structured as follows. In the first section of the chapter we describe the labelled natural deduction system under consideration and in the second section we define a translation from this system to the hybrid-logical natural deduction system given in Section 2.2. In the third section we compare reductions in the two systems. The material in this chapter is taken from Braüner(2007).

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).

In this chapter we compare and contrast the natural deduction system given in Section 2.2 to a modified version of a hybrid-logical natural deduction system given by Jerry Seligman. The chapter is structured as follows. In the first section of the chapter we describe the natural deduction systems under consideration, in particular, we define our version of Seligman’s system. In the second and third sections, we give translations of derivations backwards and forwards between the systems, and in the fourth section we devise a set of reduction rules for our version of Seligman’s system by translation of the reduction rules for the system given in Section 2.2. In the final section we discuss the results.

1 This paper is a continuation of the investigations reported in Corcoran and Weaver [1] where two logics j£Π and -CDD, having natural deduction systems based on Lewis's S5, are shown to have the usually desired properties (strong soundness, strong completeness, compactness). As in [1], we desire to treat modal logic as a clean natural deduction system with a conceptually meaningful semantics. Here, our investigations are carried out for several S4 based logics. These logics, when regarded as logistic systems (cf. Corcoran [2], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be linked. Some of the results of [l] are presupposed here and the more obvious definitions will not be repeated in detail. We consider the logics -Cl, -C2, and -£3. These logics share the same language (^DD) and deductive system (Δ'DD) but each has its own semantics system (Σl, Σ2, Σ3). Σl is an extension of the Kripke [3] semantics for S5 as modified in [l], Σ2 is largely due to Makinson [5], and Σ3 is due to Kripke [4]. -C DD is the usual modal sentential language with D, ~ and 3 as logical constants (see 2 below). Δ'DD (see 3 below), a modification of the natural deduction system given in [l], permits proofs from arbitrary sets of premises. For S a set of sentences and A & sentence, Sv-A means that A is provable from S, i.e., there is a proof (in Δ'DD) of A whose premises are among the members of S. ([-A means S\-A where S is empty.) If S\-A, we sometimes say that the argument (S, A) is demonstrable and when, in addition, S is empty we say that A is provable.

Towards a canonical classical natural deduction system

We introduce a Hyper Natural Deduction system as an extension of Gentzen's Natural Deduction system. A Hyper Natural Deduction consists of a finite set of derivations which may use, beside typical Natural Deduction rules, additional rules providing means for communication between derivations. We show that our Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron's Hyper sequent Calculus. We also provide conversions for normalisation and prove the existence of normal forms for our Hyper Natural Deduction system.

This volume provides an extensive treatment of Natural Deduction and related types of proof systems, with a focus on the practical aspects of proof methods. The book has two main aims: Its first aim is to provide a systematic and historical survey of the variety of Natural Deduction systems in Classical and Modal Logics. The second aim is to present some systems of hybrid character, mixing Natural Deduction with other kinds of proof methods (including Sequent systems, Tableaux, Resolution). Such systems tend to be more universal and effective, because of the possibility of mixing strategies of proof search from different areas. All necessary background material is provided, in particular, a detailed presentation of Modal Logics, including First-Order Modal and Hybrid Modal Logics. The deduction systems presented in the book may be of interest to working logicians, researchers on automated deduction and teachers of logic.

This paper studies a new classical natural deduction system, presented as a typed calculus named \(\underline{\lambda}\mu let\). It is designed to be isomorphic to Curien-Herbelin’s \(\underline{\lambda}\mu \widetilde{\mu}\)-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 Parigot’s λμ-calculus with the idea of ”coercion calculus” due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot’s syntactic class of named terms.This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. \(\underline{\lambda}\mu let\) is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of \(\underline{\lambda}\mu \widetilde{\mu}\)-calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by \(\underline{\lambda}\mu let\). The third problem is the lack of a robust process of ”read-back” into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of λ-calculi for call-by-value. An isomorphic counterpart to the Q-subsystem of \(\underline{\lambda}\mu \widetilde{\mu}\)-calculus is derived, obtaining a new λ-calculus for call-by-value, combining control and let-expressions.

The ‘meaning is use’ approach to the theory of meaning, the approach we inherit through Dummett from Wittgenstein and which makes heavy use of natural deduction systems for the study of the meanings of logical words, has somehow to reconstruct proof-theoretically the properties of sentences and arguments that customarily are treated semantically by theorists of other schools. Another way of putting this idea is that the semantical properties of a language are the result of how it is employed socially. The main linguistic uses of the logical constants involve their roles in inferences, and Gentzen’s natural deduction systems have a strong philosophical claim to being the right codification of these uses, so this Wittgensteinian approach leads to concentration on natural deduction systems of logic for the study of the meanings of logical words. This approach to analysing semantical properties of language is seen, for instance, in Prawitz’s charge that Tarski’s analysis of validity is philosophically unenlightening [4, p. 67], cited in [2, p. 201]. The alternative is to hold an argument valid when it is provable subject to various structural constraints on the shape of the proof, i.e. when there is a canonical proof of the argument. Still, it is a mistake to think that this way of approaching questions of validity is not semantical in a broad sense. Semantical properties should be emergent properties, arising from the right constellation of proof-theoretical ones. On anyone’s account, validity has to have something to do with preserving truth (or ideal verifiability, or warranted assertibility or whatever truth-surrogate is active at the time). Likewise, if the meaning of the ampersand is fixed by its introduction and elimination rules, these must suffice to say when the conjunction is true or false, once the values of its conjuncts have been fixed.KeywordsClassical LogicIntuitionistic LogicDouble NegationLogical ConstantElimination 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.

In this chapter we introduce the proof-theory of propositional hybrid logic. The chapter is structured as follows. In the first section of the chapter we sketch the basics of natural deduction systems and in the second section we introduce a natural deduction system for hybrid logic. In the third section we sketch the basics of Gentzen systems and in the fourth section we introduce a Gentzen system corresponding to the natural deduction system for hybrid logic. In the fifth section we give an axiom system for hybrid logic. The natural deduction system and the Gentzen system are taken from Brauner (2004a) whereas the axiom system is taken from Brauner (2006).

Firstly, a natural deduction system in standard style is introduced for Nelson's para-consistent logic N4, and a normalization theorem is shown for this system. Secondly, a natural deduction system in sequent calculus style is introduced for N4, and a normalization theorem is shown for this system. Thirdly, a comparison between various natural deduction systems for N4 is given. Fourthly, a strong normalization theorem is shown for a natural deduction system for a sublogic of N4. Fifthly, a strong normalization theorem is proved for a typed λ-calculus for a neighbor of N4. Finally, it is remarked that the natural deduction frameworks presented can also be adapted for Wansing's basic connexive logic C.

It is proved that the standard sequent calculus proof system of linear logic is equivalent to a natural deduction style proof system. The natural deduction system is used to investigate the pragmatic problems of type inference and type safety for a linear lambda calculus. Although terms do not have a single most-general type (for either the standard sequent presentation or the natural deduction formulation), there is a set of most-general types that may be computed using unification. The natural deduction system also facilitates the proof that the type of an expression is preserved by any evaluation step. An execution model and implementation is described, using a variant of the three-instruction machine. A novel feature of the implementation is that garbage-collected nonlinear memory is distinguished from linear memory, which does not require garbage collection and for which it is possible to do secure update in place.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>