We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using these axiomatizations, we prove interpolation theorems for basic propositional logic.

We introduce a Gentzen-style sequent calculus axiomatization for Basic Predicate Calculus. Our new axiomatization is an improvement of the previous axiomatizations, in the sense that it has the subformula property. In this system the cut rule is eliminated.

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for (...) classical logic is formalized based on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word (...) ‘for’ has taken on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with thededucibility relationsof the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzenrelations. By aHilbert relationwe mean any substitution-invariant consequence relation onformulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By aGentzen relationwe mean the fully fledged generalization of this notion in whichsequentstake the place of single formulas. In the literature, Hilbert relations are often referred to assentential logics. Gentzen relations as defined here are their exactsequentialcounterparts. (shrink)

We consider four natural deduction systems: Fitch-style systems, Gentzen-style systems (in the form of dags), general deduction Frege systems and nested deduction Frege systems, as well as dag-like Gentzen-style sequent calculi. All these calculi soundly and completely formalize classical propositional logic. -/- We show that general deduction Frege systems and Gentzen-style natural calculi provide at most quadratic speedup over nested deduction Frege systems and Fitch-style natural calculi and at most cubic speedup over (...) Gentzen-style sequent calculi. (shrink)

We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke (...) models. The cut-elimination theorem isproved in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics. (shrink)

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by (...) augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory. Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof. (shrink)

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 1 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on (...) the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. (shrink)

We define the notions of a canonical inference rule and a canonical system in the framework of single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and prove that such a canonical system is non-trivial iff it is coherent (where coherence is a constructive condition). Next we develop a general non-deterministic Kripke-style semantics for such systems, and show that every constructive canonical system (i.e. coherent canonical single-conclusion system) induces a class of non-deterministic Kripke-style frames (...) for which it is strongly sound and complete. We use this non-deterministic semantics to show that all constructive canonical systems admit a strong form of the cut-elimination theorem. We also use it for providing a decision procedure for every such system. These results identify a large family of basic constructive connectives, each having both a proof-theoretical characterization in terms of a coherent set of canonical rules, as well as a semantic characterization using non-deterministic frames. The family includes the standard intuitionistic connectives, together with many other independent connectives. (shrink)

We introduce a cyclic proof system for the two-way alternation-free modal μ-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the (...) opponent’s strategy. The idea for trace atoms comes from Vardi’s reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms. (shrink)

The paper introduces Hilbert– and Gentzen-style calculi which correspond to systems ${\mathsf{C}_{n}}$ from Gupta and Belnap [3]. Systems ${\mathsf{C}_{n}}$ were shown to be sound and complete with respect to the semantics of finite revision. Here, it is shown that Gentzen-style systems ${\mathsf{GC}_{n}}$ admit a syntactic proof of cut elimination. As a consequence, it follows that they are consistent.

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system (...) can be used with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)

Building on recent work, I present sequent systems for the non-classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.”.

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's (...) laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

Dynamic epistemic logic (DEL) is a multi-modal logic for reasoning about the change of knowledge in multi-agent systems. It extends epistemic logic by a modal operator for actions which announce logical formulas to other agents. In Hilbert-style proof calculi for DEL, modal action formulas are reduced to epistemic logic, whereas current sequent calculi for DEL are labelled systems which internalize the semantic accessibility relation of the modal operators, as well as the accessibility relation underlying the semantics of the actions. (...) We present a novel cut-free ordinary sequent calculus, called |$ \textbf{G4}_{P,A}[] $|⁠, for propositional DEL. In contrast to the known sequent calculi, our calculus does not internalize the accessibility relations, but—similar to Hilbert style proof calculi—action formulas are reduced to epistemic formulas. Since no ordinary sequent calculus for full S5 modal logic is known, the proof rules for the knowledge operator and the Boolean operators are those of an underlying S4 modal calculus. We show the soundness and completeness of |$ \textbf{G4}_{P,A}[] $| and prove also the admissibility of the cut-rule and of several other rules for introducing the action modality. (shrink)

We have here a systematic examination of systems of logic cast in "natural deduction" form, in the widest sense of that word. Naturally enough, the author begins with Gentzen-style systems, moves by means of the inversion principle in int-elim logics to deductions of classical and intuitionistic logic which have a canonical "normal form"; more briefly, but always with clarity, Prawitz examines natural deduction for second-order and modal logics, and also systems with even less usual forms of implication. Three (...) appendices cover the Sequenzen-kalkül of Gentzen, a system of set theory of Fitch, and some alternative formulations of natural deductions due to Jáskowski and others. There is a short bibliography. There are also some hilarious misprints, few serious.—P. J. M. (shrink)

In this paper two deductive systems associated with relevance logic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first one turns out to be (...) an axiomatic extension of WR. Moreover we provide WR with a natural Gentzen calculus. It is proved that both deductive systems have the same associated class of algebras but different classes of models on these algebras. The notion of model used here is an abstract logic, that is, a closure operator on an abstract algebra; the abstract logics obtained in the case of WR are also the models, in a natural sense, of the given Gentzen calculus. (shrink)

We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic ($$\textrm{CHL}$$ CHL ), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: $$\textrm{CHL}$$ CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for $$\textrm{CHL}$$ CHL (...) ; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner’s idea of superconnexivity. (shrink)

Reasoning with the maximally consistent subsets of the premises is a well-known approach for handling contradictory information. In this paper we consider several variations of this kind of reasoning, for each one we introduce two complementary computational methods that are based on logical argumentation theory. The difference between the two approaches is in their ways of making consequences: one approach is of a declarative nature and is related to Dung-style semantics for abstract argumentation, while the other approach has a (...) more proof-theoretical flavor, extending Gentzen-style sequent calculi. The outcome of this work is a new perspective on reasoning with MCS, which shows a strong link between the latter and argumentation systems, and which can be generalized to some related formalisms. As a by-product of this we obtain soundness and completeness results for the dynamic proof systems with respect to several of Dung’s semantics. In a broader context, we believe that this work helps to better understand and evaluate the role of logic-based instantiations of argumentation frameworks. (shrink)

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and (...) show that the logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. (shrink)

We combine intuitionistic logic and classical logic into a new, first-order logic called polarized intuitionistic logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzenʼs intuitionistic sequent calculus LJ. In addition, this system also bears essential (...) similarities to Girardʼs LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalinʼs multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. (shrink)

We offer a critical overview of two sorts of proof systems that may be said to characterize classical propositional logic indirectly (and non-standardly): refutation systems, which prove sound and complete with respect to classical contradictions, and rejection systems, which prove sound and complete with respect to the larger set of all classical non-tautologies. Systems of the latter sort are especially interesting, as they show that classical propositional logic can be given a paraconsistent characterization. In both cases, we consider Hilbert-style (...) systems as well as Gentzen-style sequent calculi and natural-deduction formalisms. (shrink)

Our aim is to express in exact terms the old idea of solving problems by pure questioning. We consider the problem of derivability: "Is A derivable from Δ by classical propositional logic?". We develop a calculus of questions E*; a proof (called a Socratic proof) is a sequence of questions ending with a question whose affirmative answer is, in a sense, evident. The calculus is sound and complete with respect to classical propositional logic. A Socratic proof in E* can be (...) transformed into a Gentzen-style proof in some sequent calculi. Next we develop a calculus of questions E**; Socratic proofs in E** can be transformed into analytic tableaux. We show that Socratic proofs can be grounded in Inferential Erotetic Logic. After a slight modification, the analyzed systems can also be viewed as hypersequent calculi. (shrink)

ABSTRACT Two objects o1, o2 of an information system are said to be complementary with respect to attribute a if α(o1) = -α(o2), where α(o) is the set of values of attribute a assigned to o. They are said to be complementary with respect to a set of attributes A if they are complementary with respect to each attribute α ε A. A multi-modal logical language for reasoning about complementarity relations is presented, with modalities [A] and ?A? parameterised by (...) subsets of a given set ATTR of attributes. Two complete deduction system for the language are presented: a Rasiowa-Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories. (shrink)

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements.

Different researchers use "the philosophy of automated theorem p r o v i n g " t o cover d i f f e r e n t concepts, indeed, different levels of concepts. Some w o u l d count such issues as h o w to e f f i c i e n t l y i n d e x databases as part of the philosophy of automated theorem p r o v i n g . (...) Others wonder about whether f o r m u l a s should be represented as strings or as trees or as lists, and call this part of the philosophy of automated theorem p r o v i n g . Yet others concern themselves w i t h what k i n d o f search should b e embodied i n a n y automated theorem prover, or to what degree any automated theorem prover should resemble Prolog. Still others debate whether natural deduction or semantic tableaux or resolution is " b e t t e r " , a n d c a l l t h i s a part of the p h i l o s o p h y of automated theorem p r o v i n g . Some people wonder whether automated theorem p r o v i n g should be " h u m a n oriented" or "machine o r i e n t e d " — sometimes arguing about whether the internal p r o o f methods should be " h u m a n - I i k e " or not, sometimes arguing about whether the generated proof should be output in a f o r m u n d e r s t a n d a b l e by p e o p l e , and sometimes a r g u i n g a b o u t the d e s i r a b i l i t y o f h u m a n intervention in the process of constructing a proof. There are also those w h o ask such questions as whether we s h o u l d even be concerned w i t h completeness or w i t h soundness of a system, or perhaps we should instead look at very efficient (but i n c o m p l e t e ) subsystems or look at methods of generating models w h i c h might nevertheless validate invalid arguments. A n d a l l of these have been v i e w e d as issues in the philosophy of automated theorem proving. Here, I w o u l d l i k e to step back from such i m p l e m e n t - ation issues and ask: " W h a t do we really think we are doing when we w r i t e an automated theorem prover?" My reflections are perhaps idiosyncratic, but I do think that they put the different researchers* efforts into a broader perspective, and give us some k i n d of handle on w h i c h directions we ourselves m i g h t w i s h to pursue when constructing (or extending) an automated theorem proving system. A logic is defined to be (i) a vocabulary and formation rules ( w h i c h tells us w h a t strings of symbols are w e l l - formed formulas in the logic), and ( i i ) a definition of ' p r o o f in that system ( w h i c h tells us the conditions under which an arrangement of formulas in the system constitutes a proof). Historically speaking, definitions of ' p r o o f have been given in various different manners: the most c o m m o n have been H i l b e r t - s t y l e ( a x i o m a t i c ) , Gentzen-style (consecution, or sequent), F i t c h - s t y l e (natural deduction), and Beth-style (tableaux).. (shrink)

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem''s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap''s display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or (...) its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. (shrink)

We prove that the Cutting Plane proof system based on Gomory–Chvátal cuts polynomially simulates the lift-and-project system with integer coefficients written in unary. The restriction on the coefficients can be omitted when using Krajíček’s cut-free Gentzen-style extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension.

The variety \ of semi-Heyting algebras was introduced by H. P. Sankappanavar [13] as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo :9–25, 2011) [6]. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The (...) advantage of this presentation of the logic is that we can prove a cut-elimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras. (shrink)

This volume contains finalized versions of papers presented at an international workshop on extensions of logic programming, held at the Seminar for Natural Language Systems at the University of Tübingen in December 1989. Several recent extensions of definite Horn clause programming, especially those with a proof-theoretic background, have much in common. One common thread is a new emphasis on hypothetical reasoning, which is typically inspired by Gentzen-style sequent or natural deduction systems. This is not only of theoretical significance, (...) but also bears upon computational issues. It was one purpose of the workshop to bring some of these recent developments together. The volume covers topics such as the languages Lambda-Prolog, N-Prolog, and GCLA, the relationship between logic programming and functional programming, and the relationship between extensions of logic programming and automated theorem proving. It contains the results of the first conference concentrating on proof-theoretic approaches to logic programming. (shrink)

The primary objective of this paper, which is an addendum to the author’s [8], is to apply the general study of the latter to Łukasiewicz’s n-valued logics [4]. The paper provides an analytical expression of a 2(n−1)-place sequent calculus (in the sense of [10, 9]) with the cut-elimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi in the sense of a complexity of systems of premises of introduction rules. This together (...) with a quite effective procedure of construction of an equality determinant (in the sense of [5]) for the logics involved to be extracted from the constructive proof of Proposition 6.10 of [6] yields an equally effective procedure of construction of both Gentzen-style [2] (i.e., 2-place) and Tait-style [11] (i.e., 1-place) minimal sequent calculi following the method of translations described in Subsection 4.2 of [7]. (shrink)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is (...) defined in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

We present a Hilbert style axiomatization and an equational theory for reasoning about actions and capabilities. We introduce two novel features in the language of propositional dynamic logic, converse as backwards modality and abstract processes specified by preconditions and effects, written as \({\varphi \Rightarrow \psi}\) and first explored in our recent paper (Hartonas, Log J IGPL Oxf Univ Press, 2012), where a Gentzen-style sequent calculus was introduced. The system has two very natural interpretations, one based on (...) the familiar relational semantics and the other based on type semantics, where action terms are interpreted as types of actions (sets of binary relations). We show that the proof systems do not distinguish between the two kinds of semantics, by completeness arguments. Converse as backwards modality together with action types allow us to produce a new purely equational axiomatization of Dynamic Algebras, where iteration is axiomatized independently of box and where the fixpoint and Segerberg induction axioms are derivable. The system also includes capabilities operators and our results provide then a finitary Hilbert-style axiomatization and a decidable system for reasoning about agent capabilities, missing in the KARO framework. (shrink)

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite (...) conjunctions and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional. (shrink)

In this article, a cut-free system TLMω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLMω1 is a kind of Gentzen style sequent calculus, but a sequent of TLMω1 is defined as a finite tree of sequents in a standard sense. We prove the cut-elimination theorem for TLMω1 via its Kripke completeness.

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. (shrink)

We introduce a general approach for representing and reasoning with argumentation-based systems. In our framework arguments are represented by Gentzen-style sequents, attacks between arguments are represented by sequent elimination rules, and deductions are made according to Dung-style skeptical or credulous semantics. This framework accommodates different languages and logics in which arguments may be represented, allows for a flexible and simple way of expressing and identifying arguments, supports a variety of attack relations, and is faithful to standard methods (...) of drawing conclusions by argumentation frameworks. Altogether, we show that argumentation theory may benefit from incorporating proof theoretical techniques and that different non-classical formalisms may be used for backing up intended argumentation semantics. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the (...) perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

We present a sequent calculus with the Henkin constants in the place of the free variables. By disposing of the eigenvariable condition, we obtained a proof system with a strong locality property—the validity of each inference step depends only on its active formulas, not its context. Our major outcomes are: the cut elimination via a non-Gentzen-style algorithm without resorting to regularization and the elimination of Skolem functions with linear increase in the proof length for a subclass of (...) derivations with cuts. (shrink)

Minimalism concerning truth is the view that that all there is to be said concerning truth is exhausted by a set of basic platitudes. In the first part of this thesis, I apply this methodology to the concept of knowledge. In so doing, I develop a model of inexact knowledge grounded in what I call minimal margin for error principles. From these basic principles, I derive the controversial result that epistemological internalism and internalism with respect to self-knowledge are untenable doctrines. (...) In the second part of this thesis, I develop a minimal theory of vagueness in which a rigorous but neutral definition of vagueness is shown to be possible. Three dimensions of vagueness are distinguished and a proof is given showing how two of these dimensions are equivalent facets of the same phenomenon. From the axioms of this minimal theory one can also show that there must be higher-order vagueness, contrary to what some have argued. In the final part of this thesis, I return to issues concerning the credentials of truth-minimalism. Is truth-minimalism compatible with the possibility of truth-value gaps? Is it right to say that truth-minimalism is crippled by the liar paradox? With respect to the former question, I develop a novel three-valued logical system which is both proof-theoretically and truth-theoretic ally well-motivated and compatible with at least one form of minimalism. With respect to the second question, a new solution to the liar paradox is developed based on the claim that while the liar sentence is meaningful, it is improper to even suppose that this sentence has a truth-status. On that basis, one can block the paradox by restricting the Rule of Assumptions in Gentzen-style presentations of the sentential sequent calculus. The first lesson of the liar paradox is that not all assumptions are for free. The second lesson of the liar is that, contrary to what has been alleged by many, minimalism concerning truth is far better placed than its rival theories to solve the paradox. (shrink)

This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzen-style proof theory in which there is only alimited possibility to use structural rules. Following the literture, we use an operator to mark formulas to which the extra structural rules may be applied. New in our approach is that we do not see this as a modality, but rather as themeet of the marked formula with a special typeQ. In this way we (...) can make the specific structural behaviour of marked formulas more explicit.The main motivation for our approach is that we can provide a nice, intuitive semantics for hybrid substructural logics. Soundness and completeness for this semantics are proved; besides this we consider some proof-theoretical aspects like cut-elimination and embeddings of the strong system in the hybrid one. (shrink)

We provide a non-Gentzen, though fully syntactical, cut-elimination algorithm for classical propositional logic. The designed procedure is implemented on $$\textsf{GS4}$$ GS 4, the one-sided version of Kleene’s sequent system $$\textsf{G4}$$ G 4. The algorithm here proposed proves to be more ‘dexterous’ than other, more traditional, Gentzen-style techniques as the size of proofs decreases at each step of reduction. As a corollary result, we show that analyticity always guarantees minimality of the size of $$\textsf{GS4}$$ GS 4 -proofs.

The lambdaPi-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the lambdaPi-calculus and prove the cut-elimination theorem. The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. (...) The type-theoretic setting considered here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus. (shrink)

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox (...) is presented following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)

In advanced books and courses on logic (e.g. Sm], BM]) Gentzen-type systems or their dual, tableaux, are described as techniques for showing validity of formulae which are more practical than the usual Hilbert-type formalisms. People who have learnt these methods often wonder why the Automated Reasoning community seems to ignore them and prefers instead the resolution method. Some of the classical books on AD (such as CL], Lo]) do not mention these methods at all. Others (such as Ro]) do, (...) but the connections and reasons for preference remain unclear after reading them (at least to the present author, and obviously also the authors of OS], in which a theorem-prover, based exclusively on tableaux, is described). The confusion becomes greater when the reader is introduced to Kowalski's form of a clause ( Ko], Bu]), which is nothing but a Gentzen's sequent of atomic formulae, and when he realizes that resolution is just a form of a Cut, and so that while the elimination of cuts is the principal tool in proof-theory, its use is the main technique in AD! (shrink)

Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system (...) style, and so on. The tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)

Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzen-style framework is introduced for analytic proof systems that derive admissible rules of non-classical logics. While Gentzen systems for derivability treat sequents as basic objects, for admissibility, the basic objects are sequent rules. Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic (...) IPC. With minor restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the logics. (shrink)

A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, (...) and to give a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator. (shrink)