The paper presents a compositional approach to the verification of CTL* properties over reactive systems. Both symbolic model-checking and deductive verification are considered. Both methods are based on two decomposition principles. A general state formula is decomposed into basic state formulas which are CTL* formulas with no embedded path quantifiers. To deal with arbitrary basic state formulas, we introduce another reduction principle which replaces each basic path formula, i.e., path formulas whose principal operator is temporal and which contain no embedded (...) temporal operators or path quantifiers, by a newly introduced boolean variable which is added to the system. Thus, both the algorithmic and the deductive methods are based on two statification transformations which successively replace temporal formulas by assertions which contain no path quantifiers or temporal operators. Performing these decompositions repeatedly, we remain with basic assertional formulas, i.e., formulas of the form Efp and Afp for some assertion p. In the model-checking method we present a single symbolic algorithm to verify both universal and existential basic assertional properties. In the deductive method we present a small set of proof rules and show that this set is sound and relatively complete for verifying universal and existential basic assertional properties over reactive systems. Together with two proof rules for the decompositions, we obtain a sound and relatively complete proof system for arbitrary CTL* properties. Interestingly, the deductive approach for CTL* presented here, offers a viable new approach to the deductive verification of arbitrary LTL formulas. The paper corrects a previous preliminary version of a deductive system for CTL*, in which some of the rules were unsound. The correction is based on the introduction of a new type of temporal testers which are guaranteed to be non blocking. That is, when composed with a deadlock-free system, which is a key operation in the verification process, the resulting composed system is guaranteed to remain deadlock free. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the division of (...) the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded middle.

This is a companion paper to Braüner where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order (...) hybrid logic. (shrink)

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness (...) and proof normalization. We have implemented our work in the Isabelle Logical Framework. (shrink)

Defining qualified reasoning as reasoning containing such loose qualifying words as 'probably,' 'usually,' 'probable, 'likely,' 'ceteris paribus,' and 'primafacie, Ennis argues that typical cases of qualified reasoning, though they might be good arguments, are deductively invalid, implying that such arguments fail soundness standards. He considers and rejects several possible alternative ways of viewing such cases, ending with a proposal for applying qualified soundness standards, which requires employment of sufficient background knowledge, sensitivity, experience and understanding of the situation. All of this (...) takes place as part of the process, introduced elsewhere, of successively applying different legitimate sets of standards until one is found that is satisfied, or none is found, the latter calling for rejection of the argument, the former calling for its acceptance. (shrink)

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional (...) logic, for which I show soundness and completeness for both deductions and refutations. (shrink)

I discuss three elements of Dennis Schulting’s new book on the transcendental deduction of the pure concepts of the understanding, or categories. First, that Schulting gives a detailed account of the role of each individual category. Second, Schulting’s insistence that the categories nevertheless apply ‘en bloc’. Third, Schulting’s defence of Kant’s so-called reciprocity thesis that subjective unity of consciousness and objectivity in the sense of cognition’s objective purport are necessary conditions for the possibility of one another. I endorse these (...) fascinating but unfashionable claims and sketch my own version of what they amount to, which is quite different to Schulting’s own construal. I point to some fundamental limitations and problems for Schulting’s position and argue that his project needs to be reshaped or at least reconceived in the face of them. Even if Schulting’s argument is sound, it does not provide a deduction, properly speaking, of the categories. (shrink)

Hybrid deduction-refutation systems are presented for four first degree entailment based logics. The hybrid systems are shown to deductively and refutationally sound and complete with respect to their logics. The proofs of completeness are presented in a uniform way. This paper builds on work in [6], where Goranko presented a deductively and refutationally sound and complete hybrid system for classical logic.

We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over (...) some point on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic (...) versions of the connectives in question. (shrink)

In this paper we present labelled sequent calculi and labelled natural deduction calculi for the counterfactual logics CK + {ID, MP}. As for the sequent calculi we prove, in a semantic manner, that the cut-rule is admissible. As for the natural deduction calculi we prove, in a purely syntactic way, the normalization theorem. Finally, we demonstrate that both calculi are sound and complete with respect to Nute semantics [12] and that the natural deduction calculi can be (...) effectively transformed into the sequent calculi. (shrink)

The purpose of this paper is to present a formalization of the language, semantics and axiomatization of justification logic in Coq. We present proofs in a natural deduction style derived from the axiomatic approach of justification logic. Additionally, we present possible world semantics in Coq based on Fitting models to formalize the semantic satisfaction of formulas. As an important result, with this implementation, it is possible to give a proof of soundness for $\mathsf{L}\mathsf{P}$ with respect to Fitting models.

Harold Hodes in [1] introduces an extension of first-order modal logic featuring a backtracking operator, and provides a possible worlds semantics, according to which the operator is a kind of device for ‘world travel’; he does not provide a proof theory. In this paper, I provide a natural deduction system for modal logic featuring this operator, and argue that the system can be motivated in terms of a reading of the backtracking operator whereby it serves to indicate modal scope. (...) I prove soundness and completeness theorems with respect to Hodes’ semantics, as well as semantics with fewer restrictions on the accessibility relation. (shrink)

This chapter contains sections titled: Introduction Statements The Square of Opposition Figure and Mood Deduction Counterexamples Independence Soundness Completeness: Syllogistic Arguments Completeness: Categorical Arguments Completeness: Arguments in General Note Bibliography.

We can formalize judgments as logical formulas. Judgment aggregation deals with judgments of several agents, which need to be aggregated to a collective judgment. There are several logical formalizations of judgment aggregation. This paper focuses on a modal formalization which nicely expresses classical properties of judgment aggregation rules and famous results of social choice theory, like Arrow’s impossibility theorem. A natural deduction system for modal logic of judgment aggregation is presented in this paper. The system is sound and (...) complete. As an example of derivation, a formal proof of Arrow’s impossibility theorem is given. (shrink)

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.

This print edition of Read's account of logical thought includes the original publication's diagrams and tables. In this excellent book, Read commences by offering an overview of past attitudes and definitions of logic. Individual chapters consider the various means by which logical processes are conceived and developed in the mind. Philosophical arguments, spatial reasoning and mathematical forms of logic are discussed in great depth, with illustrations appended where deemed necessary. Read, an academic and philosopher, employs his decades long experience of (...) teaching students about the logical disciplines to write a detailed and comprehensive accounts of logic's aspects. Throughout this work, he attempts to retain a clarity of expression which makes the concepts discussed understandable for readers previously unversed in the types of logical expression. Read also sets out the principles behind setting up a hypothesis, and highlights the common logical fallacies which crop up in human reasoning. Extensive attention is given to the process of syllogism, by which sound conclusions are drawn from the premises offered. Throughout the book, Read is attentive in defining and explaining the esoteric terminology native to logic. As well as allowing the reader swift comprehension of the various subtopics, this explanatory style confers improved flow to the book. At the conclusion we are offered many questions corresponding to each chapter: these are intended to test the reader's understanding, and enforce the mental habits of utilising logical and critical thought. First published in 1898 and subject to several editions with revisions and refinements thereafter, Logic Deductive and Inductive retains a good reputation among academics and logicians. Its successful balance between striking a concise introductory tone with an embrace of the core elements of the subject has served to preserve its popularity into the modern day. (shrink)

We present a multiple-assumption multiple-conclusion system for bi-intuitionistic logic. Derivations in the systems are graphs whose edges are labelled by formulas and whose nodes are labelled by rules. We show how to embed both the standard intuitionistic and dual-intuitionistic natural deduction systems into the proposed system. Soundness and completeness are established using translations with more traditional sequent calculi for bi-intuitionistic logic.

This essay presents deductive arguments to an introductory-level audience via a discussion of Aristotle's three types of rhetoric, the goals of and differences between deductive and non-deductive arguments, and the major features of deductive arguments (e.g., validity and soundness).

Defining qualified reasoning as reasoning containing such loose qualifying words as 'probably,' 'usually,' 'probable, 'likely,' 'ceteris paribus,' and 'primafacie, Ennis argues that typical cases of qualified reasoning, though they might be good arguments, are deductively invalid, implying that such arguments fail soundness standards. He considers and rejects several possible alternative ways of viewing such cases, ending with a proposal for applying qualified soundness standards, which requires employment of sufficient background knowledge, sensitivity, experience and understanding of the situation. All of this (...) takes place as part of the process, introduced elsewhere, of successively applying different legitimate sets of standards until one is found that is satisfied, or none is found, the latter calling for rejection of the argument, the former calling for its acceptance. (shrink)

In 1944 Hans Reichenbach developed a three-valued propositional logic (RQML) in order to account for certain causal anomalies in quantum mechanics. In this logic, the truth-value _indeterminate_ is assigned to those statements describing physical phenomena that cannot be understood in causal terms. However, Reichenbach did not develop a deductive calculus for this logic. The aim of this paper is to develop such a calculus by means of First Degree Entailment logic (FDE) and to prove it sound and complete with (...) respect to RQML semantics. In Section 1 we explain the main physical and philosophical motivations of RQML. Next, in Sections 2 and 3, respectively, we present RQML and FDE syntax and semantics and explain the relation between both logics. Section 4 introduces \(\varvec{\mathcal {Q}}\) calculus, an FDE-based tableaux calculus for RQML. In Section 5 we prove that \(\varvec{\mathcal {Q}}\) calculus is sound and complete with respect to RQML three-valued semantics. Finally, in Section 6 we consider some of the main advantages of \(\varvec{\mathcal {Q}}\) calculus and we apply it to Reichenbach’s analysis of causal anomalies. (shrink)

Basin-Matthews-Viganò approach to construction of labelled deduction systems for normal modal logics is adapted to „Fitch proof-format“, and it is applied to the language of deontic-praxeological logic. Segerberg's suggestion on how to asses the adequacy of a logic for Kanger's theory of rights is being formally explicated and it is proved that herewith proposed system of labelled deduction satisfies Segerberg's criteria of adequacy. For the purpose of building the proof a semantics is given, which connects „the simplest semantics (...) of action“ with standard semantics of deontic logic. Soundness and completeness of the proposed labelled deduction with respect to aforementioned semantics is proved. (shrink)

THE INDICATIVE CONDITIONAL. A PROBABILISTIC CRITERION OF SOUNDNESS FOR DEDUCTIVE INFERENCES Our objective in this section is to establish a prima facie case ...

We introduce a natural deduction system for the until-free subsystem of the branching time logic Although we work with labelled formulas, our system differs conceptually from the usual labelled deduction systems because we have no relational formulas. Moreover, no deduction rule embodies semantic features such as properties of accessibility relation or similar algebraic properties. We provide a suitable semantics for our system and prove that it is sound and weakly complete with respect to such semantics.

An inference is standardly said to be sound just in case it is deductively valid and it has only true assumptions. The importance of a coherent concept of soundness to proof theory is obvious, in that it is only sound derivations, and not merely deductively valid arguments, that advance knowledge by providing proofs of theorems in logic and mathematics. The soundness paradox is informally albeit impredicatively formulated as argument : Argument is unsound, therefore, argument is unsound. This paper (...) introduces and explains the importance of the soundness paradox, formally demonstrates how to avoid superficial impredication via Gödelization, and compares it with the similar but significantly different liar and validity or Pseudo-Scotus paradoxes. Although there are similarities in this family of semantic diagonalizations, the soundness paradox is not just a hybrid of the liar and validity paradoxes, but is more fundamental, belonging to a special category that resists the most powerful received solutions to the liar and validity paradoxes. (shrink)

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems.

We present a multi-conclusion natural deduction calculus characterizing the dynamic reasoning typical of Adaptive Logics. The resulting system AdaptiveND is sound and complete with respect to the propositional fragment of adaptive logics based on CLuN. This appears to be the first tree-format presentation of the standard linear dynamic proof system typical of Adaptive Logics. It offers the advantage of full transparency in the formulation of locally derivable rules, a connection between restricted inference-rules and their adaptive counterpart, and the (...) formulation of abnormalities as a subtype of well-formed formulas. These features of the proposed calculus allow us to clarify the relation between defeasible and multiple-conclusion approaches to classical recapture. (shrink)

In this paper we study natural deduction for the intuitionistic and classical (normal) modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a (...) decision procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula ; and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches. (shrink)

In this paper a uniform methodology to perform natural\ndeduction over the family of linear, relevance and intuitionistic\nlogics is proposed. The methodology follows the Labelled\nDeductive Systems (LDS) discipline, where the deductive process\nmanipulates {\em declarative units} -- formulas {\em labelled}\naccording to a {\em labelling algebra}. In the system described\nhere, labels are either ground terms or variables of a given {\em\nlabelling language} and inference rules manipulate formulas and\nlabels simultaneously, generating (whenever necessary)\nconstraints on the labels used in the rules. A set of natural\ndeduction style (...) inference rules is given, and the notion of a\n{\em derivation} is defined which associates a labelled natural\ndeduction style ``structural derivation'' with a set of generated\nconstraints. Algorithmic procedures, based on a technique called\n{\em resource abduction\/}, are defined to solve the constraints\ngenerated within a structural derivation, and their termination\nconditions discussed. A natural deduction derivation is then\ndefined to be {\em correct} with respect to a given substructural\nlogic, if, under the condition that the algorithmic procedures\nterminate, the associated set of constraints is satisfied with\nrespect to the underlying labelling algebra. Finally, soundness\nand completeness of the natural deduction system are proved with\nrespect to the LKE tableaux system \cite{daga:LAD}.\footnote{Full\nversion of a paper presented at the {\em 3rd Workshop on Logic,\nLanguage, Information and Computation\/} ({\em WoLLIC'96\/}), May\n8--10, Salvador (Bahia), Brazil, organised by Univ.\ Federal de\nPernambuco (UFPE) and Univ.\ Federal da Bahia (UFBA), and\nsponsored by IGPL, FoLLI, and ASL.}. (shrink)

This paper is concerned with a natural deduction system for First Degree Entailment (FDE). First, we exhibit a brief history of FDE and of combined systems whose underlying idea is used in developing the natural deduction system. Then, after presenting the language and a semantics of FDE, we develop a natural deduction system for FDE. We then prove soundness and completeness of the system with respect to the semantics. The system neatly represents the four-valued semantics for FDE.

We propose the use of variable declarations in natural deduction. A variable declaration is a line in a derivation that introduces a new variable into the derivation. Semantically, it can be regarded as declaring that the variable denotes an element of the universe of discourse. Undeclared variables, in contrast, do not denote anything, and may not occur free in any formula in the derivation. Although most natural deduction systems in use today do not have variable declarations, the idea (...) can be traced back to one of the first papers on natural deduction. We show how the use of variable declarations in natural deduction leads to a formal system that has a number of desirable features: It is simple, easy to use and understand, and corresponds closely to ordinary informal reasoning. Soundness and completeness of the system are easily proven. Furthermore, the system clarifies the role of the existential instantiation rule in natural deduction. (shrink)

We study prefixed tableaux for first-order multi-modal logic, providing proofs for soundness and completeness theorems, a Herbrand theorem on deductions describing the use of Herbrand or Skolem terms in place of parameters in proofs, and a lifting theorem describing the use of variables and constraints to describe instantiation. The general development applies uniformly across a range of regimes for defining modal operators and relating them to one another; we also consider certain simplifications that are possible with restricted modal theories and (...) fragments. (shrink)

For each truth-functionally complete set of connectives, we construct a sound and complete natural deduction system containing no axioms and the smallest possible number of inference rules, namely one.

Charles S. Peirce attempted to develop his semiotic theory of cognitive signs interpretation, which are originated in our basic perceptual operations that quasi-prove the truth of perceptual judgment representing reality. The essential problem was to explain how, by a cognitive interpretation of the sequence of perceptual signs, we can represent external physical reality and reflectively represent our cognitive mind’s operations of signs. With his phaneroscopy introspection, Peirce shows how, without going outside our cognitions, we can represent external reality. Hence Peirce (...) can avoid the Berkeleyian, Humean, and Kantian phenomenologies, as well as the modern analytic philosophy and hermeneutic phenomenology. Peirce showed that with the trio of semiotic interpretation – abductive logic of discovery of hypotheses, deductive logic of necessary inference, and inductive logic of evaluation – we can reach a complete proof of the true representation of reality. This semiotic logic of reasoning is the epistemic logic representing human confrontation in reality, with which we can achieve knowledge and conduct our behavior. However, Peirce did not complete his realistic revolution to eliminate previously accepted nominalistic and idealistic epistemologies of formal logic and pure mathematics. Here, I inquire why Peirce did not complete his historical realist epistemological revolution and following that inquiry I attempt to reconstruct it. (shrink)

Kant's A-Edition objective deduction is naturally (and has traditionally been) divided into two arguments: an " argument from above" and one that proceeds " von unten auf." This would suggest a picture of Kant's procedure in the objective deduction as first descending and ascending the same ladder, the better, perhaps, to test its durability or to thoroughly convince the reader of its soundness. There are obvious obstacles to such a reading, however; and in this chapter I will argue (...) that the arguments from above and below constitute different, albeit importantly inter-related, proofs. Rather than drawing on the differences in their premises, however, I will highlight what I take to be the different concerns addressed and, correspondingly, the distinct conclusions reached by each. In particular, I will show that both arguments can be understood to address distinct specters, with the argument from above addressing an internal concern generated by Kant’s own transcendental idealism, and the argument from below seeking to dispel a more traditional, broadly Humean challenge to the understanding’s role in experience. These distinct concerns also imply that these arguments yield distinct conclusions, though I will show that they are in fact complementary. (shrink)

This monograph provides a new account of justified inference as a cognitive process. In contrast to the prevailing tradition in epistemology, the focus is on low-level inferences, i.e., those inferences that we are usually not consciously aware of and that we share with the cat nearby which infers that the bird which she sees picking grains from the dirt, is able to fly. Presumably, such inferences are not generated by explicit logical reasoning, but logical methods can be used to describe (...) and analyze such inferences. Part 1 gives a purely system-theoretic explication of belief and inference. Part 2 adds a reliabilist theory of justification for inference, with a qualitative notion of reliability being employed. Part 3 recalls and extends various systems of deductive and nonmonotonic logic and thereby explains the semantics of absolute and high reliability. In Part 4 it is proven that qualitative neural networks are able to draw justified deductive and nonmonotonic inferences on the basis of distributed representations. This is derived from a soundness/completeness theorem with regard to cognitive semantics of nonmonotonic reasoning. The appendix extends the theory both logically and ontologically, and relates it to A. Goldman's reliability account of justified belief. This text will be of interest to epistemologists and logicians, to all computer scientists who work on nonmonotonic reasoning and neural networks, and to cognitive scientists. (shrink)

We describe a relational framework that uniformly supports formalization and automated reasoning in varied propositional modal logics. The proof system we propose is a relational variant of the classical Rasiowa-Sikorski proof system. We introduce a compact graph-based representation of formulae and proofs supporting an efficient implementation of the basic inference engine, as well as of a number of refinements. Completeness and soundness results are shown and a Prolog implementation is described.

We provide a natural extension of the intuitionistic natural deduction system NJ with at most two consequences, which we call a binary conclusion natural deduction system. The classical natural deduction system NK is defined as NJ together with the law of the excluded middle or the double negation principle. However, in contrast to NK, the new BCN system has no such classical rule, but has two kinds of rules managing two consequences. One is to eliminate one consequence (...) and the other is to introduce another consequence. Except on this point, BCN proofs are derived in exactly the same way as those in NJ. It is shown that with respect to the propositional fragment of NK, the BCN system is sound and sufficiently complete to prove classical propositional theorems. Moreover, as a corollary we obtain a general form of Glivenko's theorem and disjunction property for one of the two consequences. We also show the normalization theorem for BCN. Finally, we study some extensions of BCN to classical predicate logic. (shrink)

The tetravalent modal logic is one of the two logics defined by Font and Rius :481–518, 2000) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so (...) good as in $${{\mathcal {TML}}}^N$$ TML N, but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus :481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al., we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic. (shrink)

This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we present the (...) two systems of natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)

This text examines the principles of sound reasoning concerning deduction, induction, and observation brought to bear upon the activity of value judgements. It has practical exercises, to aid application, and avoids technical jargon.

What is the philosophical significance of the soundness and completeness theorems for first-order logic? In the first section of this paper I raise this question, which is closely tied to current debate over the nature of logical consequence. Following many contemporary authors' dissatisfaction with the view that these theorems ground deductive validity in model-theoretic validity, I turn to measurement theory as a source for an alternative view. For this purpose I present in the second section several of the key ideas (...) of measurement theory, and in the third and central section of the paper I use these ideas in an account of the relation between model theory, formal deduction, and our logical intuitions. (shrink)

Consider two competing interpretations of Kant's Critique of Pure Reason: the epistemic and cognitive interpretations. The epistemic interpretation presents the first Critique as a work of epistemology, but what is more, it sees Kant as an early proponent of anti-psychologism—the view that descriptions of how the mind works are irrelevant for epistemology.2 Even if Kant does not always manage to purge certain psychological-sounding idioms from his writing, the epistemic interpretation has it, he is perfectly clear that he means his evaluation (...) of knowledge to be carried out independently of psychology.3 In contrast, the cognitive interpretation presents the first Critique as a description of the operation of human .. (shrink)

Instantiation Theory presents a new, general unification algorithm that is of immediate use in building theorem provers and logic programming systems. Instantiation theory is the study of instantiation in an abstract context that is applicable to most commonly studied logical formalisms. The volume begins with a survey of general approaches to the study of instantiation, as found in tree systems, order-sorted algebras, algebraic theories, composita, and instantiation systems. A classification of instantiation systems is given, based on properties of substitutions, degree (...) of type strictness, and well-foundedness of terms. Equational theories and the use of typed variables are studied in terms of quotient homomorphisms and embeddings, respectively. Every instantiation system is a quotient system of a subsystem of first-order term instantiation. The general unification algorithm is developed as an application of the basic theory. Its soundness is rigorously proved, and its completeness and efficiency are verfied for certain classes of instantiation systems. Appropriate applications of the algorithm include unification of first-order terms, order-sorted terms, and first-order formulas modulo alpha-conversion, as well as equational unification using simple congruences. (shrink)