The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. -- Godel [11].

The notion of efficient proof-checking has always been central to complexity theory, and it gave rise to the definition of the class NP. In the last 15 years there has been a number of exciting, unexpected and deep developments in complexity theory that exploited the notion of randomized and interactive proof-checking. Results developed along this line of research have diverse and powerful applications in complexity theory, cryptography, and the theory of approximation algorithms for combinatorial optimization problems. (...) In this paper we survey the main lines of developments in interactive and probabilistic proof-checking, with an emphasis on open questions. (shrink)

The proof paradox results from conflicting intuitions concerning different types of fallible evidence in a court of law. We accept fallible individual evidence but reject fallible statistical evidence even when the conditional probability that the defendant is guilty given the evidence is the same, a seeming inconsistency. This paper defends a solution to the proof paradox, building on a sensitivity account of checking and settling a question. The proposed sensitivity account of legal proof not only requires (...) sensitivity simpliciter but sensitivity of each relevant step of the proof procedure and/or sensitivity concerning all relevant alternatives. This account avoids problems that have been identified for other sensitivity views of legal proof. Moreover, it is argued that sensitivity, rather than safety, is the crucial modal condition for legal proof. It is, finally, shown that the provided account can fruitfully support very different existing views on the relationship between knowledge and legal proof. (shrink)

__Language Proof and Logic_ is available as a physical book with the software included on CD and as a downloadable package of software plus the book in PDF format. The all-electronic version is available from Openproof at ggweb.stanford.edu._ The textbook/software package covers first-order language in a method appropriate for first and second courses in logic. An on-line grading services instantly grades solutions to hundred of computer exercises. It is designed to be used by philosophy instructors teaching a logic course (...) to undergraduates in philosophy, computer science, mathematics, and linguistics. Introductory material is presented in a systematic and accessible fashion. Advanced chapters include proofs of soundness and completeness for propositional and predicate logic, as well as an accessible sketch of Godel's first incompleteness theorem. The book is appropriate for a wide range of courses, from first logic courses for undergraduates to a first graduate logic course. The software package includes four programs: Tarski's World 5.0, a new version of the popular program that teaches the basic first-order language and its semantics; Fitch, a natural deduction proof environment for giving and checking first-order proofs; Boole, a program that facilitates the construction and checking of truth tables and related notions ; Submit, a program that allows students to submit exercises done with the above programs to the Grade Grinder, the automatic grading service. Grade reports are returned to the student and, if requested, to the student's instructor, eliminating the need for tedious checking of homework. All programs are available for Windows, Macintosh and Linux systems. Instructors do not need to use the programs themselves in order to be able to take advantage of their pedagogical value. More about the software can be found at lpl.stanford.edu. The price of a new text/software package includes one Registration ID, which must be used each time work is submitted to the grading service. Once activated, the Registration ID is not transferable. (shrink)

Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi (...) is proved, and used to conclude that the considered logical systems are PSPACE-complete. (shrink)

In our previous work we have introduced loop-type sequent calculi for propositional linear discrete tense logic and proved that these calculi are sound and complete. Decision procedures using the calculi have been constructed for the considered logic. In the present paper we restrict ourselves to the logic with the unary temporal operators “next” and “henceforth always”. Proof-theory of the sequent calculus of this logic is considered, focusing on loop specification in backward proof-search. We describe cyclic sequents and prove (...) that any loop consists of only cyclic sequents. A class of sequents for which backward proof-search do not require loop-check is presented. It is shown how sequents can be coded by binary strings that are used in backward proof-search for the sake of more efficient loop-check. (shrink)

Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. Formal logic allows you to check a logical claim without considering what the claim means. This highly abstracted idea is an essential and practical part of computer science. The idea of a formal system-a collection of rules and axioms, which define a universe of logical proofs-is what gives us programming languages and modern-day programming. This (...) book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses-natural deduction-is very small and very simple; working with it helps you see how large mathematical universes can be built on small foundations. The book is divided into four parts: Part I "Basics" gives an introduction to formal logic with a short history of logic and explanations of some technical words. Part II "Formal Syntactic Proof" show you how to do calculations in a formal system where you are guided by shapes and never need to think about meaning. Your experiments are aided by Jape, which can operate as both inquisitor and oracle. Part III "Formal Semantic Disproof" shows you how to construct mathematical counterexamples to shoe that proof is impossible. Jape can check the counterexamples you build. Part IV " Program Specification and Proof" describes how to apply your logical understanding to a real computer science problem, the accurate description and verification of programs. Jape helps, as far as arithmetic allows. Aimed at undergraduates and graduates in computer science, logic, mathematics and philosophy, the text includes reference to and exercises based on the computer software package Jape, an interactive teaching and research tool designed and hosted by the author that is freely available on the web. (shrink)

In the last 2500 years, since the beginnings in ancient Greece of the literary genre we call “history,” the relationship between history and law has been very close. True, the Greek word historia is derived from medical language, but the argumentative ability it implied was related to the judicial sphere. History, as Arnaldo Momigliano emphasized some years ago, emerged as an independent intellectual activity at the intersection of medicine and rhetoric. Following the example of the former, the historian analyzed specific (...) cases and situations looking for their natural causes; following the prescriptions of the latter—a technique, or an art, born in tribunals—he communicated the results of his inquiry.2Within the classical tradition, historical writing had to display a feature the Greeks called enargeia, and the Romans, evidential in narrtatione: the ability to convey a vivid representation of characters and situations. The historian, like the lawyer, was expected to make a convincing argument by communicating the illusion of reality, not by exhibiting proofs collected either by himself or by others.3 Collecting proofs was, until the mid-eighteenth century, an activity practiced by antiquarians and erudite, not by historians.4 When, in his Traité des differentes sortes de preuves qui servent à établir la vérité he l’histoire , the erudite Jesuit Henri Griffet compared the historian to a judge who carefully evaluates proofs and witnesses, he was expressing a still-unaddressed intellectual need. Only a few years later Edward Gibbon published his Decline and Fall of the Roman Empire, the first work that effectively combined historical narrative with an antiquarian approach.5 2. See Arnaldo Momigliano, ”History between Medicine and Rhetoric,” Ottavo contributo alla storia degli studi classici e del mondo antico, trans. Riccardo Di Donato , pp. 14-25.3. See Ginzburg, “Montrer et citer.”4. See Momigliano, “Ancient History and the Antiquarian,” Contributo alla storia degli studi classici , pp. 67-106.5. See Henri Griffet, Traité des differentes sortes de preuves qui servent à établir la vérité de l’histoire, 2d ed. . Allen Johnson, in his Historian and Historical Evidence , speaks of the Traité as “the most significant book on method after Mabillon’s De re diplomatic” . See also Momigliano, “Ancient History and the Antiquarian,” p. 81, and Ginzburg, “Just One Witness.” On Gibbon, see Momilgiano, Sesto contributo alla storia degli studi classici e del mondo antico , pp. 231-84. Carlo Ginzburg is Franklin D. Murphy Professor of Italian Renaissance Studies at the University of California, Los Angeles. His two most recent books are Ecstasies: Deciphering the Witches’ Sabbath and Il giudice e lo storico. (shrink)

This textbook/software package covers first-order language in a method appropriate for first and second courses in logic. The unique on-line grading services instantly grades solutions to hundred of computer exercises. It is specially devised to be used by philosophy instructors in a way that is useful to undergraduates of philosophy, computer science, mathematics, and linguistics. The book is a completely rewritten and much improved version of The Language of First-order Logic. Introductory material is presented in a more systematic and accessible (...) fashion. Advanced chapters include proofs of soundness and completeness for propositional and predicate logic, as well as an accessible sketch of Godel's first incompleteness theorem. The book is appropriate for a wide range of courses, from first logic courses for undergraduates to a first graduate logic course. The package includes four pieces of software: Tarski's World 5.0, a new version of the popular program that teaches the basic first-order language and its semantics; Fitch, a natural deduction proof environment for giving and checking first-order proofs; Boole, a program that facilitates the construction and checking of truth tables and related notions ; Submit, a program that allows students to submit exercises done with the above programs to the Grade Grinder, the automatic grading service. Grade reports are returned to the student and, if requested, to the student's instructor, eliminating the need for tedious checking of homework. All programs are available for Windows, Macintosh and Linux systems. Instructors do not need to use the programs themselves in order to be able to take advantage of their pedagogical value. More about the software can be found at lpl.stanford.edu. The price of a new text/software package includes one Registration ID, which must be used each time work is submitted to the grading service. Once activated, the Registration ID is not transferable. (shrink)

Tetravalent modal logic (TML) was introduced by Font and Rius in 2000. It is an expansion of the Belnap-Dunn four-valued logic FOUR, a logical system that is well-known for the many applications found in several fields. Besides, TML is the logic that preserves degrees of truth with respect to Monteiro’s tetravalent modal algebras. Among other things, Font and Rius showed that TML has a strongly adequate sequent system, but unfortunately this system does not enjoy the cut-elimination property. However, in a (...) previous work we presented a sequent system for TML with the cut-elimination property. Besides, in this same work, it was also presented a sound and complete natural deduction system for this logic. In the present article we continue with the study of TML under a proof-theoretic perspective. In the first place, we show that the natural deduction system that we introduced before admits a normalization theorem. In the second place, taking advantage of the contrapositive implication for the tetravalent modal algebras introduced by A. V. Figallo and P. Landini, we define a decidable tableau system adequate to check validity in the logic TML. Finally, we provide a sound and complete tableau system for TML in the original language. These two tableau systems constitute new (proof-theoretic) decision procedures for checking validity in the variety of tetravalent modal algebras. (shrink)

Deductive proof checking programs are the most popular form of logic CAI. Whatever the reason for their widespread use, the proliferation and continuous development of these programs is evident. Contemporary proof checkers cover a wider variety of texts and rule sets, and offer more helpful editing, diagnostic, and remedial features than were once provided. These programs appear to be prime candidates for developing in the direction of "intelligent" CAI (ICAI). The primary thrust of ICAI is to build (...) programs that make use of information about learner strengths and weaknesses, the content of the subject matter being taught, and techniques for teaching various kinds of subject matter. This is a tall order by any standard, but there are signs that some initial progress is being made in the area of logic CAI. In particular, the expert system approach for offering strategic advice during proof construction is being explored by some projects. (shrink)

Model checking, a prominent formal method used to predict and explain the behaviour of software and hardware systems, is examined on the basis of reflective work in the philosophy of science concerning the ontology of scientific theories and model-based reasoning. The empirical theories of computational systems that model checking techniques enable one to build are identified, in the light of the semantic conception of scientific theories, with families of models that are interconnected by simulation relations. And the mappings (...) between these scientific theories and computational systems in their scope are analyzed in terms of suitable specializations of the notions of model of experiment and model of data. Furthermore, the extensively mechanized character of model-based reasoning in model checking is highlighted by a comparison with proof procedures adopted by other formal methods in computer science. Finally, potential epistemic benefits flowing from the application of model checking in other areas of scientific inquiry are emphasized in the context of computer simulation studies of biological information processing. (shrink)

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that there should be a canonical function from sequent proofs to proof nets, it should be possible to check (...) the correctness of a net in polynomial time, every correct net should be obtainable from a sequent calculus proof, and there should be a cut-elimination procedure which preserves correctness.Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley [22], the author presented a calculus of proof nets satisfying and ; the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of . That sequent calculus, called LK⁎LK⁎ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper [22]), we give a full proof of for expansion nets with respect to LK⁎LK⁎, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria. (shrink)

Proof and Consequence is a rigorous, elegant introduction to classical first-order natural deductive logic; it provides an accurate and accessible first course in the study of formal systems. The text covers all the topics necessary for learning logic at the beginner and intermediate levels: this includes propositional and quantificational logic (using Suppes-style proofs) and extensive metatheory, as well as over 800 exercises. Proof and Consequence provides exclusive access to the software application Simon, an easily downloadable program designed to (...) facilitate an intuitive understanding of classical logic through the generation and analysis of proofs. It also aids with the representation of natural language sentences in the formal language. Equipped with nearly all the exercises found in the text, Simon helps students work efficiently and effectively by detecting and explaining errors in solutions as they proceed. Students can also submit assignments, view their own records, and check their standing in the class. The complete logic package includes: * The logic textbook, Proof and Consequence *A very helpful study guide to the textbook, containing extra exercises, Simple Simon *Access, through Simon, to the grading software, Simon Says, that allows students to submit assignments and track their grades. (shrink)

Complex frameworks for defining programming languages aim to generate various tools (e.g. interpreters, symbolic execution engines, deductive verifiers, etc.) using only the formal definition of a language. When used at an industrial scale, these tools are constantly updated, and at the same time, it is required to be trustworthy. Ensuring the correctness of such a framework is practically impossible. A solution is to generate proof objects as correctness artefacts that can be checked by an external trusted checker. A logic (...) suitable for developing such frameworks is matching logic. K framework is a canonical example having matching logic-based foundation. Since the (symbolic) configurations of the programs are represented by matching logic patterns, the algorithms computing the dynamics of these configurations can be seen as pattern transformers and a proof object should be generated for the relationship between these patterns. In this paper, we show that conjunctions and disjunctions of patterns, produced by semantics or analysis rules, can be safely normalized using unification and antiunification algorithms. We also provide a prototype implementation of our proof object generation technique and a checker for certifying the generated objects. (shrink)

The logic of single-conclusion proofs () is introduced. It combines the verification property of proofs with the single valuedness of proof predicate and describes the operations on proofs induced by modus ponens rule and proof checking. It is proved that is decidable, sound and complete with respect to arithmetical proof interpretations based on single-valued proof predicates. The application to arithmetical inference rules specification and -admissibility testing is considered. We show that the provability in gives the (...) complete admissibility test for the rules which can be specified by schemes in -language. The test is supplied with the ground proof extraction algorithm which eliminates the admissible rules from a -proof by utilizing the information from the corresponding -proofs. (shrink)

We describe a proof by a computer program of the unsolvability of the halting problem. The halting problem is posed in a constructive, formal language. The computational paradigm formalized is Pure LISP, not Turing machines. The machine was led to the proof by the authors, who suggested certain function definitions and stated certain intermediate lemmas. The machine checked that every suggested definition was admissible and the machine proved the main theorem and every lemma. We believe this is the (...) first instance of a machine checking that a given problem is not solvable by machine. (shrink)

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of Constructions. The book (...) also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material. (shrink)

The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several (...) celebrated theorems in metamathematics including those of Gödel and Church-Rosser. The computer verification using the Boyer-Moore theorem prover yields precise and rigorous proofs of these difficult theorems. It also demonstrates the range and power of automated proof checking technology. The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction. (shrink)

In this paper, we provide a general setting under which results of normalization of proof trees such as, for instance, the logicality result in equational reasoning and the cut-elimination property in sequent or natural deduction calculi, can be unified and generalized. This is achieved by giving simple conditions which are sufficient to ensure that such normalization results hold, and which can be automatically checked since they are syntactical. These conditions are based on basic properties of elementary combinations of inference (...) rules which ensure that the induced global proof tree transformation processes do terminate. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs (...) of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

Proof-theoretic validity has proven a useful tool for proof-theoretic semantics, because it explains the harmony found in the introduction and elimination rules for the intuitionistic calculus. However, the demonstration that a rule of proof is proof-theoretically valid requires checking an infinite number of cases, which raises the question of whether proof-theoretic validity is decidable. It is proven here that it is for the most prominent formulations in the literature for propositional logic.

In this section, we apply the notions obtained above to a famous historical example of a false proof. Our goal is to demonstrate that this proof shows a sufficient degree of distinctiveness for a formalization in a Naproche-like system and hence that automatic checking could indeed have contributed in this case to the development of mathematics. This example further demonstrates that even incomplete distinctivication can be sufficient for automatic checking and that actual mistakes may occur already (...) in the margin between the degree of distinctiveness necessary for formalization and complete distinctiveness. (shrink)

This paper is concerned with the structure of texts in which aproof is presented. Some parts of such a text are assumptions, otherparts are conclusions. We show how the structural organisation of thetext into assumptions and conclusions helps to check the validity of theproof. Then we go on to use the structural information for theformulation of proof rules, i.e., rules for the (re-)construction ofproof texts. The running example is intuitionistic propositional logicwith connectives , and. We give new proofs of (...) some familiar results aboutthe proof theory of this logic to indicate how the new techniques workout. (shrink)

If we want to understand why mathematical knowledge is extraordinarily reliable, we need to consider both the nature of mathematical arguments and mathematical practice as a social practice. Mathematical knowledge is extraordinarily reliable because arguments in mathematics take the form of deductive mathematical proofs. Deductive mathematical proofs are surveyable in the sense that they can be checked step by step by different experts, and a purported proof is only accepted as a proof by the mathematical community once a (...) number of experts have checked the proof. Hence, the reliability of the body of mathematical knowledge is in part obtained through the surveyability of proofs and the social process of proof validation. This chapter reviews work relating to (1) the norm in the mathematical community of only counting as argument deductive mathematical proof, and (2) the norm of only counting as deductive mathematical proof an argument that has been confirmed to be a deductive mathematical proof by a number of experts. The chapter also presents cases of unusual mathematical proofs or arguments that challenge these norms. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents (...) genuine meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

We study satisfiability and infinite-state model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2EXPTIME, thus 2EXPTIME-complete by an existing lower bound, and (ii) infinite-state model checking of basic process algebras and pushdown systems is also 2EXPTIME-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular tree satisfiability in ICPDL, a reasoning problem that we (...) introduce specifically for this purpose. This problem is then reduced to the emptiness problem for alternating two-way automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2EXPTIME. We prove the lower bound(s) for infinite-state model checking using an encoding of alternating Turing machines. (shrink)

Ideas on meaning, rules and mathematical proofs abound in Wittgenstein’s writings. The undeniable fact that they are present together, sometimes intertwined in the same passage of Philosophical Investigations or Remarks on the Foundations of Mathematics, does not show, however, that the connection between these ideas is necessary or inextricable. The possibility remains, and ought to be checked, that they can be plausibly and consistently separated. I am going to examine two views detectable in Wittgenstein’s works: one about proofs, the other (...) about meaning and rules. The first is the denial of the objectivity of proof. The second is a conception of meaning stemming from the rule-following considerations. I shall argue that, though Wittgenstein seems to conjoin the two views, they can be, and should be, separated1. (shrink)

Recently I asked a well-known ID sympathizer what shape he thought the ID movement was in. I raised the question because, after some initial enthusiasm on his part three years ago, his interest seemed to have flagged. Here is what he wrote: An enormous amount of energy has been expended on "proving" that ID is bogus, "stealth creationism," "not science," and so on. Much of this, ironically, violates the spirit of science. The proof of the pudding is in the (...) eating. But on the other side, too much stuff from the ID camp is repetitive, imprecise and immodest in its claims, and otherwise very unsatisfactory. The "debate" is mostly going around in circles. The real work needs to go forward. There is a tremendous ferment right now in the "evo/devo" field, for instance. Some bright postdocs sympathetic to ID (and yes, I know how hard a time they would have institutionally at many places) should plunge right into the thick of that. Maybe they are at this very moment: I hope so! Every now and again we need to take a good, hard look in the mirror. The aim of this talk is to help us do just that. Intelligent design has made tremendous inroads into the culture at large. Front page stories featuring our work have appeared in the New York Times, L.A. Times, Wall Street Journal, San Francisco Chronicle, and so on. Television, radio, and weeklies like Time Magazine are focusing the spotlight on us as well. This publicity is at once useful and seductive. It useful because it helps get the word out and attract talent to the movement. It is seductive because it can deceive us into thinking that we have accomplished more than we actually have. (shrink)

In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account does not accord with actual mathematical practice with respect to computability theory. We argue instead (...) for an alternative account in which imagination, anticipation, and interpretations of natural language play roles in establishing mathematical rigor. (shrink)

The paper provides a conceptual distinction between evidence assessment criteria and standards of proof. Evidence must be assessed in order to check whether it satisfies a relevant standard of proof, and the assessment is operated with some criterion; so both criteria and standards are necessary for fact-finding. In addition to this conceptual point, the article addresses three main questions: Why do some scholars and decision-makers take assessment criteria as standards of proof and vice versa? Why do systems (...) differ as to criteria and standards? How can a system work if it neglects one of these things? The answers to the first and second question come from the historical and procedural differences between the systems. The answer to the third focuses on the functional connection between criteria and standards. (shrink)

This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL. As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

This paper presents a new correctness criterion for marked Danos-Reginer graphs of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL. As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

The director of Courage International talks about the work of the apostolate in addressing homosexuality according to the mind and heart of the Church, which he calls “one of the most demanding aspects of education, formation, and pastoral care today.” But it is also an opportunity to attend to the often acute and persistent wounds of those who need healing within what Pope Francis calls the “field hospital” of the Church. The author points out that the work of Courage is (...) not first about homosexuality but about what it means to be human. It is an invitation to consider the question of the fulfillment of the human heart according to God’s gracious plan. National Catholic Bioethics Quarterly 15.2 : 221–230. (shrink)

It is widely accepted that in fallible reasoning potential error necessarily increases with every additional step, whether inferences or premises, because it grows in the same way that the probability of a lengthening conjunction shrinks. As it stands, this is disappointing but, I will argue, not out of keeping with our experience. However, consulting an expert, proof-checking, constructing gap-free proofs, and gathering more evidence for a given conclusion also add more steps, and we think these actions have the (...) potential to improve our reliability or justifiedness. Thus, the received wisdom about the growth of error implies a skepticism about the possibility of improving our reliability and level of justification through effort. Paradoxically, and even more implausibly, taking steps to decrease your potential error necessarily increases it. I will argue that the self-help steps listed here are of a distinctive type, involving composition rather than conjunction. Error grows differently over composition than over conjunction, I argue, and this dissolves the apparent paradox. (shrink)