Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that (...) none of these arguments withstands scrutiny, and so there is no reason to believe that computer-assisted proofs are not a priori. Thanks are due to Michael Levin, David Corfield, and an anonymous referee for Philosophia Mathematica for their helpful comments. Earlier versions of this paper were presented at the Hofstra University Department of Mathematics colloquium series, and at the 2005 New Jersey Regional Philosophical Association; I am grateful to both audiences for their comments. CiteULike Connotea Del.icio.us What's this? (shrink)

In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In this paper, I present some examples of computer-assisted proofs, including Appel and Haken’s work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. (...) In the conclusion, I talk about whether the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them. (shrink)

It has been claimed that computer-assisted proof utilizes empirical evidence in a manner unheard of in traditional mathematics and therefore its employment forces us to modify our conception of proof. This paper provides a critical survey of some arguments for this claim. It starts by revisiting a well known paper by Thomas Tymoczko on the computer proof of the Four-Color Theorem. Drawing on some ideas from the works of Tyler Burge and others, it then (...) considers a way to see the philosophical significance of computer proof that casts doubts on the claim. (shrink)

The authors present the main ideas of the computer-assisted proof of Mischaikow and Mrozek that chaos is really present in the Lorenz equations. Methodological consequences of this proof are examined. It is shown that numerical calculations can constitute an essential part of mathematical proof not only in the discrete mathematics but also in the mathematics of continua.

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. (...) There are “verifications” of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

This article aims to evaluate the purported empirical character of computer-assisted proof, as suggested by Thomas Tymoczko and others. Tymoczko famously argued that the proof of the Four-Color Theorem introduced a new, empirical method of proof, forcing us to modify the traditional conception of mathematical argument as a priori reasoning. Detlefsen and Luker contended that Tymoczko’s suggestion entailed that typically mathematical proofs were empirical. My chief interest is to raise some objections to a line of (...) thought common to both of these arguments, with a view to outlining an account of the a priori which allows thepossibility of a priori knowledge obtained by appeal to computers or through testimony. Drawing on some recent discussions by Tyler Burge, this account gives a broad construal of the non-justificatory, ‘enabling’ role that experience is held to play in knowledge and cognition, allowing us to argue that the purported empirical character of the appeal to computers pertains only to the role experience plays in enabling our access to the a priori warrant provided by computer proof. (shrink)

I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore (...) not so novel as he maintains. Finally, without attempting to give a full account of the notion of empirical mathematical evidence, I sketch a view showing how the use of calculation injects an empirical ingredient into proof. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have (...) been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

In the article, the philosophical significance of quantum computation theory for philosophy of mathematics is discussed. In particular, I examine the notion of “quantum-assisted proof” (QAP); the discussion sheds light on the problem of the nature of mathematical proof; the potential empirical aspects of mathematics and the realism-antirealism debate (in the context of the indispensability argument). I present a quasi-empiricist account of QAP’s, and discuss the possible impact on the discussions centered around the Enhanced Indispensabity Argument (EIA).

In this programmatic paper we renew the well-known question “What is a proof?”. Starting from the challenge of the mathematical community by computer assisted theorem provers we discuss in the first part how the experiences from examinations of proofs can help to sharpen the question. In the second part we have a look to the new challenge given by “big proofs”.

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical (...) truths of which we cannot have a priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical (...) truths of which we cannot have a priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language ar- gument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding (...) light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argu- ment’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools ef- fectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments. (shrink)

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too (...) numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions. (shrink)

The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by (...) surveyable proofs, and can be sensibly regarded as a priori. We also show that the aforementioned impression is mistaken because it fails to distinguish between proof search (the context of discovery) and proof checking (the context of justification). By using mechanized proof assistants capable of producing certificates that can be independently checked, it is possible to carry out complex proofs without the need to trust arbitrary custom-written code. We only need to trust one fixed, small, and simple piece of software: the proof checker. This is not only possible in principle, but is in fact becoming a viable methodology for performing complicated mathematical reasoning. This is evinced by a new proof of the four-color theorem that appeared in 2005, and which was developed and checked in its entirety by a mechanical proof system. (shrink)

This paper shows in a case study that for the development, the presentation, and the computer-assisted verification of formal ontologies the usage of higher-order languages and associated proof ass...

This paper outlines the development of a proof-of-concept workflow using machine learning and computer vision techniques to unlock the data within digitized handwritten US Census forms from the 1950s. The 1950s US Census includes over 6.5 million page images and was only recently made available to the public on April 1, 2022, following a 72-year access restriction period. Our project uses computational treatments to assist researchers in their efforts to recover and preserve the history of the erased Sacramento (...) Japantown. Sacramento once housed the fourth largest Japantown in the United States before experiencing WWII Japanese American Incarceration and the 1950s US Government program of urban renewal. The goal is to augment a researcher’s work in selecting a subset of Census pages for further transcription and analysis. We demonstrate a workflow for extracting demographic information using computer vision for image segmentation, and machine learning for handwritten character recognition. The workflow consists of a computational filtering process for Census records and a user interface for page review. These computational techniques are suitable for other cities, states, and communities, and demonstrate new strategies to unlock vital demographic information. The approach highlights the potential benefits of computational techniques for the analysis of form-based historical records of the twentieth century that can have an impact on social justice. (shrink)

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical proving (...) process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings. (shrink)

We prove that every computably enumerable random real is provable in Peano Arithmetic to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefix-free Turing machine” can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the (...) case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the Kraft-Chaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle. (shrink)

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine (...) learning results in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (shrink)

RESUMENPresentamos la contribución de Th. Tymoczko a la filosofía de la matemática y analizamos y evaluamos las demostraciones asistidas por ordenador y los razonamientos no rigurosos en la matemática experimental, con particular referencia al Teorema de los Cuatro Colores.PALABRAS CLAVETYMOCZKO – CUASI-EMPIRISMO – MATEMÁTICA EXPERIMENTAL – RAZONAMIENTO NO RIGUROSO – DEMOSTRACIONES ASISTIDAS POR ORDENADORABSTRACTWe present Th. Tymoczko’s contribution to the philosophy of mathematics, and we analyze and evaluate the computer-assisted proofs and the non-rigorous reasonings in the experimental mathematics, (...) particularly in reference to the Four-Colour Theorem.KEYWORDTYMOCZKO – QUASI-EMPIRICISM – EXPERIMENTAL MATHEMATICS – NON-RIGOROUS REASONING – COMPUTER-ASSISTED PROOF. (shrink)

In this paper we summarize observations bridging the declared aspirations of pictorial semiotics and its real achievements. Pictorial semiotics is here understood as the general study of pictures as signs and it constituted a fundamental step beyond the art historical captivation with individual images. In the first part of our contribution we present a review of the most important methods that have been proposed as an answer to deal with several pictorial problems (multiple instances, segmentation, non-figurative meaning). In the second (...) part, we offer some positive and on-going implementations designed to remedy the shortcomings observed. What is suggested is the proof of concept that human researchers need the assistance of computing methodologies. However, computers can only do their job once an adequate phenomenology of human experience is fed into the process. (shrink)

Computational tools might tempt us to renounce complete cer- tainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correct- ness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will (...) be too tired or too busy to scrutinize our putative proofs, computer proof assistants could help. But feeding a mathematical argument to a computer is hard. Still, we might be willing to undertake the endeavor in view of the extra perks that formalization may bring—chiefly among them, an enhanced mathematical understanding. (shrink)

The article is an attempt at collecting and systematising views on the role and place of computers in mathematics, in particular the views on the consequences of using computers in proving mathematical theorems. The following issues are considered in the article: the problem connected with the concept of mathematical proof and its features; the attempts at answering the question whether computer proofs are genuine mathematical proofs; the problems with methods of checking the correctness of classical and computer- (...) class='Hi'>assisted proofs; and finally the problem of the reliability of proofs and a related problem of the validity of mathematical conjectures. The main issue under discussion is the problem of philosophical consequences of regarding computer-assisted proofs as rightful in mathematics and especially its possible influence on the opinion that mathematical knowledge is a priori. (shrink)

This paper attempts to look at the contemporary mathematical practice through the lenses of the distributed cognition approach. The ubiquitous use of personal computers and the internet as a key attribute of the digital society is interpreted here as a means to achieve a more effective distribution of the human cognitive activity. The major challenge that determines the transformation of mathematical practice is identified as ‘the problem of complexity’. The computer-assisted complete formalization of mathematical proofs as a current (...) tendency is viewed as one of the strands along which the mathematical community responds to the challenge. It is shown that this tendency gives live to the project calling to revisit and rebuild the very foundations of mathematics to secure more effective communication and thus guarantee the reliability of contemporary mathematics. (shrink)

Computer-Assisted Argument Mapping (CAAM) is a new way of understanding arguments. While still embryonic in its development and application, CAAM is being used increasingly as a training and development tool in the professions and government. Inroads are also being made in its application within education. CAAM claims to be helpful in an educational context, as a tool for students in responding to assessment tasks. However, to date there is little evidence from students that this is the case. This (...) paper outlines the use of CAAM as an educational tool within an Economics and Commerce Faculty in a major Australian research university. Evaluation results are provided from students from a CAAM pilot within an upper-level Economics subject. Results indicate promising support for the use of CAAM and its potential for transferability within the disciplines. If shown to be valuable with further studies, CAAM could be included in capstone subjects, allowing computer technology to be utilised in the service of generic skill development. (shrink)

Examples in the history of Automated Theorem Proving are given, in order to show that even a seemingly ‘mechanical’ activity, such as deductive inference drawing, involves special cultural features and tacit knowledge. Mechanisation of reasoning is thus regarded as a complex undertaking in ‘cultural pruning’ of human-oriented reasoning. Sociological counterparts of this passage from human- to machine-oriented reasoning are discussed, by focusing on problems of man-machine interaction in the area of computer-assisted proof processing.

This article reviews the strengths and limitations of five major paradigms of medical computer-assisted decision making (CADM): (1) clinical algorithms, (2) statistical analysis of collections of patient data, (3) mathematical models of physical processes, (4) decision analysis, and (5) symbolic reasoning or artificial intelligence (Al). No one technique is best for all applications, and there is recent promising work which combines two or more established techniques. We emphasize both the inherent power of symbolic reasoning and the promise of (...) artificial intelligence and the other techniques to complement each other. Keywords: Diagnosis, Computer Assisted Decision Making, Artificial Intelligence * Current address: Intelligenetics, 124 University Avenue, Palo Alto, CA. 94301, U.S.A. ** Dr. Shortliffe is a Henry J. Kaiser Family Foundation Faculty Scholar in General Internal Medicine and recipient of research career development award LM0048 from the National Library of Medicine. CiteULike Connotea Del.icio.us What's this? (shrink)

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization (...) corresponds to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning. (shrink)

A study is described in which the effectiveness of a computer program (Hermes) on improving argumentative writing is tested. One group of students was randomly assigned to a control group and the other was assigned to the experimental group where they are asked to use the Hermes program. All students were asked to write essays on controversial topics to an opposed audience. Their essays were content-analysed for dialectical traits. Based on this analysis, it was concluded that the experimental group (...) wrote more dialectically effective essays than the control group, and the amount of difference between the control and experimental groups was related to the students' intellectual developmental level, as assessed by the Measure of Epistemological Reflection (MER). It is concluded that argumentative writing, operationalized here as dialectical writing, can be improved by computer-assisted instruction, but that attempts to teach such forms of thinking and writing need to take into account students' capacity to benefit from such instruction. Such capacity is defined here as intellectual development. (shrink)

Following the removal of fuel subsidy in Nigeria on 29th of May 2023, residents of Lagos State have witnessed a significant spike in cost of living. Incidentally, this has also turned into a serious challenge in the educational sector as this has affected the continued sustenance of the traditional mode of learning where students and lecturers meet on campus on a daily basis. Hence, the present research offers distance computer-assisted learning via Open and Distance Learning (ODL) as a (...) complement to traditional or conventional learning in tertiary institutions in Nigeria. Distance computer-assisted learning boasts mainly of synchronous, asynchronous or blended means of learning all of which have been used in the past to complement face-to-face interactions between lecturers and students of Lagos State University. It is on the basis of the economic inconveniences of fuel subsidy removal with the declaration of the three-day work week by the Lagos State University that this research proposes distance computer-assisted learning to mediate the gap created between lecturers and students of Lagos State University. However, the present research concerns with students’ readiness and awareness of this mode of learning. The methodology adopted for this research is quantitative analysis which involves the use of questionnaires offered to 380 students of Lagos State University in February 2024. The result shows that whereas students are willing to adopt distance computer-assisted learning, the challenges of power and telecommunications appear as serious challenges. It is the submission of this study that when the recommendations offered in this research are explored, ODL may serve as a complement to conventional learning. (shrink)

Equational reasoning arises in many areas of mathematics and computer science. It is a cornerstone of algebraic reasoning and results essential in tasks of specification and verification in functional programming, where a program is mainly a set of equations. The usual manipulation of identities while conducting informal proofs obviates many intermediate steps that are neccesary while developing them using a formal system, such as the equationally complete Birkhoff calculus ${\mathcal{B}}$. This deductive system does not fit in the common manner (...) of doing mathematical proofs, and it is not compatible with the mechanisms of proof assistants. The aim of this work is to provide a deductive system ${\mathcal{B}}^{\textrm{GOAL}}$ for equality, equivalent to ${\mathcal{B}}$ but suitable for constructing equational proofs in a backward fashion. This feature makes it adequate for interactive proof-search in the approach of proof assistants. This will be achieved by turning ${\mathcal{B}}^{\textrm{GOAL}}$ into a transition system of formal tactics in the style of Edinburgh LCF, such transformation allows us to give a direct formal definition of backward proof in equational logic. (shrink)

278 non-freshman university students taking a l2-week critical thinking course in a large single-section class, with computer-assisted guided practice as a replacement for small-group discussion, and all testing in machine-scored multiple-choice format, improved their critical thinking skills, as measured by the California Critical Thinking Skills Test, by half a standard deviation, a moderate improvement. The improvement was more than that reported with a traditional format without computer-assisted instruction, but less than that reported with a format using (...) both computer-assisted instruction and essay-type assignments. Further studies are needed to test hypotheses suggested by these results. (shrink)

This article reviews the strengths and limitations of five major paradigms of medical computer-assisted decision making (CADM): (1) clinical algorithms, (2) statistical analysis of collections of patient data, (3) mathematical models of physical processes, (4) decision analysis, and (5) symbolic reasoning or artificial intelligence (Al). No one technique is best for all applications, and there is recent promising work which combines two or more established techniques. We emphasize both the inherent power of symbolic reasoning and the promise of (...) artificial intelligence and the other techniques to complement each other. (shrink)

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully (...) adopted in larger scale by philosophers. (shrink)

Argument mapping is a way of diagramming the logical structure of an argument to explicitly and concisely represent reasoning. The use of argument mapping in critical thinking instruction has increased dramatically in recent decades. This paper overviews the innovation and provides a procedural approach for new teaches wanting to use argument mapping in the classroom. A brief history of argument mapping is provided at the end of this paper.

This article generalises the refined A-translation method for extracting programs from classical proofs [U. Berger,W. Buchholz, H. Schwichtenberg, Refined program extraction from classical proofs, Annals of Pure and Applied Logic 114 3–25] to the scenario where additional assumptions such as choice principles are involved. In the case of choice principles, this is done by adding computational content to the ‘translated’ assumptions, an idea which goes back to [S. Berardi, M. Bezem, T. Coquand, On the computational content of the axiom of (...) choice, JSL 63 600–622] and has been further elaborated in [U. Berger, P. Oliva, Modified bar recursion and classical dependent choice, in: M. Baaz, S.D. Friedman, J. Kraijcek , Logic Colloquium ’01, in: Lecture Notes in Logic, vol. 20, Springer, Berlin, 2005, pp. 89–107]. We further investigate the applicability of this extension by means of a small but non-trivial example, and discuss the formalisation as well as some optimisations necessary in order to extract terminating programs. Both formalisation and term extraction have been implemented in the interactive proof assistant Minlog. (shrink)

This paper introduces the special issue on Logic and Religion of the journal Logica Universalis (Springer). The issue contains the following articles: Logic and Religion, by Jean-Yves Beziau and Ricardo Silvestre; Thinking Negation in Early Hinduism and Classical Indian Philosophy, by Purushottama Bilimoria; Karma Theory, Determinism, Fatalism and Freedom of Will, by Ricardo Sousa Silvestre; From Logic in Islam to Islamic Logic, by Musa Akrami; Leibniz’s Ontological Proof of the Existence of God and the Problem of Impossible Objects, by (...) Wolfgang Lenzen; A Logical Analysis of the Anselm’s Unum Argumentum (from Proslogion), by Jean-Pierre Desclés; Monotonic and Non-monotonic Embeddings of Anselm’s Proof, by Jacob Archambault; Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy, by C. Benzmüller, L. Weber and B. Woltzenlogel Paleo. (shrink)

This paper describes formalizations of Tait's normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of (...) added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers. (shrink)

Certainty (and the lack thereof) is a major issue in mathematics and computer science. Mathematicians strongly believe in a special kind of certainty for their theorems.

This article addresses the use of computer-assisted career guidance systems (CACGS) in career interventions. Major CACGS developed in the past decades were based on the trait-factor or person-environment fit approaches in their conceptualization and design. The strengths and limitations of these CACGS in addressing the career development needs of individuals are discussed. The Career Construction Theory (CCT) is a promising paradigm to guide the development of new generations of CACGS. The narrative tradition, career adaptability model, and life-design interventions (...) of CCT offer rich conceptual and practical applications that could expand the scope and breadth of career exploration and identity construction through using CACGS. A digital system developed in Hong Kong called Infinity is a case-in-point of a CACGS where users could learn about career planning, engage in self and career construction through using the quantitative and qualitative assessment applications and gamified tools, take career planning actions over time, and communicate with their social supportive systems. Initial findings suggested that users of the system reported lesser decision-making difficulties, higher levels of decision clarity, and better understanding of what is career planning than non-users. Users from high academic achievement schools reported higher levels of career adaptability than their counterparts in schools of similar academic background. Users from low achievement schools reported higher intention to pursue government-supported universities than non-users from schools of similar background. Research and practice implications in schools and organizational settings are discussed. (shrink)

The comments accompanying Proposition (Prop.) 11 (“God... necessarily exists”) in Part I of Spinoza’s The Ethics contain sketches of what appear to be at least three more or less distinct ontological arguments. The first of these is problematic even on its own terms. More is true: even the proposition “God exists” (GE), a consequence of Prop. 11, cannot be derived from the definitions and axioms of Part I (the “DAPI”) of The Ethics; thus, Prop. 11 cannot be derived from the (...) DAPI, either. To prove these claims, I use an automated deduction system (ADS) to show that Prop. 11 is independent of the DAPI. I then augment the DAPI with some auxiliary assumptions I believe Spinoza would accept and that sustain an automated derivation of (GE). The results illustrate how an ADS can facilitate the analysis of arguments and yield an apparently novel argument cast in the style of Spinoza. (shrink)

In her 2010 study of the Shi zhu duan jie jing T309, Jan Nattier found that several passages in T309 were copied from earlier Chinese Buddhist texts. She thus proposed that T309 is not a translation from an Indian text, but a “forgery” by Zhu Fonian. Extending Nattier’s analysis with the help of TACL, a tool for computational textual analysis, we conducted a more thorough analysis of Zhu Fonian’s four Mahayana texts, namely, T309, the Pusa chu tai jing T384, the (...) Zhongyin jin T385, and the Pusa yingluo jing T656, and found in T309 and T656 additional content deriving from earlier Chinese texts. On the basis of this analysis of these features of the texts, we propose that all four were likely compiled by Zhu Fonian himself. (shrink)

There has been increasing use of argument-based approaches in the development of safety-critical systems. Within this approach, a safety case plays a key role in the system development life cycle. The key components in a safety case are safety arguments, which are designated to demonstrate that the system is acceptably safe. Inappropriate reasoning in safety arguments could undermine a system's safety claims which in turn contribute to safety-related failures of the system. The review of safety arguments is therefore a crucial (...) step in the development of safety-critical systems. Reviews are conducted using dialogues where elements of the argument and their relations are proposed and scrutinised. This paper investigates an approach of conducting argument review using dialectical models. After studying five established dialectical models with varying strengths and drawbacks, a new dialectical model specially designed to support persuasion and information-seeking dialogues has been proposed to suit the requir... (shrink)