Showing papers on "Formal system published in 2020"

A Formal System of Axiomatic Set Theory in Coq

Abstract: Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

Theorizing about the Syntax of Human Language: A Radical Alternative to Generative Formalisms

Abstract: Linguists standardly assume that a grammar is a formal system that ‘generates’ a set of derivations. But this is not the only way to formalize grammars. I sketch a different basis for syntactic theory: model-theoretic syntax (MTS). It defines grammars as finite sets of statements that are true (or false) in certain kinds of structure (finite labeled graphs such as trees). Such statements provide a direct description of syntactic structure. Generative grammars do not do this; they are strikingly ill-suited to accounting for certain familiar properties of human languages, like the fact that ungrammaticality is a matter of degree. Many aspects of linguistic phenomena look radically different when viewed in MTS terms. I pay special attention to the fact that sentences containing invented nonsense words (items not in the lexicon) are nonetheless perceived as sentences. I also argue that the MTS view dissolves the overblown controversy about whether the set of sentences in a human language is always infinite: many languages (both Brazilian indigenous languages and others) appear not to employ arbitrarily iterative devices for embedding or coordination, but under an MTS description this does not define them as radically distinct in typological terms.

Consistency and incompleteness in general equilibrium theory

Abstract: We consider the implications for general equilibrium theory of the problems of consistency and completeness as shown in the Godel-Rosser theorems of the 1930s. That a rigorous consistent formal system is incomplete poses serious problems for dealing with unresolved problems in a fully formal system such as general equilibrium theory. We review the underlying mathematical issues and apply them to this problem for general equilibrium theory. We also consider alternative approaches such as agent-based modeling founded on empirically estimated behavioral assumptions for agents that may allow for a better way to model non-equilibrium evolutionary economic dynamics.

FASTEN.Safe: A Model-Driven Engineering Tool to Experiment with Checkable Assurance Cases

Abstract: The Goal Structuring Notation (GSN) is popular among safety engineers for modeling assurance cases. GSN elements are specified using plain natural language text, this giving safety engineers great flexibility to express their arguments. However, pure textual arguments introduce ambiguities and prevent automation. Currently, assurance cases are verified by manual reviews, which are error prone, time consuming, and not adequate for today’s systems complexity and agile development methodologies. In this paper we present our research tool FASTEN.Safe, which extends GSN with a set of higher-level modeling language constructs capturing recurring argumentation patterns and integrating formal system models. This allows automatically checking 1) the intrinsic consistency of assurance models, 2) the consistency of arguments with system models and 3) the verification of safety claims themselves by using external verification tools. FASTEN.Safe is open source and allows experimenting with language abstractions to bridge the world of GSN-based arguments that are common among safety engineers and the world of formal methods that enable automation. Last but not least, we report on the preliminary experience gained with FASTEN.Safe.

Factors impacting on effective implementation of land title registration – a perspective from Ghana

Abstract: Many governments across Sub Saharan Africa are in the process of introducing or improving land registration and formal titling systems. One of the stated aims is to achieve modern land information management in order to facilitate the development of the land market. It is often assumed that, because formal systems and institutions have enjoyed some positive outcomes in terms of realising wealth in developed countries, they will succeed equally well in developing economies. However, findings from empirical studies across several developing countries show that the performance of formal land registration systems has been mixed. Relying on empirical data from two major cities in Ghana, this paper examines the operations of land registration system with particular reference to its land information management aspects. The analysis shows that a divergence in the implementation of principles of the legal framework and organisational challenges are major contributory factors to deficiencies in the land information regime of the land registration system. Hence, there is a need for effective implementation of well-crafted and functional legal frameworks for land registration, to ensure that the principles and operations of land registration are locally relevant and sensitive. To address the inadequate organisational capacity there is a need to improve the capacity of the human resource base of the officials of the formal land administration sector. The procedure for land registration must also be streamlined in order to eliminate unnecessary requirements and thereby reduce the transaction time, costs of registration and frustration of clients.

Socio-hydrological modelling to assess reliability of an urban water system under formal-informal supply dynamics

Abstract: Increasing water scarcity in developing world cities combined with poor performance of water supply systems has led to an increasing reliance on informal water supply systems. Although the availability of informal supply provides a coping mechanism that enables water consumers to be resilient to failures in water supply, the longer-term effects on formal water supply systems (FWSS) are uncertain, with a potential reduction of tariff recovery (RT), and in turn a service provider’s financial sustainability. This motivates an analysis of the coevolving dynamics and feedbacks involved in water systems where formal and informal components co-exist. Investigating Hyderabad, Pakistan as a case study, a dynamic socio-hydrologic system model is built, comprised of a formal system’s water and fund balance, consumer behaviour and infrastructure conditions. Simulations are executed on a monthly basis at a household level and for a 100-year period (2007–2107) using data available from years 2007–2017. Demand shift to informal is observed to be weakly associated with lower recovery rates, with household income as a major predictor. The FWSS’s financial balance, predominantly driven by infrastructure condition, appears to be less sensitive to recovery of a tariff to generate sufficient revenue.

A Structured Argumentation Framework for Modeling Debates in the Formal Sciences

Abstract: Scientific research in the formal sciences comes in multiple degrees of formality: fully formal work; rigorous proofs that practitioners know to be formalizable in principle; and informal work like rough proof sketches and considerations about the advantages and disadvantages of various formal systems This informal work includes informal and semi-formal debates between formal scientists, eg about the acceptability of foundational principles and proposed axiomatizations In this paper, we propose to use the methodology of structured argumentation theory to produce a formal model of such informal and semi-formal debates in the formal sciences For this purpose, we propose ASPIC-END, an adaptation of the structured argumentation framework ASPIC+ which can incorporate natural deduction style arguments and explanations We illustrate the applicability of the framework to debates in the formal sciences by presenting a simple model of some arguments about proposed solutions to the Liar paradox, and by discussing a more extensive—but still preliminary—model of parts of the debate that mathematicians had about the Axiom of Choice in the early twentieth century

In which sense (if any) can it be said that Hegel’s Logic is formal?

Abstract: According to some commentators, Hegel's Logic is not a kind of formal logic, but a material one. This statement has documental support, because Hegel says formal logic is empty. Never the less there are some texts in Hegelian Opus, which suggests that Hegel thinks of logic as a formal subject. Because of that, also, there are commentator which say Hegel’s logic is a formal system. I propose that it must be realized that the term "formal" has had several senses along the history of philosophy. According to some of this senses, it can be said that Hegel's Logic is formal.

Shifting Our Gaze: Relational Space in Professional Learning Network Research

Abstract: Reflecting on professional learning networks (PLN) in rural and equity-seeking spaces, the authors foreground the importance of “relational space” in studying PLNs in this commentary. The authors argue that while the complexity of taking a relational approach is challenging, it offers an important and necessary perspective, one which is often implicit in the studies featured in this book but not explicitly considered. The chapter is organized around three broad concepts from social network theory – boundedness, connectedness, and mutuality – which serve as starting points for shifting our gaze from formal system structures to more deeply interrogating the informal relational spaces within PLNs. The authors conclude with a call to make use of network theory and methods on their own, and in complement to other literatures, to do so.

Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes

Abstract: We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, in our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

How Do Mental Processes Preserve Truth? Husserl’s Discovery of the Computational Theory of Mind

Abstract: Hubert Dreyfus once noted that it would be difficult to ascertain whether Edmund Husserl had a computational theory of mind. I provide evidence that he had one. Both Steven Pinker and Steven Horst think that the computational theory of mind must have two components: a representational-symbolic component and a causal component. Bearing this in mind, we proceed to a close-reading of the sections of “On the Logic of Signs” wherein Husserl presents, if I’m correct, his computational theory of mind embedded in a language of thought. My argument goes like this: the computational theory of mind is the idea, following Haugeland, that the mind comes prepackaged as, or is endogenously constrained to be (with respect to certain domains), an automatic formal system; this explains, according to Husserl, why automatic trains of thought without logical intent resemble arguments exhibiting deductive structure with logical intent. In general, an automatic formal system yields true results provided that (1) the syntactic symbols with which they compute are univocal and are semantically evaluable, and (2) the mechanized inferences they perform are valid and preserve truth. These two conditions describe a computational (as opposed to an associative) cognitive process: the first condition connects representations to syntax (corresponding to Pinker and Horst’s first component), and the second condition uses the syntax, in inauthentic judging, to arrive at true conclusions through blind causality (corresponding to Pinker and Horst’s second component). Now, in point of textual fact, these are the conditions which Husserl attributes to our “natural psychological mechanism of symbolic inference” which typically yields true results. Since a formal system attributed to the “internal structure” of the mind, and guided by blind causality, just is the computational theory of mind, it follows, I think, that Husserl had a computational theory of mind. This computational theory is, moreover, embedded in a language of thought, since Husserl attributes a language-like form to our thoughts so that they may be mechanically processed. I conclude with a discussion of my results.

Some cautions regarding the phonological continuity hypothesis

Abstract: We consider the Phonological Continuity Hypothesis (PCH) of Fitch (2018) in light of a broader range of formal systems. A consideration of the learning and generalization of simple patterns such as AAB from Marcus (Marcus 2000 Curr. Dir. 9, 145-147(doi:10.1111/1467-8721.00080)) shows that finite-state automata defined in the standard way fail to generalize in a compatible fashion. However, pushdown automata with finite-memory limits do show compatible generalization capabilities. The third class of formal systems-tree automata-provide yet another possibility for the processing of words within sentences. We conclude that there are additional possible formal differences between sound patterns and sentence patterns, which will make testing the PCH even more difficult. This article is part of the theme issue 'What can animal communication teach us about human language?'

Computing with Known and Unknown

Abstract: The current science for developing artificial systems is limited in three aspects: the incompleteness of formal systems, the bounded rationality of humans and the basic assumption of science.

Formal and Logic Models of Knowledge and the Procedure for Logical Inference in the Problems of Recognition of Radio-Emitting Objects and their States

Abstract: The problem of formalizing the computational- logical problem of recognizing radio-emitting objects and their states on the basis of the formal-logical approach is considered. As a formal system for creating logical models of knowledge and a logical inference procedure, calculus of first-order predicates is used. A new method for formalizing the problem was developed, which ensured the creation of an intelligent expert system for recognizing radio-emitting objects and their states. Experimental studies were carried out, which confirmed their effectiveness and high reliability of the results.

Comparison of Customary and Formal Tenure Systems in Benishangul Gumuz Regional State: A case of Assosa District

Abstract: Though the global trend is to manage land through formal systems, still there is debate among scholars on importance of customary tenure systems and its treatment in modern land formalization. The study done in the study area also indicates the contradicting debate on both tenure systems, whether customary land tenure activities should recognized as it is or selectively in modern land formalization. However, their argument was not after studying and comparing both tenure systems simultaneously, making major problem needed to be identified in this study. Therefore, the study in this thesis aimed to assess and compare customary and formal tenure systems. Both descriptive and explanatory case study type of research was used in this study. Similarly, both qualitative and quantities data were collected from primary and secondary data sources. The primary data were collected by using different data collection tools like interview question, In-depth-key informant interview, focal group discussion and field observation. The survey data obtained from 360 house hold heads were analyzed by using SPSS (IBM-21) in which descriptive cross-tabulation model was mainly used. To compare means of continues variables, one –way ANOVA and descriptive system were also used. The study findings from House hold survey indicate significant variation between formal and customary tenure systems at (P 0.05);land related explanatory variables like land accessibility, defining right, ensuring tenure security, legalization of land transfer and conflict reduction and resolution; and Local farmers ‘perception on importance of modern land formalization -that was positive in formalized tenure and negative in customary tenure. These findings were triangulated with data collected with other tools. Based on the findings, the researcher recommends recognition of non formal customary tenure selectively in modern land formalization policies, rules, regulations and laws. For place where formalization coverage takes longer time to achieve, customary systems should have legally accepted bylaws and effective institution.

On the Formal Incompleteness of Reductive Logic

Abstract: A proof of formal incompleteness is spelled-out in relation to ‘bottom-up’ reductive Logic and three stacked abstract reductive formal system models composing a three-level hierarchy of complexity. Undecidable reductive propositions appear at the ‘upper’ boundary of each inter-related formal system model of ‘sufficient complexity’. The consequence of formal reductive incompleteness demands necessary meta-consideration in the determination of reductive logical consistency. The further implications of formal reductive incompleteness predict that identified undecidable reductive propositions might be decided in adapted reductive formal system models using modified reductive Logic and slightly different axioms and rules that can handle undecidable dynamics. The adapted Logic and reductive formal system models must accept the inevitably of reductive incompleteness. Reductive incompleteness further predicts the exploration of an adjacent possible abstract domain in which ‘bottom-up’ formal reductive Logic can be preserved as a ‘special case’; while multiple adapted forms of reductive Logic as well as multiple inter-related adapted reductive formal system models, develop a deeper understanding of how to abstractly manage undecidable dynamics and reductive incompleteness. The insights, outcome and implications arising from the exploration of the abstract domain, could instigate further scientific work determining whether modified reductive Logic and reductive formal system models sensitive to undecidable dynamics, provides a closer approximation of natural incompleteness driven, novelty generating evolutionary processes. Any abstract or applied, mathematical, computational or information grounded system model, equivalent to a reductive formal system model, will predictably be shown to manifest undecidable dynamics and incompleteness driven novelty generation. The implications of formal reductive incompleteness can be extended to the study of non-linear systems, Chaos Theory, Cellular Automata, Complex Physical Systems and Complex Adaptive Systems. Subsequently, future scientific and mathematical thought may derive incompleteness driven novelty generating formulations of reductive scientific philosophy, epistemology, methods for theoretical modeling and experimental methodology; each of which will unravel the implicit or explicit intention of the Reductive Scientific Paradigm to compose an integrated, unified, closed, resolved and complete ‘bottom-up’ Reductive Scientific Narrative describing our Universe. An encompassing incompleteness driven Meta-Reductive Scientific Paradigm is a possibly, wherein novel approaches to previously unresolvable reductive scientific problems may reveal a unique path toward consilience and integration of the Reductive Sciences and the developing Complexity Sciences.

The importance of non-formal education for the economy of Serbia

Abstract: Adult learning and education are an important component of the lifelong learning process, integrating different forms and programs of non-formal and informal learning. This area of learning encompasses a variety of content, from literacy and general skills, through vocational education and training content, to family, civic, environmental, media, leisure education, and many other areas whose priorities depend on the specific needs of each country . The formal system does not meet the needs of the individual or community to learn. Due to the inability of formal education to monitor changes in education resulting from the rapid development of science and technology, as well as economic and social changes, informal education has emerged. This education is an essential and necessary complement to formal education. Non-formal education is defined as any organized educational activity outside the existing formal system, which serves specific users and learning objectives. Non-formal education is provided through activities such as courses, seminars, lectures, conferences, workshops, various types of training, as well as volunteering. The field of non-formal education is very important in Serbia, it enables persons who have not completed formal education or have finished school with which they cannot find a job in the profession, to actually retrain in the process of non-formal education to perform some of the more demanding jobs, or to apply for a job in Serbia, or looking for a job abroad.

Social and economic aspects of recognition and validation of non-formal adult learning system in greece

Abstract: Non-formal and informal aspects of education are nowadays related to the concepts of recurrent and lifelong learning, as literature significantly supports the importance of education, learning and training that takes place outside typical educational institutions. As lifelong learning is the master concept that could shape educational systems and economies, non-formal education includes any organized educational activity outside the established formal systems that serves identifiable learning objectives, while informal education refers to the lifelong process where every individual acquires knowledge, attributes, skills, values and behaviors from daily life’s experience. Furthermore, as there is currently a strong trend worldwide, to include similar practices in internal policy strategies, new alternative terms arise such as community learning, community education, etc. together with many pilot initiatives. During the recent years, such initiatives have already taken place in many countries, proving that similar actions could greatly contribute to individuals’ knowledge and skills enhancement as well as help in mitigating social inequalities, tackle unemployment, achieve a better match between jobs and skills, and thus improve employment through economic development by supporting human capital productivity. Due to the promising advantages of the organized establishment of non-formal and informal education, the present paper focuses on a thorough analysis of the aforementioned concepts and describes the initiative of a relevant research in Greece, conducted by the authors. The research consists of two parts, one that includes a reliable and representable sample of educational organizations (bodies), examining the current ways used to support and certify certain fields of informal education, noting the typical ways currently used to recognize non-formal and informal learning as well as an additional sample of individuals (beneficiaries) that are interested in further support, validation and certification of non-formal education and informal acquired learning. The results are expected to contribute to the process of highlighting information on the intensity of the demand for recognition of prior knowledge through mediation certification procedures by adult education organizations, as well as to the submission of proposals for the operation of the relevant national mechanisms under development in Greece, according to the European Directive 2012. JEL: I25, I21, I30, O31, E24, A20.

Embracing undecidability: Cognitive needs and theory evaluation

Abstract: There are many ways we can not know. Even in systems that we created ourselves, as, for example, systems in mathematical logic, Goedel and Tarski's theorems impose limits on what we can know. As we try to speak of the real world, things get even harder. We want to compare the results of our mathematical theories to observations, and that means the use of inductive methods. While we can demonstrate how an ideal probabilistic induction should work, the requirements of such a method include a few infinities. Furthermore, it would not be even enough to be able to compute those methods and obtain predictions. There are cases where underdeterminacy might be unavoidable, such as the interpretation of quantum mechanics or the current status of string theory. Despite that, scientists still behave as if they were able to know the truth. As it becomes clear that such behavior can cause severe cognitive mistakes, the need to accept our limits, both our natural human limits and the limits of the tools we have created, become apparent. This essay will discuss how we must accept that knowledge is almost only limited to formal systems. Moreover, even in those, there will always be undecidable propositions. We will also see how those questions influence the evaluation of current theories in physics.

Statistical patterns of word frequency suggesting the probabilistic nature of human languages.

Abstract: Traditional linguistic theories have largely regard language as a formal system composed of rigid rules. However, their failures in processing real language, the recent successes in statistical natural language processing, and the findings of many psychological experiments have suggested that language may be more a probabilistic system than a formal system, and thus cannot be faithfully modeled with the either/or rules of formal linguistic theory. The present study, based on authentic language data, confirmed that those important linguistic issues, such as linguistic universal, diachronic drift, and language variations can be translated into probability and frequency patterns in parole. These findings suggest that human language may well be probabilistic systems by nature, and that statistical may well make inherent properties of human languages.

Calculus CL as a Formal System

Abstract: In recent years CL diagrams inspired by Lange’s Cubus Logicus have been used in various contexts of diagrammatic reasoning. However, whether CL diagrams can also be used as a formal system seemed questionable. We present a CL diagram as a formal system, which is a fragment of propositional logic. Syntax and semantics are presented separately and a variant of bitstring semantics is applied to prove soundness and completeness of the system.

A Survey of Languages for Formalizing Mathematics

Abstract: In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce human-readable documents. These goals are challenging to combine and state-of-the-art tools typically have to make difficult compromises. In this paper we discuss languages that have been created for this purpose, including logical languages of proof assistants and other formal systems, semi-formal languages, intermediate languages for exchanging mathematical knowledge, and language frameworks that allow building customized languages. We evaluate their advantages based on our experience in designing and applying languages and tools for formalizing mathematics. We reach the conclusion that no existing language is truly good enough yet and derive ideas for possible future improvements.

Incompleteness for stably consistent formal systems

Abstract: We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second Godel incompleteness theorem to stably $1,2$-consistent formal systems. Our argument in particular re-proves the original incompleteness theorems from first principles, using Turing machine language to (computably) construct our "Godel sentence" directly, in particular we do not use the diagonal lemma, nor any meta-logic, with the proof naturally formalizable in set theory. In practice such a stably consistent formal system could be meant to represent the mathematical output of humanity evolving in time, so that the above gives a formalization of a famous disjunction of Godel, obstructing computability of intelligence.

Axioms and Formalisms

Abstract: The Fregean work sought to free the theory of numbers from all roots in psychology and intuition to link it to logic alone and, more precisely, to a system of pure laws of thought. The paradoxes of set theory condemned this so-called logicist approach. This opened the way to the marked will, at the beginning of the twentieth century, to establish arithmetic and mathematics on other foundations or, at the very least, to try to elucidate what, in the Fregean project, had led to failure. The names of Hilbert and Godel stand out among those attempts, which profoundly changed our understanding of the relationship between logic and mathematics, of the possibilities of formal systems and of the role of infinity in mathematics, while at the same time opening the way to new disciplines such as proof theory and making it possible to build a foundation on which computer science could then be born and develop.

Observement as Universal Measurement.

Abstract: Measurement theory is the cornerstone of science, but no equivalent theory underpins the huge volumes of non-numerical data now being generated. In this study, we show that replacing numbers with alternative mathematical models, such as strings and graphs, generalises traditional measurement to provide rigorous, formal systems (`observement') for recording and interpreting non-numerical data. Moreover, we show that these representations are already widely used and identify general classes of interpretive methodologies implicit in representations based on character strings and graphs (networks). This implies that a generalised concept of measurement has the potential to reveal new insights as well as deep connections between different fields of research.

Machine proof of Equivalence between Axiom of Choice and Product Theorem of Standard Family Sets

Abstract: Based on the computer proof assistant Coq, the establishment of a formal system of naive set theory is realized, referring to Morse-Kelley set theory. On this basis, the formal descriptions of the axiom of choice and the product theorem of standard family sets are given, and the proof code for the equivalence between axiom of choice and the product theorem of the standard set family is completed and verified in Coq. All process fully embodies that machine proof of mathematical theorem based on Coq has the characteristics of readability and interactivity.

Математическое мышление: концептуальное доказательство или логический вывод?

Abstract: The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents and opponents of the Thesis, "conceptualists" and "formalists", is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of "interesting" mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat's Last Theorem ( Y. Rav). Formalists say that a conceptual proof "points" to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an "interesting" theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of "mentalists" and "mechanists" on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself.

On the formal justification of informal proofs

Abstract: When modeling the informal proofs of Euclid’s Elements using the sound logical system E, we go from proofs seen as somewhat nonrigorous – even having gaps to be filled – to rigorous proofs. According to the ‘standard view’, the correctness of an informal proof is underwritten by the existence of a corresponding formal derivation. However, metalogic grounds the soundness of the logical system E, and proofs in metalogic are not like formal proofs and look suspiciously like informal proofs. In our view, they are informal proofs. This brings about what we are calling here the groundedness problem: how can we show with certainty that our metalogical proofs are correct and sustain our logical system? According to the ‘standard view’, we cannot. In this way, we would have to doubt the soundness of the formal system E. This in turn might lead us to doubt the justification of Euclidean informal proofs in terms of the corresponding formal proofs in E.

Demystifying dependence

Abstract: Programmers are told "depend on interfaces, not implementations." But, given a program, is it possible even to assess objectively whether such advice has been followed? Programmers frequently talk in ways like this about dependence, but the very term, like many used in software engineering, has hitherto eluded precise definition. In this work, we resolve a variety of confusions about dependence, and present a formal definition unifying multiple varieties of software dependence, grounded in Halpern and Pearl's theory of actual causation. This definition is parameterized by the formal system characterizing the property of interest, and by constraints on "reasonable changes" to the program. By picking different choices of formal system, one can specialize our definition to characterize several notions of dependence, including build, correctness, and performance dependences. Overall, our work provides a path to making conversations about software dependence fully objective, and might serve as a basis for future work that automatically checks forms of dependence that were previously too abstract or high-level to be candidates for tools.

Begriffsschrift’s Logic

Abstract: In Begriffsschrift, Frege presented a formal system and used it to formulate logical definitions of arithmetical notions and to deduce some noteworthy theorems by means of logical axioms and inference rules. From a contemporary perspective, Begriffsschrift’s deductions are, in general, straightforward; it is assumed that all of them can be reproduced in a second-order formal system. Some deductions in this work present—according to this perspective—oddities that have led many scholars to consider it to be Frege’s inaccuracies which should be amended. In this paper, we continue with the analysis of Begriffsschrift’s logic undertaken in an earlier work and argue that its deductive system must not be reconstructed as a second-order calculus. This leads us to argue that Begriffsschrift’s deductions do not need any correction but, on the contrary, can be explained in coherence with a global reading of this work and, in particular, with its fundamental distinction between function and argument.