Logics

It is astonishing that we humans are able to have, act on and express moral beliefs. This dissertation aims to provide a better philosophical understanding of why and how this is possible especially when we assume metaethical expressivism. Metaethical expressivism is the combination of expressivism and noncognitivism. Expressivism is the view that the meaning of a sentence is explained by the mental state it is conventionally used to express. Noncognitivism is the view that the mental state expressed by a moral (...) sentence is a noncognitive, desire-like state. The central question of this dissertation is this: "How can metaethical expressivists explain that we are able to express moral beliefs?" The shortest answer this dissertation will give is the following: "by becoming `dispositional expressivists´." Dispositional expressivists hold that moral beliefs are dispositions to be in certain mental states that resemble desires. More precisely, a central thesis of this study is that to hold a moral belief is to be disposed to be in a desire-like, noncognitive state where this state has some specific structure. This dissertation arrives at and justifies this thesis by making a number of steps. It starts with preparing the ground by developing a general theory of what it is for a sentence to express a mental state. Roughly, this theory states that for a sentence to express a mental state is for that sentence to be conventionally used to perform a speech act which counts as sincere only if the user or speaker is in that mental state. This theory of expressing satisfies all the constraints that must be satisfied for it to be compatible with the purposes of metaethical expressivism. The next step is to provide a solution to the most notorious semantic problem confronting metaethical expressivism, the problem of explaining how sentences containing moral words can have compositional meaning and stand in logical relations also known as the Frege-Geach or embedding problem. Building on the research of Mark Schroeder, a new expressivist semantics, called `Attitude Semantics´, will be developed for a first-order language. The motivating idea underlying attitude semantics is that expressivists can avoid the problematic feature of so-called `biforcation´ if they treat the structures of the mental states expressed by moral sentences as well as that expressed by non-moral sentences as being isomorphic. In order to give those structures a philosophically interesting interpretation, it will be suggested that we treat sentences as expressing dispositional states. As a toy example we will treat the sentence `murder is wrong´ as expressing the disposition to be for blaming for murder. After dealing with the Frege-Geach problem, I turn to the metaethical expressivists' project of quasi-realism. Two major goals of this project are to explain how moral sentence can be truth-apt and how the states expressed by them can be regarded as moral beliefs. In order to establish those goals I critically examine a number of theories of truth-aptness by providing counterexamples to them. I then present my own theory which says that a sentence is truth-apt iff and because there is a conventional way to use it to express a belief. What is new about this theory is that it treats expressing a belief as being explanatorily more basic than being truth-apt. Belief, according to this theory, explains truth-aptness, not vice versa. This has the important consequence that it shows that what I call the `orthodox strategy´ of quasi-realists is mistaken and needs to be inverted: expressivists must `earn the right´ to moral belief before they can establish that moral sentences are truth-apt, and not the other way round. The second part of this dissertation is then concerned with the question of what it is to hold a moral belief and how they can figure in rational reasoning. In short: it is concerned with earning the right to moral belief. After discussing the major problems confronting a noncognitivist theory of belief, most importantly what I call the `tightrope problem´, I argue for a dispositional form of noncognitivism about moral belief that combines elements from Eric Schwitzgebel's dispositionalism about belief and Sebastian Köhler's `conceptual role expressivism´. This dispositional noncognitivism perfectly matches and justifies the dispositionalist interpretation of the expressivist semantics developed before. In order to fully earn the right to moral belief, however, metaethical expressivists must also deal with several epistemological objections concerning the rationality of moral belief. The first objection I deal with comes from Derek Baker and is directed against Mark Schroeder's specific development of expressivism. I deal with this objection because my own theory bears close resemblance to Schroeder's theory and so it might be worried that it confronts the same issues. Fortunately, we will see that Baker's objection rest on a number of mistaken assumptions and so neither applies to Schroeder's expressivism nor to the version which I defend in this study. The other epistemological objection to metaethical expressivism I deal with is Cian Dorr's famous wishful-thinking objection. In contrast to previous replies to this objection I attack it by directly questioning the central assumption on which it rests, namely that reasoners can never be justified in basing a factual belief on a desire-like state. I argue that this is false, and that under certain conditions reasoning from desire-like mental states to factual beliefs is justified. My argument shows that even if holding a moral belief is being in a desire-like noncognitive state, basing factual belief on a moral belief can give one an epistemic reason for the factual belief. Hence, even if moral thoughts are wishes, moral thinking is not wishful thinking. I take my arguments to earn the dispositional expressivist the right to moral belief. By the previously defended theory of truth-aptness in terms of belief, this allows the dispositional expressivists to conclude that moral sentences are also truth-apt. This finishes the quasi-realist project of this dissertation. In the last part of this dissertation I switch gears and present two unnoted problems for metaethical expressivists having to do with explaining moral motivation. Both arguments are surprisingly simple. The first one is this: if motivation requires a belief as well as a desire (Humeanism), and moral judgments motivate by themselves (internalism), then moral judgments cannot be desires only. If this argument is sound, it shows that the famous `motivation argument´ not only entails the falsity of cognitivism, but also the falsity of noncognitivism. The second argument is the following: moral judgments when combined with suitable external desires motivate to act (externalism), but if motivation requires belief as well as desire (Humeanism), then moral judgments cannot be desires only. I discuss several ways in which noncognitivists might want to deal with those problems, but leave it as an open question whether those solutions will ultimately be viable. Overall, this dissertation makes serious contributions to the research program of metaethical expressivism. It makes proposals for how to solve a number of notorious problems, such as the Frege-Geach problem in semantics and the wishful-thinking problem in epistemology. Besides this, however, it also raises two new problems for metaethical expressivism in the philosophy of action. For those reasons this dissertation will be of interest for anyone who wants to defend or criticize expressivism in metaethics, and more generally, anyone who wants to better understand our astonishing ability to express moral beliefs. (shrink)

A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.

This short article discusses the fact that the word ‘ancestor’ features in certain arguments that a) are apparently logically valid, b) contain infinitely many premises, and c) are such that none of their finite sub-arguments are logically valid. The article's aim is to motivate, within its brief compass, the study of infinitary logics.

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

Las máquinas aceleradas de Turing (ATMs) son dispositivos capaces de ejecutar súper-tareas. Sin embargo, el simple ejercicio de definirlas ha generado varias paradojas. En el presente artículo se definirán las nociones de súper-tarea y ATM de manera exhaustiva y se aclarará qué debe entenderse en un contexto lógico-formal cuando se pregunta por la existencia de un objeto. A partir de la distinción entre posibilidades lógicas y físicas se disolverán las paradojas y se concluirá que las ATMs son posibles y existen (...) como objetos abstractos. (shrink)

Reasoning over our knowledge bases and theories often requires non-deductive inferences, especially – but by no means only – when commonsense reasoning is the case, i.e. when practical agency is called for. This kind of reasoning can be adequately formalized via the notion of supraclassical consequence, a non-deductive consequence tightly associated with default and non-monotonic reasoning and featuring centrally in abductive, inductive, and probabilistic logical systems. In this paper, we analyze core concepts and problems of these systems in the light (...) of supraclassical consequence. (shrink)

The paper argues that logical reasoning is what constitutes critical thinking because reasoning is normative, and logic provides a norm for reasoning. If a logical way of thinking is simply the same thing as reasoning, then the only limit or constraint of reasoning is the nature of the (logical) “rule” the kind of which is engaged with the practice of reasoning; rather than with either a component of social context or a com- ponent of a purely non-inferential or “immediate” perception (...) of a situation which is called sensitivity toward an object alone. However, such components are built into my analysis of what “reasoning” is. An approach to critical thinking being employed here is “an agent-based approach”. This paper also aims to pave the way for a further discussion on the limit of reasoning in a way that is relevant to the question of how critical thinking or reasoning should be taught in higher education. (shrink)

Persuasive technologies can adopt several strategies to change the attitudes and behaviors of their users. In this work I synthesize the lessons learned from three empirical case studies on automated persuasion that have been carried out in the last decade in the contexts of: persuasive news recommendations, social robotics, and e-commerce, respectively. In particular, such studies have assessed, in the technological domain, the effects of nudging techniques relying on well known persuasive argumentation schemas and on framing strategies. In discussing the (...) main findings, I will argue that the obtained persuasive effects are due to the fact that such techniques leverage on cognitive mechanisms that refer to the “system 1” types of automatic processes hypothesized in the context of the dual process theory of reasoning. As a consequence of this state of affairs, any automated persuasive systems (used for ethical purposes: e.g. suggesting good health habits) should be able to re-use these types of system 1 strategies in order to gradually take the users in an argumentation territory where system 2 processes can take place. -/- . (shrink)

There is a long tradition of logic, from Aristotle to Gödel, of understanding a proof from the concepts of necessity and causality. Gödel's attempts to define provability in terms of necessity led him to the distinction of formal and absolute (abstract) provability. Turing's definition of mechanical procedure by means of a Turing machine (TM) and Gödel's definition of a formal system as a mechanical procedure for producing formulas prompt us to understand formal provability as a mechanical causality. We propose a (...) formalism which makes explicit the mechanical causal nature of a TM's work. We claim that Gödel's axiomatised ontotheology and his ontological proof give a clue for the understanding of the concept of absolute provability and the pattern of the corresponding absolute completeness proof, respectively. (shrink)

En The paradox of Addition and its dissolution (1969), Mario Bunge presenta algunos argumentos para mostrar que la Regla de Adición puede ocasionar paradojas o problemas semánticos. Posteriormente, Margáin (1972) y Robles (1976) mostraron que las afirmaciones de Bunge son insostenibles, al menos desde el punto de vista de la lógica clásica. Aunque estamos de acuerdo con las críticas de Margáin y Robles, no estamos de acuerdo en el diagnóstico del origen del problema y tampoco con la manera en la (...) que se proponen solucionarlo. En esta medida, en este texto mostraremos cómo la Regla de Adición sí trae problemas semánticos que pueden ser planteados desde diferentes perspectivas contemporáneas, como la extensionalidad del condicional, los principios proscriptivos y la validez conexiva. (shrink)

Сборник эссе о логике и логическом мышлении.

In this work we present NERVOUS, an intelligent recommender system exploiting a probabilistic extension of a Description Logic of typicality to dynamically generate novel contents in AllMusic, a comprehensive and in-depth resource about music, providing data about albums, bands, musicians and songs. The tool can be used for both the generation of novel music genres and styles, described by a set of typical properties characterizing them, and the reclassification of the available songs within such new genres.

This paper establishes a sound and complete semantics for the impure logic of ground. Fine (Review of Symbolic Logic, 5(1), 1–25, 2012a) sets out a system for the pure logic of ground, one in which the formulas between which ground-theoretic claims hold have no internal logical complexity; and it provides a sound and complete semantics for the system. Fine (2012b) [§§6-8] sets out a system for an impure logic of ground, one that extends the rules of the original pure system (...) with rules for the truth-functional connectives, the first-order quantifiers, and λ-abstraction. However, no semantics has yet been provided for this system. The present paper partly fills this lacuna by providing a sound and complete semantics for a system GG containing the truth-functional operators that is closely related to the truth-functional part of the system of Fine (2012b). (shrink)

در تکوین نظریه محاسبات از اوایل قرن بیستم پارادکسها و خود ارجاعی نقش ویژه ای را بازی کرده اند. هر چند نظریه محاسبات عام بر اساس تعریف ماشین تورینگ، فرض تورینگ_چرچ و کاربردهای آن بنا شده ،اما از همان ابتدا تا به امروز منطق و حوزه های مختلف این علم در ارتباط تنگاتنگ با این تیوری و در ابتدا نظریه محاسبات خاص بوده و این ارتباط روز به روز گسترده و گسترده تر گشته است. از تاثیر پارادوکس دروغگو و پارادکس (...) بری در تمایز میان مفهوم محاسبه پذیر و محاسبه پذیر شمارایانه، قضایای گودل و قضیه چیتین همه جا شاهد این تاثیر و تاثر متقابل هستیم. در این مقال می خواهیم بعد از معرفی اجمالی مطالب باال، تاثیر یک پارادوکس دیگر، پارادوکس اعدام غیر منتظره، را بر این نظریه بررسی نماییم. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

This paper provides an exposition of the structuralist approach to underdetermination, which aims to resolve the underdetermination of theories by identifying their common theoretical structure. Applications of the structuralist approach can be found in many areas of philosophy. I present a schema of the structuralist approach, which conceptually unifies such applications in different subject matters. It is argued that two classic arguments in the literature, Paul Benacerraf’s argument on natural numbers and W. V. O. Quine’s argument for the indeterminacy of (...) translation, can be analyzed as instances of the structuralist schema. These two applications illustrate different kinds of conclusions that can be drawn through the structuralist approach; Benacerraf’s argument shows that we can derive an ontological conclusion about the given subject matter, while Quine’s structuralist approach leads to a semantic conclusion about how to determine linguistic meanings given radical translation. Then, as a case study, I review a recent debate in metaphysics between Shamik Dasgupta, Jason Turner, and Catharine Diehl to consider the extent to which different instances of the structuralist schema are conceptually unified. Both sides of the debate can be interpreted as utilizing the structuralist approach; one side uses the structuralist approach for an ontological conclusion, while the other side relies on a semantic conclusion. I argue that this has a strong dialectical consequence, which sheds light on the conceptual unity of the structuralist approach. (shrink)

We discuss the role of perceptron (or threshold) connectives in the context of Description Logic, and in particular their possible use as a bridge between statistical learning of models from data and logical reasoning over knowledge bases. We prove that such connectives can be added to the language of most forms of Description Logic without increasing the complexity of the corresponding inference problem. We show, with a practical example over the Gene Ontology, how even simple instances of perceptron connectives are (...) expressive enough to represent learned, complex concepts derived from real use cases. This opens up the possibility to import concepts learnt from data into existing ontologies. (shrink)

Types are a crucial concept in conceptual modelling, logic, and knowledge representation as they are an ubiquitous device to un- derstand and formalise the classification of objects. We propose a logical treatment of types based on a cognitively inspired modelling that ac- counts for the amount of information that is actually available to a cer- tain agent in the task of classification. We develop a predicative modal logic whose semantics is based on conceptual spaces that model the ac- tual information (...) that a cognitive agent has about objects, types, and the classification of an object under a certain type. In particular, we ac- count for possible failures in the classification, for the lack of sufficient information, and for some aspects related to vagueness. (shrink)

In the present article, I offer a new reading of Hannah Arendt’s The Origins of Totalitarianism, specifically her argument that ideologies such as racism engender totalitarianism when the lonely and disenfranchised laborers of modern society develop a pathological fixation on formal logic, which I term “logomania.” That is, such logical deductions, from horrifically false premises, are the closest thing to thinking that individuals can engage in after their psyches, relationships, and communities have broken down. And it is only thus that (...) totalitarianism can achieve power, since it offers at least some form of connectedness and meaning, regardless how terrifying and violent. The danger persists, clearly, with the resurgence of the far Right, including in the extraordinary regime of Trump in the United States. From this I conclude that, along with the admirable calls to fight loneliness and rebuild our communities, we should also supplement all formal logical instruction and community education with instruction in creative thinking (including aesthetics), thereby discouraging the monomaniac reliance on formal logic as inadvertent weapon of totalitarianism. (shrink)

Discuto la tesi di Micheletti secondo la quale ogni spiegazione fattuale presuppone premesse di ordine superiore rispetto alla spiegazione (M.Micheletti, “Radical Divine Alterity and the God-World Relationship”). Nella prima sezione del testo introdurrò la tesi, muovendo dalla analisi di alcuni esempi di spiegazione, ed elencherò le ragioni che (apparentemente) richiedono la postulazione di higher-degree propositions per rendere conto di factual propositions. Nella seconda sezione regimenterò logicamente la tesi di Micheletti. Nella terza sezione discuterò la validità della logica della spiegazione così (...) introdotta, con un’attenzione particolare alla analisi semantica della nozione di spiegazione, allo statuto modale delle ragioni che costituiscono una spiegazione, e all’approccio grammaticale alle higher order propositions. Nella quarta e ultima sezione avanzerò brevissime considerazioni conclusive. (shrink)

In a recent paper by Tranchini (Topoi, 2019), an introduction rule for the paradoxical proposition ρ∗ that can be simultaneously proven and disproven is discussed. This rule is formalized in Martin-Löf’s constructive type theory (CTT) and supplemented with an inferential explanation in the style of Brouwer-Heyting-Kolmogorov semantics. I will, however, argue that the provided formalization is problematic because what is paradoxical about ρ∗ from the viewpoint of CTT is not its provability, but whether it is a proposition at all.

Este texto presenta, y en cierta medida analiza, ambigüedades existentes en textos de lógica y filosofía de la lógica (como la interpretación de los llamados principios, postulados, leyes o verdades lógicas, la coexistencia de la tesis de que toda relación presupone la existencia de al menos dos relata y la de que una cosa puede relacionarse consigo misma, o la llamada "paradoja del mentiroso") bajo el supuesto de que, dado que la lógica no es anterior a la semántica, un análisis (...) semántico puede mostrar no solo que tal polisemia sugiere que ni la disciplina ni las teorías lógicas son tan rigurosas como se ha afirmado, sino también que varias afirmaciones de esas disciplinas carecen de sentido. (shrink)

‎A universal schema for diagonalization was popularized by N. S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fit more theorems in the universal‎ ‎schema of diagonalization‎, ‎such as Euclid's proof for the infinitude of the primes and new proofs (...) of G. Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset‎. ‎Then‎, ‎in Linear Temporal Logic‎, ‎we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox (1985‎, ‎1993)‎. ‎Thus‎, ‎Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic‎. ‎Again the diagonal schema of the paper is used in this proof; and it is also shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal / fixed-point schema‎. ‎We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions‎, ‎such as primitive recursive functions) in the schema. (shrink)

Textbook for students in mathematical logic. Part 1. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem.

Acceptable analyticities, i.e. contradictions or tautologies, constitute problematic evidence for the idea that language includes a deductive system. In recent discussion, two accounts have been presented in the literature to explain the available evidence. According to one of the accounts, grammatical analyticities are accessible to the system but a pragmatic strengthening repair mechanism can apply and prevent the structures from being actually interpreted as contradictions or tautologies. The proposed data, however, leaves it open whether other versions of the meaning modulation (...) operation are required. Novel evidence we present argues that a loosening version of the repair mechanism must be available. Our observation concerns acceptable lexical contradictions that cannot be rescued if only a strengthening version of the pragmatic strategy is available. (shrink)

This paper discusses the theoretical assumptions behind the conception of the logic of faith and deed (LF&D) and outlines its formal-axiomatic frame and its method of construction, which enable us to understand it as a kind of deductive science. The paper is divided into several sections, starting with the logical analysis of the ambiguous terms of 'faith’ and 'action', and focusing in particular on the concepts of religious faith and deed as a type of conscious activity relating to a matter (...) or matters of social importance. After outlining the main ideas and basic assumptions of the theoretical conception of the LF&D as an axiomatic theory, the author introduces some axiom systems for: 1) the logics of faith LF (doxastic logics), 2) the logic of deed LD, and 3) certain logics of norms DL (deontic logics) connected with "duties" and concerning actions/deeds. Lastly, the paper outlines the scientific LF&D based on the three types of logic 1)-3). (shrink)

Mind, according to cognitive neuroscience, is a set of brain functions. But, unlike sets, our minds are cohesive. Moreover, unlike the structureless elements of sets, the contents of our minds are structured. Mutual relations between the mental contents endow the mind its structure. Here we characterize the structural essence and the logical form of the mind by focusing on thinking. Examination of the relations between concepts, propositions, and syllogisms involved in thinking revealed the reflexive graph structure of the conceptual mind. (...) Objective logic of the conceptual mind is calculated from its structure. Noteworthy features of the logic of conceptual mind are: degrees of truth, varieties of negation, admission of contradiction, and the failure of a de Morgan's law. Furthermore, cohesion of the conceptual mind follows from its reflexive graph structure. Our characterization of the structure and logic of mind constitutes a substantial refinement of the contemporary cognitive neuroscientific conceptualization of the mind as a set. (shrink)

Analytic theologians have proposed numerous “solutions” to the Logical Problem of the Trinity (LPT), mostly versions of Social Trinitarianism (ST) and Relative Identity Trinitarianism (RI). Both types of solution are controversial, but many hold out hope that further “Trinitarian theorizing” may yield some as yet unimagined, and somehow importantly different, solution to the LPT. I first give a precise definition of the LPT and of what would count as a solution to it. I then show how, though there are infinitely (...) many possible solutions, all solutions can be grouped together into a finite, exhaustive taxonomy, based precisely on those features which make them either controversial, heretical, or inconsistent. The taxonomy reveals why ST and RI have been the major proposed solutions, and also proves that there can be no importantly different, new solutions to the LPT. (shrink)

In this paper we propose a computational framework aimed at extending the problem solving capabilities of cognitive artificial agents through the introduction of a novel, goal-directed, dynamic knowledge generation mechanism obtained via a non monotonic reasoning procedure. In particular, the proposed framework relies on the assumption that certain classes of problems cannot be solved by simply learning or injecting new external knowledge in the declarative memory of a cognitive artificial agent but, on the other hand, require a mechanism for the (...) automatic and creative re-framing, or re-formulation, of the available knowledge. We show how such mechanism can be obtained trough a framework of dynamic knowledge generation that is able to tackle the problem of commonsense concept combination. In addition, we show how such a framework can be employed in the field of cognitive architectures in order to overcome situations like the impasse in SOAR by extending the possible options of its subgoaling procedures. (shrink)

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

The purpose of this paper is to provide a new system of logic for existence and essence, in which the traditional distinctions between essential and accidental properties, abstract and concrete objects, and actually existent and possibly existent objects are described and related in a suitable way. In order to accomplish this task, a primitive relation of essential identity between different objects is introduced and connected to a first order existence property and a first order abstractness property. The basic idea is (...) that possibly existent objects are completely determinate and that essentially identical objects are just different individuations of the same individual essence. Accordingly, essential properties are defined as properties that are invariant with respect to this kind of identity, while abstract objects are determined by being characterized by essential properties only. Once such ideas are implemented, a number of classical intuitions about objects, their essence, and their way of existence can be consistently interpreted. (shrink)

Anthropic principle, Perspectival ontology, Hard problem, Why something rather than nothing, Life after death, Buddhism, God.

Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In this paper, we (...) propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them. (shrink)

We argue that a cognitive semantics has to take into account the possibly partial information that a cognitive agent has of the world. After discussing Gärdenfors's view of objects in conceptual spaces, we offer a number of viable treatments of partiality of information and we formalize them by means of alternative predicative logics. Our analysis shows that understanding the nature of simple predicative sentences is crucial for a cognitive semantics.

A concept is traditionally defined via the necessary and sufficient conditions that clearly determine its extension. By contrast, cognitive views of concepts intend to account for empirical data that show that categorisation under a concept presents typicality effects and a certain degree of indeterminacy. We propose a formal language to compactly represent concepts by leveraging on weighted logical formulas. In this way, we can model the possible synergies among the qualities that are relevant for categorising an object under a concept. (...) We show that our proposal can account for a number of views of concepts such as the prototype theory and the exemplar theory. Moreover, we show how the proposed model can overcome some limitations of cognitive views. (shrink)

The main subject of Cusanus’ investigations was the name of God. He claimed to have achieved the best possible one, Not-Other. Since Cusanus stressed that these two words do not mean the corresponding affirmative word, i.e. the same, they represent the failure of the double negation law and there￾fore belong to non-classical, and above all, intuitionist logic. Some of his books implicitly applied intuitionist reasoning and the corresponding organization of a theory which is governed by intuitionist logic. A comparison of (...) two of Cusanus’ short writings shows that throughout his life he substantially improved his use of this kind of logic and ultimately was able to reason consistently within such a logic and recognize some of its basic laws. One important idea developed by him was that of a proposition composed of a triple repetition of “not-other” expressing “the Tri-unity of concordance” i.e. the “best name for the Trinity”. I complete his application of intuitionist logic to theological subjects by charac￾terizing the inner relationships within the Trinity in such a way that there are no longer contradictions in the notion. Generally speaking, the notion of the Trin￾ity implies a translation from intuitionist to classical logic, to which Cusanus closely approximated. Moreover, I show that the main aspects of Christian rev￾elation, including Christ’s teachings, are represented both by this translation and by some doubly negated propositions of intuitionist logic. Hence, intuition￾ist logic was introduced into the history of Western theological thinking with Christian revelation, as only Cusanus partly recognized. Appendix 1 summa￾rizes a detailed analysis of Cusanus’ second short writing. Appendix 2 shows that the Athanasian creed regarding the Christian Trinity is a consistent se￾quence of intuitionist propositions provided that some verbal emendations are added, showing that ancient trinitarian thinking was also close to intuitionist reasoning. (shrink)

It has been recently argued that the well-known square of opposition is a gathering that can be reduced to a one-dimensional figure, an ordered line segment of positive and negative integers [3]. However, one-dimensionality leads to some difficulties once the structure of opposed terms extends to more complex sets. An alternative algebraic semantics is proposed to solve the problem of dimensionality in a systematic way, namely: partition (or bitstring) semantics. Finally, an alternative geometry yields a new and unique pattern of (...) oppositions that proceeds with colored diagrams and an increasing set of bitstrings. (shrink)

Russell prétend qu’un examen des croyances est indispensable pour définir nos raisonnements quotidiens et comprendre ce que les philosophes entendent par la notion de vérité. Cela étant, l’auteur considère qu’une étude de ces croyances n’a aucun rapport avec la logique, laquelle concerne uniquement le vrai et le faux. En d’autres termes, Russell associe croyance et psychologie tout en réservant le domaine de la logique au thème de la proposition, vraie ou fausse par définition. Une certaine théorie de la vérité sous-tend (...) son rejet primordial d’une logique épistémique ; si la croyance a un intérêt philosophique, c’est parce que Russell se sert de son examen pour discerner la théorie de la vérité que défendent plusieurs philosophes : la théorie de la vérité-cohérence est une cible principale du Britannique, qui défend pour sa part une théorie de la vérité-correspondance en caractérisant la proposition par sa relation de correspondance ou de non-correspondance avec un fait. (shrink)

This paper considers the nature of the changes that took place in logic teaching at the University of Oxford from the beginning of the sixteenth century, when students attended university lectures on Aristotle’s texts as well as studying short works dealing with specifically medieval developments, to the beginning of the eighteenth century when teaching was centred in the colleges, the medieval developments had largely disappeared, and manuals summarizing Aristotelian logic were used. The paper also considers the reasons for these changes, (...) including changes in English society, and the effect of humanism and the more scholarly Aristotelianism that it produced. (shrink)

A first-order theory T has the Independence Property provided deduction of a statement of type (quantifiers) (P -> (P1 or P2 or .. or Pn)) in T implies that (quantifiers) (P -> Pi) can be deduced in T for some i, 1 <= i <= n). Variants of this property have been noticed for some time in logic programming and in linear programming. We show that a first-order theory has the Independence Property for the class of basic formulas provided it (...) can be axiomatized with Horn sentences. The existence of free models is a useful intermediate result. The independence Property is also a tool to decide that a sentence cannot be deduced. We illustrate this with the case of the classical Caratheodory theorem for Pasch-Peano geometries. (shrink)

The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to natural deduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of a specific sequence of (...) reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (shrink)

Proceeding of the first public presentation of the work of the mexican logical group "Deduktor" under the direction of the mexican professor Hugo Padilla Chacón. This work was in fact the first whole and stand alone arithmetization of logics.

Why is there something rather than nothing? This is probably the most profound question that can be asked. In this paper, a rather unexpected simple solution is provided. The solution comes from analysing the truth value of the proposition “I exist.” It will be shown that this proposition is always true, so our existence is a logical necessity. Speculations about the implications over the universe as a whole are then provided.

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. This lecture expands points which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Hybrid languages are introduced in order to evaluate the strength of “minimal” mereologies with relatively strong frame definability properties. Appealing to a robust form of nominalism, I claim that one investigated language \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document} is maximally acceptable for nominalistic mereology. In an extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {m}}$\end{document}, a modal analog (...) for the classical systems of Leonard and Goodman and Leśniewski is introduced and shown to be complete with respect to 0-deleted Boolean algebras. We characterize the formulas of first-order logic invariant for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}_{\textsf {gem}}$\end{document}-bisimulations. (shrink)

The present text provides a logical theory which originated in the unification of a number of well-known philosophical logics as well as the introduction and study of new operators. Further M-logic contains an object theory. With both the logical part and the object part we achieve a formal calculus that is able to express many metaphysical dogmas.

This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...) this lecture. To repeat, a subregular polyhedron has congruent faces and congruent [polyhedral] angles. A subregular polyhedron whose faces are all regular polygons is regular—using standard terminology. -/- Geometers before Euclid showed that there are “essentially” only five regular polyhedra: every regular polyhedron is a tetrahedron (4 faces), a hexahedron or cube (6 faces), an octahedron (8 faces), a dodecahedron (12 faces), or an icosahedron (20 faces). -/- The first question is whether there are subregular polyhedra that are not regular. For example, are there tetrahedra having congruent angles and congruent triangular faces but whose faces are not equilateral triangles? -/- Another question is the classification of subregular polyhedra if they exist. For example, considering the fact that the regular tetrahedra all have equilateral triangles as faces, we ask which triangles other than equilaterals are faces of subregular tetrahedra. Similarly, considering the fact that the regular hexahedra all have squares as faces, we ask which quadrangles other than squares are faces of subregular hexahedra. -/- After introductory remarks that include historical and philosophical points, we concentrate on tetrahedra. A triangle that is congruent to each of the four faces of a tetrahedron is called a generator of the tetrahedron. The main result proved is that every acute triangle is a generator of a subregular tetrahedron. The proof includes an algorithm –implementable with scissors and paper –that constructs from any given acute triangle a subregular tetrahedron whose faces are congruent to the given triangle. -/- Algorithm: Given any acute triangle. Construct a similar triangle whose sides are double the sides of the given triangle. Draw the three lines connecting the three midpoints of the sides (making four triangles congruent to the given triangle—a central triangle surrounded by three peripheral triangles). Make three “hinges” along the lines connecting the midpoints. “Fold” the peripheral triangles together (into a tetrahedron). [LIGHTLY EDITED VERSION OF PRINTED ABSTRACT] Acknowledgements: William Lawvere, Colin McLarty, Irvin Miller, Frango Nabrasa, Lawrence Spector, Roberto Torretti, and Richard Vesley. -/- . (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)