Logics

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)

In “Some Remarks on Extending and Interpreting Theories with a Partial Truth Predicate”, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke–Feferman in analogy to Hilbert’s program. Reinhardt suggested to view $\mathrm {KF}$ as a tool for generating “the significant part of $\mathrm {KF}$ ”, that is, as a tool for deriving sentences of the form $\mathrm{Tr}\ulcorner {\varphi }\urcorner $. The constitutive question of Reinhardt’s program was whether it was possible “to justify the use of nonsignificant sentences (...) entirely within the framework of significant sentences”. This question was answered negatively by Halbach & Horsten [10] but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from $\mathrm {KF}$ -provably true sentences to $\mathrm {KF}$ -provably true inferences, that is, we shall identify the significant part of $\mathrm {KF}$ with the set of pairs $\langle {\Gamma, \Delta }\rangle $, such that $\mathrm {KF}$ proves that if all members of $\Gamma $ are true, at least one member of $\Delta $ is true. In way of addressing Reinhardt’s question we show that the provably true inferences of suitable $\mathrm {KF}$ -like theories coincide with the provable sequents of matching versions of the theory Partial Kripke–Feferman. (shrink)

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (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)

This volume of the Logica Yearbook series brings together articles presented at the annual international symposium Logica 2012, Hejnice, the Czech Republic. The articles range over mathematical and philosophical logic, history and philosophy of logic, and the analysis of natural language.

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)

Extended revised edition of "Logica" (2003).

Italian translation of "Schaum's Outline of Theory and Problems of Logic" (1988).

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)

In this paper we consider the modal logic with both \ and \ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra \. We provide an axiomatic system extending the one from Caicedo and Rodriguez :37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given (...) too. We also prove that in the studied logic it is not possible to get \ as an abbreviation of \, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature. (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 is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne, who seek to establish that mathematics is committed to its own necessity. I demonstrate that their assumptions collapse the counterfactual conditional into the material conditional. This collapse entails the success of counterfactual strengthening, which is controversial within counterfactual logic, and which has counterexamples within pure and applied mathematics. I close (...) by discussing the dispensability of counterfactual conditionals within the language of mathematics. (shrink)

This paper presents two versions of opposition relations for prophetical statements, the first one is an application of “Ockham’s thesis” in Classical propositional Logic. The second one is a reinterpretation of that thesis in the logic MRSP.

Providing more human-like concept learning in machines has always been one of the most significant goals of machine learning paradigms and of human-machine interaction techniques. This article attempts to provide a logical specification of conceptual mappings from humans’ minds into machines’ knowledge bases. We will focus on the representation of the mappings (transformations) relying on First-Order Predicate Logic. Additionally, the structure of concepts in the common ground between humans and machines will be analysed. It seems quite necessary to pay attention (...) to the philosophy of constructivism and constructivist models of knowing. This research constructs a conceptual ground for expressing and analysing concepts in the common ground between humanistic and informatics sciences and in the context of human-machine interplays. (shrink)

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the strict-tolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external logic and (...) argue that while ST+’s nonclassicality can be granted, its self-dual character does not tie it to LP+ more closely than to K3+. (shrink)

We present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing and Odintsov & Wansing, as well as the modal logic KN4 with strong implication introduced in Goble. In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system (...) and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown. (shrink)

El objetivo de este artículo es analizar qué quiere decir capturar una teoría (se utilizará la aritmética como ejemplo paradigmático) por medio de un sistema axiomático formal. Se considerarán para ello dos enfoques, que pueden denominarse “semántico” y “sintáctico”, tanto en términos de sus ventajas como de sus limitaciones. El enfoque semántico (presupuesto por Barrio y Da Ré en este volumen y expuesto en la primera sección), entiende la expresabilidad como una restricción de la clase de los modelos; se muestra (...) que esta concepción lleva a tres clases distintas de limitaciones expresivas. En la segunda sección se examinan -y finalmente rechazan- algunos posibles caminos de solución a estos problemas. En la última sección se ofrecen objeciones a las alternativas restantes y en base a ellas se elabora el enfoque sintáctico que liga la noción de captura a la de prueba de las oraciones relevantes. También se argumenta que desde este marco pueden evitarse algunas de las limitaciones del enfoque semántico. (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)

The minimal weakening \ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of \ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of \. These logics have the finite model property and they are decidable.

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)

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz (...) and the Maltsev hierarchies. (shrink)

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation (...) for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it. (shrink)

State spaces are, in the most general sense, sets of entities that contain information. Examples include states of dynamical systems, processes of observations, or possible worlds. We use domain theory to describe the structure of positive and negative information in state spaces. We present examples ranging from the space of trajectories of a dynamical system, over Dunn’s aboutness interpretation of fde, to the space of open sets of a spectral space. We show that these information structures induce so-called hype models (...) which were recently developed by Leitgeb. Conversely, we prove a representation theorem: roughly, hype models can be represented as induced by an information structure. Thus, the well-behaved logic hype is a sound and complete logic for reasoning about information in state spaces. As application of this framework, we investigate information fusion. We motivate two kinds of fusion. We define a groundedness and a separation property that allow a hype model to be closed under the two kinds of fusion. This involves a Dedekind–MacNeille completion and a fiber-space like construction. The proof-techniques come from pointless topology and universal algebra. (shrink)

In his Berlin Lectures of the 1820s, the German philosopher Arthur Schopenhauer (1788–1860) used spatial logic diagrams for philosophy of language. These logic diagrams were applied to many areas of semantics and pragmatics, such as theories of concept formation, concept development, translation theory, clarification of conceptual disputes, etc. In this paper we first introduce the basic principles of Schopenhauer’s philosophy of language and his diagrammatic method. Since Schopenhauer often gives little information about how the individual diagrams are to be understood, (...) we then make the attempt to reconstruct, specify and further develop one diagram type for the field of conceptual analysis. (shrink)

An important link between model theory and proof theory is to construe a deductive disproof of S as an attempted construction of a countermodel to it. In the function logic outlined here, this idea is implemented in such a way that different kinds of individuals can be introduced into the countermodel in any order whatsoever. This imposes connections between the length of the branches of the tree that a disproof is and their number. If there are already n individuals in (...) the countermodel that is being constructed, the next individual has to be considered in its relations to each of the n old ones, creating 2^n different cases and accordingly at least 2^n different branches. Hence a disproof procedure of a polynomial length is normally not equivalent with an exponential one. Because every computation can be represented as a deduction with the same number of constant terms, the same holds for nondeterministic computations. Apparent exceptions seem to come about if a branch created by a new individual i is redundant. But when the disproof is a shortest one (contains the minimum number of different constant terms) then not introducing that idle term at all would result in an even shorter disproof, violating the shortness assumption. (shrink)

This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

We consider the propositional logic equipped with Chellas stit operators for a finite set of individual agents plus the historical necessity modality. We settle the question of whether such a logic enjoys restricted interpolation property, which requires the existence of an interpolant only in cases where the consequence contains no Chellas stit operators occurring in the premise. We show that if action operators count as logical symbols, then such a logic has restricted interpolation property iff the number of agents does (...) not exceed three. On the other hand, if action operators are considered to be nonlogical symbols, then the restricted interpolation fails for any number of agents exceeding one. It follows that unrestricted Craig interpolation also fails for almost all versions of stit logic. (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)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

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)

Satisfiability or Sat\ is the metatheoretic statementEvery formally intuitionistically consistent set of first-order sentences has a model.The models in question are the Tarskian relational structures familiar from standard first-order model theory, but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat\ to be false outright. Following the lead of Carter :75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic. These metatheorems (...) distinguish the intuitionistic foundational significance of Sat\ sharply from that of Sat\, the satisfiability claim for intuitionistic propositional logic. At the same time, they establish intuitionistic connections with and between Test, COMP\, and \. Here, Test is the scheme of Testability, and COMP\ and \ are completeness for intuitionistic propositional logic and predicate logic, respectively. (shrink)

L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a \-semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is (...) proved that every lattice effect algebra is an interval in a right \-group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description. (shrink)

Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators “t : ”, indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We (...) replace this classical boolean base with Gödel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval [0, 1], which are intended to model inference under vagueness. We extend the canonical possible-world semantics for justification logic to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic justification logic. (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)

The standard Leibnizian view of identity allows for substitutivity of identicals and validates transitivity of identity within classical semantics. However, in a series of works, Graham Priest argues that Leibnizian identity invalidates both principles when formalized in paraconsistent semantics. This paper aims to show the Leibnizian view of identity validates substitutivity of identicals and transitivity of identity whether the logic is classical or paraconsistent. After presenting Priest's semantics of identity, I show what a semantic expression of Leibnizian identity does amount (...) to. Then, I argue that Priest's semantic definition of identity is not Leibnizian. Finally, I offer a semantics characterization of identity in paraconsistent logic that is truly Leibnizian. I demonstrate that the correct formalization of Leibnizian identity in paraconsistent logic also validates substitutivity of identicals and transitivity of identity. (shrink)

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

In this article, we deal with the intelligible world of the angels in the Areopagetic tradition and we compose references found in the De divinis nominibus to form, as far as possible, a complete definition of them. This systematic approach to the Areopagetic corpus takes into consideration Pseudo-Dionysius the Areopagite’s text and George Pachymeres’ Paraphrasis of this treatise. We also offer a methodological proposal on how we can structure theoretically general concepts that refer to objective realities, which however cannot be (...) proved by the tools of formal Logic. (shrink)

This research employs Description Logics in order to focus on logical description and analysis of the phenomenon of ‘concept understanding’. The article will deal with a formal-semantic model for figuring out the underlying logical assumptions of ‘concept understanding’ in knowledge representation systems. In other words, it attempts to describe a theoretical model for concept understanding and to reflect the phenomenon of ‘concept understanding’ in terminological knowledge representation systems. Finally, it will design an ontology that schemes the structure of concept understanding (...) based on the proposed semantic model. (shrink)

In the court of law, a person can be punished for attempting to commit a crime. An open issue in the study of Artificial Intelligence and Law is whether the law of attempts could be formally modelled. There are distinct legal rules for determining attempted crime whereas the last-act rule (also called proximity rule) represents the strictest approach. In this paper, we provide a formal model of the last-act rule using structured argumentation.

In this paper we investigate splitting algebras in varieties of logics, with special consideration for varieties of BL-algebras and similar structures. In the case of the variety of all BL-algebras a complete characterization of the splitting algebras is obtained.