In the course of proving a tenability result about the probabilities of conditionals, van Fraassen (1976) introduced a semantics for conditionals based on ω-sequences of worlds, which amounts to a particularly simple special case of ordering semantics for conditionals. On that semantics, ‘If p, then q’ is true at an ω-sequence just in case q is true at the first tail of the sequence where p is true (if such a tail exists). This approach has become increasingly popular in recent (...) years. However, its logic has never been explored. We axiomatize the logic of ω-sequence semantics, showing that it is the result of adding two new axioms to Stalnaker’s logic C2: one, Flattening, which is prima facie attractive, and, and a second, Sequentiality, which is complex and difficult to assess, but, we argue, likely invalid. But we also show that when sequence semantics is generalized from ω-sequences to arbitrary (transfinite) ordinal sequences, the result is a more attractive logic that adds only Flattening to C2. We also explore the logics of a few other interesting restrictions of ordinal sequence semantics. Finally, we address the question of whether sequence semantics is motivated by probabilistic considerations, answering, pace van Fraassen, in the negative. (shrink)

I introduce a proof system for the logic of relative fundamentality, as well as a natural semantics with respect to which the system is both sound and complete. I then “modalise” the logic, and finally I discuss the properties of grounding given a suggested account of this notion in terms of necessity and relative fundamentality.

A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence (...) are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games. (shrink)

When we say “I know why he was late”, we know not only the fact that he was late, but also an explanation of this fact. We propose a logical framework of “knowing why” inspired by the existing formal studies on why-questions, scientific explanation, and justification logic. We introduce the Kyi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}{}\textit{y}}_i$$\end{document} operator into the language of epistemic logic to express “agent i knows why φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}” and propose a Kripke-style semantics of such expressions in terms of knowing an explanation of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}. We obtain two sound and complete axiomatizations w.r.t. two different model classes depending on different assumptions about introspection. Finally we connect our logic with justification logic technically by providing an alternative semantics and an in-depth comparison on various design choices. (shrink)

We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine’s (...) semantics. Finally, we prove that evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic. (shrink)

This article introduces and defends a probabilistic measure of the explanatory power that a particular explanans has over its explanandum. To this end, we propose several intuitive, formal conditions of adequacy for an account of explanatory power. Then, we show that these conditions are uniquely satisfied by one particular probabilistic function. We proceed to strengthen the case for this measure of explanatory power by proving several theorems, all of which show that this measure neatly corresponds to our explanatory intuitions. Finally, (...) we briefly describe some promising future projects inspired by our account. (shrink)

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent textbf{LRA} cite{LRA}. We call our logic $ bf LIRA$. In contrast to textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic cite{ZS}. This logic may be useful in a situation when according (...) to an agent's point of view her/his reasoning is rational, while from the external one it might be not the case. One may use textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations of textbf{LRA}, two more possibility operators for textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $bf LIRA$. Finally, we formalize all the truth-functional $ n $-ary extensions of the negation fragment of $bf LIRA$ as well as their basic modal extension via linear-type natural deduction systems. (shrink)

Firstly, the authors analyzed the properties of primary-onditionals and secondary-conditionals, establishthe minimum system $C2L_{m}$ of primary-conditionals and secondary-conditionals, and then prove some of the formal theorems of the system which have important intuitive meanings. Secondly, the authors constructed the neighborhood semantics, prove the soundness of $C2L_{m}$ , introduce a general concept of canonical model by the neighborhood semantics, and then prove the completeness of $C2L_{m}$ by the canonical model. Finally, according to the technical results of the minimum system $C2L_{m}$ , (...) the authors discuss some of the important problems concerning primary-conditionals and secondary- onditionals. (shrink)

This paper introduces a comprehensive logical framework to reason about threshold-driven diffusion and threshold-driven link change in social networks. It considers both monotonic dynamics, where agents can only adopt new features and create new connections, and non-monotonic dynamics, where agents may also abandon features or cut ties. Three types of operators are combined: one capturing diffusion only, one capturing link change only, and one capturing both at the same time. We first characterise the models on which diffusion of a unique (...) feature and link change stabilise, whilst discussing salient properties of stable models with multiple spreading features. Second, we show that our operators (and any combination of them) are irreplaceable, in the sense that the sequences of model updates expressed by a combination of operators cannot always be expressed using any other operators. Finally, we analyse classes of models on which some operators can be replaced. (shrink)

This paper articulates an account of imperatives that sensibly supports the idea of a logic of imperative inferences. We rebuke common objections to the very possibility of such a logic, from a perspective based on recent linguistic work on the morphosyntax of imperatives. Specifically, we develop the notion that the content of an imperative sentence includes both a force operator alongside an imperational content to which the force applies. We further argue that this account of the content of (...) imperatives constitutes a plausible and flexible framework to develop a logic of imperative by examining a number of reconstructions of this idea using semantical analogs of widespread modal semantics. After studying the performance of those approaches, we conclude that progress in imperative logic has been hindered by the failure to adopt conflict-tolerant and resource-sensitive semantics, but suggest that such considerations can be incorporated in this flexible framework. Finally, we also propose a simple account of the difference between operators applying to the content of imperatives and imperative operators, in a way that sheds light on some of the issues underlying the usual antinomies. (shrink)

Can human acts be causally explained in the same way as the rest of nature? If so, causal explanation in the manner of the Hempelian model shouldn’t the human sciences and the natural sciences equally. This is not so much a question of whether the Hempelian model is a completely adequate account of causal explanation, but about whether it is adequate or inadequate in the same way for each: if there is some unique feature of human acts that dictates that (...) they are to be explained differently from natural events, then it is reasonable to suppose that this feature will be revealed by consideration of Hempelian explanations whether this is our final account of explanation or not, and if no such feature is revealed then it is reasonable to suppose that there are no fundamental differences between the human and natural sciences in how causal explanations proceed, whether this explanation is Hempelian or of some other kind. Two arguments have been given for there being such a feature. One - the well-known "Logical Connection Argument" - states that there cannot be a causal relation between a human action and the intention to perform that action. If this argument succeeds, then our ordinary explanations of human acts in terms of psychological states like intentions are not causal explanations at all. Although this does not mean that no causal explanation is true, we have the problem of finding appropriate causal antecedents of the acts. The other - the anti-predictionist argument - concludes from the fact that at least some human acts cannot be predicted that they cannot be causally explained. I wish to disarm the force of both of these arguments, and thus argue for explanatory monism: we do not need to adopt a mode of explanation for human sciences distinct from that of natural sciences. (shrink)

By performing an abduction of Bateson and Peirce, we come to understand the logic of the sacred as a semiotic and phenomenological communicative phenomenon. First, I compare and contrast their ideas concerning ontology, epistemology, and logic. Next, I articulate how both theorists construct their epistemologies within a triadic frame of relations that successfully accounts for a communicative logic that activates the integration of body and Mind. Finally, I bring Bateson’s triadic relations of aesthetics,consciousness (mental process), and the (...) sacred in line with Peirce’s existential semiotic categories of Firstness, Secondness, and Thirdness. We discover the sacred is constituted within a communicative logic that is abductive. As such, I argue that Bateson’s epistemology of the sacred becomes more accessible as a postmodern philosophy when viewed as a condition of phenomenological semiosis. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

In this paper we sketch a logic of message and reply. The logic is intended for application in a wide variety of situations, not restricted to the two-person, turn-taking situation. Each message has a body and a vector; the vector specifies the from, to, and the like. To reply to a message, it suffices to give either (1) a complete reply to the body or (2) a corrective reply to at least one presumption derivable from the vector. We (...) discuss the problems of achieving effectiveness and completeness with respect to certain aspects of communication. The results are mixed. In section 9 we argue semi-formally that, in a certain sense, dialogue is necessary. Finally we note that this logic is not a rival of other approaches but may be combinable with them. (shrink)

The overriding concern of theRepublicis the relation between morally right action and happiness. To settle the matter Plato says he must first discover the formula for politically just action for individuals, and that it will be easier to do this if we first take justice to be a virtue of a city as a whole. He develops a model of an ideal city, hardly mentioning justice and the other cardinal virtues, and then looks to that model to individuate them.Here I (...) shall first discuss the defense of his portrait of an ideal city. Second, the logic of virtueis considered, in two steps. It is argued that judgments of virtue are to be split into those of functionand power.Then there is a brief consideration of each half. Third, the implications of this for understanding and threefold classification of goods are drawn. Finally, there is a discussion of the formula for each of the cardinal virtues of a city. (shrink)

This work proposes a logical analysis for the kind of knowledge or insight provided by Buddhist enlightenment, which is often presented only in the form of contradictions and riddles. The comprehension of contradictions is perplexing to most western logic, and yet developed here is a theory demonstrating how a non truth-functional interpretation can be attached to certain naturalistic contradictions. In this way, the logical and psychological status of Enlightenment can be analysed in a manner consistent with the claims of (...) much Eastern thought. For example, it is argued that by breaking the logical principle of identity, it is possible to understand the contradiction "Everybody's friend is nobody's friend" in such a way as to learn something new about friendship, thereby causing us to modify our concept of friendship. In this way our grasp of contradictions can lead to conceptual change. -/- It is further noted that conflicts in all areas of life can be expressed by contradictions, so that our capacity in general to grasp naturalistic contradictions is also a basis for conflict-resolution -thus justifying the term 'Enlightenment'. This pragmatic comprehension of contradictions is thus shown to formalise, not only the Middle Way of Buddhism, but also Socratic dialectic. Finally, the analysis explains why gnomic utterances in every religion and Zen in particular, are sometimes expected to trigger spiritual enlightenment. (shrink)

This paper describes the adaptive logic of compatibility and its dynamic proof theory. The results derive from insights in inconsistency-adaptive logic, but are themselves very simple and philosophically unobjectionable. In the absence of a positive test, dynamic proof theories lead, in the long run, to correct results and, in the short run, sometimes to final decisions but always to sensible estimates. The paper contains a new and natural kind of semantics for S5from which it follows that a specific (...) subset of the standard worlds-models is characteristic for S5. (shrink)

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] (...) and its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

We present a class of first-order modal logics, called transformational logics, which are designed for working with sentences that hold up to a certain type of transformation. An inference system is given, and com- pleteness for the basic transformational logic HOS is proved. In order to capture ‘up to isomorphism’, we express a very weak version of higher category theory in terms of first-order models, which makes tranforma- tional logics applicable to category theory. A category-theoretical concept of isomorphism is (...) used to arrive at a modal operator nisoφ expressing ‘up to isomorphism, φ’, which is such that category equivalence comes out as literally isomorphism up to isomorphism. In the final part of the paper, we explore the possibility of using trans- formational logics to define weak higher categories. We end with two informal comparisons: one between HOS and counterpart semantics, and one between isomorphism logic, as a transformational logic, and Homo- topy Type Theory. (shrink)

This paper looks at intrinsic disputation within proposition, and specifically within propositions that offer a moderated version of the freedom of speech and expression. It begins from a consideration of what is at stake in Othello's ‘Rude am I in my speech’, a rhetorical gesture that frames an act of great eloquence, and in which the eloquence serves to formulate a quarrel by ostensibly resolving it. This example reveals that there is a conflict between empirical quarrel and articulated spoken resolution. (...) This leads the essay to explore the way in which diplomacy works, whereby we establish the pretence that there is peace between disputatious positions through the power of the logic of ‘but’, thus: ‘I agree with you, but …’. Finally, this is extended to a consideration of the limits of and/or on free speech: ‘I defend free speech, but …’, where the ‘but’ is a gesture in which the defence of free speech is modified to the point of being obliterated. (shrink)

The aim of this paper, is to provide a logical framework for reasoning about actions, agency, and powers of agents and coalitions in game-like multi-agent systems. First we define our basic Dynamic Logic of Agency ( ). Differently from other logics of individual and coalitional capability such as Alternating-time Temporal Logic (ATL) and Coalition Logic, in cooperation modalities for expressing powers of agents and coalitions are not primitive, but are defined from more basic dynamic logic operators (...) of action and (historic) necessity. We show that STIT logic can be reconstructed in . We then extend with epistemic operators, which allows us to distinguish capability and power. We finally characterize the conditions under which agents are aware of their capabilities and powers. (shrink)

The approach to decision theory floated in my 1965 book is reviewed (I), challenged in various related ways (II–V) and defended, firstad hoc (II–IV) and then by a general argument of Ellery Ells's (VI). Finally, causal decision theory (in a version sketched in VII) is exhibited as a special case of my 1965 theory, according to the Eellsian argument.

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than (...) the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given. (shrink)

The present paper, taking as a point of departure Aristotle’s dispute with the ancient physicalists in Physics II.8-9 about the role of the final cause in nature, examines the context of the problem, his theory of the causes. Aristotle assumes an analogy between nature and craft and takes the production of artifacts to be paradigmatic. With these assumptions as guiding principles, the paper attempts to motivate his causal theory and propose what may be called a “logic” of the causes. (...) It examines artefactual coming-to-be more closely, focusing on the aspects of Aristotle’s account that are highlighted in his explanation of natural coming-to-be: the basic character of the causes, the peculiar distinction between the causes and the accompanying the deeper coincidence among three of them, the complementarity between the final and moving causes, the nature and role of desire in coming-to-be, and the primacy of form. It introduces the notion of a trans-temporal objective whole and shows the need to consider the full whole — which includes the entire process of coming-to-be together with its source — as the proper context for a full understanding of coming-to-be. It also points out the importance of the distinction between the objective and subjective perspectives, especially useful in understanding final causality. (shrink)

This article offers a new understanding of populism. The argument unfolds as follows: first, the populist literature is reviewed and two main approaches are identified: ontic and logic-oriented, the more important of which is the Schmitt-Laclau logic of enmity. While the authors broadly agree with Laclau’s criticism of ontic approaches, they endorse neither his ontological understanding of enmity, nor his claim that populism is politics, and enmity is the logic of populism. Next, the origins of populism are (...) located in a paradox at the heart of democracy. Democracy defines itself as a community of inclusion, yet exclusion is constitutive of inclusion, including therefore democratic inclusion. Then is discussed what the authors believe to be the true logic of populism: resentment. Unlike enmity, which functions in Laclau’s populist theory as an ontology of non-identity, resentment operates within a rivalrous framework, which presupposes identification between the parts and refers to a set of normative commitments. Finally, the article concludes by presenting an understanding of populism as a specific logic of political action. (shrink)

We look at lying as an act of communication, where (i) the proposition that is communicated is not true, (ii) the utterer of the lie knows that what she communicates is not true, and (iii) the utterer of the lie intends the lie to be taken as truth. Rather than dwell on the moral issues, we provide a sketch of what goes on logically when a lie is communicated. We present a complete logic of manipulative updating, to analyse the (...) effects of lying in public discourse. Next, we turn to the study of lying in games. First, a game-theoretical analysis is used to explain how the possibility of lying makes such games interesting, and how lying is put to use in optimal strategies for playing the game. Finally, we give a matching logical analysis. Our running example of lying in games in liar’s dice. (shrink)

Projectible conditions are (roughly) those whose universal generalizations are con firmed by their positive instances. This paper proposes certain modifications to the above definition in order to capture the pre-analytic notion it is supposed to explicate. Then we investigate what logical operations, when performed on projectible conditionals, yield new projectible conditionals. A number of surprising theorems are proven, and these theorems indicate that few conditionals having complex antecedents and consequents are projectible. It is also shown that projectibility is not closed (...) under contraposition, and this is proposed as a solution to the Paradox of the Ravens. Finally, a general conjecture is made concerning just what conditionals are projectible, and it follows from this conjecture that although most conditionals are not projectible, they can still be dealt with inductively, but in ways more complex than recognized by classical confirmation theory. (shrink)

We propose a dynamic-epistemic analysis of the different epistemic operations constitutive of the process of interrogative inquiry, as described by Hintikka’s Interrogative Model of Inquiry (IMI). We develop a dynamic logic of questions for representing interrogative steps, based on Hintikka’s treatment of questions in the IMI, along with a dynamic logic of inferences for representing deductive steps, based on the tableau method. We then merge these two systems into a dynamic logic of interrogative inquiry which articulates a (...) joint treatment of questions and inferences, providing thereby a unified framework representing the informational dynamics of interrogative inquiry. We provide sound and complete axiomatic systems for the three dynamic logics that we introduce, we compare our framework with existing approaches, and we finally propose several directions for further work. (shrink)

George Wilhelm Friedrich Hegel has seldom been considered a major figure in the history of logic. His two texts on logic, both called The Science of Logic, both written in Hegel's characteristically dense and obscure language, are often considered more as works of metaphysics than logic. But in this highly readable book, John Burbidge sets out to reclaim Hegel's Science of Logic as logic and to get right at the heart of Hegel's thought. Burbidge (...) examines the way Hegel moves from concept to concept through every chapter of his work, and traces the origins of Hegel's effort to "think through the way thought thinks" to Plato, Kant, and Fichte. Having established the framework of Hegel's logical thought, Burbidge demonstrates how Hegel organized the rest of his system, including the Philosophy of Nature, Philosophy of Spirit and his Lectures on World History, Art, Religion and Philosophy. A final section discusses English-language interpretations of Hegel's logic from the nineteenth through twentieth centuries. Burbidge's The Logic of Hegel's 'Logic' is written with an eye to the reader of general interests, avoiding as much as possible the use of Hegel's technical vocabulary. It is an excellent introduction to an otherwise very difficult text, and has recently appeared in an Iranian translation. (shrink)

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano (...) arithmetic including primitive recursive functions, finally extending this work to that of general recursion. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

Quantum theory has played a significant role in modern philosophy both as a source of metaphysical ideas and as an important example of a 'scientific revolution'. In spite of the sixty or so years that have elapsed since its invention, a long lasting controversy concerning the interpretation and meaning of quantum theory prevails. Almost all authors, however, seem to agree on one major point, namely, that there could be no interpretation of this theory which is both realistic and local. ;The (...) purpose of this thesis is to demonstrate that this premiss is false and that a realistic, local and deterministic interpretation of quantum theory does exist, provided that we extend the classical concept of probability. ;In order to establish this a 'quasi classical' probability theory is developed based on some non Lebesgue measurable 'events', which is then applied to account for spin-statistics. Finally I note how this model reflects on the problems of physical realism, locality, the status of probability theory and the philosophical foundations of mathematics. (shrink)

The paper introduces a semantics for the language of classical first order logic supplemented with the additional operators and . This semantics understands formulas as tasks. An agent , working as a slave for its master , can carry out the task αβ if it can carry out any one of the two tasks α, β, depending on which of them was requested by the master; similarly, it can carry out xα if it can carry out α for any (...) particular value for x selected by the master; an agent can carry out α→β if it can carry out β as long as it has, as a slave , an agent who carries out α; finally, carrying out P, where P is an atomic formula, simply means making P true; in particular, is a task that no agent can carry out. When restricted to the language of classical logic, the meaning of formulas is isomorphic to their classical meaning, which makes our semantics a conservative extension of classical semantics.This semantics can claim to be a formalization of the resource philosophy associated with linear logic, if resources are understood as agents carrying out tasks. The classical operators of our language correspond to the multiplicative operators of linear logic, while and correspond to the additive conjunction and universal quantifier, respectively.Our formalism may also have a potential to be used in AI as an alternative logic of planning and action. Its main appeal is that it is immune to the frame problem and the knowledge preconditions problem.The paper axiomatically defines a logic L in the above language and proves its soundness and completeness with respect to the task semantics in the following intuitive sense: L α iff α can be carried out by an agent who has nothing but its intelligence for carrying out tasks. This logic is shown to be semidecidable in the full language and decidable when the classical quantifier is forbidden in it. (shrink)

The main aim of this paper is to introduce the logics of evidence and truth $$LET_{K}^+$$ and $$LET_{F}^+$$ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics $$LET_{K}$$ and $$LET_{F}^-$$ with rules of propagation of classicality, which are inferences that express how the classicality operator $${\circ }$$ is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more (...) values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for $$LET_{K}$$ is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for $$LET_{K}^+$$ is then obtained by imposing restrictions on the semantics of $$LET_{K}$$. These restrictions correspond exactly to the rules of propagation of classicality that extend $$LET_{K}$$. The logic $$LET_{F}^+$$ is obtained as the implication-free fragment of $$LET_{K}^+$$. We also show that the 6 values of $$LET_{K}^+$$ and $$LET_{F}^+$$ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that $$LET_{K}^+$$ is Blok-Pigozzi algebraizable and that its implication-free fragment $$LET_{F}^+$$ coincides with the degree-preserving logic of the involutive Stone algebras. (shrink)

In this paper Richard Jeffrey's 'Logic of Decision' is extended by examination of agents' attitudes to the sorts of possibilities identified by indicative conditional sentences. An expression for the desirability of conditionals is proposed and, along with Adams' thesis that the probability of a conditional equals the conditional probability of its antecedent given its consequent, is defended by informally deriving it from Jeffrey's notion of desirability and some weak constraints on rational preference for conditional possibilities. Finally a statement is (...) given of a representation theorem establishing the conditions under which a rational agent's preferences for conditionals determines the existence of unique measures (up to choice of scale) of her degrees of belief and desire. (shrink)

The connection of Hermann Сohen’s “The Logic of Pure Knowledge” with the revolutionary transformations in physics and mathematics at the end of the 19th century is shown. Сohen criticised Kant’s answer to the question “How is mathematics possible”? If Kant refers to a priori forms of pure intuition, Сohen sees in it a restriction of freedom of mathematical thinking by limits of intuition. It has been shown that Cohen's position is in accordance with the main development of mathematics in (...) the last decades of the 19th century, in particular, with K. Weierstrass’ striving to get rid of geometrical or mechanical images and intuitions in the mathematical analysis. Cohen was also well informed about the latest ideas in physics of his time. They are also discussed in “The Logic of Pure Knowledge.” The revolutionary spirit of physics and mathematics was appealing to Cohen, and he felt a corresponding enthusiasm for it. The ongoing scientific revolution is consonant with Cohen’s assertion that the foundations of science are hypotheses. The purity of pure thinking does not guarantee the correctness of any of its constructions. Each step in the development of science requires a critique of existing notions. The development of knowledge from the naive to the critical one shows a movement from a picture of the world as a set of stable things to a picture of continuous movement and change, where change is more important than what changes. Cohen sees such a development in an evolution of the concept of substance in modern physics and he welcomes the replacement of material substance by energy, seeing this movement as a confirmation of critical idealism. Finally, it is discussed whether we can speak of the actuality of Cohen’s logic of pure knowledge. (shrink)

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how (...) one can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

A topological space is hereditarilyk-irresolvable if none of its subspaces can be partitioned into k dense subsets. We use this notion to provide a topological semantics for a sequence of modal logics whose n-th member K4Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}_n$$\end{document} is characterised by validity in transitive Kripke frames of circumference at most n. We show that under the interpretation of the modality ◊\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Diamond $$\end{document} as the (...) derived set operation, K4Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}_n$$\end{document} is characterised by validity in all spaces that are hereditarily n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}-irresolvable and have the TD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_D$$\end{document} separation property. We also identify the extensions of K4Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}_n$$\end{document} that result when the class of spaces involved is restricted to those that are crowded, or densely discrete, or openly irresolvable, the latter meaning that every non-empty open subspace is 2-irresolvable. Finally, we give a topological semantics for K4M, where M is the McKinsey axiom. (shrink)

Belnap–Dunn’s relevance logic, \(\textsf{BD}\), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. \(\textsf{BD}\) is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori, while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion \(\textsf{BD2}\) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce (...) a four-valued expansion of BD called \({\textsf{BD}^\copyright }\), obtained by adding an unary connective \({\copyright }\,\ \) which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to \(\textsf{BD}\) the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, \({\textsf{BD}^\copyright }\) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of \({\textsf{BD}^\copyright }\) obtained by means of twist-structures semantics, we propose a first-order version of \({\textsf{BD}^\copyright }\) called \({\textsf{QBD}^\copyright }\), with semantics based on an appropriate notion of four-valued Tarskian-like structures called \(\textbf{4}\) -structures. We show that in \({\textsf{QBD}^\copyright }\), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for \({\textsf{QBD}^\copyright }\) is presented, proving the corresponding soundness and completeness theorems. (shrink)

We present the first-order logic of change, which is an extension of the propositional logic of change $\textsf {LC}\Box $ developed and axiomatized by Świętorzecka and Czermak. $\textsf {LC}\Box $ has two primitive operators: ${\mathcal {C}}$ to be read it changes whether and $\Box $ for constant unchangeability. It implements the philosophically grounded idea that with the help of the primary concept of change it is possible to define the concept of time. One of the characteristic axioms for (...) ${\mathcal {C}}$ is ${\mathcal {C}} A \to {\mathcal {C}}\neg A$ and one of the primitive rules is $\omega $-rule introducing $\Box $. It turns out that the next operator is definable in $\textsf {LC}\Box $. Logic $\textsf {LC}\Box $ with next is equivalent to certain fragment of $\textsf {LTL}$ extended by the appropriate definition of ${\mathcal {C}}$. Recently, $\textsf {LC}\Box $ has also been modified to the propositional logic of ‘branching’ changes $\textsf {BTC}$ by Łyczak and the logic of ‘parallel’ changes $\textsf {LC}\Box $ ↳ by Świętorzecka and Łyczak. Extended in an appropriate manner, $\textsf {BTC}$ is equivalent to a certain fragment of logic $\textsf {CTL}$ to which definitions of two kinds of changes have been added. Here we propose an extension of $\textsf {LC}\Box $ to first-order logic in which, again, the only primitive modal operators are ${\mathcal {C}}$ and $\Box $. We interpret our logic in the semantics of histories of changes. We give an axiomatic system $\textsf {LC}\Box Q$ for the considered logic, and we show selected theses about some relationships between ${\mathcal {C}}$, $\Box $ and $\forall $. Then we prove the completeness of $\textsf {LC}\Box Q$. Finally, we compare our logic with $\textsf {FOLT}$ and show the relation between $\textsf {LC}\Box Q$ and a certain fragment of the Kröger system $\varSigma $ to which the definition of ${\mathcal {C}}$ was added. (shrink)

Abstract.Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C∀) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C∀) is recursively axiomatizable, and we show that this is the case iff C only consists of the Gödel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C∀) is not even arithmetical. Finally we consider predicate monadic (...) logics TautM(C∀) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable. (shrink)

We shall introduce in this paper a language whose formulas will be interpreted by games of imperfect information. Such games will be defined in the same way as the games for first-order formulas except that the players do not have complete information of the earlier course of the game. Some simple logical properties of these games will be stated together with the relation of such games of imperfect information to higher-order logic. Finally, a set of applications will be outlined.

The topic of history-of-science explanation is first briefly introduced as a generally important one for the light it may shed on action theory, on the logic of discovery, and on philosophy''s relations with historiography of science, intellectual history, and the sociology of knowledge. Then some problems and some conclusions are formulated by reference to some recent relevant literature: a critical analysis of Laudan''s views on the role of normative evaluations in rational explanations occasions the result that one must make (...) aconceptual distinction between evaluations and explanations of belief, and that there are at leastthree subclasses of the latter, rational, critical, and theoretical; I then discuss the problem of whether explanations of discoveries are self-evidencing and predictive by focusing on views of Hempel and Nickles, and I attempt a formalization of some aspects of the problem. Finally, a more systematic and concrete analysis is undertaken by using as an example the explanation of Galileo''s rejection of space-proportionality, and it is argued that the historical explanation of scientific beliefs is a type of logical analysis. (shrink)

We study the propositional logic associated with the variety of distributive nearlattices. We prove that the logic coincides with the assertional logic associated with the variety and with the order‐based logic associated with. We obtain a characterization of the reduced matrix models of logic. We develop a connection between the logic and the ‐fragment of classical logic. Finally, we present two Hilbert‐style axiomatizations for the logic.

Against recent commentators such as Annstrong, D’Arcy, Copleston, O’Connor, Bourke, and Grisez, I argue that the logic referred to by Thomas in his “Treatise on Law” should not be understood metaphorically. Instead, it involves a chain of syllogisms, beginning with the synderesis principle, followed by primary, secondary, and tertiary principles, and ends with a practical syllogism. In showing this, I attack the view that the synderesis principle, “good ought to be done and evil avoided,” is tautological. Second, I show (...) the syllogistic relation between this and the more subordinate moral principIes. Finally, I argue that the practical syllogism also involves a logical deduction, where the minor premise is a propositional attitude of perception, and the conclusion is an action which expresses a proposition. What emerges is a more precise account of how actions are related to natural law. (shrink)

This paper presents and discusses Henri Poincaré’s view on the logical non-derivability of imperatives from indicatives. Firstly, his point of view is compared with David Hume’s. Then Poincaré’s thesis is shown to have been implicitly or explicitly essential to attempts to build a logic of imperatives (norms) as a special linguistic category as well as a deontic logic. The paper shows deontic logic enables us to formulate general principles concerning the logical separability of deonticals (sentences with deontic (...) operators) and non-deontic sentences. Finally, the paper includes remarks on the reception of Poincaré’s discussed ideas in later logical and philosophical literature with the second issue illustrated by legal philosophy. (shrink)

Among the quantificational notions neglected by classical logic are "many," "few," and "nearly all." Despite the apparent vagueness associated with these terms in ordinary discourse, in specific contexts we can and do draw strict inferences from statements in which they occur. In this pioneering work, Altham has attempted to uncover something of the formal logic that justifies such inferences. He begins by showing the mutual interdefinability of the three terms. If negation and any one of them are taken (...) as primitive, the other two are definable in terms of these. Selecting as his basic semantic notion the idea of a manifold or many-membered-set, he shows it is only necessary to fix upon a number n, greater than one, to represent the least number of elements to constitute a specific manifold and then one can construct a natural deductive system which uses the standard predicate logic. He develops at great length a weak system that yields principles of inference which are valid in every non-empty domain and in which the classical predicate logic appears as a limiting case. One point that emerges is that the three plurality operators may all be exhibited as involving a double quantification not characteristic of "some," "none," or "all." Next, two stronger systems are developed more briefly which are valid only in many-membered domains. The strongest can represent the largest set of informal arguments and seems to be the most natural of the three. Nevertheless, since these three systems admittedly do not give an exhaustive account of the use of these quantifiers in informal discourse, Altham explains three further systems which correspond more closely to the natural language use of these terms. Finally an appendix indicates what modifications would be required for an intuitionistic logic like Heyting's or the minimal logic of Kolmogorov-Johansson. The text is clearly written and requires no greater logical sophistication to understand than would be needed to follow Lemmon's Beginning Logic or Mates' Elementary Logic.--A. B. W. (shrink)

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two (...) distinct positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual. (shrink)

Protein post‐translational modifications (PTMs) play a crucial role in all cellular functions by regulating protein activity, interactions and half‐life. Despite the enormous diversity of modifications, various PTM systems show parallels in their chemical and catalytic underpinnings. Here, focussing on modifications that involve the addition of new elements to amino‐acid sidechains, I describe historical milestones and fundamental concepts that support the current understanding of PTMs. The historical survey covers selected key research programmes, including the study of protein phosphorylation as a regulatory (...) switch, protein ubiquitylation as a degradation signal and histone modifications as a functional code. The contribution of crucial techniques for studying PTMs is also discussed. The central part of the essay explores shared chemical principles and catalytic strategies observed across diverse PTM systems, together with mechanisms of substrate selection, the reversibility of PTMs by erasers and the recognition of PTMs by reader domains. Similarities in the basic chemical mechanism are highlighted and their implications are discussed. The final part is dedicated to the evolutionary trajectories of PTM systems, beginning with their possible emergence in the context of rivalry in the prokaryotic world. Together, the essay provides a unified perspective on the diverse world of major protein modifications. (shrink)