Europe's long sixteenth century—a period spanning the years roughly from the voyages of Columbus in the 1490s to the English Civil War in the 1640s—was an era of power struggles between avaricious and unscrupulous princes, inquisitions and torture chambers, and religious differences of ever more violent fervor. Ideas of Liberty in Early Modern Europe argues that this turbulent age also laid the conceptual foundations of our modern ideas about liberty, justice, and democracy. Hilary Gatti shows how these ideas emerged (...) in response to the often-violent entrenchment of monarchical power and the fragmentation of religious authority, against the backdrop of the westward advance of Islam and the discovery of the New World. She looks at Machiavelli's defense of republican political liberty, and traces how liberty became intertwined with free will and religious pluralism in the writings of Luther, Erasmus, Jean Bodin, and Giordano Bruno. She examines how the St. Bartholomew's Day Massacre and the clash of science and religion gave rise to concepts of liberty as freedom of thought and expression. Returning to Machiavelli and moving on to Jacques Auguste de Thou, Paolo Sarpi, and Milton, Gatti delves into debates about the roles of parliamentary government and a free press in guaranteeing liberties. Drawing on a breadth of canonical and lesser-known writings, Ideas of Liberty in Early Modern Europe reveals how an era stricken by war and injustice gave birth to a more enlightened world. (shrink)

Nella storia della fortuna di Cicerone, l’età della Controriforma è un periodo non specificamente indagato, benché la presenza dell’oratore sia fondamentale nella formazione di molti protagonisti del rinnovamento cattolico e poi nei nuovi sistemi educativi elaborati per formare un clero più qualificato. L’articolo intende mettere in luce il ruolo centrale assunto da Cicerone in tale delicata fase storica, concentrandosi in particolare sulla figura esemplare di Girolamo Ragazzoni, commentatore delle Epistulae ad Familiares e volgarizzatore delle Filippiche. Ragazzoni, da giovane umanista in (...) stretto contatto con Paolo Manuzio e Carlo Sigonio, parteciperà al Concilio, tenendone l’orazione conclusiva, per poi diventare nel 1577 vescovo di Bergamo: lì, operando in collaborazione con Carlo Borromeo, si rivelerà uno dei più solerti esecutori dell’opera di riforma interna alla Chiesa, in particolare nel settore pedagogico e scolastico. (shrink)

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

The literature on indicative conditionals contains two appealing views. The first is the selectional view: on this view, conditionals operate by selecting a single possibility, which is used to evaluate the consequent. The second is the informational view: on this view, conditionals don’t express propositions, but rather impose constraints on information states of speakers. Both views are supported by strong arguments, but they are incompatible on their standard formulations. Hence it appears that we have to choose between mutually exclusive options. (...) But the options are not exclusive. This paper develops a theory of the semantics and assertability of conditionals that is both selectional and informational. The theory vindicates the signature inferences of both selectional and informational theories, including so-called Conditionals Excluded Middle and principles about the interplay between conditionals and ‘might’. It also predicts some interesting and puzzling facts about the assertion of conditionals. (shrink)

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Suppose that, yesterday at noon, Maria considered flipping a fair coin, but didn't. What probability do you assign to "If Maria had flipped the coin, the coin would have landed heads"? Now suppose that, contrary to fact, Maria did indeed flip the coin. In that counterfactual scenario, what is the probability of "The coin will land tails"? The two questions sound strikingly similar. I argue that they sound similar because they are equivalent. The chance of a counterfactual "If A, would (...) C" equals the chance of C, in the counterfactual scenario that A (and a similar principle holds for credence). This principle does better than similar principles that have been defended (like Skyrms' Thesis), avoids triviality, and gives us important clues for a semantics for counterfactuals. (shrink)

This paper investigates the link between informational consequence and credence. I first suggest a natural constraint, namely that informational consequence should preserve certainty: on any rational credence distribution, when the premises of an informational inferences have credence 1, the conclusion also has credence 1. Then I show that the certainty‐preserving constraint leads to triviality. In particular, the following three claims are incompatible: (i) informational consequence is extensionally distinct from classical consequence; (ii) informational inferences preserve certainty; (iii) credences obey (a subset (...) of) classical Bayesian constraints. The proof is straightforward, but the theoretical implications are substantial. Informational theorists need to either give up the idea that credence applies to epistemic discourse, or develop a nonclassical theory of credence and credal update. Moreover, the result also suggests that there is a connection between informational consequence and triviality results, including classical triviality results like Lewis's. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, and points to the philosophical pay-offs to (...) be expected by deepening our understanding of the topic. (shrink)

This paper investigates what happens when we merge two different lines of theorizing about counterfactuals. One is the comparative closeness view, which was developed by Stalnaker and Lewis in the framework of possible worlds semantics. The second is the interventionist view, which is part of the causal models framework developed in statistics and computer science. Common lore and existing literature have it that the two views can be easily fit together, aside from a few details. I argue that, on the (...) contrary, transplanting causal-models-inspired ideas in a possible worlds framework yields a new semantics. The difference is grounded in different algorithms for handling inconsistent information, hence it touches on issues that are at the very heart of a semantics for contrary-to-fact conditionals. Roughly, Stalnaker/Lewis semantics requires us to evaluate the consequent of a counterfactual at all closest antecedent-verifying possibilities. Causal-models-based semantics also does this, but in addition uses the information contained in the antecedent, together with background causal information, to shift what worlds count as closest. This makes systematically different predictions and generates a new logic. The upshot is that we have a new semantics to study, and a substantial theoretical choice to make. (shrink)

According to the orthodox account developed by Kaplan, indexicals like I, you, and now invariably refer to elements of the context of speech. This essay argues that the orthodoxy is wrong. I, you, and the like are shifted by certain modal operators and hence can fail to refer to elements of the context, for example, I can fail to refer to the speaker. More precisely, indexicals are syntactically akin to logical variables. They can be free, in which case they work, (...) roughly, on the Kaplan model. But they can also be bound: this happens, in a systematic fashion, when they are in the scope of epistemic modals or attitude verbs. The new view has two interesting philosophical consequences. First, it vindicates a broadly Fregean perspective on referential expressions, essentially refuting the idea that indexicals are rigid designators. Second, it suggests a new picture of the interaction between context and linguistic meaning: compositional semantics does not need to look at the context and hence has no need for a context parameter. (shrink)

Know-how and expressivism are usually regarded as disjoint topics, belonging to distant areas of philosophy. This paper argues that, despite obvious differences, the two debates have important similarities. In particular, semantic and conceptual tools developed by expressivists can be exported to the know-how debate. On the one hand, some of the expressivists' semantic resources can be used to deflect Stanley and Williamson's influential argument for factualism about know-how: the claim that knowing how to do something consists in knowing a fact. (...) On the other, expressivism provides the resources to formulate a nonfactualist account of know-how. On this account, know-how has a kind of nonpropositional content and plays the role of guiding performance of action, rather than recording information from the environment. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...) in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

A growing body of literature documents the important role played by moral outrage or moral anger in stakeholders’ reactions to cases of corporate social irresponsibility. Existing research focuses more on the consequences of moral outrage than a systematic analysis of how appraisals of irresponsible corporate behavior can lead to this emotional experience. In this paper, we develop and test, in two field studies, an extended model of moral outrage that identifies the cognitions that lead to, and are associated with, this (...) emotional experience. This research contributes to the existing literature on reactions to corporate social irresponsibility by explaining how observers’ evaluation of irresponsible corporate behavior leads to reactions of moral anger. The paper also helps clarify the difference between moral outrage and other types of anger and offers useful insights for managers who have to confront public outrage following cases of irresponsible corporate behavior. Finally, the analysis of the causes of stakeholders’ anger at irresponsible corporations opens important avenues for future research that are presented in the paper. (shrink)

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a (...) characterization of visualization that is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into disrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( \sqsubseteq (...) \) defined by $a \sqsubseteq b$ if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation $\sqsubseteq$. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with (...) some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role. (shrink)

In a recent paper, Christian List (2014) has argued for the compatibilism of free will and determinism. Drawing on a distinction between physical possibility (used in defining determinism) and agential possibility (used in defining free will), List constructs a formal two-level model in which the two concepts are consistent. This paper's first contribution is to show that though List's model is formally consistent, philosophically it falls short of establishing a satisfactory compatibilist position. Ensuingly, an analysis of the shortcomings of the (...) model leads to the identification of a controversial epistemological assumption implicit in the statements of both compatibilist and incompatibilist positions. Arguing that this assumption is not currently satisfied, the paper's second contribution is to show that neither the compatibilist nor the incompatibilist position is presently well-founded. (shrink)

A version of the Church-Turing Thesis states that every effectively realizable physical system can be simulated by Turing Machines (‘Thesis P’). In this formulation the Thesis appears to be an empirical hypothesis, subject to physical falsification. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, analog, quantum, and retrocausal computation. The conclusions are that these models reduce to supertasks, i.e. infinite computation, and that even supertasks are no solution for recursive incomputability. This yields that the realization (...) of hypercomputing devices is implausible, and that Thesis P is not essentially different from the standard Church-Turing Thesis. (shrink)

In this article I describe and evaluate the debate that surrounds the proper interpretation of Tarski’s account of logical consequence given in his classic 1936 article ‘On the concept of logical consequence’. In the late 1980s Etchemendy argued that the familiar model theoretic account of logical consequence is not to be found in Tarski’s original article. Whereas the contemporary account of logical consequence is a variable‐domain conception – in that it calls for a reinterpretation of the domain of variation of (...) the quantifiers when evaluating logical consequence –, no such reinterpretation is found in Tarski’s original account, which was rather based on a ‘fixed‐domain’ conception. Etchemendy’s claims have sparked a debate on Tarski’s conception of logical consequence with important contributions by, among others, Bach, Bays, Corcoran, Gómez‐Torrente, Mancosu, Ray, Sagüillo, and Sher. (shrink)

Open peer commentary on the article “Perception-Action Mutuality Obviates Mental Construction” by Martin Flament Fultot, Lin Nie & Claudia Carello. Upshot: I contrast enactivist and ecological perspectives on some of the themes raised by the authors. I discuss some of their worries about the notion of sense-making and other epistemological aspects of enactivism.

It is reported that in reply to John Wisdom’s request in 1944 to provide a dictionary entry describing his philosophy, Wittgenstein wrote only one sentence: “He has concerned himself principally with questions about the foundations of mathematics”. However, an understanding of his philosophy of mathematics has long been a desideratum. This was the case, in particular, for the period stretching from the Tractatus Logico-Philosophicus to the so-called transitional phase. Marion’s book represents a giant leap forward in this direction. In the (...) preface, Marion provides a diagnosis for why it has taken such a long time to obtain an accurate picture of Wittgenstein’s ideas in the foundations of mathematics. When the Remarks on the Foundations of Mathematics came out in 1956 the reception by specialists, such as Kreisel, was negative. This led many Wittgenstein scholars to set aside the issues in the foundations of mathematics and go on with the business of the day, focusing on Wittgenstein’s philosophy of language and psychology. However, these early negative appraisals were severely limited by two facts. First of all, the nature of the Remarks was a hindrance to an understanding of what Wittgenstein was up to. The book was edited by Wittgenstein’s literary executors by collating together passages from several manuscripts dating from 1937 to 1944, cutting, however, many passages in between remarks. Second, many of the original manuscripts and typescripts of the transitional phase were not available and thus it was virtually impossible to obtain a balanced picture of Wittgenstein’s development. Finally, a stark contraposition between the early and late Wittgenstein did not encourage people to work seriously on the development of Wittgenstein’s position from the Tractatus through the transitional period to the later writings. (shrink)

The temporality of intentions and actions in situations of social interaction can sometimes be paradoxical. I argue that in these situations it may sometimes be possible to conceive of individual acts that can, in a strong sense, be intended retroactively. This could happen when the relational patterns in social interaction literally alter the virtual structure of a participant's past corporeal intentions resulting in an odd experience of having intended something all along without knowing it. I propose that this possibility should (...) be interpreted as more than just a narrowly epistemic phenomenon. Examining this claim involves clarifying the enactive perspective on intentionality, which I do here. The enactive approach rejects the model of a causal relation between intention and action for one of an intrinsic qualitative relation between the two as facets of sense-making. I develop this idea and compare it with Merleau-Ponty's Fundierung model of the mutual relation between corporeal and reflexive intentionality to show that co-regulated moves/affections during social interaction may modulate both arcs of this relation, creating the possibility of a re-signification that alters not the actuality but the virtual tendencies that preceded the social act. (shrink)

In December 2017, Law 219/2017, ‘Provisions for informed consent and advance directives’, was approved in Italy. The law is the culmination of a year-long process and the subject of heated debate throughout Italian society. Contentious issues are addressed in the law. What emerges clearly are concepts such as quality of life, autonomy, and the right to accept or refuse any medical treatment – concepts that should be part of an optimal relationship between the patient and healthcare professionals. The law maximizes (...) the value of the patient’s time to decide. Every patient is allowed to make choices for the present as well as for the future, conceived as a continuation of the present, and to decide what comes next, based on what he/she already knows. The law identifies three distinct but converging paths towards the affirmation of a care relationship based on reciprocal trust and respect: the possibility to consent to or refuse treatment, the shared care planning, and advance directives. The fundamental point to emerge from the new Italian law is that consensus is an essential connotation of the treatment relationship. Consensus is not limited to the acceptance/rejection of medical treatment but is ongoing. It is projected into the future through shared care planning and advance directives which act as tools for self-determination and the manifestation of the beliefs and preferences of persons unable to express their will. These principles are in line with the idea of appropriate care as evaluated from two different perspectives, one of scientific adequacy and the other commensurate with the individual’s resources, fragility, values, and beliefs. Surely, however, the new law is not the end of the matter on issues such as conscientious objection, which is deeply rooted within the Italian cultural and political debate. In this regard, healthcare institutions and policymakers will be called upon to develop and implement organizational policies aimed at the management of foreseeable conscientious objection in this field. (shrink)

On a traditional view, the semantics of natural language makes essential use of a context parameter, i.e. a set of coordinates that represents the situation of speech. In classical semantic frameworks, this parameter plays two key roles: first, context contributes to determining the content of utterance; second, it is crucial for defining logical consequence. I point out that recent empirical proposals about context shift in natural language (in particular, context-shifting semantics in the style of Anand and Nevins 2004) are incompatible (...) with the traditional view of context. At the same time, there is increasing cross-linguistic support for this brand of context shift. Hence I suggest that context has no place in semantic theory proper. We should revert back to so-called multiple-indexing frameworks that were developed by Montague and others, and relegate context to the postsemantic stage. (shrink)

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a widespread debate in Germany (...) on ‘directions’ as a central notion in the theory of parallels. (shrink)

Recent developments in generalized probability theory have renewed a debate about whether regularity (i.e., the constraint that only logical contradictions get assigned probability 0) should be a necessary feature of both chances and credences. Crucial to this debate, however, are some mathematical facts regarding the interplay between the existence of regular generalized probability measures and various cardinality assumptions. We improve on several known results in the literature regarding the existence of regular generalized probability measures. In particular, we give necessary and (...) sufficient conditions for the existence of regular generalized probability measures defined on the whole powerset of any sample space. (shrink)

This article explores the concept of landscape through the lens of performativity, challenging the traditional visual-centric understanding rooted in Western art and culture but without denying the visual and representational character of landscape. It examines the evolution of landscape representation, from its origins in linear perspective and Cartesian dualism to contemporary approaches that integrate performative practices. The analysis highlights the dialectical tension between visual representation and immersive, multisensory experiences, arguing for a more integrated view that acknowledges the performative aspects of (...) the visual. By re-evaluating the role of distance, vision, and representation, the article advocates for a nuanced understanding of landscape that balances the visual with embodied practices, ultimately proposing that landscape should be seen as a dynamic interplay between seeing and performing. (shrink)

The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources (...) I will describe how Hilbert, Behmann, and Bernays,among others, were influenced by and reacted to the technical and philosophical thesespresented in Principia Mathematica. I also argue that there are some elements of continuity between Russell's approach and Hilbert's program as it was presented inthe early twenties. (shrink)

In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...) by Cantor’s cardinality assignments. In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The talk has four main parts. The first takes a historical look at 19th-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions. (shrink)

In the last decades, Systems Biology (including cancer research) has been driven by technology, statistical modelling and bioinformatics. In this paper we try to bring biological and philosophical thinking back. We thus aim at making diferent traditions of thought compatible: (a) causality in epidemiology and in philosophical theorizing—notably, the “sufcient-component-cause framework” and the “mark transmission” approach; (b) new acquisitions about disease pathogenesis, e.g. the “branched model” in cancer, and the role of biomarkers in this process; (c) the burgeoning of omics (...) research, with a large number of “signals” and of associations that need to be interpreted. In the paper we summarize frst the current views on carcinogenesis, and then explore the relevance of current philosophical interpretations of “cancer causes”. We try to ofer a unifying framework to incorporate biomarkers and omic data into causal models, referring to a position called “evidential pluralism”. According to this view, causal reasoning is based on both “evidence of diference-making” (e.g. associations) and on “evidence of underlying biological mechanisms”. We conceptualize the way scientists detect and trace signals in terms of information transmission, which is a generalization of the mark transmission theory developed by philosopher Wesley Salmon. Our approach is capable of helping us conceptualize how heterogeneous factors such as micro and macro-biological and psycho-social—are causally linked. This is important not only to understand cancer etiology, but also to design public health policies that target the right causal factors at the macro-level. (shrink)

Mascolo successfully defends a relational, developmental approach to emotions. I draw parallels between his perspective and the enactive approach, in particular with the concept of participa...

Logical forgetting is NP-complete as a decision problem even in the simple case of propositional Horn formulae, and may exponentially increase their size. A way to forget is to replace each variable to forget with the body of each clause whose head is the variable. It takes polynomial time in the single-head case: each variable is the head of at most a clause. Some formulae are not single-head but can be made so to simplify forgetting. They are called single-head equivalent. (...) The first contribution of this article is the study of a semantical characterization of single-head equivalence. Two necessary conditions are given. They are sufficient when the formula is interequivalent, that is, the formula makes two sets of variables equivalent only if they are also equivalent to their intersection. All acyclic formulae are interequivalent. The second contribution of this article is an incomplete algorithm for turning a formula single-head. In case of success, forgetting becomes possible in polynomial time and produces a polynomial-size formula, none of which is otherwise guaranteed. The algorithm is complete on interequivalent formulae. (shrink)

Hypercomputation—the hypothesis that Turing-incomputable objects can be computed through infinitary means—is ineffective, as the unsolvability of the halting problem for Turing machines depends just on the absence of a definite value for some paradoxical construction; nature and quantity of computing resources are immaterial. The assumption that the halting problem is solved by oracles of higher Turing degree amounts just to postulation; infinite-time oracles are not actually solving paradoxes, but simply assigning them conventional values. Special values for non-terminating processes are likewise (...) irrelevant, since diagonalization can cover any amount of value assignments. This should not be construed as a restriction of computing power: Turing’s uncomputability is not a ‘barrier’ to be broken, but simply an effect of the expressive power of consistent programming systems. (shrink)