This is a title on the foundations of defeasible logic, which explores the formal properties of everyday reasoning patterns whereby people jump to conclusions, reserving the right to retract them in the light of further information. Although technical in nature the book contains sections that outline basic issues by means of intuitive and simple examples. This book is primarily targeted at philosophers interested in the foundations of defeasible logic, logicians, and specialists in artificial intelligence and theoretical computer science.
This paper investigates the “general” semantics for first-order logic introduced to Antonelli, 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. :217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.
This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of "non-existing" objects. The models here introduced have only one (possibly empty) domain, and a partial reference function for the singular terms (that might be undefined at some arguments). Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non (...) unlike the notion of "truth at world" found in modal logic. A model theory is developed, establishing compactness, interpolation (implying a strong form of Beth definability), and completeness (with respect to a particular axiomatization). (shrink)
In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a classical construction due (...) to Kleene, and indicate how they can be employed to reconstruct the class of the general recursive functions. We also point at how Revision Rules can be employed to access non-minimal fixed points of partially defined computing procedures. (shrink)
The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...) of each. (shrink)
In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege’s debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege’s critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be (...) other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege’s skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege’s metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege’s conception of logic, especially in relation to contemporary conceptions. (shrink)
Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of (...) a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system. (shrink)
The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference , i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. This is (...) in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be "undone" by new information. (shrink)
This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
While second-order quantifiers have long been known to admit nonstandard, or interpretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretationgeneral” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further constraints that the interpretation is to satisfy.
Due programmi diversi si intersecano nel lavoro di Frege sui fondamenti dell’aritmetica: • Logicismo: l’aritmetica `e riducibile alla logica; • Estensionalismo: l’aritmetica `e riducibile a una teoria delle estensioni. Sia nei Fondamenti che nei Principi, Frege articola l’idea che l’aritmetica sia riducibile a una teoria logica delle estensioni.
In this paper we show that the Gupta-Belnap systems S# and S* are П12. Since Kremer has independently established that they are П12-hard, this completely settles the problem of their complexity. The above-mentioned upper bound is established through a reduction to countable revision sequences that is inspired by, and makes use of a construction of McGee.
A propositional system of modal logic is second-order if it contains quantiﬁers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantiﬁers, K, B, T, K4 and S4 all become eﬀectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.
We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the (...) game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)
The present paper analyzes the dialogue between Gilles Deleuze and François Jullien. We focus on three axes articulated by the idea of immanence. Firstly, we shall compare the usage of the expression “absolutization of immanence”, coined by the sinologist in order to play down transcendence in Chinese thought, which was eventually applied by Deleuze within the realm of belief in this world. Secondly, we shall examine Jullien’s critic to Deleuze on the relation between philosophy and wisdom. We sustain that even (...) if the deleuzian description presents ambivalence in the “pre-philosophical” plane of immanence, it does not entail the superiority of philosophy. Thirdly, we shall confront the divergent perspectives on immanence: Deleuze’s treatment as plan and Jullien’s conception as fonds. Finally, we shall expose the hardship to think the idea of immanence present in the work of both authors. (shrink)
_The Philosophy of Brentano_ has as its goal to explore the significance and impact of Franz Brentano’s thought, to promote a deepening of the ongoing renaissance of interest in Brentano, and to advance the project of understanding Brentano’s actual philosophical positions and correcting entrenched misunderstandings.
In this paper we apply the idea of Revision Rules, originally developed within the framework of the theory of truth and later extended to a general mode of deﬁnition, to the analysis of the arithmetical hierarchy. This is also intended as an example of how ideas and tools from philosophical logic can provide a diﬀerent perspective on mathematically more “respectable” entities. Revision Rules were ﬁrst introduced by A. Gupta and N. Belnap as tools in the theory of truth, and they (...) have been further developed to provide the foundations for a general theory of (possibly circular) deﬁnitions. Revision Rules are non-monotonic inductive operators that are iterated into the transﬁnite beginning with some given “bootstrapper” or “initial guess.” Since their iteration need not give rise to an increasing sequence, Revision Rules require a particular kind of operation of “passage to the limit,” which is a variation on the idea of the inferior limit of a sequence. We then deﬁne a sequence of sets of strictly increasing arithmetical complexity, and provide a representation of these sets by means of an operator G(x, φ) whose “revision” is carried out over ω2 beginning with any total function satisfying certain relatively simple conditions. Even this relatively simple constraint is later lifted, in a theorem whose proof is due to Anil Gupta. (shrink)
This article seeks to reconstruct the phrase τὸν ἄρτον ἡμῶν τὸν ἐπιούσιον in the light of an African predicament with the Ewe-Ghanaian context in focus. The article posits that the various interpretations of the phrase throughout the epochs of Christianity have arisen as a result of the ambiguity associated with ἐπιούσιος and the quest to make the Lord’s Prayer in general relevant to the life situation of the recipient communities. Although the Lord’s Prayer is still regarded as a prayer par (...) excellence in the Ewe-Ghanaian Christian community, its central theme in popular Ewe-Ghanaian spirituality has been demonological instead of eschatological. The demonological interpretation is premised on the primal Ewe belief that successful spiritual warfare against the evil forces believed to be militating against one’s destiny in life can restore one’s fortunes and lead to the blessing of material prosperity. Thus, the phrase τὸν ἄρτον ἡμῶν τὸν ἐπιούσιον in popular Ewe-Ghanaian Christian spirituality is a call on God to ‘grant us the blessing of material prosperity, good health and longevity’. The demonological approach towards material prosperity, however, is discontinuous with the evangelisation approach, which was introduced into Ewe-Ghanaian spirituality through missionary activities in the mid-19th century. The missionaries identified the cardinal Ewe-Ghanaian predicament – poverty of the mind and spirit – and addressed them holistically through the message of the Gospel and the establishment of schools, hospitals, and agriculture to guarantee food security. This holistic approach to alleviating the poverty of the spirit and mind laid the foundation for the socio-economic development of their Ewe-Ghanaian Christian converts and the communities in which they practise their faith.Contribution: This article forms part of the researcher’s contribution to the academic knowledge on the Lord’s Prayer and inspires the use of Mother Tongue Biblical hermeneutics in the development of theological materials for the Ewe-Ghanaian Christian communities in Ghana, Togo, and Benin. (shrink)
The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
In this article I raise empirical challenges for the claim tha area MT/V5 is the neural correlate for visual experience as of motion (Block 2005). In particular, I focus on the claim that there is matching content between area MT, on one hand, and visual experience as of motion, on the other hand (Chalmers 2000, Block 2007). I survey two lines of empirical evidence which challenge the claim of matching content in area MT. The first line of evidence covers new (...) results in neuroscience which emphasize the ongoing dynamics in cortical activity. The second line of evidence focuses on results regarding area MT in particular (Maier et al. 2007 and Cohen and Newsome 2008). Together, the empirical results indicate that neural processing is context sensitive in a way that challenges the attribution of content to local areas of cortex, to area MT in particular. In the final part of the article I explore alternative approaches and discuss remaining issues. (shrink)
Henry Leonard and Karel Lambert first introduced so-called presupposition-free (or just simply: free) logics in the 1950’s in order to provide a logical framework allowing for non-denoting singular terms (be they descriptions or constants) such as “the largest prime” or “Pegasus” (see Leonard  and Lambert ). Of course, ever since Russell’s paradigmatic treatment of definite descriptions (Russell ), philosophers have had a way to deal with such terms. A sentence such as “the..
This paper presents a bivalent extensional semantics for positive free logic without resorting to the philosophically questionable device of using models endowed with a separate domain of “non-existing” objects. The models here introduced have only one domain, and a partial reference function for the singular terms. Such an approach provides a solution to an open problem put forward by Lambert, and can be viewed as supplying a version of parametrized truth non unlike the notion of “truth at world” found in (...) modal logic. A model theory is developed, establishing compactness, interpolation, and completeness. (shrink)
Ausgehend von Franz Brentanos berühmter Intentionalitätspassage aus der Psychologie vom empirischen Standpunkt wird dargelegt, daß die vorherrschende ontologische Deutung seines sogenannten frühen Intentionalitätsgedankens unhaltbar ist. Unter Berücksichtigung von Brentanos Quellen, vor allem Aristoteles' Wahmehmungslehre und Theorie der Relativa, wird die Auffassung des sogenannten intentionalen bzw. immanenten Objektes als bewußtseinsimmanenter Entität abgelehnt und die Kontinuität hervorgehoben, die zwischen Brentanos früher und späterer, sogenannter reistischer Intentionalitätsauffassung besteht.
This paper introduces a generalization of Reiter’s notion of “extension” for default logic. The main difference from the original version mainly lies in the way conﬂicts among defaults are handled: in particular, this notion of “general extension” allows defaults not explicitly triggered to pre-empt other defaults. A consequence of the adoption of such a notion of extension is that the collection of all the general extensions of a default theory turns out to have a nontrivial algebraic structure. This fact has (...) two major technical fall-outs: ﬁrst, it turns out that every default theory has a general extension; second, general extensions allow one to deﬁne a well-behaved, skeptical relation of defeasible consequence for default theories, satisfying the principles of Reﬂexivity, Cut, and Cautious Monotonicity formulated by D. Gabbay. (shrink)
The purpose of this note is to acknowledge a gap in a previous paper — “The Complexity of Revision”, see  — and provide a corrected version of argument. The gap was originally pointed out by Francesco Orilia (personal communication and ), and the ﬁx was developed in correspondence with Vann McGee.
The reflection elaborated in these pages, fleeing all submission to the now abused rhetoric of the prevailing economism, traces in the works of René Girard - the most serious pretender to the legacy of the masters of suspicion - and Jacques Derrida - the last great philosopher of the twentieth century - the constituent elements of a critical paradigm with which to interpret the present time. The volume investigates the multiple correspondences between the different legacies of deconstruction and the most (...) recent developments of mimetic theory through the articulated reconstruction of the reflections, still little known in Italy, of some of the most brilliant contemporary exponents of those traditions. Formulating the features of an ontology of actuality hinged on the notions of mimesis and trace, the author proposes a philosophical diagnosis that recognizes in our time an age in the grip of panic. (shrink)
La thèse de l’« inexistence intentionnelle » formulée par Brentano a été traditionnellement interprétée comme une théorie de la « relation intentionnelle », autrement dit de la relation entre l’acte mental et son « objet immanent » ou « intentionnel », c’est-à-dire interne à la conscience. Se fondant sur la lecture du fameux passage sur l’intentionnalité de la Psychologie du point de vue empirique , le présent article démontre que l’interprétation ontologique de la théorie de l’intentionnalité du premier Brentano est (...) insoutenable, toute dominante qu’elle est. Pour ce faire, nous partirons des sources de la pensée de Brentano, en particulier de la théorie de la perception et des relatifs d’Aristote, pour rejeter la conception de l’objet immanent ou intentionnel comme entité immanente à la conscience et mettre en évidence la continuité qui existe entre la première conception de Brentano de l’intentionnalité et la seconde, consécutive à ce qu’on définit comme le tournant réiste de sa pensée.Brentano’s thesis of “intentional inexistence” has been traditionally interpreted as a theory of “intentional relationships”, i.e., of the relation between a mental act and its “immanent” or “intentional object”, within consciousness. Starting from the famous passage on intentionality in Psychology from an Empirical Standpoint , the present paper shows that the dominant ontological interpretation of Brentano’s former theory of intentionality is untenable. Proceeding from the sources of Brentano’s thought, in particular from Aristotle’s theory of perception and of relatives, the conception of the immanent or intentional object as an immanent entity to consciousness is rejected. Instead, the continuity between Brentano’s former conception of intentionality and the subsequent one, following the so-called reistic turning-point in his thought, is highlighted. (shrink)
Der Triestiner Vittorio Benussi , Mitglied der Grazer gegenstandstheoretischen und psychologischen Schule um Alexius Meinong, war einer der bedeutendsten Experimentalpsychologen seiner Zeit. Seine Pionierleistungen auf dem Gebiet der experimentellen Gestaltpsychologie gerieten jedoch bald durch die fortschreitende Durchsetzung der Berliner Schule der Gestalttheorie in Vergessenheit, so daß sein Werk bis heute weitgehend unbekannt geblieben ist.Benussis wissenschaftliche Tätigkeit, die sich durch eine streng experimentelle Vorgangsweise auszeichnet, erweist sich rückblickend als fruchtbarer Anknüpfungspunkt für die zeitgenössische Kognitionswissenschaft. Dies ermöglicht eine Neubewertung seiner wissenschaftlichen Arbeit (...) und jener "aktpsychologischen" Tradition, der er angehörte. Benussis Untersuchungen nehmen ihren Ausgangspunkt in der Voraussetzung der Intentionalität des Psychischen. Seine Arbeiten thematisieren das Problem des subjektiven Eingriffs in die Konstitution der Erfahrungsgegebenheiten, indem sie die Vorherrschaft des Subjektes über die elementaren Sinnesbedingungen betonen. In Benussis Gesamtwerk läßt sich ein einheitliches Programm erkennen: Ausgehend von der Gegenstands- und Produktionstheorie der Grazer Schule, entwickelt Benussi seinen eigenen theoretischen Standpunkt, der ihn - bewußt - der Phänomenologie Husserls nähebringt. Benussi verlagert schrittweise den Schwerpunkt seiner Forschungen von der gegenstandstheoretischen Perspektive zur Hervorhebung der latenten Subjektivität, die im Konstitutionsprozeß des Gegebenen wirksam ist.Leben und Werk Vittorio Benussis rückblickend zu betrachten, heißt sich mit einem Klassiker der Psychologie zu beschäftigen, dessen Modernität erst in der heutigen Zeit erkannt werden kann. (shrink)
Many diﬀerent modes of deﬁnition have been proposed over time, but none of them allows for circular deﬁnitions, since, according to the prevalent view, the term deﬁned would then be lacking a precise signiﬁcation. I argue that although circular deﬁnitions may at times fail uniquely to pick out a concept or an object, sense still can be made of them by using a rule of revision in the style adopted by Anil Gupta and Nuel Belnap in the theory of truth.
With the aid of a non-standard (but still ﬁrst-order) cardinality quantiﬁer and an extra-logical operator representing numerical abstraction, this paper presents a formalization of ﬁrst-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantiﬁer and the abstraction operator.
In this paper, I explore a new way of understanding Christian ethics by critically interconnecting the theological meanings of the Aqedah ("binding") narrative of Mt. Moriah and the Passion story of Mt. Golgotha. Through an in-depth critical-theological investigation of the relation between these two biblical events, I argue that Christian ethics is possible not so much as a moralization or as a literalistic divine command theory, but rather as a "covenantal-existential" response to God's will in the impossible love on Mt. (...) Moriah as well as in the Son's willing embrace of God's will on Mt. Golgotha. (shrink)
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