Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are (...) closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Pure Type Systems, PTSs, introduced as a generalisation of the type systems of Barendregt's lambda-cube, provide a foundation for actual proof assistants, aiming at the mechanic verification of formal proofs. In this paper we consider simplifications of some of the rules of PTSs. This is of independent interest for PTSs as this produces more flexible PTS-like systems, but it will also help, in a later paper, to bridge the gap between PTSs and systems of Illative Combinatory Logic. First we consider (...) a simplification of the start and weakening rules of PTSs, which allows contexts to be sets of statements, and a generalisation of the conversion rule. The resulting Set-modified PTSs or SPTSs, though essentially equivalent to PTSs, are closer to standard logical systems. A simplification of the abstraction rule results in Abstraction-modified PTSs or APTSs. These turn out to be equivalent to standard PTSs if and only if a condition (*) holds. Finally we consider SAPTSs which have both modifications. (shrink)

It is well known that the simple types of closed lambda terms or combinators can be interpreted as the theorems of intuitionistic implicational logic . Venneri, using an equivalence between the intersection type system for lambda calculus, without the universal type ω, TA∧λ, and a similar system for combinators, TA∧, shows that the types of TA∧λ are the theorems of a Hilbert-style sublogic of the → ∧ fragment of H→.In this paper we fill a gap in the equivalence proof and (...) introduce a new system of intersection types for lambda calculus that is weaker than TA∧λ, but has the same set of types. The new system has the advantage that the logic of types can be obtained directly from the rules - just as H→ can be obtained from simple type theory. (shrink)

Because the main difference between combinatory weak equality and λβ-equality is that the rule \begin{equation*}\tag{\xi} X = Y \vdash \lambda x.X = \lambda x.Y\end{equation*} is valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the (...) more common abstraction algorithms, the result will be an equality, = ξ , that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than β-equality. This paper will study the relations = ξ for several commonly used abstraction algorithms, distinguish between them, and axiomatize them. (shrink)

Chakraborty and Banerjee have introduced a rough consequence logic based on the modal logic S5. This paper shows that rough consequence logics, with many of the same properties, can be based on modal logics as weak as K, with a simpler formulation than that of Chakraborty and Banerjee. Also provided are decision procedures for the rough consequence logics and equivalences and independence relations between various systems S and the rough consequence logics, based on them. It also shows that each logic, (...) based on such an S, is theorem equivalent, but not necessarily equivalent, to the modal logic M-S. The paper also shows that rough consequence logic, which was designed to handle rough equality, is somewhat limited for that purpose. (shrink)

Chakraborty and Banerjee have introduced a rough consequence logic based on the modal logic S5. This paper shows that rough consequence logics, with many of the same properties, can be based on modal logics as weak as K, with a simpler formulation than that of Chakraborty and Banerjee. Also provided are decision procedures for the rough consequence logics and equivalences and independence relations between various systems S and the rough consequence logics, based on them. It also shows that each logic, (...) based on such an S, is theorem equivalent, but not necessarily equivalent, to the modal logic M-S. The paper also shows that rough consequence logic, which was designed to handle rough equality, is somewhat limited for that purpose. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations (...) are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of $\lambda$ -terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which (...) derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

Companies increasingly seek to strategically integrate social objectives in commercial activities to address societal challenges, yet little is known about how companies can sustain such a commitment over time. To address this question, we conduct a case-based, abductive study of two pharmaceutical companies widely considered industry leaders in facilitating access to medicine over a 20-year period (2000–2019). We identify product and operation-level integration as distinct types of integration efforts enacted by these companies. Tracing the intraorganizational dynamics associated with these efforts, (...) we theorize that sustained integration is contingent on companies’ ability to respond to and address the challenges specific to product and operation-level integration. The theoretical framework we develop contributes to an emerging debate on the potential of companies to make progress on societal challenges by strategically integrating social objectives, including but not limited to those related to global health. (shrink)

A long-standing theme in discussion of perception and thought has been that our primary cognitive contact with individual objects and events in the world derives from our perceptual contact with them. When I look at a duck in front of me, I am not merely presented with the fact that there is at least one duck in the area, rather I seem to be presented withthisthing (as one might put it from my perspective) in front of me, which looks to (...) me to be a duck. Furthermore, such a perception would seem to put me in a position not merely to make the existential judgment that there is some duck or other present, but rather to make a singular, demonstrative judgment, that that is a duck. My grounds for an existential judgment in this case derives from my apprehension of the demonstrative thought and not vice versa. (shrink)

What exactly is hope and how does it influence our decisions? In How We Hope, Adrienne Martin presents a novel account of hope, the motivational resources it presupposes, and its function in our practical lives. She contends that hoping for an outcome means treating certain feelings, plans, and imaginings as justified, and that hope thereby involves sophisticated reflective and conceptual capacities. Martin develops this original perspective on hope--what she calls the "incorporation analysis"--in contrast to the two dominant philosophical (...) conceptions of hope: the orthodox definition, where hoping for an outcome is simply desiring it while thinking it possible, and agent-centered views, where hoping for an outcome is setting oneself to pursue it. In exploring how hope influences our decisions, she establishes that it is not always a positive motivational force and can render us complacent. She also examines the relationship between hope and faith, both religious and secular, and identifies a previously unnoted form of hope: normative or interpersonal hope. When we place normative hope in people, we relate to them as responsible agents and aspire for them to overcome challenges arising from situation or character. Demonstrating that hope merits rigorous philosophical investigation, both in its own right and in virtue of what it reveals about the nature of human emotion and motivation, How We Hope offers an original, sustained look at a largely neglected topic in philosophy. (shrink)

On Inception is a translation of Martin Heidegger's Gesamtausgabe 70. This work belongs to the crucial period, before and during WWII, when Heidegger was at work on a series of treatises that begins with "Contributions to Philosophy" and includes "The Event" and "The History of Beyng." These works are difficult, even hermetic, but represent a crucial development in Heidegger's thinking. On Inception deepens the investigation underway in the other volumes of the series and provides a unique perspective on Heidegger's (...) thinking of Being and of Event. Here, Heidegger asks, with a greater insistence than anywhere else in his work, what it might mean to think of being as event, and not as presence. Event cannot be thought without the sense of a beginning - an inception - and so, Heidegger insists, we must try to think of being as inception, as fundamentally inceptive. On Inception pursues rigorously the difficult and puzzling implications of this speculation. It does not merely extend work already undertaken but also opens doors onto wholly other pathways. (shrink)

Is it OK to lie to Siri? Is it bad to mistreat a robot for our own pleasure? Under what condition should we grant a moral status to an artificial intelligence (AI) system? This paper looks at different arguments for granting moral status to an AI system: the idea of indirect duties, the relational argument, the argument from intelligence, the arguments from life and information, and the argument from sentience. In each but the last case, we find unresolved issues with (...) the particular argument, which leads us to move to a different one. We leave the idea of indirect duties aside since these duties do not imply considering an AI system for its own sake. The paper rejects the relational argument and the argument from intelligence. The argument from life may lead us to grant a moral status to an AI system, but only in a weak sense. Sentience, by contrast, is a strong argument for the moral status of an AI system—based, among other things, on the Aristotelian principle of equality: that same cases should be treated in the same way. The paper points out, however, that no AI system is sentient given the current level of technological development. (shrink)

Over the past three decades, the philosophy of biology has emerged from the shadow of the philosophy of physics to become a respectable and thriving philosophical subdiscipline. The authors take a fresh look at the life sciences and the philosophy of biology from a strictly realist and emergentist-naturalist perspective. They outline a unified and science-oriented philosophical framework that enables the clarification of many foundational and philosophical issues in biology. This book will be of interest both to life scientists and philosophers.

Eric Martin and Daniel N. Osherson present a theory of inductive logic built on model theory. Their aim is to extend the mathematics of Formal Learning Theory to a more general setting and to provide a more accurate image of empirical inquiry. The formal results of their study illuminate aspects of scientific inquiry that are not covered by the commonly applied Bayesian approach.

Martin Heidegger's Contributions to Philosophy reflects his famous philosophical "turning." In this work, Heidegger returns to the question of being from its inception in Being and Time to a new questioning of being as event.

A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and (...) axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete. (shrink)

A ‘lottery belief’ is a belief that a particular ticket has lost a large, fair lottery, based on nothing more than the odds against it winning. The lottery paradox brings out a tension between the idea that lottery beliefs are justified and the idea that that one can always justifiably believe the deductive consequences of things that one justifiably believes – what is sometimes called the principle of closure. Many philosophers have treated the lottery paradox as an argument against the (...) second idea – but I make a case here that it is the first idea that should be given up. As I shall show, there are a number of independent arguments for denying that lottery beliefs are justified. (shrink)

A landmark discussion between two great thinkers, vital to an understanding of twentieth-century philosophy and intellectual history.

Martin Heidegger is one of the most important and influential philosophers of the 20th Century. A major figure in the development of phenomenology, his work also profoundly influenced many of the intellectual movements that followed in his wake, from Sartre's Existentialism to Derrida's deconstructionism. Towards the Definition of Philosophy brings together two seminal lectures that mark a breakthrough moment in Heidegger's thought and introduces the major themes that he would develop in his opus Being and Time.

This book provides the first systematic presentation of anthropology's 'ontological turn', placing it in the landscape of contemporary social theory.

Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds (...) of PTSs. Most importantly PTSs are about statements of the form M:A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. (shrink)

This paper proposes a view of time that takes passage to be the most basic temporal notion, instead of the usual A-theoretic and B-theoretic notions, and explores how we should think of a world that exhibits such a genuine temporal passage. It will be argued that an objective passage of time can only be made sense of from an atemporal point of view and only when it is able to constitute a genuine change of objects across time. This requires that (...) passage can flip one fact into a contrary fact, even though neither side of the temporal passage is privileged over the other. We can make sense of this if the world is inherently perspectival. Such an inherently perspectival world is characterized by fragmentalism, a view that has been introduced by Fine in his ‘Tense and Reality’ (2005). Unlike Fine's tense-theoretic fragmentalism though, the proposed view will be a fragmentalist view based in a primitive notion of passage. (shrink)