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 inion-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)

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)

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)

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. 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)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants 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)

BB′IW logic (or T→ is known to be D-complete. This paper shows that there are infinitely many weaker D-complete logics and it also examines how certain D-incomplete logics can be made complete by altering their axioms using simple substitutions.

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 bracket abstraction algorithm is a means of translating $\lambda$-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each $\lambda$-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schonfinkel [4] algorithm plus two mappings which convert the Schonfinkel abstract into the new abstract. This paper shows how the new (...) abstraction can be done more directly, in fact, using only 2n - 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered. (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)

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)

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.

We provide a battery of examples of delusions against which theoretical accounts can be tested. Then, we identify neuropsychological anomalies that could produce the unusual experiences that may lead, in turn, to the delusions in our battery. However, we argue against Maher’s view that delusions are false beliefs that arise as normal responses to anomalous experiences. We propose, instead, that a second factor is required to account for the transition from unusual experience to delusional belief. The second factor in the (...) aetiology of delusions can be described superficially as a loss of the ability to reject a candidate for belief on the grounds of its implausibility and its inconsistency with everything else that the patient knows. But we point out some problems that confront any attempt to say more about the nature of this second factor. (shrink)

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)

In this paper a tentative neurophysiologically framed approach of the Freudian unconscious that would function on the basis of linguistic (phonological) organizing principles, is proposed. A series of arguments, coming from different fields, are taken together. First, clinical reports indicate that in a state of high emotional arousal linguistic fragments are treated in a decontextualized way, and can lead to the isolation of phoneme sequences which, independently of their actual meaning, are able to resort emotional effects. Second, phonological and neurophysiological (...) arguments are given to make the case that language processing – be it producing, receiving or imagining language – is a motor event. A crucial distinction is proposed: while contextualized processing correlates on a neurophysiological level with action understanding, and is psychoanalytically akin to secondary processing, decontextualized language processing has a neurophysiological counterpart in object understanding and is psychoanalytically akin to primary processing. Third, isolated speech fragments are therefore considered as objects which, similarly to non-linguistic objects, undergo emotional conditioning and establish an idiosyncratic linguistically structured emotional memory. Phoneme sequences which in this way come to carry high emotional valences are thought to be more readily subject to threaten the bodily integrity, and therefore more readily inhibited. When this inhibition leads to the prevention of effective realization of the voluntary motor output, this is thought to result in the sustained high levels of neuronal activation which seek for release by attracting substitutes that are phonemically similar to the censored speech fragments though they are cognitively non threatening. As a result, the speech of the subject would be particularly concerned with the verbalizations of these substitutive phoneme sequences. In summary, the Freudian unconscious is conceived as the instantiation of a linguistic action space which would be idiosyncratically organized by particular phonemic “phantoms” operating as attractors for the subject’s (linguistic) actions. (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)

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)