Disturbance-generated biomass results from endogenous and exogenous natural and cultural disturbances that affect the health and productivity of forest ecosystems. These disturbances can create large quantities of plant biomass on predictable cycles. A systems analysis model has been developed to quantify aspects of system capacities, spatial aspects of the biomass generation process, and deterioration impacts on biomass quality in the various inventory states. Optimal decision alternatives can be used to guide responses to reclamation, utilization, mitigation, and control. This is particularly (...) advantageous in insect and disease outbreaks, in which the process may last several years, with varying levels of intensity. The prescriptive system description, assuming capacities are fixed, results in a linear programming model. The time-dependent capacity decision model results in a mixed-integer programming model. The analytical model is developed in detail in this analysis. (shrink)

Why and under what conditions are individuals altruistic to family and friends in their social networks? Evolutionary psychology suggests that such behaviour is primarily the product of adaptations for kin- and reciprocal altruism, dependent on the degree of genetic relatedness and exchange of benefits, respectively. For this reason, individuals are expected to be more altruistic to family members than to friends: whereas family members can be the recipients of kin and reciprocal altruism, friends can be the recipients of reciprocal altruism (...) only. However, there is a question about how the effect of kinship is implemented at the proximate psychological level. One possibility is that kinship contributes to some general measure of relationship quality (such as ‘emotional closeness’), which in turn explains altruism. Another possibility is that the effect of kinship is independent of relationship quality. The present study tests between these two possibilities. Participants (N= 111) completed a self-report questionnaire about their willingness to be altruistic, and their emotional closeness, to 12 family members and friends at different positions in their extended social networks. As expected, altruism was greater for family than friends, and greater for more central layers of the network. Crucially, the results showed that kinship made a significant unique contribution to altruism, even when controlling for the effects of emotional closeness. Thus, participants were more altruistic towards kin than would be expected if altruism was dependent on emotional closeness alone – a phenomenon we label a ‘kinship premium’. These results have implications for the ongoing debate about the extent to which kin relations and friendships are distinct kinds of social relationships, and how to measure the ‘strength of ties’ in social networks. (shrink)

In this paper we 1. provide a natural deduction system for full first-order linear logic, 2. introduce Curry-Howard-style terms for this version of linear logic, 3. extend the notion of substitution of Curry-Howard terms for term variables, 4. define the reduction rules for the Curry-Howard terms and 5. outline a proof of the strong normalization for the full system of linear logic using a development of Girard's candidates for reducibility, thereby providing an alternative to Girard's proof using (...) proof-nets. (shrink)

“Curry’s paradox”, as the term is used by philosophers today, refers to a wide variety of paradoxes of self-reference or circularity that trace their modern ancestry to Curry (1942b) and Löb (1955). The common characteristic of these so-called Curry paradoxes is the way they exploit a notion of implication, entailment or consequence, either in the form of a connective or in the form of a predicate. Curry’s paradox arises in a number of different domains. Like Russell’s (...) paradox, it can take the form of a paradox of set theory or the theory of properties. But it can also take the form of a semantic paradox, closely akin to the Liar paradox. Curry’s paradox differs from both Russell’s paradox and the Liar paradox in that it doesn’t essentially involve the notion of negation. Common truth-theoretic versions involve a sentence that says of itself that if it is true then an arbitrarily chosen claim is true, or—to use a more sinister instance—says of itself that if it is true then every falsity is true. The paradox is that the existence of such a sentence appears to imply the truth of the arbitrarily chosen claim, or—in the more sinister instance—of every falsity. In this entry, we show how the various Curry paradoxes can be constructed, examine the space of available solutions, and explain some ways Curry’s paradox is significant and poses distinctive challenges. (shrink)

Reminiscences of Peter, by P. Oppenheim.--Natural kinds, by W. V. Quine.--Inductive independence and the paradoxes of confirmation, by J. Hintikka.--Partial entailment as a basis for inductive logic, by W. C. Salmon.--Are there non-deductive logics?, by W. Sellars.--Statistical explanation vs. statistical inference, by R. C. Jeffre--Newcomb's problem and two principles of choice, by R. Nozick.--The meaning of time, by A. Grünbaum.--Lawfulness as mind-dependent, by N. Rescher.--Events and their descriptions: some considerations, by J. Kim.--The individuation of events, by D. Davidson.--On properties, by (...) H. Putnam.--A method for avoiding the Curry paradox, by F. B. Fitch.--Publications (1934-1969) by Carl G. Hempel (p. [266]-270). (shrink)

Metaphysics and language: Quine, W. V. O. On the individuation of attributes. Körner, S. On some relations between logic and metaphysics. Marcus, R. B. Does the principle of substitutivity rest on a mistake? Van Fraassen, B. C. Platonism's pyrrhic victory. Martin, R. M. On some prepositional relations. Kearns, J. T. Sentences and propositions.--Basic and combinatorial logic: Orgass, R. J. Extended basic logic and ordinal numbers. Curry, H. B. Representation of Markov algorithms by combinators.--Implication and consistency: Anderson, A. R. Fitch (...) on consistency. Belnap, N. D., Jr. Grammatical propaedeutic. Thomason, R. H. Decidability in the logic of conditionals. Myhill, J. Levels of implication.--Deontic, epistemic, and erotetic logic: Bacon, J. Belief as relative knowledge. Wu, K. J. Believing and disbelieving. Kordig, C. R. Relativized deontic modalities. Harrah, D. A system for erotetic sentences. (shrink)

This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this (...) constructive method to non-classical fragments equipped with weaker implications (that avoid Curry's objection). This is the case, for example, of Jan Lukasiewicz's n-valued logics and Wilhelm Ackermann's logic of restricted implication. Besides such problems, both Henkin's method and the triviality phenomenon enable us to propose a new positive tableau proof system which uses only positive meta-linguistic resources, and to mo- tivate a new discussion concerning the role of negation in logic proposing the concept of paratriviality. In this way, some relations between positive reasoning and infinity, the possibilities to obtain a ¯first-order positive logic as well as the philosophical connection between truth and meaning are dis- cussed from a conceptual point of view. (shrink)

Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is that paradox (...) arises already at the first stage. Accordingly, blocking the second or third stages won't suffice for resolving the paradox. (shrink)

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, (...) but not jointly, lack the problematic feature. (shrink)

Bullot and Ludwig are advisory editors. Contributors inclue N. Carroll, G. Curry & J. Jureidini, M. DeBellis, M. Rollins, D. McIver Lopes, R. Hopkins, and J. Hyman.

Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds (...) to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning. (shrink)

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, (...) the substructure—of the validity relation itself. (shrink)

CHAPTER Addenda to Pure Combinatory Logic This chapter will treat various additions to, and modifications of, the subject matter of Chapters-7. ...

Black males have been characterized as violent, misogynist, predatory rapists by gender theorists dating back to mid-nineteenth–century ethnologists to contemporary intersectional feminists. These caricatures of Black men and boys are not rooted in any actual studies or empirical findings, but the stereotypes found throughout various racist social scientific literatures that held Black males to be effeminate while nonetheless hyper-masculine and delinquent. This paper argues that contemporary gender theories not only deny the peculiar sexual oppression of racialized outgroup males under patriarchy, (...) but theories like intersectional invisibility actually perpetuates the idea that racialized males are disposable. To remedy the imperceptibility of sexual oppression and violence under the male category, the author gives an historical account of the development of racist misandry throughout the centuries and proposes a theory of phallicism to describe the seemingly contradictory constructions of Black men as sexually predatory as in the case of the rapist, but nonetheless sexually vulnerable and raped under patriarchy. (shrink)

“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.

We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.

Many psychologists studying lay belief attribution and behavior explanation cite Donald Davidson in support of their assumption that people construe beliefs as inner causes. But Davidson’s influential argument is unsound; there are no objective grounds for the intuition that the folk construe beliefs as inner causes that produce behavior. Indeed, recent experimental work by Ian Apperly, Bertram Malle, Henry Wellman, and Tania Lombrozo provides an empirical framework that accords well with Gilbert Ryle’s alternative thesis that the folk construe beliefs as (...) patterns of living that contextualize behavior. (shrink)

This article reconsiders the relationship between interpretivism about belief and normative standards. Interpretivists have traditionally taken beliefs to be fixed in relation to norms of interpretation. However, recent work by philosophers and psychologists reveals that human belief attribution practices are governed by a rich diversity of normative standards. Interpretivists thus face a dilemma: either give up on the idea that belief is constitutively normative or countenance a context-sensitive disjunction of norms that constitute belief. Either way, interpretivists should embrace the intersubjective (...) indeterminacy of belief. (shrink)

A pluralistic approach to folk psychology must countenance the evaluative, regulatory, predictive, and explanatory roles played by attributions of intelligence in social practices across cultures. Building off of the work of the psychologist Robert Sternberg and the philosophers Gilbert Ryle and Daniel Dennett, I argue that a relativistic interpretivism best accounts for the many varieties of intelligence that emerge from folk discourse. To be intelligent (in the sense invoked in folk psychological practices) is to be comparatively good at solving intellectual (...) problems that an interpreter deems worth solving. (shrink)

This is the collection of essays presented to Bochenski on his 60th birthday, and it contains, as a mirror of Bochenski's own work, a broad spectrum of studies ranging from formal logic and history of logic, to the philosophy of logic and language, and to the methodology of explanation in Greek philosophy. Of the seventeen articles, these are some of the more important to the reviewer: "Betrachtungen zum Sequenzen Kalkül" by Paul Bernays, which is an extensive study of Gentzen-type formulations (...) of logic; "Remarks on Formal Deduction," H. B. Curry, a further discussion of sequenzen-logics; "Marginalia on Gentzen's Sequenzen Kalkül" by Hughes Leblanc; "Method and Logic in Presocratic Explanation," Jerry Stannard; "On the Logic of Preference and Choice," H. S. Houthakker, a suggestive presentation of decision and utility theory in logical form; "Leibniz's Law in Belief Contexts," Chisholm; "On Ontology and the Province of Logic," R. M. Martin; and "N. A. Vasilev and the Development of Many-valued Logics," G. L. Kline, an important addition to the history of logic. Other contributors are: Storrs McCall, Albert Menne, E. W. Beth, Benson Mates, Ivo Thomas, J. F. Staal, F. R. Barbò, A.-T. Tymieniecka, and N. M. Luyten. There is a bibliography of Bochenski's writings through 1962.—P. J. M. (shrink)

Curry's paradox for "if.. then.." concerns the paradoxical features of sentences of the form "If this very sentence is true, then 2+2=5". Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2+2=5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. -/- Once we (...) have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of "if... then...". This paper argues that the closest-worlds approach outlined in Nolan 1997 offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related "contraction conditionals". (shrink)

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

Imagine: A 15-year-old girl has sex with a 20-year-old man. It is her first sexual experience. Her first time having intercourse. She remembers that “he basically took it from me,” but feels an affection for the person and the event. She was not at the age of consent, but describes the experience as “just pleasure.” Was this rape or simply a man ushering a young girl into womanhood? Now imagine her as a 15-year-old boy and him to be a 20-year-old (...) woman. She “took it from him,” but he recalls it was pleasurable. Was he forced into sexual relations with an adult before the age of consent? Was he raped? Can he actually be raped?It is often difficult to conceptualize male bodies as being victims of sexual violence, and... (shrink)

Curry's paradox, so named for its discoverer, namely Haskell B. Curry, is a paradox within the family of so-called paradoxes of self-reference (or paradoxes of circularity). Like the liar paradox (e.g., ‘this sentence is false’) and Russell's paradox , Curry's paradox challenges familiar naive theories, including naive truth theory (unrestricted T-schema) and naive set theory (unrestricted axiom of abstraction), respectively. If one accepts naive truth theory (or naive set theory), then Curry's paradox becomes a direct challenge (...) to one's theory of logical implication or entailment. Unlike the liar and Russell paradoxes Curry's paradox is negation-free; it may be generated irrespective of one's theory of negation. An intuitive version of the paradox runs as follows. (shrink)

This book is a major new introduction to the field of ecological ethics. Taking issue with the common assumption that existing human ethics can be 'extended' to meet the demands of the ongoing ecological crisis, Patrick Curry shows that a new and truly ecological ethic is both possible and urgently needed. With this distinctive proposition in mind, Curry introduces and discusses all the major concepts needed to understand the full range of ecological ethics. Focussing first on the major (...) concepts of ethics - religious and secular - and value, Curry then examines the gradations of ecological ethics. He discusses light green, shallow or anthropomorphic ethics with the examples of stewardship, lifeboat ethics, and social ecology; mid-green or intermediate ethics represented by animal liberation/rights and biocentrism; and dark green, deep, or ecocentric ethics. Particular attention is given to the various kinds of ecocentric ethics, such as the Land Ethic, The Gaia Hypothesis, and Deep Ecology and its offshoots: Deep Green Theory, Left Biocentrism and the Earth Manifesto. Ecofeminism is also considered in this context. The concluding chapters discuss green ethics as post-secular, moral pluralism and pragmatism, green citizenship, and human population in the light of ecological ethics. This comprehensive and wide-ranging textbook offers a radical but critical introduction to the subject. It will be of great interest to students, activists, and to a wider public concerned with the ecocrisis, its main theories, debates, and possible solutions. (shrink)