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)

The release of Kanye West’s Yeezus was indelibly marked by the provocation of his hit song entitled “New Slaves,” which introduced a pessimistic terminology to capture the paradoxical condition whereby Black freedom from enslavement only resulted in the capturing of Black people psychically in the neo-liberal entanglements of poverty, servitude, and corporatism. His analysis, not unlike currently en vogue theories of Afro-pessimism or Critical Race Theory’s realist lens, maintains that despite all the rhetoric and symbols of progress to the contrary, (...) Black people are simply not free in America. West’s performance of “New Slaves” on Saturday Night Live was only amplified by the “Not For Sale” insignia .. (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)

“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.

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W causes. Among (...) these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (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.

“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)

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.

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.

In approaching Ch. 4 of Saving Truth from Paradox, it might be helpful first to revisit Curry’s original paper, and to revisit Lukasiewicz too, to provide more of the scenesetting that Field doesn’t himself fill in. So in §1 I’ll say something about Curry, in §2 we’ll look at what Lukasiewicz was up to in his original three-valued logic, and in §3 we’ll look at the move from a three-valued to a many-valued Lukasiewicz logic. In §4, I move (...) on to announce a theorem by H´. (shrink)

Although formal analysis provides us with interesting tools for treating Curry’s paradox, it certainly does not exhaust every possible reading of it. Thus, we suggest that this paradox should be analysed with non-formal tools coming from pragmatics. In this way, using Grice’s logic of conversation, we will see that Curry’s sentence can be reinterpreted as a peculiar conditional sentence implying its own consequent.

Ravelingien et al1 argue that we should recategorise people in a permanent vegetative state as dead. Although the dilemma they describe is very real, their solution will not work. Other respondents to this paper have advanced several powerful arguments against the attempt to describe patients in a PVS as dead. Fortunately, the original argument contains sufficient resources for developing an alternative solution to this dilemma without having to radically change the current legal or social status of patients in a PVS. (...) In fact, patients in a PVS may be enrolled in xenotransplantation experiments, provided that their prior informed consent has been sought.The motivation for the original paper was to resolve an apparently intractable ethical conflict. On the one hand, there are powerful ethical and medical reasons for proceeding with research into the transplantation of non-human organs into human patients, whereas on the other, there are equally powerful ethical reasons for blocking whole-organ transplant experiments. These reasons are unusual in that special problems with xenotransplantation will block experiments that are permitted in other cases. In particular, the unquantifiable risk of a pandemic triggered by diseases such as porcine endogenous retrovirus crossing the species barrier will either block the research or expose subjects to the risk of indefinite quarantine . The authors also argue that the restrictions placed on transplant recipients and the possibility of long-term and intrusive monitoring, restrictions on sexual contacts and reproduction, and possible confinement represent excessive and ethically unjustifiable burdens. Over time, the recipients may wish to withdraw their consent, but the risks to public health would prevent them from doing so, thereby violating the basic principle that subjects of scientific and medical experiments should be able to revise or withdraw consent . …. (shrink)

"Joan Robinson: One Woman's Story' is a cinéma vérité style record of a woman's losing struggle against ovarian cancer. The film has been shown now twice on the American Public Television Network. It has received good notices primarily from the lay press. Yet the film depicts much that is out-of-date and much that is debatable. In general, we feel that it presents a depressing picture of the cancer patient. This was not Joan Robinson's intention and her bravery only serves to (...) highlight this picture of suffering with cancer. We point to specific flaws in the film. We then go on to account for why many reviewers seem to have been blind to these flaws. It is suggested that criteria for good works of art, for good public health information, and for admirable personal traits were confused. (shrink)

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. (...) If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (shrink)

Reciprocal altruism involves foregoing an immediate benefit for the sake of a greater long-term reward. It follows that individuals who exhibit a stronger preference for future over immediate rewards should be more disposed to engage in reciprocal altruism – in other words, ‘patient’ people should be more cooperative. The present study tested this prediction by investigating whether participants’ contributions in a public-good game correlated with their ‘discount rate’. The hypothesis was supported: patient people are indeed more cooperative. The paper discusses (...) alternative interpretations of this result, and makes some suggestions for future research. (shrink)

Curry claims that the positive paradox principle, ` A ⊃ in his elementary statement presentation, ‘is valid in any normal interpretation’ . By previous definition, ‘an interpretation of a system S is a normal interpretation just when the proposition A is true when and only when ` A’ . But his argument to normal validity is interestingly, and relevantly, invalid.

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)

I am grateful to Catharine Abell and Jonathan Gilmore for their comments and for the opportunity to think again about some important questions. Before I respond.

Curry’s paradox is well known. The original version employed a conditional connective, and is not forthcoming if the conditional does not satisfy contraction. A newer version uses a validity predicate, instead of a conditional, and is not forthcoming if validity does not satisfy structural contraction. But there is a variation of the paradox which uses “external validity”. And since external validity contracts, one might expect the appropriate version of the Curry paradox to be inescapable. In this (...) paper we show that this is not the case. We consider two ways of formalising the notion of external validity, and show that in both of these the paradox is not forthcoming without the appropriate forms of contraction. (shrink)

This paper draws from critical data studies and related fields to investigate police officer-involved homicide data for Los Angeles County. We frame police officer-involved homicide data as a rhetorical tool that can reify certain assumptions about the world and extend regimes of power. We highlight the possibility that this type of sensitive civic data can be investigated and employed within local communities through creative practice. Community involvement with data can create a countervailing force to powerful dominant narratives and supplement activist (...) projects that hold local officials accountable for their actions. Our analysis examines four Los Angeles County police officer-involved homicide data sets. First, we provide accounts of the semantics, granularity, scale and transparency of this local data. Then, we describe a “counter data action,” an event that invited members of the community to identify the limits and challenges present in police officer-involved homicide data and to propose new methods for deriving meaning from these indicators and statistics. (shrink)

Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other problems which (...) they do not adequately address. (shrink)

The adaptationist framework is necessary and sufficient for unifying the social and natural sciences. Gintis's “beliefs, preferences, and constraints” (BPC) model compares unfavorably to this framework because it lacks criteria for determining special design, incorrectly assumes that standard evolutionary theory predicts individual rationality maximisation, does not adequately recognize the impact of psychological mechanisms on culture, and is mute on the behavioural implications of intragenomic conflict. (Published Online April 27 2007).

This important book provides a theory about the nature of fiction, and about the relation between the author, the reader and the fictional text. The approach is philosophical: that is to say, the author offers an account of key concepts such as fictional truth, fictional characters, and fiction itself. The book argues that the concept of fiction can be explained partly in terms of communicative intentions, partly in terms of a condition which excludes relations of counterfactual dependence between the world (...) and the text. This communicative model is then applied to the following problems: how can something be 'true in the story' without being explicitly stated in the text? In what ways does interpreting a fictional story depend upon grasping its author's intentions? Is there always a unique best interpretation of a fictional text? What is the correct semantics for fictional names? What is the nature of our emotional response to a fictional work? In answering these questions the author explores the complex interaction between author, reader, and text. This interaction requires the reader to construct a 'fictional author' - a character in the story whose personality, beliefs and emotional states must be interpreted if the reader is to grasp the meaning of the work. (shrink)

In this article, I explore how researchers’ metaphysical commitments can be conducive—or unconducive—to progress in animal cognition research. The methodological dictum known as Morgan’s Canon exhorts comparative psychologists to countenance the least mentalistic fair interpretation of animal actions. This exhortation has frequently been misread as a blanket condemnation of mentalistic interpretations of animal behaviors that could be interpreted behavioristically. But Morgan meant to demand only that researchers refrain from accepting default interpretations of (apparent) actions until other fair interpretations have been (...) duly considered. The Canon backfired largely because of Morgan’s background metaphysical commitment to a univocal, hierarchical, and anthropocentric account of cognitive architecture. I make the case that, going forward, comparative psychologists would do well to pair judicious use of Morgan’s Canon with an openness to the existence of non-humanlike animal minds comprising phenomena belonging to distinct cognitive and folk psychological ontologies. And I argue that this case gives us pragmatic reason to reconcile deep—e.g., psychofunctionalist—and superficial—e.g., dispositionalist—approaches to the metaphysics of belief. (shrink)

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.

I have proposed wedding the theories of belief known as dispositionalism and interpretivism. Krzysztof Poslajko objects that dispositionalism does just fine on its own and, moreover, is better off without interpretivism’s metaphysical baggage. I argue that Poslajko is wrong: in order to secure a principled criterion for individuating beliefs, dispositionalism must either collapse into psychofunctionalism (or some other non-superficial theory) or accept interpretivism’s hand in marriage.

Imre Lakatos' philosophical and scientific papers are published here in two volumes. Volume I brings together his very influential but scattered papers on the philosophy of the physical sciences, and includes one important unpublished essay on the effect of Newton's scientific achievement. Volume 2 presents his work on the philosophy of mathematics (much of it unpublished), together with some critical essays on contemporary philosophers of science and some famous polemical writings on political and educational issues.

A trenchant and wide-ranging look at this alarming national trend, _Disrupting the School-to-Prison Pipeline_ is unsparing in its account of the problem while pointing in the direction of meaningful and much-needed reforms. The “school-to-prison pipeline” has received much attention in the education world over the past few years. A fast-growing and disturbing development, it describes a range of circumstances whereby “children are funneled out of public schools and into the juvenile and criminal justice systems.” Scholars, educators, parents, students, and organizers (...) across the country have pointed to this shocking trend, insisting that it be identified and understood—and that it be addressed as an urgent matter by the larger community. This new volume from the _Harvard Educational Review _features essays from scholars, educators, students, and community activists who are working to disrupt, reverse, and redirect the pipeline. Alongside these authors are contributions from the people most affected: youth and adults who have been incarcerated, or whose lives have been shaped by the school-to-prison pipeline. Through stories, essays, and poems, these individuals add to the book’s comprehensive portrait of how our education and justice systems function—and how they fail to serve the interests of many young people.". (shrink)

We argue that narratives are central to the success of historical reconstruction. Narrative explanation involves tracing causal trajectories across time. The construction of narrative, then, often involves postulating relatively speculative causal connections between comparatively well-established events. But speculation is not always idle or harmful: it also aids in overcoming local underdetermination by forming scaffolds from which new evidence becomes relevant. Moreover, as our understanding of the past’s causal milieus become richer, the constraints on narrative plausibility become increasingly strict: a narrative’s (...) admissibility does not turn on mere logical consistency with background data. Finally, narrative explanation and explanation generated by simple, formal models complement one another. Where models often achieve isolation and precision at the cost of simplification and abstraction, narratives can track complex changes in a trajectory over time at the cost of simplicity and precision. In combination both allow us to understand and explain highly complex historical sequences. (shrink)

In this important and impressive book, Gregory Currie tackles several fundamental topics in the philosophy of film and says much of general interest about the nature of imagination. The first part examines the nature of film representation, rejecting the view that spectators are subject to any kind of cognitive or perceptual illusions. Currie also argues against Walton’s transparency claim, which holds that when we look at a photograph we are literally seeing the object photographed. He instead defends perceptual realism, the (...) thesis that photographs resemble the objects they depict, where the resemblance is a matter of photographs triggering the same recognitional response as would the object photographed. He also argues convincingly that films should not be thought of as a language-like form of communication. (shrink)