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)

“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.

“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)

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 (...) class='Hi'>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)

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)

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)

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 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)

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.

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)

"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)

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)

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.

The idea that fictions may educate us is an old one, as is the view that they distort the truth and mislead us. While there is a long tradition of passionate assertion in this debate, systematic arguments are a recent development, and the idea of empirically testing is particularly novel. Our aim in this chapter is to provide clarity about what is at stake in this debate, what the options are, and how empirical work does or might bear on its (...) resolution. We distinguish between merely influencing people’s opinions and providing genuine learning, where the latter requires that the source of the change in opinion should be reliable. Other important ideas here are the extent to which authors of fiction may be considered to provide testimony, or something like it, and the possibility of recognizing an unstated purpose in the project of the fictional work. We ask whether fictions can furnish us not merely with ideas but with reasons for believing them, perhaps by constituting thought experiments. We consider whether the focus should be on understanding rather than on knowledge, and whether fictions can inform us about the qualities of another’s experiences. We briefly describe some experimental work of recent decades and suggest that the evidence for learning from fiction is currently meagre. (shrink)

An argument that we should be optimistic about the capacity of “methodologically omnivorous” geologists, paleontologists, and archaeologists to uncover truths about the deep past. -/- The “historical sciences”—geology, paleontology, and archaeology—have made extraordinary progress in advancing our understanding of the deep past. How has this been possible, given that the evidence they have to work with offers mere traces of the past? In Rock, Bone, and Ruin, Adrian Currie explains that these scientists are “methodological omnivores,” with a variety of strategies (...) and techniques at their disposal, and that this gives us every reason to be optimistic about their capacity to uncover truths about prehistory. Creative and opportunistic paleontologists, for example, discovered and described a new species of prehistoric duck-billed platypus from a single fossilized tooth. Examining the complex reasoning processes of historical science, Currie also considers philosophical and scientific reflection on the relationship between past and present, the nature of evidence, contingency, and scientific progress. -/- Currie draws on varied examples from across the historical sciences, from Mayan ritual sacrifice to giant Mesozoic fleas to Mars's mysterious watery past, to develop an account of the nature of, and resources available to, historical science. He presents two major case studies: the emerging explanation of sauropod size, and the “snowball earth” hypothesis that accounts for signs of glaciation in Neoproterozoic tropics. He develops the Ripple Model of Evidence to analyze “unlucky circumstances” in scientific investigation; examines and refutes arguments for pessimism about the capacity of the historical sciences, defending the role of analogy and arguing that simulations have an experiment-like function. Currie argues for a creative, open-ended approach, “empirically grounded” speculation. (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)

Do not ask for the definition of deconstruction; ask for its history. What needs and desires did it meet at the time of its emergence? What kind of threat did it represent? How has our understanding of deconstruction changed over time? This book offers an account of the invention and reinvention of deconstruction in literary studies and the humanities more generally. Focusing on the work of Jacques Derrida and Paul de Man, it argues that the early impact of deconstruction was (...) connected to its perceived assault upon truth. After de Man's death there is a steady insistence in Derrida's work on questions about time - invention, advent, event - and on the distance between them. This book tells the story of this transition from truth to time against a background of some of the most divisive debates of the late-twentieth and early twenty-first century, about politics, history and ethics. (shrink)

Thought experiments play a role in science and in some central parts of contemporary philosophy. They used to play a larger role in philosophy of science, but have been largely abandoned as part of the field’s “practice turn”. This chapter discusses possible roles for thought experimentation within a practice-oriented philosophy of science. Some of these roles are uncontroversial, such as exemplification and aiding discovery. A more controversial role is the reliance on thought experiments to justify philosophical claims. It is proposed (...) that if we adopt an underlying empiricist view of concepts, then thought experiments can be seen as affording us contact with scientific practice, despite their seemingly a priori character. The advantages and drawbacks of thought experiments are discussed via comparison with case studies, on the one, and simulations on the other hand. The chapter closes with some remarks on how to combine thought experiments with other methods. (shrink)

Thought experiments play a role in science and in some central parts of contemporary philosophy. They used to play a larger role in philosophy of science, but have been largely abandoned as part of the field’s “practice turn”. This chapter discusses possible roles for thought experimentation within a practice-oriented philosophy of science. Some of these roles are uncontroversial, such as exemplification and aiding discovery. A more controversial role is the reliance on thought experiments to justify philosophical claims. It is proposed (...) that if we adopt an underlying empiricist view of concepts, then thought experiments can be seen as affording us contact with scientific practice, despite their seemingly a priori character. The advantages and drawbacks of thought experiments are discussed via comparison with case studies, on the one, and simulations on the other hand. The chapter closes with some remarks on how to combine thought experiments with other methods. (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)

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).

‘Gouldian arguments’ appeal to the contingency of a scientific domain to establish that domain’s autonomy from some body of theory. For instance, pointing to evolutionary contingency, Stephen Jay Gould suggested that natural selection alone is insufficient to explain life on the macroevolutionary scale. In analysing contingency, philosophers have provided source-independent accounts, understanding how events and processes structure history without attending to the nature of those events and processes. But Gouldian Arguments require source-dependent notions of contingency. An account of contingency is (...) source-dependent when it is indexed to some pattern and some process. Positions like Gould’s do not turn on the mere fact of life’s contingency—that life’s shape could have been different due to its sensitivity to initial conditions, path-dependence or stochasticity. Rather, Gouldian arguments require that the contingency is due to particular kinds of processes: in this case, those which microevolutionary theory cannot account for. This source-dependent perspective clarifies both debates about the nature and importance of contingency, and empirical routes for testing Gould’s thesis. (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 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)

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.