Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic (...) and non-symbolic approximate division. Subjects were presented with non-symbolic or symbolic dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children and adults were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction. (shrink)

Responding to serious environmental problems, requires urgent and fundamental shifts in our day-to-day lifestyles. This paper employs a qualitative, cross-cultural approach to explore people’s subjective self-reflections on their experiences of pro-environmental behavioral spillover in three countries; Brazil, China, and Denmark. Behavioral spillover is an appealing yet elusive phenomenon, but offers a potential way of encouraging wider, voluntary lifestyle shifts beyond the scope of single behavior change interventions. Behavioral spillover theory proposes that engaging in one pro-environmental action can catalyze the performance (...) of others. To date, evidence for the phenomenon has been mixed, and the causal processes governing relationships between behaviors appear complex, inconsistent and only partly understood. This paper addresses a gap in the literature by investigating accounts of behavioral spillover in three diverse cultural settings using qualitative semi-structured interviews. The analysis shows that while around half of participants overall who were questioned recalled spillover effects, the other half had not consciously experienced spillover. There were few significant differences across cultures, though some forms of spillover effects were reported more in some cultures than others. More environmentally-engaged participants across all three countries were significantly more likely to experience spillover than those who were less engaged. Accounts of within-domain spillovers were most commonly reported, mainly comprising waste, resource conservation and consumption-related actions. Accounts of between-domain spillover were very rare. Recollection of contextual and interpersonal spillover effects also emerged from the interviews. Our findings suggest that more conscious behavioral spillover pathways may be limited to those with pre-existing environmental values. Behavioral spillover may comprise multiple pathways incorporating conscious and unconscious processes. We conclude that targeting behavioral catalysts that generate more socially diffuse spillover effects could offer more potential than conventional spillover involving a single individual. (shrink)

Public acceptance is often seen as a key reason why water-recycling technology is rejected. A common assumption is that projects fail because the general public is unable to comprehend specialist information about risk and the belief that if the public were better informed, they would accept change more readily. This article suggests that rhetoric about acceptance is counterproductive in progressing sustainability as it does not address issues relating to institutional arrangements and reinforces a dichotomy between expert and lay groups. Instead, (...) it is argued that institutional change is needed to build opportunities for constructive public engagement. The failure to implement sustainable water use through recycling can be understood as the result of several factors including present cost structures for water, institutional conservatism, administrative fragmentation, and inadequate involvement of communities in planning. Achieving sustainable water use through recycling may require better coordination between agencies and integrated government policies. (shrink)

Trait emotional intelligence concerns people’s perceptions of their emotional functioning. Two studies investigated this construct in surgeons and comparison occupations. We hypothesized that trait EI profiles would differ both within surgical specialties as well as between them and other professions. Study 1 compared the trait EI profiles of four different surgical specialties. There were no significant differences amongst these specialties or between consultant surgeons and trainees in these specialties. Accordingly, the surgical data were combined into a single target sample that (...) was compared against samples of engineers, executives and senior managers, lawyers, junior military managers, nurses, and salespeople. Surgeons scored significantly higher on global trait EI than junior military managers, but lower than executives and senior managers, salespeople, and nurses. There were no significant differences vis-à-vis engineers or lawyers. A MANOVA confirmed a similar pattern of differences in the four trait EI factors. Global trait EI scores correlated strongly with single-question measures of job satisfaction and job performance in the surgical sample. These findings suggest that interventions to optimize the trait EI profiles of surgeons can be helpful in relation to job satisfaction, job performance, and overall psychological wellbeing. (shrink)

Noise in behavior is often viewed as a nuisance: while the mind aims to take the best possible action, it is let down by unreliability in the sensory and response systems. How researchers study cognition reflects this viewpoint – averaging over trials and participants to discover the deterministic relationships between experimental manipulations and their behavioral consequences, with noise represented as additive, often Gaussian, and independent. Yet a careful look at behavioral noise reveals rich structure that defies easy explanation. First, both (...) perceptual and preferential judgments show that sensory and response noise may potentially only play minor roles, with most noise arising in the cognitive computations. Second, the functional form of the noise is both non-Gaussian and non-independent, with the distribution of noise being better characterized as heavy-tailed and as having substantial long-range autocorrelations. It is possible that this structure results from brains that are, for some reason, bedeviled by a fundamental design flaw, albeit one with intriguingly distinctive characteristics. Alternatively, noise might not be a bug but a feature: indeed, we suggest that noise is fundamental to how cognition works. Specifically, we propose that the brain approximates probabilistic inference with a local sampling algorithm, one that uses randomness to drive its exploration of alternative hypotheses. Reframing cognition in this way explains the rich structure of noise and leads to a surprising conclusion: that noise is not a symptom of cognitive malfunction but plays a central role in underpinning human intelligence. (shrink)

We invite systematic consideration of the metaphors of cycles and circulation as a long-term theme in the history of the life and environmental sciences and medicine. Ubiquitous in ancient religious and philosophical traditions, especially in representing the seasons and the motions of celestial bodies, circles once symbolized perfection. Over the centuries cyclic images in western medicine, natural philosophy, natural history and eventually biology gained independence from cosmology and theology and came to depend less on strictly circular forms. As potent ‘canonical (...) icons’, cycles also interacted with representations of linear and irreversible change, including arrows, arcs, scales, series and trees, as in theories of the Earth and of evolution. In modern times life cycles and reproductive cycles have often been held to characterize life, in some cases especially female life, while human efforts selectively to foster and disrupt these cycles have harnessed their productivity in medicine and agriculture. But strong cyclic metaphors have continued to link physiology and climatology, medicine and economics, and biology and manufacturing, notably through the relations between land, food and population. From the grand nineteenth-century transformations of matter to systems ecology, the circulation of molecules through organic and inorganic compartments has posed the problem of maintaining identity in the face of flux and highlights the seductive ability of cyclic schemes to imply closure where no original state was in fact restored. More concerted attention to cycles and circulation will enrich analyses of the power of metaphors to naturalize understandings of life and their shaping by practical interests and political imaginations. (shrink)

onald Dworkin says he does not believe in the metaphysics of morality. He is a 'quietist' about this issue. He thinks that there are no coherent 'external' or 'archimedian' questions that we can raise about the whole discipline of moral thought and talk, and that the only questions we can raise are 'internal' ones about what moral thoughts we should think. Dworkin thinks that some metaphysical debates can go ahead, it is just the metaphysics of morality that is ill-gotten. This (...) is because those other areas of thought involve causal claims about the origins of our thought, as part of their content; but moral thought does not (p. 119). So Dworkin thinks that religious and astrological thought can be assessed in terms of the causal claims they make, but moral thought is not the kind of thought which can be assessed in that way. But is moral thought really answerable to nothing except itself? How convenient for it! Furthermore, does this unanswerability generalize to any form of thought which does not involve causal claims? For example, is the attempt to give mathematics a foundation misguided and worthy of the derision that Dworkin heaps on meta-ethics? Are morality and mathematics as it were free-floating modes of thought held up by nothing? Even if Dworkin is right that some forms of thought are not to be measured in causal terms, the idea that they are answerable to nothing is bizarre. (shrink)

1 Evidence-based policy requires researchers to provide the answers to ecological questions that are of interest to policy makers. To find out what those questions are in the UK, representatives from 28 organizations involved in policy, together with scientists from 10 academic institutions, were asked to generate a list of questions from their organizations. 2 During a 2-day workshop the initial list of 1003 questions generated from consulting at least 654 policy makers and academics was used as a basis for (...) generating a short list of 100 questions of significant policy relevance. Short-listing was decided on the basis of the preferences of the representatives from the policy-led organizations. 3 The areas covered included most major issues of environmental concern in the UK, including agriculture, marine fisheries, climate change, ecosystem function and land management. 4 The most striking outcome was the preference for general questions rather than narrow ones. The reason is that policy is driven by broad issues rather than specific ones. In contrast, scientists are frequently best equipped to answer specific questions. This means that it may be necessary to extract the underpinning specific question before researchers can proceed. 5 Synthesis and applications. Greater communication between policy makers and scientists is required in order to ensure that applied ecologists are dealing with issues in a way that can feed into policy. It is particularly important that applied ecologists emphasize the generic value of their work wherever possible. (shrink)

Williamson (2000) appeals to considerations about when it is natural to say that a hypothesis is consistent with one’s evidence in order to motivate the claim that all and only knowledge is evidence. It is argued here that the relevant considerations do not support this claim, and in fact conflict with it.

In this article, we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modelling for classical propositional and modal logics. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms (...) under comparison both semantically and syntactically at the level of derivations. Moreover, our analysis clarifies the interplay between the complexity of translations and axiomatizations of the corresponding logics: adding axioms to the target bimodal logic simplifies translations, or vice versa, complex translations can simplify frame conditions. We also investigate a transfer of first-order correspondence theory between possibility semantics and its bimodal counterpart. Finally, we discuss the conceptual trade-off between giving translations and giving new semantics for logical systems, and we identify a number of further research directions to which our analysis gives rise. (shrink)

This note examines the case of a group of gay men who, having engaged in consensual sexual acts together, became known as the `Bolton Seven' following their conviction in 1998 for offences of buggery and/or gross indecency. More particularly the note scrutinises the implications of the ages of the participants (one of whom, at 17 $\tfrac{1}{2}$ , was unable to give lawful consent to sexual intercourse with a man) in the light of the enactment of Part I of the Sex (...) Offenders Act 1997 which introduces a system of compulsory registration by some convicted and cautioned sex offenders with the police (including men convicted of, or cautioned for, buggery or gross indecency). The note explores the justification for inclusion of these offences within the remit of the1997 Act together with the cultural construction of gay men as predatory and as constituting a risk to younger members of society. It also analyses some of the effects of the registration requirement in terms of it constituting a potential violation of fundamental rights such as equality and respect for private life. This discussion is located particularly within the context of the jurisprudence of the European Court of Human Rights and the introduction of the Human Rights Act 1998 in the U.K. (shrink)

This commentary on Professor Kasher's and General Yadlin's article employs a bit of violence. It transforms and broadens some of the ideas presented in their article. I argue that committing these acts of violence are justified because, if their article is left as written, it is difficult to tell at what point the Kasher/Yadlin (K/Y) theory corresponds with just war theory and at what points it does not. This commentary alters K/Y theory, and alters classical just war theory as well, (...) so as to place the two theories in parallel as much as possible. Once the alterations are complete, it becomes more evident that the two theories share much in common. What also becomes more evident is that the two theories differ in important respects since the one deals with conventional war and the other with unconventional (asymmetric) war. The differences vindicate Kasher's and Yadlin's claim that it is necessary to spend time to re-think just war theory, especially when dealing with terrorism. (shrink)

I had my accident on the rugby field on July 29, 2000 about 2.00 p.m. during a simple line - out, even before the ball was thrown in. I t just felt like another simple case of concussion , I staggered to the sideline, the coach asked me “what ’s wrong”? He said I told him I just felt sick and to put me back on the field in 10 minutes. Then I collapsed, eventually blacked out and then was rushed (...) to hospital unconscious in an ambulance with them struggling drastically to keep me alive. Af ter three days of being in there, they thought I was alright and were going to send me home. Then it star ted happening. First I nearly collapsed again tak ing a shower . For days the spe - cialists didn’t k now what was wrong with me. M y girlfriend at the time, who had rushed down from Wanak a when she had heard it had happened went mad at the specialists to do something. Af ter six days of going in and out of seizures, finally af ter what seemed like all the tests k nown to man, they said I had suffered several brain stem strokes then one massive major one, which altogether had lef t me diagnosed with the ex tremely rare and unk nown condition only k nown to a few as “ locked in syndrome”. (shrink)

In this essay we attempt to accomplish two things related to the work of G.K. Chesterton. The first is to use one of his favorite ploys to articulate the nature of play. We discuss several paradoxical characteristics of play and attempt to show how seemingly contradictory features actually help us to understand play’s allure and other values. We introduce the second topic of theological analyses of work and play with a review of the Christian literature on these subjects. We then (...) employ Chesterton’s paradoxical theology to see how these two aspects of living might come to life and how people of faith should experience them. (shrink)

A central problem for the international governance of heritable germline gene editing is that there are important differences in attitudes and values as well as ethical and health care considerations around the world. These differences are reflected in a complicated and diverse regulatory landscape. Several publications have discussed whether reproductive uses would be legally permissible in individual countries and whether clinical applications could emerge in the context of regulatory gaps and gray areas. Systematic comparative studies that explore issues related to (...) the governance of this technology from different national and international perspectives are needed to address the lack of knowledge in this area. In this research report, we contribute to filling this gap by presenting views of stakeholders in the United Kingdom on challenges to the governance of heritable genome editing. We present findings from a multistakeholder study conducted in the United Kingdom between October 2016 and January 2018 and funded by the Wellcome Trust. This research included interviews, literature analysis, and a workshop. We involved leading U.K. scientists, in vitro fertilization clinicians, and representatives from regulatory bodies, patient organizations, and other civil societal organizations, as well as fertility companies. Part one of this article explores stakeholder perceptions of possible global developments in heritable genome editing and associated risks and governance challenges. Part two presents a range of policy options that were generated during the workshop in relation to the challenges discussed in part one. (shrink)

We recently argued that—contrary to what we call the dominant view— physicians ought to avoid engaging patients on religious grounds.1 The six responses to our article present an array of concerns and have provided us with the opportunity to consider further aspects of our view. While we cannot reply to all the relevant issues, our aim here is to reply to the most significant concerns. Against our Public Reason Argument, Nick Colgrove maintains that physicians are not relevantly akin to (...) public officials and should therefore not be constrained by public reasons. Colgrove does not take issue with the idea that it may be reasonable for public officials to justify their policymaking according to public reasons because their policies affect a diverse population. But since ‘the [medical] decision being made is not binding for other patients,’ physicians need not be constrained by public reasons.2 Colgrove’s criticism hinges on the claim that deliberation ought to be constrained by public reason only if the outcome of such deliberation is a policy or a rule that a diverse group of people must follow. We disagree with this claim. We do not believe, nor did we argue, that this is a necessary condition for when we are morally required to cite public reasons. Instead we believe that three facts together provide sufficient reason for why physicians should generally be constrained by public reasons: physicians are directly and indirectly paid by the state via Medicare and Medicaid; physicians allocate public resources; and physicians are regulated by the state. Colgrove takes particular issue with our ‘Dr Chatterjee case’, which is meant to illustrate why citing religious reasons is inappropriate. And our explanation for why we find this case jarring …. (shrink)

We present data from an exploratory qualitative interview-based pedagogical research project on the development of student agency in higher education. Our aim was to respond to Nick Zepke’s claim that what is often missing from the current neoliberal discourse of higher education ‘is students having a voice in what and how they learn and how they can action their voice in the wider community as agentic citizens.’ Informed by Lacanian discourse analysis, our project investigated the opportunities and threats facing (...) some of our undergraduate students as they struggled to exercise agency and develop autonomy in the marketised university. Repeat interviews with final year students focussed on the psychosocial categories of power, affect, intersubjectivity and desire. The analysis was guided by Lacan’s theory of the four discourses, an account of the vicissitudes of agency. We found that students can move between discourses depending on the extent to which their agency was enabled or thwarted. Our critique of the metaphor of the ‘student journey’ addresses the implications for learning and teaching and the university’s mission to develop its students in light of perceived commercial pressures. (shrink)

Modern types of social reality require updated ways of comprehending them. The research is devoted to a new analytical form of understanding modernity that has recently emerged - accelerationism, still rarely discussed in Russian philosophy. The representatives of accelerationism call for a radical and rapid acceleration of socio-economic and technological processes in capitalist societies. The article reflects some ideas of the Manifesto for an Accelerationist Politics by Alex Williams and Nick Srnicek, after which the accelerationist trend in philosophy and (...) social sciences intensified and gained clear theoretical guidelines. The Manifesto’s ideas about accelerating technological evolution as a means of resolving social conflicts, about unleashing all the latent forces of capitalist production to achieve a state of post-capitalism, denying a return to the Fordist type of production and calling for the restoration of the future as such, are highlighted. The Manifesto and the works of Nick Land, the founder and the most prominent representative of accelerationism, present the position of creating a new program and the very style of thinking with regard to changing the capitalist system along the vector of acceleration. The article pays attention to the interpretation of Gilles Deleuze’s and Félix Guattari’s concept of “deterritorialization” in Land’s works. It emphasizes the focus of accelerationism on the future as a kind of realization of the paradoxical thesis of “looking back from the future.” The content of Land’s accelerationist theory shows the fundamental concepts of K-space (cyberspace), K-war (cyberwar), time and reality, technocratic future of society as Techno-Capital Singularity, expansion of capital as opposed to its reterritorialization. The meaning of Land’s idea of an acceleration of capitalism and the transition to a more progressive future through the collapse of outmoded structures and phenomena of the existing system of capitalism and its technological basis is deduced. (shrink)

Remarkable progress in the mathematics and computer science of probability has led to a revolution in the scope of probabilistic models. In particular, ‘sophisticated’ probabilistic methods apply to structured relational systems such as graphs and grammars, of immediate relevance to the cognitive sciences. This Special Issue outlines progress in this rapidly developing field, which provides a potentially unifying perspective across a wide range of domains and levels of explanation. Here, we introduce the historical and conceptual foundations of the approach, explore (...) how the approach relates to studies of explicit probabilistic reasoning, and give a brief overview of the field as it stands today. (shrink)

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...) algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety (...) theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1. (shrink)

The purpose of this investigation is to develop a ranking of academic business ethics journals. For this, a revealed preference approach, also known as a citation impact method, was employed. The citation data were generated by using Google Scholar; h-index, g-index and hc-index were utilised to obtain a ranking. It was observed that the scores of these three indices correlated almost perfectly. This study also demonstrates that business ethics is a well-established discipline that should have its own set of recognised (...) academic outlets. (shrink)

Recent research suggests that language evolution is a process of cultural change, in which linguistic structures are shaped through repeated cycles of learning and use by domain-general mechanisms. This paper draws out the implications of this viewpoint for understanding the problem of language acquisition, which is cast in a new, and much more tractable, form. In essence, the child faces a problem of induction, where the objective is to coordinate with others (C-induction), rather than to model the structure of the (...) natural world (N-induction). We argue that, of the two, C-induction is dramatically easier. More broadly, we argue that understanding the acquisition of any cultural form, whether linguistic or otherwise, during development, requires considering the corresponding question of how that cultural form arose through processes of cultural evolution. This perspective helps resolve the “logical” problem of language acquisition and has far-reaching implications for evolutionary psychology. (shrink)

An influential line of thinking in behavioral science, to which the two authors have long subscribed, is that many of society's most pressing problems can be addressed cheaply and effectively at the level of the individual, without modifying the system in which the individual operates. We now believe this was a mistake, along with, we suspect, many colleagues in both the academic and policy communities. Results from such interventions have been disappointingly modest. But more importantly, they have guided many (though (...) by no means all) behavioral scientists to frame policy problems in individual, not systemic, terms: To adopt what we call the “i-frame,” rather than the “s-frame.” The difference may be more consequential than i-frame advocates have realized, by deflecting attention and support away from s-frame policies. Indeed, highlighting the i-frame is a long-established objective of corporate opponents of concerted systemic action such as regulation and taxation. We illustrate our argument briefly for six policy problems, and in depth with the examples of climate change, obesity, retirement savings, and pollution from plastic waste. We argue that the most important way in which behavioral scientists can contribute to public policy is by employing their skills to develop and implement value-creating system-level change. (shrink)

Probabilistic methods are providing new explanatory approaches to fundamental cognitive science questions of how humans structure, process and acquire language. This review examines probabilistic models defined over traditional symbolic structures. Language comprehension and production involve probabilistic inference in such models; and acquisition involves choosing the best model, given innate constraints and linguistic and other input. Probabilistic models can account for the learning and processing of language, while maintaining the sophistication of symbolic models. A recent burgeoning of theoretical developments and online (...) corpus creation has enabled large models to be tested, revealing probabilistic constraints in processing, undermining acquisition arguments based on a perceived poverty of the stimulus, and suggesting fruitful links with probabilistic theories of categorization and ambiguity resolution in perception. (shrink)

morality? I want to pursue these questions by examining an argument against moral realism that Simon Blackburn has developed.' In parts 1 and 2, I consider..

ABSTRACTIn ‘The Principles of Art’, R. G. Collingwood argues that art is the imaginative expression of emotion. So much the worse, then, for Collingwood. The theory seems hopelessly inadequate to the task of capturing art’s extension: of encompassing all the works we generally suppose should be rounded up under the concept. A great number of artworks, and several art forms, have nothing to do with emotion. But it would be surprising were Collingwood philistine enough to think that art is only (...) ever concerned with communicating quotidian affective states, like anger, fear or love. Surely he has some more sophisticated notion of emotion in mind, and quite probably of expression too. And it turns out that those sophisticated notions can be pushed towards a version of the expression theory that meets the extensional challenge. If we interpret Collingwood as saying that expression is a particular application of imagination, and that imagination is the faculty that refines ideas of emotions, and that ‘emotions’... (shrink)

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are the formulas preserved in monotonic images (...) of frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013. (shrink)