Recently in this journal Sherrilyn Roush (2004) defends positive relevance as a necessary (albeit not a sufficient) condition for evidence by rejecting two of the counterexamples from my earlier (2001) work. In this reply I argue that Roush's critique is not successful.

This paper addresses two examples due to Peter Achinstein purporting to show that the positive relevance view of evidence is too strong, that is, that evidence need not raise the probability of what it is evidence for. The first example can work only if it makes a false assumption. The second example fails because what Achinstein claims is evidence is redundant with information we already have. Without these examples Achinstein is left without motivation for his account of evidence, (...) which uses the concept of explanation in addition to that of probability. (shrink)

The author gentzenizes the positive fragments T₊ and R₊ of relevant T and R using formulas with prefixes (subscripts). There are three main Gentzen formulations of $S_{+}\in \{T_{+},R_{+}\}$ called W₁ S₊, W₂ S₊ and G₂ S₊. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible in S₊ but it is needed in the proof of a cut elimination theorem for G₂ S₊. (...) W₁ S₊ has a weak rule of weakening W₁ and it is not closed under a general transitivity rule. This allows the proof that $\vdash A$ in S₊ iff $\vdash A$ in W₁ S₊. From the cut elimination theorem for G₂ S₊ it follows that if $\vdash A$ in S₊, then $\vdash A$ in G₂ S₊. In order to prove the converse, W₂ S₊ is needed. It contains modus ponens, transitivity, and a restricted weakening rule. G₂ S₊ is contained in W₂ S₊ and there is a proof that $\vdash A$ in W₂ S₊ iff $\vdash A$ in W₁ S₊. (shrink)

The author gentzenizes the positive fragmentsT + andR + of relevantT andR using formulas with, prefixes (subscripts). There are three main Gentzen formulations ofS +{T+,R +} calledW 1 S +,W 2 S + andG 2 S +. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible inS + but it is needed in the proof of a cut elimination theorem forG 2 (...) S +.W 1 S + has a weak rule of weakeningW 1 and it is not closed under a general transitivity rule. This allows the proof that A inS + iff A inW 1 S +. From the cut elimination theorem forG 2 S + it follows that if A inS +, then A inG 2 S +. In order to prove the converse,W 2 S + is needed. It contains modus ponens, transitivity, and a restricted weakening rule.G 2 S + is contained inW 2 S + and there is a proof that A inW 2 S + iff A inW 1 S +. (shrink)

This paper presents a sequent calculus for the positive relevant logic with necessity and a proof that it admits the elimination of cut.

[10] offers two (cut-free) subscripted Gentzen systems, G 2 T + and G 2 R +, which are claimed to be equivalent in an appropriate sense to the positive relevant logics T + and R +, respectively. In this paper we show that that claim is false. We also show that the argument in [10] for the further claim that cut and/or modus ponens is admissible in two other subscripted Gentzen systems, G 1 T + and G 1 R (...) +, is unsound. (shrink)

Volume 34, Issue 8, December 2020, Page 1632-1645.

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

I show that the lattice of the positive integers ordered by division is characteristic for Urquhart’s positive semilattice relevance logic; that is, a formula is valid in positive semilattice relevance logic if and only if it is valid in all models over the positive integers ordered by division. I show that the same frame is characteristic for positive intuitionistic logic, where the class of models over it is restricted to those satisfying a heredity (...) condition. The results of this article highlight deep connections between intuitionistic and semilattice relevance logic. (shrink)

The goal of the current paper is to provide an introduction to and survey of the diverse landscape of relevant alternatives theories of knowledge. Emphasis is placed throughout both on the abstractness of the relevant alternatives approach and its amenability to formalization through logical techniques. We present some of the important motivations for adopting the relevant alternatives approach; briefly explore the connections and contrasts between the relevant alternatives approach and related developments in logic, epistemology and philosophy of science; provide a (...) schema for classifying and studying relevant alternatives theories at different levels of abstraction; and present a sample of relevant alternatives theories (contrasting what we call question-first and topic-first theories) that tie our discussion to on-going debates in the philosophical literature, as well as showcasing techniques for formalizing some of the important positions in these debates. (shrink)

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared (...) between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

Can facts about subpersonal states and events be constitutively relevant to personal-level phenomena? And can knowledge of these facts inform explanations of personal-level phenomena? Some philosophers, like Jennifer Hornsby and John McDowell, argue for two negative answers whereby questions about persons and their behavior cannot be answered by using information from subpersonal psychology. Knowledge of subpersonal states and events cannot inform personal-level explanation such that they cast light on what constitutes persons’ behaviors. In this paper I argue against this position. (...) After having distinguished between enabling and constitutive relevance, I defend the claim that at least some facts about subpersonal states and events are constitutively relevant to some personal-level phenomenon, and therefore can, and sometimes should, inform personal-level explanations. I draw some of the possible consequences of my claim for our understanding of personal-level behavior by focusing on the phenomenon of addiction. (shrink)

Conditionals frequently convey that the antecedent is relevant to the consequent. Recently many authors have argued that this relevance is part of the conventional meaning of conditionals, but this approach fails to account for many examples where a conditional is used to conveyirrelevance of antecedent to consequent. Both types of conditionals are best explained by a conventional meaning with no relevance requirement, and a separate process of coherence establishment among successive clauses in discourse. This account is supported by (...) the distribution of discourse particles and is able to account for experimental studies used to support the conventionalist position. (shrink)

This paper contributes to the growing literature in social epistemology of diagnosing the epistemically problematic features of fake news. I identify two novel problems: the problem of relevant alternatives; and the problem of the degradation of the epistemic environment. The former arises among individual epistemic transactions. By making salient, and thereby relevant, alternatives to knowledge claims, fake news stories threaten knowledge. The problem of the degradation of the epistemic environment arises at the level of entire epistemic communities. I introduce the (...) notion of an epistemic environment, roughly the totality of resources and circumstances relevant to assessing the epistemically interesting statuses, such as knowledge. Fake news degrades our epistemic environment by undermining confidence in epistemic institutions and altering epistemic habits, thereby making the environment less conducive to achieving positive epistemic statuses. This is problematic even if the decrease in confidence and the altering of habits are rational. I end by considering solutions to these problems, stressing the importance of reproaching each other for proliferating fake news. I argue that we should reproach even faultless purveyors of fake news. This is because fake news typically arises in abnormal epistemic contexts, where there is widespread ignorance of, and noncompliance with, correct epistemic norms. (shrink)

This study aims to analyze what drives and prevents the purchasing of eco-friendly products across different consumer groups and develops a conceptual model embracing the positive altruistic, positive ego-centric, and negative ego-centric antecedents of eco-friendly product purchase intention and behavior. We empirically validate the conceptual model for green and non-green consumers. Data are analyzed using structural equation modeling and multi-group analysis of the two groups. The results confirm the relevance of the determining factors in the model and (...) show significant differences in eco-friendly product purchasing patterns between green and non-green consumers. Altruistic motives are more important for green than for non-green consumers. Negative ego-centric motives affect the purchase intentions of non-green consumers more than the intentions of green consumers, whereas the impact of negative motives on behavior is stronger for green than for non-green consumers. The first contribution of this paper is the development and testing of a parsimonious model of eco-friendly products purchasing that embraces both positive and negative antecedents, which have been theoretically suggested in the past but have rarely been empirically tested together. The second contribution of this study is that it develops insight into the specific antecedents of eco-friendly products purchasing for green and non-green consumers to assess potential similarities and differences in eco-friendly products purchasing process, the hypothesized antecedents, their impact on eco-friendly products purchase intention and behavior, and the intention–behavior relation. (shrink)

Can facts about subpersonal states and events be constitutively relevant to personal-level phenomena? And can knowledge of these facts inform explanations of personal-level phenomena? Some philosophers, like Jennifer Hornsby and John McDowell, argue for two negative answers whereby questions about persons and their behavior cannot be answered by using information from subpersonal psychology. Knowledge of subpersonal states and events cannot inform personal-level explanation such that they cast light on what constitutes persons? behaviors. In this paper I argue against this position. (...) After having distinguished between enabling and constitutive relevance, I defend the claim that at least some facts about subpersonal states and events are constitutively relevant to some personal-level phenomenon, and therefore can, and sometimes should, inform personal-level explanations. I draw some of the possible consequences of my claim for our understanding of personal-level behavior by focusing on the phenomenon of addiction. (shrink)

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in (...) R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T (...) theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader. (shrink)

Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), Sperber (...) and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (shrink)

S CIENTISTS often claim that an experiment or observation tests certain hypotheses within a complex theory but not others. Relativity theorists, for example, are unanimous in the judgment that measurements of the gravitational red shift do not test the field equations of general relativity; psychoanalysts sometimes complain that experimental tests of Freudian theory are at best tests of rather peripheral hypotheses; astronomers do not regard observations of the positions of a single planet as a test of Kepler's third law, even (...) though those observations may test Kepler's first and second laws. Observations are regarded as relevant to some hypotheses in a theory but not relevant to others in that same theory. There is another kind of scientific judgment that may or may not be related to such judgments of relevance: determinations of the accuracy of the predictions of some theories are not held to provide tests of those theories, or, at least, positive results are not held to support or confirm the theories in question. There are, for example, special relativistic theories of gravity that predict the same phenomena as does general relativity, yet the theories are regarded as.. (shrink)

The present paper focuses on the heuristic selection process preceding the actual transitive reasoning process. A part of the difficulty of transitive reasoning lies in the selection of the relevant problem aspects. Two experiments are presented using the paradigm introduced by Markovits, Dumas, and Malfait (1995), in which children were asked to make “higher than” inferences about arrays of coloured blocks. In order to discriminate between genuine transitive inference and a simple strategy of relative position, Markovits et al. interspersed white (...) blocks with the coloured blocks, such that the relative position strategy leads to erroneous responses. However, we argue that the white blocks cause confusion due to their ambiguity, which interferes with the heuristic selection process. Two methodological adaptations were introduced, which are hypothesised to facilitate the selection process and improve transitive reasoning: (1) the white blocks were replaced by coloured blocks, and (2) a less abstract context was added to the experimental design. The colour manipulation leads to a clear increase in the use of a transitive strategy by 9-year-old children; 8-year-old children mainly used the relative position strategy. When adding a context story, 9-year-old children used the transitive strategy regardless of the colour of the interspersed blocks. The overall performance of 8-year-olds improved slightly. These results are interpreted as support for a dual-process model of transitive reasoning. (shrink)

This paper shows that model structures for R+, the system of positive relevant implication, can be constructed from ordered geometries. This extends earlier results building such model structures from projective spaces. A final section shows how such models can be extended to models for the full system R.

The aim of this paper is to explore a new framework for personality assessment that may function as sanity nosology of personality traits: the Positive Personality Model. The recent publication of DSM-5 created the opportunity to assess personality traits as dimensional constructs (American Psychiatric Association, 2013). In Section III, five maladaptive personality traits are proposed as the maladaptive versions of Five Factor Model (FFM) traits (Costa and McCrae, 1985). This approach draws on the existing idea of conceptualizing pathological and (...) typical personality traits as part of a continuum. It places DSM-5’s maladaptive traits in a sickness pole and FFM’s traits in a “typical” pole. This spectrum, however, does not include a positive perspective that represents healthy behavior: a sanity nosology. The Positive Traits Inventory-5 (PTI-5; de la Iglesia and Castro Solano, 2018) is a measure designed to assess the positive reverse of the Personality Inventory for DSM-5-Adult (PID-5; Krueger, Derringer, Markon, Watson and Skodol, 2013). The 220 positive personality criteria were studied psychometrically using a sample of 1902 Argentinean adults from the general population (Mage = 39.10, SD = 13.81, Min = 18, Max = 83; 50.1% females, 49.9% males). Exploratory and confirmatory factor analyses resulted in a five-factor solution. The dimensions were labeled Sprightliness, Integrity, Serenity, Moderation, and Humanity and subsumed under the denomination of Positive Personality Model (PPM). Analyses of convergent validity provided some grounds for interpreting the five positive traits as positive versions of the pathological traits and the typical traits. When tested for its predictive capability on mental health, the PPM outperformed the variance explained by the FFM. It is concluded that the PPM may constitute a positive pole in the continuum of personality traits –possibly functioning as a sanity nosology– and that it is somewhat more related to optimal functioning than to typical trait models. The PPM should be confirmed in other populations, its predictive capability ought to be tested with other relevant variables, and longitudinal studies should be done to analyze the stability of the traits over time. (shrink)

To explain why we should be structurally rational – or mentally coherent – is notoriously difficult. Some philosophers argue that the normativity of structural rationality can be explained in terms of substantive rationality, which is a matter of correct response to reason. I argue that the psychological phenomena – positive illusions – are counterexamples to the substantivist approach. Substantivists dismiss the relevance of positive illusions because they accept evidentialism that reason for belief must be evidence. I argue (...) that their evidentialist stance would imply that we are caught in unsolvable dilemmas arising from positive illusions. (shrink)

Magna Carta Libertatum is one of the few documents that continuously imply thorough discussions about fundamental principles of the law. In 2011, Lord McNelly, Justice Minister of UK at the time, has emphasized the core and everlasting principles that derived from this document: ᠅ that the power of the state is not absolute ᠅ that whoever governs the state must obey the law ᠅ and that whoever governs the state must take account of the views of those who are governed. (...) These are the fundamental principles of any government that strives to be distinguished as democratic, these are the self-evident truths that have been developed in the theory of social contract that established the modern day democracies. It is very common that article 39 of Magna Carta that provides for the right to due process, as well as article 40 that provides for the right to access to justice and justice itself, to be usually analyzed from the point of view of the rights of the person accused of a crime. However, it must be taken into consideration, that failure to guarantee these two very important human rights makes the accused person a victim of abuse of power. This article aims to analyze the relevance of Magna Carta in the rise of the concept of rights of victims of abuse of power. Although it is a concept developed later in history, the clauses of Magna Carta that remain in power can be directly linked to this category of victims. The thirteenth century provides a very important perspective on the position of the victim of crime and can be analyzed in a comparative aspect regarding the Common Law and the Civil Law historical development. The article will briefly explain the evolution of the concept of victims’ rights throughout these eight centuries to the modern times when these rights have become a crucial part of the national legislations of Western Balkan countries. (shrink)

Heidegger’s thoughts on modern technology have received much attention in many disciplines and fields, but, with a few exceptions, the influence has been sparse in biomedical ethics. The reason for this might be that Heidegger’s position has been misinterpreted as being generally hostile towards modern science and technology, and the fact that Heidegger himself never subjected medical technologies to scrutiny but was concerned rather with industrial technology and information technology. In this paper, Heidegger’s philosophy of modern technology is introduced and (...) then brought to bear on medical technology. Its main relevance for biomedical ethics is found to be that the field needs to focus upon epistemological and ontological questions in the philosophy of medicine related to the structure and goal of medical practice. Heidegger’s philosophy can help us to see how the scientific attitude in medicine must always be balanced by and integrated into a phenomenological way of understanding the life-world concerns of patients. The difference between the scientific and the phenomenological method in medicine is articulated by Heidegger as two different ways of studying the human body: as biological organism and as lived body. Medicine needs to acknowledge the priority of the lived body in addressing health as a way of being-in-the-world and not as the absence of disease only. A critical development of Heidegger’s position can provide us with a criterion for distinguishing the uses of medical technologies that are compatible with such an endeavor from the technological projects that are not. (shrink)

Language has been shown to influence the ability to form categories. Nevertheless, in most prior work, the effects of language could have been bolstered by the fact that linguistic labels were introduced by the experimenter prior to the categorization task in ways that could have highlighted their relevance for the task. Here, we compared the potency of labels to that of other non‐linguistic cues on how people categorized novel, perceptually ambiguous natural kinds (e.g., flowers or birds). Importantly, we varied (...) whether these cues were explicitly presented as relevant to the categorization task. In Experiment 1, we compared labels, numbers, and symbols: One group of participants was told to pay attention to these cues because they would be helpful (Relevant condition), a second group was told that the cues were irrelevant and should be ignored (Irrelevant condition), and a third group was told nothing about the cues (Neutral condition). Even though task relevance affected overall reliance on cues during categorization, participants were more likely to use labels to determine category boundaries, compared to numbers or symbols. In Experiments 2 and 3, we replicated and fine‐tuned the advantage of labels in more stringent categorization tasks. These results offer novel evidence for the position that labels offer unique indications of category membership, compared to non‐linguistic cues. (shrink)

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where (...) negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. (shrink)

There are a number of relevant alternatives accounts of knowledge in the literature, including those by contextualists (like Lewis and Cohen), and invariantists (like Dretske). Despite widespread discussion of such views, an explication of the notion of relevance is conspicuously absent from the literature. Without a careful explication of that notion, relevant alternatives accounts resist evaluation. This paper attempts to aid in the evaluation of those accounts, by providing an account of relevance. The account rejects two common presuppositions (...) about the notion of relevance. The account holds that worlds, rather than alternatives, are relevant, and that distant worlds can be relevant. Relevant worlds turn out to be those worlds at which an alternative to one’s belief obtains, and is such that one’s epistemic position (with respect to what one believes at the actual world) is worse than it is at the actual world. (shrink)

In this article, I call into question the relevance of emergence theories as presently used by thinkers in the science–religion discussion. Specifically, I discuss theories of emergence that have been used by both religious naturalists and proponents of panentheism. I argue for the following conclusions: (1) If we take the background theory to be metaphysical realism, then there seems to be no positive connection between the reality of emergent properties and the validity of providing reality with a religious (...) interpretation, though one could perhaps construe an argument for the positive ontological status of emergence as a negative case for a religious worldview. (2) To be considered more plausible, religious naturalism should interpret religious discourse from the perspective of pragmatic realism. (3) Panentheistic models of divine causality are unable to avoid ontological dualism. (4) It is not obvious that emergent phenomena and/or properties are nonreducible in the ontological sense of the terms; indeed, the tension between weak and strong emergence makes it difficult for the emergentist to make ontological judgments. My general conclusion is that the concept of emergence has little metaphysical significance in the dialogue between science and theology. (shrink)

I address the issue of whether differences in ethical theory have any relevance for the practical issues of environmental management and policy making. Norton’s answer, expressed as a convergence hypothesis, is that environmentalists are evolving toward a consensus in policy even though they remain divided regarding basic values. I suggest that there are good reasons for rejecting Norton’s position.I elaborate on these reasons, first, by distinguishing between different forms of anthropocentrism and nonanthropocentrism, second, by contrasting the different goals that (...) anthropocentrists, biocentrists, and ecocentrists set up for environmental policy making, and, lastly, by identifying three important policy areas (population growth, wilderness preservation, and wildlife management) where differences in basic values generate divergent policies. (shrink)

The larger philosophical world has on the whole turned from a mix of averted gaze and outright antipathy toward x‐phi, to a mix of grudging acceptance and enthusiastic embrace. This chapter explains that the experimental philosophy is relevant, and that it is dangerous, and explains some ways that people can do more to remain both. Experimental philosophy's semi‐official sigil of the burning armchair has advertised its dangerousness for the past decade and a half as well. The chapter explains that it (...) is a threat that is every bit as potently present in positive program x‐phi as it is in negative program work. To explore the idea that experimental philosophy's positive contributions will still be primarily negative in nature, the chapter interrogates the passage from Timothy Williamson. It focuses on the wheat‐from‐chaff project of separating signal from noise in the evidence produced by intuitions and the like. (shrink)

This article investigates the meaning of doing good and doing well in positive business. It examines the relationship between the two expressions and discusses their relevance, shedding new light on the significance of ‘positive’ in positive business and positive organizational scholarship. Thus, this article illuminates the ultimate end of positive states and practices. ‘Positive’ primarily represents values and assumptions. These lead to the creation of beneficial situations and marked improvements, which put individuals and (...) organizations on an upward trajectory toward achieving excellent functioning, assuring profitability in addition to sustainability and social well-being. Doing well, with reference to economic performance and profits, is partly constitutive of doing good. This presents a new approach regarding the role of economic outcomes in positive business and POS. It means that financial criteria do not have priority. Profits are considered vital and necessary, but the final ‘raison d'être’ of positive states and practices is the overall well-being of the stakeholders. (shrink)

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) (...) in relevant arithmetic. (shrink)