Papers presented at the Kierkegaard: Resources and Results Conference, held at McGill University, June 6-8, 1980.

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize (...) nicely to languages containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

In the present study, inspired by the studies in the literature, we consider Min matrix and its Hadamard exponential matrix family whose elements are Chebyshev polynomials of the first kind. Afterwards, we examine their various linear algebraic properties and obtain some inequalities. Furthermore, we shed light on the results we obtained to boost the clarity of our paper with the illustrative examples. In addition to all these, we give two MATLAB-R2023a codes that compute the Min matrix and the Hadamard (...) exponential matrix with Chebyshev polynomials of the first kind entries, as well as calculate some matrix norms. (shrink)

It is almost unanimously accepted that Kant denies resultant moral luck—that is, he denies that the lucky consequence of a person’s action can affect how much praise or blame she deserves. Philosophers often point to the famous good will passage at the beginning of the Groundwork to justify this claim. I argue, however, that this passage does not support Kant’s denial of resultant moral luck. Subsequently, I argue that Kant allows agents to be morally responsible for certain kinds of lucky (...) consequences. Even so, I argue that it is unclear whether Kant ultimately endorses resultant moral luck. The reason is that Kant does not write enough on moral responsibility for consequences to determine definitively whether he thinks that the lucky consequence for which an agent is morally responsible can add to her degree of praiseworthiness or blameworthiness. The clear upshot, however, is that Kant does not deny resultant moral luck. (shrink)

Let \({\mathcal {R}}\) be a polynomially bounded o-minimal expansion of the real field. Let _f_(_z_) be a transcendental entire function of finite order \(\rho \) and type \(\sigma \in [0,\infty ]\). The main purpose of this paper is to show that if ( \(\rho ) or ( \(\rho =1\) and \(\sigma =0\) ), the restriction of _f_(_z_) to the real axis is not definable in \({\mathcal {R}}\). Furthermore, we give a generalization of this result for any \(\rho \in [0,\infty )\).

Psychiatry is rapidly adopting digital phenotyping and artificial intelligence/machine learning tools to study mental illness based on tracking participants’ locations, online activity, phone and text message usage, heart rate, sleep, physical activity, and more. Existing ethical frameworks for return of individual research results (IRRs) are inadequate to guide researchers for when, if, and how to return this unprecedented number of potentially sensitive results about each participant’s real-world behavior. To address this gap, we convened an interdisciplinary expert working group, (...) supported by a National Institute of Mental Health grant. Building on established guidelines and the emerging norm of returning results in participant-centered research, we present a novel framework specific to the ethical, legal, and social implications of returning IRRs in digital phenotyping research. Our framework offers researchers, clinicians, and Institutional Review Boards (IRBs) urgently needed guidance, and the principles developed here in the context of psychiatry will be readily adaptable to other therapeutic areas. (shrink)

Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This (...) generalizes Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$.2.There is $\lambda \in [\mu,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case:(a) $\mathbf {K}$ has the uniqueness of $(\mu,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu,\geq \chi )$ -limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$ ;(b)any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38].Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting.Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank. (shrink)

Inquiry into the meaning of logical terms in natural language (‘and’, ‘or’, ‘not’, ‘if’) has generally proceeded along two dimensions. On the one hand, semantic theories aim to predict native speaker intuitions about the natural language sentences involving those logical terms. On the other hand, logical theories explore the formal properties of the translations of those terms into formal languages. Sometimes, these two lines of inquiry appear to be in tension: for instance, our best logical investigation into conditional connectives may (...) show that there is no conditional operator that has all the properties native speaker intuitions suggest if has. Indicative conditionals have famously been the source of one such tension, ever since the triviality proofs of both Lewis (1976) and Gibbard (1981) established conclusions which are in prima facie tension with ordinary judgments about natural language indicative conditionals. In a recent series of papers, Branden Fitelson has strengthened both triviality results (Fitelson 2013, 2015, 2016), revealing a common culprit: a logical schema known as IMPORT-EXPORT. Fitelson’s results focus the tension between the logical results and ordinary judgments, since IMPORT-EXPORT seems to be supported by intuitions about natural language. In this paper, we argue that the intuitions which have been taken to support IMPORT-EXPORT are really evidence for a closely related, but subtly different, principle. We show that the two principles are independent by showing how, given a standard assumption about the conditional operator in the formal language in which IMPORT-EXPORT is stated, many existing theories of indicative conditionals validate one, but not the other. Moreover, we argue that once we clearly distinguish these principles, we can use propositional anaphora to show that IMPORT-EXPORT is in fact not valid for natural language indicative conditionals (given this assumption about the formal conditional operator). This gives us a principled and independently motivated way of rejecting a crucial premise in many triviality results, while still making sense of the speaker intuitions which appeared to motivate that premise. We suggest that this strategy has broad application and an important lesson: in theorizing about the logic of natural language, we must pay careful attention to the translation between the formal languages in which logical results are typically proved, and natural languages which are the subject matter of semantic theory. (shrink)

In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument (...) fails in an interesting and revealing way. And it leaves a remainder: if some computer does in fact simulate all our mathematical reasoning, then, in principle, we cannot fully grasp how it works. Gödel’s result also points out a certain essential limitation of self-reflection. The resulting picture parallels, not accidentally, Davidson’s view of psychology, as a science that in principle must remain “imprecise”, not fully spelt out. What is intended here by “fully grasp”, and how all this is related to self-reflection, will become clear at the end of this comment. (shrink)

Does one’s causal responsibility increase the degree of one’s moral responsibility? The proponents of resultant moral luck hold that it does. Until quite recently, the causation literature has almost exclusively been interested in the binary question of whether one factor is a cause of an outcome. Naturally, the debate over resultant moral luck also revolved around this binary question. However, we have seen an increased interest in the question of degrees of causation in recent years. And some philosophers have already (...) explored various implications of a graded notion of causation on resultant moral luck. In this paper, I will do the same. But the implications that I will draw attention to are bad news for resultant moral luck. I will show that resultant moral luck entails some implausible results that leave resultant moral luck more indefensible than it was previously thought be. I will also show that what is typically taken to be the positive argument in favor of resultant moral luck fails. I will conclude that we should reject resultant moral luck. (NOTE: Email me for a copy.). (shrink)

Conflicting results undermine making inferences from the empirical literature. So far, the replication crisis is mainly seen as resulting from honest errors and questionable research practices such as p-hacking or the base-rate fallacy. We discuss the malleability (researcher degrees of freedom) of quantitative research and argue that conflicting results can emerge from two studies using different but plausible designs (e.g., eligibility criteria, operationalization of concepts, outcome measures) and statistical methods. We also explore how the choices regarding study design (...) and statistical techniques bias results in a way that makes them more or less relevant for a given policy or clinical question. (shrink)

In recent years, a number of theorists have claimed that beliefs about probability are transparent. To believe probably p is simply to have a high credence that p. In this paper, I prove a variety of triviality results for theses like the above. I show that such claims are inconsistent with the thesis that probabilistic modal sentences have propositions or sets of worlds as their meaning. Then I consider the extent to which a dynamic semantics for probabilistic modals can (...) capture theses connecting belief, certainty, credence, and probability. I show that although a dynamic semantics for probabilistic modals does allow one to validate such theses, it can only do so at a cost. I prove that such theses can only be valid if probabilistic modals do not satisfy the axioms of the probability calculus. (shrink)

Do extreme, unethical governmental policies often produce results opposite to those intended? This book considers the ironic outcomes of recent global events and concludes that there is a 'rule of reverse results' at work. While not a hard and fast law, the rule points out the increased probability that a policy will backfire if it is immoral while ethical policies, even if extreme, are unlikely to produce reverse results. The issue here is that of increased likelihood but (...) not of certainty. Governments can never be sure as to the effects of their actions: to some extent they are always working in the dark. But if the motivation is right, moral and humane the policies will not often produce adverse results the opposite of those intended. Based on events in global history in the Twentieth and Twenty-First centuries the chapters can each be read individually, as well as being part of the argument. (shrink)

The result model of precedent holds that a legal precedent controls a fortiori cases—those cases, that is, that are at least as strong for the winning side of the precedent as the precedent case itself. This paper defends the result model against some objections by Larry Alexander, drawing on ideas from the field of Artificial Intelligence and Law in order to define an appropriate strength ordering for cases.

Bd. 1. Wissenschaftliche Erklärung und Begründung.

Objectives Prompt Mental Health Care (PMHC) is the Norwegian version of the England’s Improving Access to Psychological Therapies (IAPT). Both programs have been associated with substantial symptom reductions from pre- to post-treatment. The present study extends these findings by investigating symptom levels at 12 months post-treatment, as well as treatment outcome in relation to low- vs. high-intensity treatment forms. Design and Outcome Measures A prospective cohort design was used. All participants ( n = 1530) were asked to complete the Patient (...) Health Questionnaire-9 (PHQ-9) and the Generalized Anxiety Disorder-7 questionnaire (GAD-7) at baseline, before each session during treatment, at final treatment, and at 12 months post-treatment. Cohen’s d was used as effect size measure. Sensitivity analyses were conducted to examine the impact of the high missing data rates at post-treatment (≈44%) and 12 months post-treatment (≈58%). Results A large symptom reduction was seen from baseline to 12 months post-treatment for both PHQ ( d = −0.98) and GAD ( d = −0.94). Improvements observed at post-treatment were largely maintained at 12 months post-treatment (PHQ (Δ d = 0.10) and GAD (Δ d = 0.09). Recovery rates decreased only slightly from 49.5% at post-treatment to 45.0% at follow-up. Both low- and high-intensity treatment forms were associated with substantial and lasting symptoms reductions (−1.26 ≤ d ≤ −0.73). Sensitivity analyses did not substantially alter the main results. Conclusion The findings suggest long-lasting effects of the PMHC program and encourage the use of low-intensity treatment forms in PMHC like settings. (shrink)

In this paper, I investigate whether we can use a world-involving framework to model the epistemic states of non-ideal agents. The standard possible-world framework falters in this respect because of a commitment to logical omniscience. A familiar attempt to overcome this problem centers around the use of impossible worlds where the truths of logic can be false. As we shall see, if we admit impossible worlds where “anything goes” in modal space, it is easy to model extremely non-ideal agents that (...) are incapable of performing even the most elementary logical deductions. A much harder, and considerably less investigated challenge is to ensure that the resulting modal space can also be used to model moderately ideal agents that are not logically omniscient but nevertheless logically competent. Intuitively, while such agents may fail to rule out subtly impossible worlds that verify complex logical falsehoods, they are nevertheless able to rule out blatantly impossible worlds that verify obvious logical falsehoods. To model moderately ideal agents, I argue, the job is to construct a modal space that contains only possible and non-trivially impossible worlds where it is not the case that “anything goes”. But I prove that it is impossible to develop an impossible-world framework that can do this job and that satisfies certain standard conditions. Effectively, I show that attempts to model moderately ideal agents in a world-involving framework collapse to modeling either logical omniscient agents, or extremely non-ideal agents. (shrink)

Hans van Ditmarsch and Barteld Kooi (2008). Semantic results for ontic and epistemic change. In: G. Bonanno, W. van der Hoek and M. Wooldridge (editors). Logic and the Foundations of Game and Decision Theory (LOFT 7). Texts in Logic and Games 3, pp. 87-117, Amsterdam University Press, Amsterdam.

There are two ways of representing rational belief: qualitatively as yes-or-no belief, and quantitatively as degrees of belief. Standard rationality conditions are: consistency and logical closure, for qualitative belief, satisfaction of the probability axioms, for quantitative belief, and a relationship between qualitative and quantitative beliefs in accordance with the Lockean thesis. In this paper, it is shown that these conditions are inconsistent with each of three further rationality conditions: fallibilism, open-mindedness, and invariance under independent conceptual expansions. Restrictions of the Lockean (...) thesis that have been suggested in the literature cannot remove the inconsistency. In the outlook we discuss two alternative ways of dealing with this problem: restricting conjunctive closure or going for a dual system account. (shrink)

Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere . The completeness theorems are used to conclude results asserting that (...) certain logics are conservatively interpretable in others. (shrink)

Generalizing on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought.

Resultatives raise important questions for the syntax-semantics interface, and this is why they have occupied a prominent place in recent linguistic theorizing. What is it that makes this construction so interesting? Resultatives are submitted to a cluster of not obviously related constraints, and this fact calls out for explanation. There are tough constraints for the verb, for example.

It is a classical result of Mortimer that $L^2$ , first-order logic with two variables, is decidable for satisfiability. We show that going beyond $L^2$ by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Härtig (or equicardinality) quantifier¶– a choice construct known as Hilbert's $\epsilon$ -operator.In fact all these extensions of $L^2$ prove to be undecidable both for satisfiability, (...) and for satisfiability in finite structures. Moreover most of them are hard for $\Sigma^1_1$ , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic. (shrink)

Community gardens are of increasing interest to scholars, policymakers, and community organizations but there has been little systematic study of community garden management at a broad scale. This study complements case study research by revealing shared experiences of community garden management across different contexts. In partnership with the American Community Gardening Association, we developed an online questionnaire. Results from 445 community garden organizations across the US and Canada reveal common themes as well as differences that are particularly significant across (...) different organizational sizes. The findings suggest that organizers see multiple benefits, and respondents confirmed recent expansion of gardening efforts. Analysis then focuses on challenges, which are closely related to garden management. We address garden losses as well as challenges to routine operation. Key challenges included funding, participation, land, and materials. We developed a typology based on organization size, to reveal distinctions between small organizations , medium-sized organizations , large organizations and very large organizations . These categories shed light on different needs for funding, land, material, and participation. Together, this analysis suggests that community gardens can be linked through the work it takes to sustain them rather than specific causes or outcomes. Community gardens can be better integrated into local food systems through analysis of how people involved with this work navigate these shared processes. (shrink)

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a theory called the operator theory and that the (...) rival restrictor theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

This paper argues that a successful philosophical analysis of models and simulations must accommodate an account of mathematically rigorous results. Such rigorous results may be thought of as genuinely model-specific contributions, which can neither be deduced from fundamental theory nor inferred from empirical data. Rigorous results provide new indirect ways of assessing the success of models and simulations and are crucial to understanding the connections between different models. This is most obvious in cases where rigorous results (...) map different models on to one another. Not only does this put constraints on the extent to which performance in specific empirical contexts may be regarded as the main touchstone of success in scientific modelling, it also allows for the transfer of warrant across different models. Mathematically rigorous results can thus come to be seen as not only strengthening the cohesion between scientific strategies of modelling and simulation, but also as offering new ways of indirect confirmation. (shrink)

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach (...) in Dunn et al. (2005) and Gehrke et al. (2005). (shrink)

We provide a Kripke semantics for a STIT logic with the "next" operator. As the atemporal group STIT is undecidable and unaxiomatizable, we are interested in strict fragments of atemporal group STIT. First we prove that the satisfiability problem of a formula of the fragment made up of individual coalitions plus the grand coalition is also NEXPTIME-complete. We then generalize this result to a fragment where coalitions are in a given lattice. We also prove that if we restrict the language (...) to nested coalitions the satisfiability problem is NP-complete if the number of agents is fixed and PSPACE-complete if the number of agents is variable. Finally we embed individual STIT with the "next" operator into a fragment of atemporal group STIT. (shrink)

We prove independence results concerning the number of nonisomorphic models (using the S-chain condition and S-properness) and the consistency of "ZCF + 2 ℵ 0 = ℵ 2 + there is a universal linear order of power ℵ 1 ". Most of these results were announced in [Sh 4], [Sh 5]. In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of ω 1 , investigate D-properness for various filters and (...) prove the consistency with G.C.H. of an axiom implying SH (for ℵ 1 ), and connected results. (shrink)

Resultant moral luck occurs whenever aspects of an agent’s moral responsibility are affected by luck pertaining to the outcomes of their actions. Many authors reject the existence of moral luck in this sense, but they do so in different ways. Michael Zimmerman argues that resultant luck affects the scope of moral responsibility, but not its degree. That is, it affects what agents are responsible for, but not how responsible they are. Andrew Khoury takes a more resolute approach, arguing that both (...) the scope and the degree of moral responsibility are free from resultant luck.In this paper, I criticize both accounts and develop an alternative. I argue, first, that Khoury’s approach leads to an implausibly far-reaching error-theory about moral responsibility. Second, Zimmerman’s account cannot account for all the ways in which moral responsibility comes in degrees. Third, these problems can be overcome by introducing a distinction between two concepts of responsibility that both come with scope and degree. The first concept I call internal responsibility, as it applies exclusively to agent-internal factors. The second concept I call external responsibility, as it applies to (partly) agent-external factors such as actions and their outcomes. Given this distinction, we can avoid the problems of Khoury’s as well as Zimmerman’s accounts while preserving the central intuition behind the rejection of resultant moral luck. (shrink)

We prove that compactness is equivalent to the amalgamation property, provided the occurrence number of the logic is smaller than the first uncountable measurable cardinal. We also relate compactness to the existence of certain regular ultrafilters related to the logic and develop a general theory of compactness and its consequences. We also prove some combinatorial results of independent interest.

We gather the following miscellaneous results in proof theory from the attic.1. 1. A provably well-founded elementary ordering admits an elementary order preserving map.2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21.3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic.4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE.5. 5. Intuitionistic (...) fixed point theories which are conservative extensions of HA.6. 6. Proof theoretic strengths of classical fixed points theories.7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣn.8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions.Each section can be read separately in principle. (shrink)

Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.

Unprecedented levels of access to adolescents' time and attention provide opportunities to convert traditional character and socioemotional competencies interventions into behavioral intervention technologies. However, these new tools must be evaluated rather than assuming previously validated activities will be efficacious when converted to a mobile platform. Thus, we sought to design and provide initial data on the effectiveness of the CharacterMe smartphone app to build self-control and patience, which are built on underlying social-emotional regulation competencies, in a sample of 618 adolescents. (...) We also sought to examine whether framing the app activities as having a transcendent rather than instrumental purpose would increase engagement and change in self-control, patience, and emotion regulation. Finally, we tested the impact of framing activities as building strengths vs. fixing weaknesses. Results highlight the difficulty of translating psychological interventions to behavioral intervention technologies. Overall, the CharacterMe smartphone app was unsuccessful in increasing self-control, patience, or emotion regulation in adolescents, with analyses showing no significant mean changes over time. Framing conditions and user engagement were largely not significant predictors of change in self-control, patience, and emotion regulation. (shrink)

It is known that evidential support, on the Bayesian definition of this notion, is intransitive. According to some, however, the Bayesian definition is too weak to be materially adequate. This paper investigates whether evidential support is transitive on some plausible probabilistic strengthening of that definition. It is shown that the answer is negative. In fact, it will appear that even under conditions under which the Bayesian notion of evidential support is transitive, the most plausible candidate strengthenings are not.

It follows from Bell’s theorem and quantum mechanics that the detection of a particle of an entangled pair can (somehow) “force” the other distant particle of the pair into a well-defined state (which is equivalent to a reduction of the state vector): no property previously shared by the particles can explain the predicted quantum correlations. This result has been corroborated by experiment, although some loopholes still remain. However, it has not been experimentally proved—and it is far from obvious—that the absence (...) of detection, as in null-result (NR) experiments could have the very same effect. In this paper a way to try to bridge this gap is suggested. (shrink)

We study the expressive power of independence-friendly quantifier prefixes composed of universal$\left$, existential$\left$, and majority quantifiers$\left$. We provide four quantifier prefixes that can express NP hard properties and show that all quantifier prefixes capable of expressing NP-hard properties embed at least one of these four quantifier prefixes. As for the quantifier prefixes that do not embed any of these four quantifier prefixes, we show that they are equivalent to a first-order quantifier prefix composed of$\forall x$,$\exists x$, and Mx. In unison, (...) our results imply a dichotomy result: every independence-friendly quantifier prefix is either decidable in LOGSPACE or NP hard. (shrink)

We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six (...) classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide. (shrink)

We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let \(\bf Ord\) denote the class of all ordinals. Let \(\langle \mathbf{K}_{\alpha}: \alpha\in \bf Ord\rangle\) be a system of varieties definable by a schema. Given any ordinal \(\alpha\), we define an operator \(\mathsf{Nr}_{\alpha}\) that acts on \(\mathbf{K}_{\beta}\) for any \(\beta>\alpha\) giving an algebra in (...) \(\mathbf{K}_{\alpha}\), as an abstaction of taking \(\alpha\)-neat reducts for cylindric algebras. We show that for any positive \(k\), and any infinite ordinal \(\alpha\) that \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k+1}\) cannot be axiomatized by a finite schema over \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k}\) given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of \(L_{\omega,\omega}\). (shrink)

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of (...) the form p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic. (shrink)