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

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)

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)

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)

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

Fictionalists maintain that possible worlds, numbers or composite objects exist only according to theories which are useful but false. Hale, Divers and Woodward have provided arguments which threaten to show that fictionalists must be prepared to regard the theories in question as contingently, rather than necessarily, false. If warranted, this conclusion would significantly limit the appeal of the fictionalist strategy rendering it unavailable to anyone antecedently convinced that mathematics and metaphysics concern non-contingent matters. I try to show that their arguments (...) can be resisted by developing and defending a strategy suggested by Rosen, Nolan and Dorr, according to which the fiction-operator is to be analysed in terms of a counterfactual that admits of non-trival truth-values even when the antecedent is impossible. (shrink)

We give a general proof showing that if the evolution from one state to another is not reversible, then the projective measurements on the superposition of these two states are impossible. Applying this no-go result to the Schrödinger’s cat paradox implies that if something is claimed to be a real Schrödinger’s cat, there will be no measurable difference between it and a trivial classical mixture of ordinary cats in any physically implementable process, unless raising the dead becomes reality. Other similar (...) macroscopic quantum superpositions cannot be observed either, due to the lack of non-commuting measurement bases. Our proof does not involve any quantum interpretation theory and hypothesis. (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 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)

What do philosophers think about arguments for the existence of God? To find out, I launched a survey among professional philosophers. This is a short summary of the results.

This Alternate View column marks three milestones: This is the 3rd anniversary of my start as an AV columnist for Analog, this is the 20th AV column I've written, and it is also the 7th anniversary of my first publication in Analog. I enjoy writing these columns on scientific subjects, but it can be frustrating. Science is continually changing as new experimental results and observations are made, as new ideas and theories are conceived and old ideas are rejected. Often (...) by the time an Alternate View column on some new development in physics or astronomy appears in these pages the field has already progressed further and there is more to be said. So this third anniversary column will be used as an occasion for a backward look at some of the subjects covered in previous columns and articles, giving an update on more recent developments. An index of my first 20 AV columns and articles is provided below. This can be used as a reference guide to indicate which topics were discussed in which columns and articles. (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)

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)

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.

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)

We obtain 2 models in which AC is false and in which there are long sequences of consecutive large cardinals.

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)

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.

The neutrino is one of nature's most peculiar particles. It has 1/2 unit of spin but no electric charge, a near-zero rest-mass, and it interacts with other particles only through gravity and the weak interaction. It can pass through light years of lead without an interaction. There is good experimental evidence that the Earth receives only about 1/3 of the neutrinos that the Sun should be producing and sending in our direction.

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)

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.

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)

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.

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating (...) properties. Although these computability results had been announced earlier, their proofs have been deferred until this paper. Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima. (shrink)

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)

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)

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)

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 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)

The DAMA project is an observatory for rare processes and it is operative deep underground at the Gran Sasso National Laboratory of the I.N.F.N. In particular, the DAMA/LIBRA (Large sodium Iodide Bulk for RAre processes) set-up consists of highly radiopure NaI(Tl) detectors for a total sensitive exposed mass of ≃250 kg. Recent results, obtained by this set-up by exploiting the model independent annual modulation signature of Dark Matter (DM) particles, have confirmed and improved those obtained by the former DAMA/NaI (...) experiment. A model independent evidence for the presence of Dark Matter particles in the galactic halo is cumulatively obtained at 8.2 σ C.L. No systematics or side reactions able to account for the measured modulation amplitude and to contemporaneously satisfy all the many specific requirements of the signature have been found or suggested by anyone over more than a decade. An example of one of the many possible model dependent corollary quests for the candidate particles and for the related astrophysical, nuclear and particle physics scenarios is presented considering the whole cumulative exposure. Future perspectives are shortly addressed. (shrink)

The American Journal of Bioethics, Volume 12, Issue 10, Page 20-22, October 2012.

In this paper we study numeral systems in the -calculus. With one exception, we assume that all numerals have normal form. We study the independence of the conditions of adequacy of numeral systems. We find that, to a great extent, they are mutually independent. We then consider particular examples of numeral systems, some of which display paradoxical properties. One of these systems furnishes a counterexample to a conjecture of Böhm. Next, we turn to the approach of Curry, Hindley, and Seldin. (...) We dwell with the general problem of obtaining their results with the additional requirement of nonconvertibility of numerals. In particular we solve a problem that they left open. Finally, we give the first example of an adequate unsolvable numeral system without a test for zero in the usual sense, thus solving a problem of Barendregt and Barendsen. (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.