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

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 \(\rho \in [0,\infty )\).

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)

Anti-criterialists argue that there are no criteria that ensure personal persistence. However, this perspective is criticized for resulting in an unacceptable consequence that undermines our intuitions, daily beliefs, and direct introspective awareness of personal persistence. I defend anti-criterialism by responding to this objection and arguing that none of these aspects are undermined by anti-criterialism. The flawed objection against anti-criterialism reveals the excessive ambition of criterialism in seeking criteria for personal persistence with metaphysical necessity, which goes beyond our abilities and needs. (...) As beings living in the actual world, we are not able to consider what happens to personal persistence in remote possible worlds. Moreover, to judge whether a person persists or not in daily life, we do not need a criterion that ensures personal persistence in all situations. Anti-criterialism acknowledges the complexity and variability of personal persistence and offers a more realistic and practical approach to understanding personal identity. (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)

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)

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.

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 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)

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)

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.

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 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.

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)

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.

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)

The analysis, with the use of two state vectors, of a quantum system, during the time interval between two measurements, leads to some amazing results, which seem to contradict our usual “quantum common sense.” We explore the questions of compatibility with the conventional quantum theory, uniqueness of pre- and post-selected ensembles, commutativity, simultaneity and reality of strong and weak values in the intermediate time, and the meaning of the weak value. Common criticisms are shown to be unfounded.

David Hyman and Charles Silver are quite right. Opinion 6.01 in the American Medical Association's Code of Medical Ethics is difficult to defend. Ties between compensation and outcomes need not mislead patients into thinking that results are guaranteed; they are widely used in other fields with considerable success, even if they have some disadvantages; they can potentially bring patients more actively into decision-making about whether and from whom to purchase which medical care; and, if carefully tuned, they can promote quality (...) by aligning providers’ welfare more closely with patients’.The purpose of this commentary is thus not to disagree with the fundamental thesis that result-based compensation arrangements can be appropriate and useful in the health-care setting. Rather, the objective is to “cut the next swath,” so to speak. While Hyman and Silver are right that RBCAs have potential to do good, they may be overly optimistic about benefits, while underestimating the potential limitations and hazards. (shrink)

This paper reports the findings of a study comparing the public examination results at GCE advanced and advanced supplementary levels of pupils with assisted places in the independent sector and pupils in the state sector of similar ability. The examination entries and results of pupils with APs were compared with those of pupils who had gained an AP at the same school but had not attended that school; they had, instead, taken their A levels in the state sector. After controlling (...) for ability it was found that the AP pupils achieved a significantly higher total point score in their A/AS levels than the pupils in the state sector and that the average point score per examination entry was higher for AP than state pupils. The results suggest that there is an advantage in taking up an AP but the possibility cannot be ruled out that the advantage for AP pupils arises from factors other than their education in the independent sector. (shrink)

This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does not force iΣ2. We prove i∀1 ⊬ i∃1, i∃1 ⊬ i∀1, iΠ2 ⊬ iΣ2 and iΣ2 ⊬ (...) iΠ2. We use Smorynski's operation Σ′ to show HA ⊬ lΠ1. (shrink)

David Hyman and Charles Silver are quite right. Opinion 6.01 in the American Medical Association's Code of Medical Ethics is difficult to defend. Ties between compensation and outcomes need not mislead patients into thinking that results are guaranteed; they are widely used in other fields with considerable success, even if they have some disadvantages; they can potentially bring patients more actively into decision-making about whether and from whom to purchase which medical care; and, if carefully tuned, they can promote quality (...) by aligning providers’ welfare more closely with patients’.The purpose of this commentary is thus not to disagree with the fundamental thesis that result-based compensation arrangements can be appropriate and useful in the health-care setting. Rather, the objective is to “cut the next swath,” so to speak. While Hyman and Silver are right that RBCAs have potential to do good, they may be overly optimistic about benefits, while underestimating the potential limitations and hazards. (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)

According to Jonathan Dancy's moral particularism, the way in which a given moral reason functions as a reason for or against an action can vary from case to case. Dancy also asserts that reasons are resultance bases. But a reason why something ought to be done is that in virtue of which it is something that ought to be done. If the function of a reason can vary, then resultance bases cannot be reasons. Perhaps the particularist might conceive (...) a reason not as a resultance base, but as a specific type of which a resultance base is a token. But this picture of reasons cannot be correct. (shrink)

The aim of this abstract ie to present the latest results concerning a system of nonsense-logic, known as the system W. Thus, we recall only those of previous results which are indispensable for our present considerations. A more exhaustive survey and bibliography can be found in [1]; our notation and terminology are those of [1], too.

Kripke bundle and C-set semantics are known as semantics which generalize standard Kripke semantics. In [4] and in [1, 2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics . Modal predicate logic Q-S4.1 is not Kripke bundle complete of the present paper). This is shown by using difference of Kripke bundle semantics and C-set semantics. In this paper, (...) by using the same idea we show that incompleteness results in Kripke bundle semantics which are extended versions of [2]. (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)

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)

Investigators and institutional review boards should integrate plans about the appropriate disclosure of individual genetic results when designing research studies. The ethical principles of beneficence, respect, reciprocity, and justice provide justification for routinely offering certain results to research participants. We propose a result-evaluation approach that assesses the expected information and the context of the study in order to decide whether results should be offered. According to this approach, the analytic validity and the clinical utility of a specific result determine whether (...) it should be offered routinely. Different results may therefore require different decisions even within the same study. We argue that the threshold of clinical utility for disclosing a result in a research study should be lower than the threshold used for clinical use of the same result. The personal meaning of a result provides additional criteria for evaluation. Finally, the context of the study allows for a more nuanced analysis by addressing the investigators' capabilities for appropriate disclosure, participants' alternative access to the result, and their relationship with the investigators. This analysis shows that the same result may require different decisions in different contexts. (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 (serving 1 garden), medium-sized organizations (2–3 gardens), large organizations (4–30 gardens) and very large organizations (31 or more gardens). 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 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)

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)

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)

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.

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)

Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that (...) every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees.In this thesis, we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics.The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees, that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions.This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss.The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about whether it is provable in $\mathsf {ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of $\mathsf {ZF}$, this holds for all partial orders of height two, but not for all partial orders of height three. Our proof also yields an analogous result for Borel partial orders and Borel embeddings in $\mathsf {ZF}$, which shows that the Borel version of Sacks’ question has a negative answer.We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research.prepared by Patrick Lutz.E-mail: [email protected]. (shrink)

Moral particularism is a promising new approach which understands itself as a subchapter of holism in the theory of reasons. So particularism may be extended to other areas, such as metaphysics. One of the bases for this kind of move is elaborated by particularism itself as resultance, a strategy for providing the relevant basis that is opposed to various forms of generalism (the thin property of goodness is constituted by several thick properties, such as being good humoured, being pleasant; (...) the property of this being a table is constituted from properties of there being four legs, a plate, a certain arrangement). It is claimed that resultance or emergence needs a background structure in order to get off the ground. (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)