We provide a general framework for constructing natural consequence relations for paraconsistent and plausible nonmonotonic reasoning. The framework is based on preferential systems whose preferences are based on the satisfaction of formulas in models. We show that these natural preferential In the research on paraconsistency, preferential systems systems that were originally designed for for paraconsistent reasoning fulfill a key condition (stopperedness or smoothness) from the theoretical research of nonmonotonic reasoning. Consequently, the nonmonotonic consequence relations that they induce fulfill the desired (...) conditions of plausible consequence relations. Hence our framework encompasses different types of preferential systems that were developed from different motivations of paraconsistent reasoning and non-monotonic reasoning, and reveals an important link between them. (shrink)

In _Abductive Analysis_, Iddo Tavory and Stefan Timmermans provide a new navigational map for constructing empirically based generalizations in qualitative research. They outline an accessible way to think about observations, methods, and theories that nurtures theory-formation without locking it into predefined conceptual boxes. The authors view research as continually moving back and forth between a set of observations and theoretical generalizations. To craft theory is to then pitch one’s observations in relation to other potential cases, both within and without (...) one’s field. The book provides novel ways to approach the challenges that plague qualitative researchers across the social sciences—how to think about the relation between methods and theories, how to conceptualize causality, how to construct axes of variation, and how to leverage the researcher’s community of inquiry. _Abductive Analysis_ is a landmark work that shows how a pragmatist approach provides a more productive and fruitful way to conduct qualitative research. (shrink)

This study examined students' reasoning about simple repeated choices. Each choice involved ''betting'' on two events, differing in probability. We asked subjects to generate or evaluate alternative strategies such as betting on the most likely event on every trial, betting on it on almost every trial, or employing a ''probability matching'' strategy. Almost half of the college students did not generate or rank strategies according to their expected value, but few subjects preferred a strategy of strict probability matching. High-school students (...) showed greater deviations from expected value than college students. Similar misunderstandings were observed in a choice task involving real (not hypothetical) repeated trials. Large gender differences in prediction strategies and in related computational skills were observed. Subjects who understand the optimal strategy usually do so in terms of independence of successive trials rather than calculation. Some subjects understand the concept of independence but fail to bring it to bear, thinking it can be overridden by intuition or local balancing (representativeness). (shrink)

Micro-sociological theory has traditionally stressed interactional pressures towards alignment: actors’ attempts to co-construct a shared definition of the situation. We argue that this model provides an insufficient account of the coordination of action and of the emergence of intersubjectivity among actors. To complement the focus on alignment, we develop a theory of disruption—a perceived misalignment of the dramaturgical structure of interaction in coordinating expected lines of action. We develop a theory of the interaction order that takes the interplay between interactional (...) alignment and disruption as a foundational challenge both for sociology and for actors in their everyday lives. We focus on the practical ways in which actors negotiate both interactional breaches and wider relational ruptures, and how they differentiate between disruptions-of relations and disruptions-for them. By doing so, we connect the interaction order to a wider relational order, providing a bridge between micro-level interactionism and the sociology of culture. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Iddo Landau responds:I believe that there is much to learn from Gilead's arguments, and that his paper adds to the understanding of the themes presented in the original discussion. However, in the end I do not think that the claims I made are rebuffed.Gilead should be commended for expanding the discussion of the Mandarin thought experiment (henceforth: Mandarin) from the existentialist context, to which it was limited in (...) my original article, to the Kantian one. This is a fruitful move. However, we should keep in mind that in Kantianism—unlike existentialism—to be autonomous is to be moral. Thus, some of my claims may be understood differently in existentialist and Kantian contexts. I wrote, "I would like to argue, however, that it [the Mandarin thought experiment] also casts doubt on an important and central existentialist notion: the value of being an authentic, autonomous individual, who behaves as he or she does not because of fear of what others would say, but because he or she genuinely chooses this or that course of action" ("To Kill a Mandarin," p. 95). But to claim, in a Kantian context, that the Mandarin casts doubt on the value of being autonomous is to suggest that the Mandarin casts doubt on the value of being moral, which it certainly does not. Similarly, in a Kantian context it cannot be the case that "if we were true, autonomous individuals... we would have killed an unknown person," since a Kantian autonomous individual is moral and would not kill an innocent person. In the existentialist context, however, things are different; there, autonomy is not tantamount to morality, and the suggestion that we should be autonomous creates many risks. When reading Gilead, I at times felt that he was examining certain expressions that had originally been set in an existentialist context, where autonomy does not equal morality, as though they had been presented in a Kantian one.Gilead also seems to overstate my position on the ability to behave morally when social supervision is revoked. He suggests that, in my [End Page 158] view, "in order to behave morally, respecting the moral law or value is certainly less than enough" and that I regard as "self-deception or simply wishful thinking... any attempt at thinking... that other motives (such as the respect for the moral law) are sufficient to behave morally." However, I do not make such strong claims. I do not think that respecting the moral law is insufficient for behaving morally, only that many (even if not all) will not choose to mind the moral law. Nor do I think that, when there is no external social supervision, it is impossible to remain moral, or that no one can successfully withstand the temptation that the Mandarin presents, or that any view that people are able to withstand the temptation and act morally without social supervision is self-deception. However, I do believe that the number of people who can withstand the temptation, and act morally when there is no social supervision, is significantly smaller than we like to think. Unfortunately, if complete secrecy is guaranteed and the return value for wrongdoing is sufficiently high, many, including "nice" people like you and me, who condemn immoral behavior and are considered by others and themselves as perfectly decent, will either kill an innocent person or have significant difficulties in refraining from doing so.But I should point out, first, that this is an empirical, objective question, and thus not in line with the subjectivist, personal character of much of the original paper. Second, I should note that while I cannot present a satisfactory empirical proof for my assessment, there is nevertheless some basis for it: too many people, upstanding people like you and me, told me time and again that they would accept the Mandarin deal, or at least find it difficult to resist it. Gilead thinks that only very few would so respond, but does not present any empirical evidence for this claim, not even as limited and unscientific as mine, stating that he simply believes "the opposite to be the case." He also feels the suggestion to be annoying. I agree that it is... (shrink)

First English-language translation of Lev Shestov's early writings on faith, ancient philosophy and biblical revelation.

This article develops the concept of experiential careers, drawing theoretical attention to the routinization and de-routinization of specific experiences as they unfold over social career trajectories. Based on interviews and ethnographic fieldwork in two religious communities, we compare the social-temporal patterning of religious experience among newly religious Orthodox Jews and converted Muslims in two cities in the United States. In both cases, we find that as newly religious people work to transform their previous bodily habits and take on newly prescribed (...) religious acts, the beginning of their religious careers becomes marked by what practitioners describe as potent religious experiences in situations of religious practice. However, over time, these once novel practices become routinized and religious experiences in these situations diminish, thus provoking actors and institutions in both fields to work to re-enchant religious life. Through this ethnographic comparison, we demonstrate the utility of focusing on experiential careers as a sociological unit of analysis. Doing so allows sociologists to use a non-reductive phenomenological approach to chart the shifting manifestations of experiences people deeply care about, along with the patterned enchantments, disenchantments, and possible re-enchantments these social careers entail. As such, this approach contributes to the analysis of social careers and experiences of “becoming” across both religious and non-religious domains. (shrink)

In his “Reply to Iddo Landau,” Edmund Wall responds to the author’s critique of some of the views expressed in his “Sexual Harassment and Wrongful Communication.” The present article concentrates on what the author takes to be the main problem in Wall’s definition: by requiring that any act, even if intentional and cruel in nature, needs to be repeated to count as sexual harassment, Wall allows too much leeway and renders permissible a wide range of intentional, mean, and harmful (...) actions that most, including, the author believes, Wall himself, would like to outlaw. The article considers Wall’s linguistic and nonlinguistic responses to this critique and finds them problematic. (shrink)

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity Π1 0 and, similarly, for any class Π n 0 . We provide a more general version of the (...) fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣ n , IΣ n − , IΠ n − and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem. (shrink)

The article presents a social phenomenology of naturalism. Starting from Stefan Bargheer’s Moral entanglements (2018), it argues that to understand the transformations of naturalist practices, we have to focus both on the shifting typifications of activity and their organizational moorings, but also on the experiential affordances of practice. Drawing on the work of Schutz and Merleau-Ponty, I focus on the transformation of animals from background into figure, the peculiar province of meaning that naturalist practice entails, as well as the experience (...) of “play” that Bargheer highlights. Doing so, I argue that the affordances of experience are a constitutive aspect of any historical account of practice: that a phenomenological approach is crucial not only for the micro-sociology of interaction and experience, but for understanding larger historical processes and transformations. (shrink)

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has been (...) done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists. (shrink)

The variety of feminist thought has produced many fruitful discussions and debates. Liberal, radical, postmodern, psychoanalytic, and other feminists have criticized each others' work and underlying presuppositions. The aim of this paper is to point out a prejudice which has not yet received sufficient attention, although it lies at the base of a fair amount of feminist research: the bias that whereas men are bad and aggressive, women are good and peaceful. Although as an explicit view this contention has been (...) debated, its distortive influence on feminist research has so far remained undiscussed. In what follows I present examples of the effect of this bias on feminist research . Then I suggest why the bias has not yet been discussed . Finally I argue that feminist research should be as critically received and treated as any other. (shrink)

We set ourselves ends and strive to achieve them. We hope that their attainment will improve our condition. The closer we get to our goals, the happier we feel. Paradoxically, however, when we finally do achieve them our joy is sometimes diminished. We have a sense of insignificance and emptiness, and we feel that in attaining our goal we have lost the meaningfulness and balance we experienced while we were striving towards it. In some ways, it seems to us, the (...) struggle is more gratifying than the achievement of the end. (shrink)

Revolutions often retain more characteristics of the pre-revolutionary state than their makers like to admit. Characterizing the pre-revolutionary state as bad , and wishing to accentuate the greatness of their doings, revolutionaries like to stress the differences between the previous state of affairs and the new one, and prefer to see the similarities as few and insignificant. They are frequently wrong. 1.

Based on interviews with climate-change activists and NGO workers in Finland and Malawi, this article reconsiders the ways in which the coordination of identity projects and action is approached in social movement scholarship. Rather than beginning with personal and collective identities, we take our cue from recent work by Laurent Thévenot and trace actors’ forms of engagement—the various ways actors produce commonality. As we show, doing so in vastly different social contexts allows us to see permutations in such forms afforded (...) by participation in a transnational social movement and to identify patterns of collective action that we would otherwise be apt to miss. Finnish activists narrated their activities by way of engaging in the forms of the common good driving the climate movement, but coordinated various situations also through engagement in familiarity, comfort, and ease. Malawian activists and NGO employees also spoke of the common good the movements worked to achieve, but principally created common ground by engaging in shared individual choices and projects, which were jointly consecrated by fellow NGO participants. Ultimately, we argue that tracing forms of engagement enables more in-depth understanding of what is at stake when people act together in social movement organizations: moving away from collective and personal identity to patterns of engagement allows a vantage point into the processes through which commonality is created and generates new hypotheses regarding the coordination of action in social movement organizations. (shrink)

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n<ω corresponds to its restriction to arithmetical Πn+1-formulas. This (...) calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time. (shrink)

The Jehovah’s Witnesses cases of the late 1930s and early 1940s presented some of the first instances of American Supreme Court’s attempts to grapple with the challenges of a multicultural society. Taken as a whole, these cases represented a favorable position towards minorities’ claims, even to some extent a path breaking one. The Jehovah’s Witnesses cases were a precursor of the Court’s growing involvement in the protection of minorities’ rights, which colored the entire second half of the 20th century. They (...) further introduced a new language, and new judicial forms into constitutional jurisprudence—the language of balancing and balancing tests. In all these aspects the Jehovah’s Witnesses cases seem to have shown the early sings of multicultural ideology in Supreme Court jurisprudence. However, not all Jehovah’s Witnesses cases showed the same kind of judicial willingness to protect minorities’ interests from the will of the majority, and not all involved the new judicial rhetoric of balancing. What explains these different judicial responses in cases which are similar in their facts and close to each other in time? In this Article I will attempt to distinguish between three types of Jehovah’s Witnesses cases and argue that the different judicial responses in each of them indicates a different structure of the multicultural conflict, and a different structure of the multicultural claims in each of them. (shrink)

The Israeli Supreme Court has become increasingly polarized between liberal and conservative judges. This phenomenon is relatively new to the Israeli Supreme Court and follows the much older and more well-known example of the U.S. Supreme Court. This article surveys both U.S. and Israeli court polarization and shows the history, reasons, and special features of polarization of both courts, including the important differences between them. It also adds a distinction to existing literature on court polarization—the distinction between court polarization and (...) politicization, and especially the distinction between “regular” polarization and “one-side” polarization. Regular court polarization happens when the court is divided roughly equally between the two main political contenders. One-side polarization happens when there is a clear advantage to one side of the political map in the make-up of the court. The current U.S. Supreme Court, with six conservative justices and only three liberal justices, is an example of one-side court polarization that is tilted to the right. The current Israeli Supreme Court is an example for one-side court polarization tilted to the left; there are 10 liberal judges and only five conservative judges on the court. The article’s main argument is that one-side court polarization presents a particularly vexing problem for the legitimacy of the court, even more so than “regular” polarization. The article concludes with a survey of possible solutions to help Israel retract itself from one-side court polarization. (shrink)

_ Source: _Volume 26, Issue 1, pp 67 - 93 The talmudic sages granted the legal status of _sefer_ to five texts: the Torah, _tefillin_, the _get_, the _mezuzah_, and the Scroll of Esther. These texts share two features: they have a ritualistic format and use, and they are the only sacred texts that demonstrate _mise en abyme_—the trait of literary self-containing. These two traits turn the rabbinic book into a radical case of “open work”: the _sefer_ consists of both (...) textual signs and the actual body of an empirical reader; its pragmatic level is not bracketed out in favor of the semiotic one. I argue that Deleuze’s reception theory best accounts for the _sefer_. (shrink)

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...) T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (shrink)

Prominent thinkers such as Jurgen Habermas and Michael Sandel are warning that biomedical enhancements will undermine fundamental political values. Yet whether biomedical enhancements will undermine such values depends on how biomedical enhancements will function, how they will be administered and to whom. Since only few enhancements are obtainable, it is difficult to tell whether these predictions are sound. Nevertheless, such warnings are extremely valuable. As a society we must, at the very least, be aware of developments that could have harmful (...) consequences. Indeed, if important values were to be jeopardized, we should take appropriate measures to protect them. This paper focuses on four central values: solidarity, personal responsibility, equality and autonomy. It delineates the conditions under which biomedical enhancements would undermine these values. It also details the circumstances under which these values would be unaffected by enhancements as well as those under which they would be promoted. Specifying these conditions is valuable; it would enable society to prepare appropriate ethical guidelines and policy responses in advance. (shrink)

ABSTRACTIn this paper I shall tackle the problem of differentiating Deleuze and Derrida. Various writers have done so, comparing these philosophers’ conceptions of repetition and difference. I shall attempt to enrich, sharpen and sometimes criticize these writers by exploring the paradigm through which Deleuze and Derrida have reflected upon repetition and difference in the first place: the mise en abyme, a literary concept designating a work that doubles itself within itself. I shall argue that Derrida applied to his theory of (...) difference a degenerated picture of the mise en abyme, which I shall term the “lacunal”, and which would cause him, against himself, to grasp the concept of difference “logocentrically”. On the other hand, Deleuze – attending to the actual emblem and to what literary theorists had to say about it – drew its correct lesson: the Other is less that which challenges Totality than the very coexistence between the Other-than-Totality and the Totality it challenges. (shrink)

Analyzes principal forces that determined the direction of Camus' thought on ethics and political values.

The main focus of the book is the presentation of the 'inertial' view of population growth. This view provides a rather simple model for complex population dynamics, and is achieved at the level of the single species without invoking species interactions. An important part of this account is the maternal effect. Investment of mothers in the quality of their daughters makes the rate of reproduction of the current generation depend not only on the current environment, but also on the environment (...) experienced by the previous generation. (shrink)

Is life meaningless? Does life have enough meaning to make it feel worthwhile? If we think our lives lack meaning, what can we do about it? Finding Meaning in an Imperfect World answers these and other difficult questions, while confronting head-on famous, recurrent theories that insist on life's meaninglessness. Landau shows us how to single out what is meaningful, explains why we sometimes fail to recognize meaning, and suggests ways in which we can resensitize ourselves to it.

I will argue that the underlying rationale for the talmudic list of trades disqualified from legal testimony is aesthetic. These trades involved professional mimicry, which as such incapacitated what R. Neis has termed “homovisuality” or self-referential witnessing in the Talmud. Reading talmudic laws of conjoined testimony and the induction of witnesses in light of Deleuze’s and Blanchot’s philosophy, I will argue that homovisuality entailed the witness’s reincarnation as the subject of the event, thus re-signifying rather than reporting the event. The (...) judge, transformed into a witness, could capture the truth of the event at a glance, in a manner both prior to and irreducible to trial procedures. (shrink)

A well-known result states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑n formulas is equivalent to the uniform reflection principle for ∑n + 1 formulas . We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑n formulas is equivalent to ω times iterated ∑n reflection principle. Moreover, for k < ω, k (...) times iterated ∑n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of ∑n induction rule.The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory by k nested applications of ∑n or Πn induction rules. In particular, the closure of a theory T under just one application of ∑1 induction rule is equivalent to T together with ∑1 reflection principle for each finite Π2 axiomatized subtheory of T.These results have closely parallel ones in the theory of subrecursive function classes. The rules under study correspond, in a canonical way, to natural closure operators on the classes of provably recursive functions. Thus, ∑1 induction rule precisely corresponds to the primitive recursive closure operator, and ∑1 collection rule, introduced below, corresponds to the elementary closure operator. (shrink)

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

Mise en abyme is a term from literary theory denoting a work that doubles itself within itself, for example a story placed within a story or a play within a play. Proliferating in experimental fiction in midcentury France, this technique had a strong impact on contemporary literary theory, but also, as this book project argues, on post-Heideggerian and post-structuralist philosophy. The Little Crystalline Seed focuses on how three of these thinkers invoke the concept of mise en abyme in order to (...) establish ontologies that deviate from that of Heidegger. Iddo Dickmann demonstrates how the concept served in modeling Derrida's logic of supplementarity, Blanchot's philosophy of "ambiguity" and mechanism of Désouvrement, and Deleuze's philosophy of difference, time and repetition. Exploring the interpretative and generative potential of the mise en abyme for continental thought, Dickmann illustrates points of resonance between various philosophical topics such as aesthetics, ethics, time, logic, mirroring, play, and signification. (shrink)

I shall show that Levinas’s idea of infinite responsibility draws on Blanchot’s mechanism of “worklessness” which in turn explicitly draws on Gide’s mechanism of retroaction and the mise en abyme—a story that doubles itself within itself—which the latter accounts for. However, a false picture of mise en abyme and worklessness brought Levinas to two interrelated misconceptions. First, of the act of responsibility as inherently futile. Second, of repetition as “mechanical,” comprising instances which are allocated to pre-established loci upon a plenitude: (...) A totalitarianism which fits badly with Levinas’s pursuit of otherness. Deleuze, on the other hand, drawing the correct lesson from mise en abyme, established that an instance of repetition—reconstituting all previous ones—generates the very ground upon which it emerges. Accordingly, in a Deleuzian ethics, the absolute Other has no existence prior to the act of attending his needs. The act is therefore “fated to succeed.”. (shrink)

This article compares the sections on the qualities desirable in a wife in Abū Ḥāmid al-Ghazālī’s Iḥyāʾ ʿulūm al-dīn and Bar Hebraeus’s Ethicon, which the West Syrian writer modeled on al-Ghazālī’s work. The article first establishes that al-Ghazālī based his profile of the ideal wife on a jurisprudential discussion of the topic by his teacher, Imām al-Ḥaramayn al-Juwaynī, expanding it, however, by adding anecdotes from Sufi literature and reasoned arguments on how “the good wife” will best facilitate her husband’s devotion (...) to God. The article then moves to consider how Bar Hebraeus appropriates, reconfigures, and reframes as Christian teaching al-Ghazālī’s discussion. Finally, the article argues that the similarity between the two texts demonstrates that both carve out a particular notion of male piety intelligible and resonant in their different religious traditions and in the broader monotheistic culture of the premodern Middle East. (shrink)

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . First, we isolate a certain subsystem inlMMLBox (...) of inlMMLBox that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for inlMMLBox are defined as the limits of chains of finite expansions of models for inlMMLBox . The techniques involves unions of n -elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic. (shrink)

We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...) of GLP Λ. (shrink)

By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by Sieg (...) [23] and Hájek [9]. We also give a new proof of independence of $B\Sigma_{n+1}$ over $I\Sigma_n$ by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8]. (shrink)

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using interpolation we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on interpolants (...) in this fragment.We then apply these results to extend and improve previous results on multilinear proofs , as studied in [Ran Raz, Iddo Tzameret, The strength of multilinear proofs. Comput. Complexity ]. Specifically, we show the following:• Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations.• Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the functional pigeonhole principle. The latter are different from previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system. (shrink)