In his paper "Undecidability without arithmetization," Andrzej Grzegorczyk introduces a theory of concatenation $\mathsf{TC}$. We show that pairing is not definable in $\mathsf{TC}$. We determine a reasonable extension of $\mathsf{TC}$ that is sequential, that is, has a good sequence coding.

In this paper we study essential hereditary undecidability. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation essential tolerance, or, in (...) the converse direction, lax interpretability that interacts in a good way with essential hereditary undecidability. We introduce the class of $$\Sigma ^0_1$$ Σ 1 0 -friendly theories and show that $$\Sigma ^0_1$$ Σ 1 0 -friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarily undecidable theory. (shrink)

This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models.A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.

There is a growing interest in how people conceptualise the legal domain for the purpose of legal knowledge systems. In this paper we discuss four such conceptualisations (referred to as ontologies): McCarty's language for legal discourse, Stamper's norma formalism, Valente's functional ontology of law, and the ontology of Van Kralingen and Visser. We present criteria for a comparison of the ontologies and discuss the strengths and weaknesses of the ontologies in relation to these criteria. Moreover, we critically review the (...) criteria. (shrink)

In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for T (...) . We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A . We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA . The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing. (shrink)

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...) Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)

This paper is concerned with the logical structure of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories.

The process-dissociation procedure has been used in a variety of experimental contexts to assess the contributions of conscious and unconscious processes to task performance. To evaluate whether motivation affects estimates of conscious and unconscious processes, participants were given incentives to follow inclusion and exclusion instructions in a perception task and a memory task. Relative to a control condition in which no performance incentives were given, the results for the perception task indicated that incentives increased the participants' ability to exclude previously (...) presented information, which in turn both increased the estimate of conscious processes and decreased the estimate of unconscious processes. However, the results also indicated that incentives did not influence estimates of conscious or unconscious processes in the memory task. The findings suggest that the process-dissociation procedure is relatively immune to influences of motivation when used with a memory task, but that caution should be exercised when the process-dissociation is used with a perception task. (shrink)

A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we give a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is complete Π20.

A possible way out of the conundrum of quantum gravity is the proposal that general relativity (GR) emerges from an underlying microscopic description. Despite recent interest in the emergent gravity program within the physics as well as the philosophy community, an assessment of the general motivation for this idea is lacking at the moment. We intend to fill this gap in the literature by discussing the main arguments in favour of the hypothesis that the metric field and its dynamics are (...) emergent. First, we distinguish between microstructure inspired from GR, such as through quantization or discretization, and microstructure that is not directly motivated from GR, such as strings, quantum bits or condensed matter fields. The emergent gravity approach can then be defined as the view that the metric field and its dynamics are derivable from the latter type of microstructure. Subsequently, we assess in how far the following properties of (semi-classical) GR are suggestive of underlying microstructure: (1) the metric's universal coupling to matter fields, (2) perturbative non-renormalizability, (3) black hole thermodynamics, and (4) the holographic principle. In the conclusion we formalize the general structure of the plausibility arguments put forward. (shrink)

In this paper we study the interpretations of a weak arithmetic, like Buss’s theory $\mathsf{S}^{1}_{2}$, in a given theory $U$. We call these interpretations the arithmetics of $U$. We develop the basics of the structure of the arithmetics of $U$. We study the provability logic of $U$ from the standpoint of the framework of the arithmetics of $U$. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.

In this paper we describe a framework for the construction of entities that can serve as interpretations of arbitrary contiguous chunks of text. An important part of the paper is devoted to describing stacking cells, or the proposed meanings for bracket-structures.

A Kripke model ? is a submodel of another Kripke model ℳ if ? is obtained by restricting the set of nodes of ℳ. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Łoś-Tarski theorem for the Classical Predicate Calculus.In Appendix A we prove that for theories with decidable identity we can take (...) as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In Appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that ℋT is HA. Here ℋT is the theory of all Kripke models ℳ such that the structures assigned to the nodes of ℳ all satisfy T in the sense of classical model theory. (shrink)

In the past decade, the concept of recognition appears to have acquired an important theoretical position in the work and organization literature. While in principle recognition denotes a positive and social form of freedom, in current-day organizations recognition may be often negative or instrumental. In order to capture this ambivalence in organizational recognitive conditions, the recent work of the American philosopher Judith Butler appears particularly applicable. The purpose of this paper is to explore theoretically to what extent her views on (...) recognition shed new light on the variety of recognition patterns in current-day organizations. Towards that purpose, this paper first turns to the ‘master-slave’ episode in the work of Hegel and its influential interpretation by Kojève, which are at the heart of Butler’s reading and conception of recognition. As a second step, the ambivalent conception of recognition in the work of Butler is discussed and critically assessed, while in the final section this conception is extended to work and organizations. (shrink)

This article focuses on the role of unions and their influence on economic inequality. Section 2 reviews the literature regarding the union effect on wages and wage inequality. Section 3 considers the separate contributions of union power, membership composition, bargaining coordination, and wage policy. Section 4 introduces the distinction between membership and coverage, while the following two sections discuss the impact of union power on earnings inequality when coverage is either exclusive or inclusive. Section 7 discusses the rationale for unions (...) aiming for wage compression while Section 8 estimates how the presence of trade unions has influenced the extent and dynamics of earnings inequality across advanced economies during the 1980s and 1990s. For this purpose indicators have been developed for trends in equality and union presence from different sources. Section 9 concludes. (shrink)

In this paper we introduce a notion of context for Groenendijk & Stokhof's Dynamic Predicate Logic DPL. We use these contexts to give a characterization of the relations on assignments that can be generated by composition from tests and random resettings in the case that we are working over an infinite domain. These relations are precisely the ones expressible in DPL if we allow ourselves arbitrary tests as a starting point. We discuss some possible extensions of DPL and the way (...) these extensions interact with our notion of context. (shrink)

Vardanyan's theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum's Theorem.

This article critically evaluates the possible impact of the Charter on the relationship between the Court of Justice of the European Union and national constitutional courts. While it is premature to provide a definitive assessment of the kind of collaboration that these courts will develop, it is crucial to identify a number of features of the new landscape that will influence the direction in which the relationship between the CJEU and constitutional courts will evolve. This article discusses several reasons that (...) may result in better or a higher number of judicial interactions, as well as factors that may create tension or cause problems in the relationship between the CJEU and national constitutional courts. As such, it offers a framework that may help us to understand future post-Charter judgments by these courts setting out how they conceive their engagement with their counterpart on fundamental rights issues. (shrink)

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree (...) structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. (shrink)

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same.

In this paper, we study finitely axiomatizable conservative extensions of a theoryUin the case whereUis recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature ofUthat contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extensionαofUin the expanded language that is conservative overU, there is a conservative extensionβofUin the expanded language, such that$\alpha (...) \vdash \beta$and$\beta \not \vdash \alpha$. The result is preserved when we consider eitherextensionsormodel-conservative extensionsofUinstead ofconservative extensions. Moreover, the result is preserved when we replace$\dashv$as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to witinterpretability that identically translates the symbols of the U-language.We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions ofUordered by interpretability that preserves the symbols ofU. (shrink)

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.

In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.

Page's manifesto makes a case for localist representations in neural networks, one of the advantages being ease of interpretation. However, even localist networks can be hard to interpret, especially when at some hidden layer of the network distributed representations are employed, as is often the case. Hidden Markov models can be used to provide useful interpretable representations.

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...) Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL -preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA. (shrink)

Over the past decade land has again moved to the centre of resource conflicts, agrarian struggles, and competing visions over the future of food and farming. This renewed interest in land necessitates asking the seemingly simple, but pertinent, question ‘whatisland?’ To reach a more profound understanding of the uniqueness of land, and what distinguishes land from other resources, this symposium suggests the notion of ‘land imaginaries’ as a crucial lens in the study of current land transformations. Political-economy, and the particular (...) economic, financial, or political interests of various actors involved in land projects do not directly result in, or translate into, outcomes, such as dispossession and enclosure, increased commodification, financialization, and assetization, or mobilization and resistance. All these processes are informed by different imaginaries of land—the underlying understandings, views, and visions of what land is, can, and should be—and associated visions, hopes, and dreams regarding land. Drawing on a variety of case studies from across the world, crossing Global North/South and East/West, and including contemporary and historical instances of land transformation, this symposium addresses the multifaceted ways in which implicit, explicit, and emergent understandings of land shape current land transformations. (shrink)

According to Anselm's official definition, freedom of choice is ‘the power to preserve rectitude of will for the sake of that rectitude itself.’ From the point of view of contemporary metaphysics, this is one of the most unhelpful definitions imaginable. Does such freedom require alternative possibilities, for example? Is it compatible with causal determination? Is the exercise of such freedom a necessary and sufficient condition for moral responsibility? The definition sheds no light on these questions.And so we need to move (...) on from Anselm's definition to Anselm's account of freedom. Here, though, we encounter the opposite problem. Where Anselm's definition seems not to answer these questions at all, Anselm's account seems to answer all these questions, sometimes with a yes and sometimes with a no. Consider the question about alternative possibilities. In De libertate arbitrii, Anselm seems clearly to deny that freedom involves alternative possibilities. God, the good angels, and the blessed dead cannot do otherwise than preserve rectitude, but they are still free- freer, in fact, than those who are capable of abandoning rectitude. (shrink)