This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in (...) no better position than the axiomatic calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (shrink)

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for (...) the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLPΛ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLPΛ for the class of provability ambiances based on Icard polytopologies. (shrink)

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the (...) standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)

The purpose of this paper is to present a formalization of the language, semantics and axiomatization of justification logic in Coq. We present proofs in a natural deduction style derived from the axiomatic approach of justification logic. Additionally, we present possible world semantics in Coq based on Fitting models to formalize the semantic satisfaction of formulas. As an important result, with this implementation, it is possible to give a proof of soundness for $\mathsf{L}\mathsf{P}$ with respect to Fitting models.

The observed complexity of nature is often attributed to an intrinsic propensity of matter to self-organize under certain (e.g., dissipative) conditions. In order better to understand and test this vague thesis, we define complexity as “logical depth,” a notion based on algorithmic information and computational time complexity. Informally, logical depth is the number of steps in the deductive or causal path connecting a thing with its plausible origin. We then assess the effects of dissipation, noise, and spatial and other symmetries (...) of the initial conditions and equations of motion on the asymptotic complexity-generating abilities of statistical-mechanical model systems. We concentrate on discrete, spatially-homogeneous, locally-interacting systems such as kinetic Ising models and cellular automata. (shrink)

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has (...) a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

This paper presents a natural deduction system for orthomodular quantum logic. The system is shown to be provably equivalent to Nishimura’s quantum sequent calculus. Through the Curry–Howard isomorphism, quantum $$\lambda $$ -calculus is also introduced for which strong normalization property is established.

The paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL+, is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL+, instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different (...) introduction rule for the exponential, !I+, which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph “Natural Deduction”.It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL+; in particular one can formulate a notion of strong validity permitting a proof of strong normalization.The conversion rules for ILL explicitly mentioned in the paper by Benton et al. do not suffice for normal forms with subformula property, but we can show that this can be remedied by addition of a special permutation conversion plus some “satellite” permutation conversions.Some discussion of the categorical models which might correspond to ILL+ is given. (shrink)

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions (...) and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional. (shrink)

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid to which properties of theories result in the presence of which rules of inference, and to restrictions on the sets of formulas to which the rules may be employed, (...) restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules (...) for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but (...) also a means of modelling proofs as well as provability. (shrink)

We present a multi-conclusion natural deduction calculus characterizing the dynamic reasoning typical of Adaptive Logics. The resulting system AdaptiveND is sound and complete with respect to the propositional fragment of adaptive logics based on CLuN. This appears to be the first tree-format presentation of the standard linear dynamic proof system typical of Adaptive Logics. It offers the advantage of full transparency in the formulation of locally derivable rules, a connection between restricted inference-rules and their adaptive counterpart, and the formulation (...) of abnormalities as a subtype of well-formed formulas. These features of the proposed calculus allow us to clarify the relation between defeasible and multiple-conclusion approaches to classical recapture. (shrink)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? This (...) informal question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation. (shrink)

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness (...) and proof normalization. We have implemented our work in the Isabelle Logical Framework. (shrink)

I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model (...) on causal explanations. (shrink)

In a previous paper we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models of polymorphic type theory. A different approach to extract proof-theoretic properties of natural deduction derivations was proposed in a recent series of papers on the basis of an embedding of intuitionistic propositional logic (...) into a predicative fragment of System F, called atomic System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification in terms of parametricity. (shrink)

Conceptions deducing the rules of behaviour from nature are analysed on the basis of a model involving three assumptions: the existence of nature, the deductibility of rules from nature, and natural rules as a criterion of valuation. Three basic constructions of nature are distinguished: „real nature”, two kinds of „ethical nature”, „logical nature”.

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.This paper introduces a typed functional language Λ! and a categorical model for it.The terms of Λ! encode (...) a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of Λ! are built out of two disjoint sets of variables. Moreover, the λ-abstractions of Λ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of Λ!, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language Λ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction.The categorical model induces operational equivalences like, for example, a set of extensional equivalences.The paper presents also an untyped version of Λ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from Λ! all the good computational properties and enjoys also the Principal-Type Property. (shrink)

One of the most debated questions in psychology and cognitive science is the nature and the functioning of the mental processes involved in deductive reasoning. However, all existing theories refer to a specific deductive domain, like syllogistic, propositional or relational reasoning.Our goal is to unify the main types of deductive reasoning into a single set of basic procedures. In particular, we bring together the microtheories developed from a mental models perspective in a single theory, for which we provide a formal (...) foundation. We validate the theory through a computational model (UNICORE) which allows fine‐grained predictions of subjects' performance in different reasoning domains.The performance of the model is tested against the performance of experimental subjects—as reported in the relevant literature—in the three areas of syllogistic, relational and propositional reasoning. The computational model proves to be a satisfactory artificial subject, reproducing both correct and erroneous performance of the human subjects. Moreover, we introduce a developmental trend in the program, in order to simulate the performance of subjects of different ages, ranging from children (3–6) to adolescents (8–12) to adults (>21). The simulation model performs similarly to the subjects of different ages.Our conclusion is that the validity of the mental model approach is confirmed for the deductive reasoning domain, and that it is possible to devise a unique mechanism able to deal with the specific subareas. The proposed computational model (UNICORE) represents such a unifying structure. (shrink)

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same (...) properties of the natural deduction counterpart ? that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. (shrink)

We investigate two different broad traditions in the abstract valuational model theory for nontransitive and nonreflexive logics. The first of these traditions makes heavy use of the natural Galois connection between sets of valuations and sets of arguments. The other, originating with work by Grzegorz Malinowski on nonreflexive logics, and best systematized in Blasio et al. : 233–262, 2017), lets sets of arguments determine a more restricted set of valuations. After giving a systematic discussion of these two different traditions (...) in the valuational model theory for substructural logics, we turn to looking at the ways in which we might try to compare two sets of valuations determining the same set of arguments. (shrink)

This article outlines a theory of naive probability. According to the theory, individuals who are unfamiliar with the probability calculus can infer the probabilities of events in an extensional way: They construct mental models of what is true in the various possibilities. Each model represents an equiprobable alternative unless individuals have beliefs to the contrary, in which case some models will have higher probabilities than others. The probability of an event depends on the proportion of models in which it occurs. (...) The theory predicts several phenomena of reasoning about absolute probabilities, including typical biases. It correctly predicts certain cognitive illusions in inferences about relative probabilities. It accommodates reasoning based on numerical premises, and it explains how naive reasoners can infer posterior probabilities without relying on Bayes's theorem. Finally, it dispels some common misconceptions of probabilistic reasoning. (shrink)

Parallel lives is an ontological model of nature in which quantum mechanics and special relativity are unified in a single universe with a single space-time. Point-like objects called lives are the only fundamental objects in this space-time, and they propagate at or below c, and interact with one another only locally at point-like events in space-time, very much like classical point particles. Lives are not alive in any sense, nor do they possess consciousness or any agency to make decisions—they are (...) simply point objects which encode memory at events in space-time. The only causes and effects in the universe occur when lives meet locally, and thus the causal structure of interaction events in space-time is Lorentz invariant. Each life traces a continuous world-line through space-time, and experiences its own relative world, fully defined by the outcomes of past events along its world-line, which are encoded in its external memory. A quantum field comprises a continuum of lives throughout space-time, and familiar physical systems like particles each comprise a sub-continuum of the lives of the field. Each life carries a hidden internal memory containing a local relative wavefunction, which is a local piece of a pure universal wavefunction, but it is the relative wavefunctions in the local memories throughout space-time which are physically real in PL, and not the universal wavefunction in configuration space. Furthermore, while the universal wavefunction tracks the average behavior of the lives of a system, it fails to track their individual dynamics and trajectories. There is always a preferred separable basis, and for an irreducible physical system, each orthogonal term in this basis is a different relative world—each containing some fraction of the lives of the system. The relative wavefunctions in the lives’ internal memories govern which lives of different systems can meet during future local interactions, and thereby enforce entanglement correlations—including Bell inequality violations. These, and many other details, are explored here, but several aspects of this framework are not yet fleshed out, and work is ongoing. (shrink)

We present a version of arithmetic in all finite types based on a systematic use of an internally definable notion of observational equivalence for dealing with equalities at higher types. For this system both intensional and extensional models are possible, the deduction theorem holds and the soundness of the Dialectica interpretation is provable inside the system itself.

For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for (...) arbitrary $\Lambda$. More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough. (shrink)

In this note we study several topics related to the schema of local reflection \\) and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with \-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema we give a new model-theoretic proof of the \-conservativity of uniform \-reflection over relativized local \-reflection. We also study the proof-theoretic strength of Feferman’s theorem, i.e., the (...) assertion of 1-provability in S of the local reflection schema \\), and its generalized versions. We relate this assertion to the uniform \-reflection schema and, in particular, obtain an alternative axiomatization of \. (shrink)

In this paper it is shown that addition of certain reductions to the standard cut removing reductions of deductions in prepositional logic makes prepositional logic non-normalizable. From this follows that some provable propositions in prepositional logic has no direct proof.

This study examines the botanical and chymical investigations Nehemiah Grew conducted for his magnum opus, The Anatomy of Plants (1682), and explores how they informed his political economic theory, as documented in the unpublished manuscript The Means of a Most Ample Increase of the Wealth and Strength of England (1707). While several scholars have argued that Grew's political economy is best described as mercantilist, this article argues for a much more multifaceted and idiosyncratic reading of Grew's political economy, which aligned (...) with or prefigured not only mercantilism, but also aspects of cameralism, physiocracy, Colbertism, and some of the utopian ideologies of Samuel Hartlib and his circle. By focusing on his knowledge of plant anatomy and plant chymistry, this paper shows that Grew developed a sophisticated understanding of local English ecologies and the natural resource management necessary to maintain a financially strong state. It also reveals an economic thinker deeply skeptical of empire, who was, instead, in favor of national economic self-sufficiency through domestic mineral extraction, agrarian improvement, and a highly regulated market. This article focuses on Grew's development of saline chymistry, his geometric approach to both plant anatomy and geography, his investigations of mineral soils, and his research into tree growth and timber management to demonstrate the deep connections between his botany and his economics. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual (...) finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

How do people make deductions? The orthodox view in psychology is that they use formal rules of inference like those of a “natural deduction” system.Deductionargues that their logical competence depends, not on formal rules, but on mental models. They construct models of the situation described by the premises, using their linguistic knowledge and their general knowledge. They try to formulate a conclusion based on these models that maintains semantic information, that expresses it parsimoniously, and that makes explicit something not (...) directly stated by any premise. They then test the validity of the conclusion by searching for alternative models that might refute the conclusion. The theory also resolves long-standing puzzles about reasoning, including how nonmonotonic reasoning occurs in daily life. The book reports experiments on all the main domains of deduction, including inferences based on prepositional connectives such as “if” and “or,” inferences based on relations such as “in the same place as,” inferences based on quantifiers such as “none,” “any,” and “only,” and metalogical inferences based on assertions about the true and the false. Where the two theories make opposite predictions, the results confirm the model theory and run counter to the formal rule theories. Without exception, all of the experiments corroborate the two main predictions of the model theory: inferences requiring only one model are easier than those requiring multiple models, and erroneous conclusions are usually the result of constructing only one of the possible models of the premises. (shrink)

Boolean-valued models for first-order languages generalize two-valued models, in that the value range is allowed to be any complete Boolean algebra instead of just the Boolean algebra 2. Boolean-valued models are interesting in multiple aspects: philosophical, logical, and mathematical. The primary goal of this paper is to extend a number of critical model-theoretic notions and to generalize a number of important model-theoretic results based on these notions to Boolean-valued models. For instance, we will investigate (first-order) Boolean valuations, which are (...) class='Hi'>natural generalizations of (first-order) theories, and prove that Boolean-valued models are sound and complete with respect to Boolean valuations. With the help of Boolean valuations, we will also discuss the Löwenheim-Skolem theorems on Boolean-valued models. (shrink)

Lance Rips describes a unified theory of natural deductive reasoning and fashions a working model of deduction, with strong experimental support, that is capable of playing a central role in mental life.

Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally (...) equivalent but not bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones. (shrink)

Although the use of possible worlds in semantics has been very fruitful and is now widely accepted, there is a puzzle about the standard definition of validity in possible-worlds semantics that has received little notice and virtually no comment. A sentence of an intensional language is typically said to be valid just in case it is true at every world under every model on every model structure of the language. Each model structure contains a set of possible worlds, and models (...) are defined relative to model structures, assigning truth-values to sentences at each world countenanced by the model structure. The puzzle is why more than one model structure is used in the definition of validity. There is presumably just one class of all possible worlds and just one model structure defined on this class that does correctly the things that model structures are supposed to do. (These include, but need not be limited to, specifying the set of individuals in each world as well as various accessibility relations between worlds.) Why not define validity simply as truth at every world under every model on this one model structure? What is the point of bringing in more model structures than just this one?

We investigate these questions in some detail and conclude that for many intensional languages the puzzle points to a genuine difficulty: the standard definition of validity is insufficiently motivated. We begin (Section 1) by showing that a plausible and natural account of validity for intensional languages can be based on a single model structure, and that validity so defined is analogous in important respects to the standard account of validity for extensional languages. We call this notion of validity "validity!", and in Section 2 we contrast it with the standard notion, which we call "validity2". Several attempts are made to discover a rationale for the almost universal acceptance of validity2, but in most of these attempts we come up empty-handed. So in Section 3 we investigate validity! for some intensional languages. Our investigation includes providing axiomatizations for several propositional and predicate logics, most of which are provably complete. The completeness proofs are given in the Appendix, which also contains a sketch of a compactness proof for one of the predicate logics. (shrink)

continent. 1.1 (2011): 22-25. The Author gratefully acknowledges the financial support of CAPES (Coordenação de Aperfeiçoamento do Ensino Superior), Brazil. From the multifarious subdivisions of semiotics, be they naturalistic or culturalistic, the realm of semiotics of value is a ?eld that is getting more and more attention these days. Our entire political and economic systems are based upon structures of symbolic representation that many times seem not only to embody monetary value but also to determine it. The connection between monetary (...) and linguistic exchanges is self-evident: the former requires the latter and develops in direct relation to it. Creative experimentation and design of digital systems of value exchange are blossoming on the web. Dr. Gilson Schwartz, an economist turned media theorist and professor at the University of São Paulo, adopted the concept of Iconomics, originally created by Michael Kaplan, an author concerned with the linguistics of economic value. Empowering local communities and unlocking new levels of value creation and representation via digital technologies are the main goals of Dr. Schwartz’s projects, aiming at the re-designing of our relationship to the economic value of imagination and the social control of property. Schwartz is Assistant Professor at the School of Communication and Arts of the University of São Paulo, a former Chief Economist at BankBoston, Brazil and Advisor to the President of the National Social and Economic Development Bank (BNDES). FURTHER READING: Kaplan, Michael. “Iconomics: The Rhetoric of Speculation.” Public Culture . 15.3 (2003): 477-493. Renata Lemos-Morais: You have recently produced and directed a short documentary about Creative Currencies in Latin America. Could you tell us a bit about its process and its findings? Gilson Schwartz: The “Creative Currencies” project is a work-in-progress platform which unfolds as an action research agenda connected both to the production of audiovisual content and the development of social currency software. The initiative dates back to 2003 when I led an experimental project supported by the Presidency of Brazil. At that time, we issued paper currency in a small, touristic village in the Northeast Region which stimulated local cultural projects. But it was only in 2009 that the Central Bank of Brazil acknowledged “social currencies” as a legitimate economic agenda, calling for more debate at the I Financial Inclusion Forum. This year, the monetary authorities organized a second forum that also opened the room to discussions on mobile payment systems and new perspectives on poverty alleviation via State subsidies. The Ministry of Culture funded the “Creative Currencies” project in 2009-2010 and our next stage in this discovery process is to be supported by the National Social and Economic Development Bank (BNDES). In short, there is genuine interest among public officials in different areas and public funding for social currencies is on the rise in Brazil. However, after all these years we are still at a very early stage of research and practice. Some of the most successful initiatives (such as Banco Palmas) actually evolved out of local monetary creation to become correspondent banking operations for commercial banks and other financial groups. After eight years of price stability and social inclusion, Brazil stands out as a major opportunity for social experimentation, even the Grameen Bank is now entering the Brazilian market and many NGOs are geared towards different forms of entrepreneurialism in the base of the pyramid, riding solidarity economic models, microfinance and microcredit for local development. It is yet to be seen, however, whether these developments are just one more stage of “bankarization”, that is to say, an extension of regular banking services or actually a new form of social and symbolic self-determination at the local level. So far, the Central Bank of Brazil is open to new forms of credit and local finance as long as they are strictly territorial and very close to barter among the poor. In other words, whether the process of monetary creation could be made to fit an open source paradigm is yet to be seen. Community banking and social currencies might as well end up as just another channel for access to and use of banking services. The “Creative Currencies” project aims at promoting the discussion of more fundamental issues, such as the limits of central banking, the prospects for local financial development and the possibility of creating and managing financial icons as cultural assets. The purpose of this project is to produce short documentaries that will bear testimony to this evolving regulatory framework while inducing more discussion about the fundamental iconomic issues concealed in the process of money and wealth creation. RLM: What new potential is there in applying digital technologies to currency creation? GS: Globalization is a result of the virtualization of money, that is, the overcoming of illusions such as the “gold standard”. Money is an icon created by institutions, not on supposedly natural or material foundations but out of political and financial interests. The “dollar standard” is an outstanding evidence of this phenomenon and Kaplan´s paper, the first to use the expression “iconomics”, insists on the rhetoric and semiotic effects associated to a supposedly scientific model of money creation and management. This is a fundamental change that was perceived and discussed much before the digitalization of the world by unorthodox economists such as John Maynard Keynes since the early 20th century. The digitalization of global financial flows accelerated this immateriality and the creation of the Euro was yet another evidence of the political foundations of currencies. The banking establishment, however, is keen on the idea that scientific formulae hold the key to sound money creation. Keynesianism has been repeatedly associated to “inflationism” while the supposedly sound monetary policies of the Orthodoxy serve well the political and financial interests of a technocratic elite. Corporate media also serve this fictitious depiction of the monetary process despite the cyclical bust of monetary rules and financial regulations. The internet, however, has created numerous opportunities for the disintermediation in industries such as film, music and commodities. There is no reason to doubt that the financial industry can also be transformed by Peer-to-Peer (P2P) infrastructures. Money is media as well. Credit is but confidence. Once people realize that the distributed computing power of networks can also be the platform to weave new monetary and credit operations, there is room for grassroots emancipation out of the established owners of monetary institutions. So far, however, there has been a privatization of monetary management and there is not a clear path or model for the emergent (P2P) property democratization that is inherent to the internet. Digital assets, however, clearly have an inherent potential to escape central control and proprietary regulatory frameworks. RLM: How are social networks transforming the ways in which we exchange value? Do you believe that online influence and reputation might be translated into a new kind of currency? GS: The foundation for social currencies as envisaged by solidarity economics is the territory. Authorities are willing to concede local monetary creation as long as it is restricted to poor neighborhoods, as a proxy to barter. In this relevant but limited context, reputation, personal knowledge and informal ties form the matrix of local or “proximity” finance which are expected to keep credit and leverage to a sustainable level. Anything beyond that should and would lead to an integration of local finance into the established banking system. However, inasmuch as the internet promotes virtual territories and reputation is now subject to all sorts of digital manipulation and stardom becomes an everyday cultural process among teenagers and elders alike, the territorial “foundation” is substituted for more complex patterns of solidarity, cooperation and exchange that reach beyond the physical territory and even the “human.” Digital assets embody knowledge, technology, cultural and educational values that are exchanged at a global scale beyond the control of democratic States (Chinese-like intervention is the exception, not the rule). Hacktivism is unstoppable and so is monetary hacktivism. Especially in the Third World, mobile communications fast became conveyors of money transfers and other appropriation strategies. Banking and telecom sectors already battle for the control of these emerging markets and, as a matter of fact, have so far contributed to the containment of these processes. Governments serve these moguls and have usually adopted a wait-and-see attitude. If open source hardware and grassroots social movements combine to challenge these proprietary battles, then social networks may evolve into new forms of social capital and thus form the ground on which to design and implement social currencies. The global financial crisis, as well as the dollar demise, are an event of such a proportion that social networks may open the path for emancipator subjects to seize the opportunity. RLM: What are some of the social experiments that you think are revolutionizing the way in which value is exchanged? GS: There is a broad evolution of economic systems towards the valuation of intangible assets such as knowledge, reputation and sustainability. There are many examples of local as well as global events that call the traditional systems of exchange into question, from carbon credits to educational bonuses, digital barter frameworks and locally based solidarity and fair trade networks. These emerging examples, no matter how different in nature or scale, represent a challenge to the traditional value creation schemes and theories which are based on use and exchange value, supply and demand. This is not to say that utility and labour or scarcity are to be totally dismissed, but there is now an emerging perception that value is also a function of behavioral codes, symbolic patterns and the energy of institutional frameworks. The plasticity of digital platforms is relevant insofar as intellectual property becomes the central source of value creation. But when you discuss the property rights of public goods such as the environment or basic social rights (of minorities or localities), then a new paradigm seems to emerge. RLM: Are gaming and online storytelling digital practices that might have an impact on the way new currencies are created and developed? GS: “Play money” has been one of the early and most intriguing phenomena associated to the diffusion of virtual worlds. These virtual currencies have also been prone to “boom and bust” dynamics, speculation and pure theft (Second Life scams were common and led to “intervention” by the owners of the game). Storytelling is a major source of value, that is, the value of attention and the markets for audience. There are plenty of private monies created and managed by corporate entertainment groups, telecom operators (via mobile virtual network operators) and marketing frameworks. The casino economy, as already depicted by Keynes and other unorthodox economists, is always storytelling about future states of the world, confidence building and leverage of animal spirits. The novelty of current events is not the fictitious nature of any form of capital, but the potential for digital democratization of controls over the imaginary nature of value creation. But this is just a potential, a possibility, a figment that more often than not is appropriated by centralized and opaque private powers. The game industry is a testbed for new connections between storytelling and money creation. RLM: Monetary value was originally connected to the scarcity of precious metals, such as gold. Our entire monetary system seems therefore to function according to a logic of scarcity. Do you believe it is possible to reverse this logic into a logic of abundance? GS: The gold standard was a fiction itself, the “scarcity” is always produced by institutions that regulate confidence and access to credit. The key issue is not to fabricate abundance, but to question the institutions which produce scarcity out of any standard, gold or whatever. Central banks are at the service of private banks in the creation and management of scarcity. RLM: The promise that nanotechnology holds for our future is one of radical abundance (or so say transhumanists). The possibility of creating any kind of material substance through nanoengineering seems each day more feasible. What would this possibility entail in terms of systems of value exchange? GS: Nanotechnology, environmental concerns and grassroots appropriation of distributed knowledge are the frontiers of a new horizon of energy creation and management. Social groups in charge of managing scarcity for the sake of wealth and power concentration must be held accountable for the destruction of our future. Once this key political conundrum is accounted for, new systems of value exchange and even a non-mercantile society will come into existence. Revolutionary theorists such as Marx envisaged transformation via the extreme clash of contradictions between capital and labour. This approach has never led to a change in the systemic logic of scarcity for the sake of social control. We are once again facing the threshold of large scale societal change, but the outcome of the current crisis should come from our imagination, not from yet another paradox of abundance and scarcity. As long as we remain attached to the materiality of value, the illusion of freedom will but reinforce the massive manipulation of necessity. RLM: What do you think is the role of crowdfunding in this context? GS: Crowdsourcing as well as crowdfunding are examples of the peer to peer opportunities that come along with distributed computer power. However, the power structure behind wealth and scarcity creation may or may not be changed by crowd management. But it should not be taken at face value, so to speak. The stock market of course is an early form of crowdfunding without calling into question the proprietary framework of the companies offering their investment projects to the market. Any market is a form of crowdsourcing, that is to say, of distributed forms of supply and demand matchmaking. The wealth of networks, however, depends on the contractual ecosystem, the institutional framework, the channeling of social imagination, not on the purely quantitative distribution of bids and offers. I see economics as supply, demand and code. Once you have access to the code, once openness becomes part of the game, then there is a chance of crowd behavior becoming collective intelligence and sustainable imagination. RLM: Is the development of digital currencies headed towards a pluralistic ecology in which many micro currency systems co-exist, or is it penetrating the mainstream monetary system? If it is affecting the mainstream financial system, what would be its effect in the long-term? GS: The same issue is at stake both at local and global levels. The global system is, as a matter of fact, an unstable ecosystem of “micro currencies” whereas the dollar has so far exerted an overwhelming influence. Local currencies and private monies coexist with national currencies only to the extent that the Central Bank admits, for instance, that grassroots monetary emissions remain territorial and strictly connected to poverty. However, the question of how new developments in digital technologies and nanotechnologies might alter not only our current monetary systems but also our understanding of value in itself, is still an open question. (shrink)

This paper describes a simple, causally deterministic model of quantum measurement based on an amplitude threshold detection scheme. Surprisingly, it is found to reproduce many phenomena normally thought to be uniquely quantum in nature. To model an \(N\) -dimensional pure state, the model uses \(N\) complex random variables given by a scaled version of the wave vector with additive complex noise. Measurements are defined by threshold crossings of the individual components, conditioned on single-component threshold crossings. The resulting detection probabilities match (...) or approximate those predicted by quantum mechanics according to the Born rule. Nevertheless, quantum phenomena such as entanglement, contextuality, and violations of Bell’s inequality under local measurements are all shown to be exhibited by the model, thereby demonstrating that such phenomena are not without classical analogs. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:NARRATIVE AND EXPLANATION: EXPLAINING ANNA KARENINA IN THE LIGHT OF ITS EPIGRAPH Marina Ludwigs University ofCalifornia, Irvine In this paper, I will be examining the relation of explanation to narrative, looking briefly at the theoretical side ofthe problematic and in more detail at specific explanatory issues that arise in Tolstoy's novel Anna Karenina. Although the use itselfofthe term "explanation" is not as visible in the humanities as it is (...) in the sciences, the explanatory enterprise by other names is just as prominent. For indeed, an animating impetus behind textual interpretation is an explanation ofhowatextcan be brought into a correspondence with another text, while contemporary critical-theoretical activities of demystifying aesthetic and cultural assumptions, contextualizing historical knowledge, or laying bare ideological presuppositions are, in one way or another, engagements in the explanation ofthe workings of language. I approach humanistic explanation via a brief detour of scientific explanation, which constitutes an already established discourse with a set of defined issues and standard arguments, limiting myself to touching upon issues relevant to the narrativization of explanation. In scientific debates, explanation is defined as an answer to the question of "why?," in contradistinction to the question of "what?," which prompts a description of a phenomenon in question. Descriptive and explanatory conceptual frameworks, however, are interdependent. A physical description involves an "in-depth" understanding of underlying regularities. An explanatory answer to a question is, in its turn, an argument that logically deduces the explanandum (the description ofa phenomenon to be explained) from the explanans (well-understood regularities). This Marina Ludwigs125 formal definition of an explanation necessitates a symmetry between explanation and prediction, because the reconstructed logical sequence can also be read in the other direction, demonstrating the inevitability of what has occurred. Thus the sight of shattered glass and a baseball lying about allows me to explain the accident by invoking the law ofthe conservation ofmomentum and the low shock-resistance ofglass. But the knowledge of these regularities would have also allowed me to predict the same outcome had I seen the baseball flying in its direction ofthe window. The ontological status of "regularities" has elicited a lively debate in the scientific community pertaining to the problem ofthe "real." The standard view of regularities defines them as general laws (the kind we learn about in physics classes) that could be deductive or inductive, deterministic or statistical. A general law may be subsumed under another, even more general law, thus forming a unifying hierarchy of laws. A rival conception appeals to causal mechanisms and defines regularities as established chains oftemporal succession. Both models, the covering-law and the causal one, have been criticized: the former—for its failure to distinguish a law from an accidental generalization; the latter—for its resurrection ofthe metaphysical notion ofcausal influence and its inability to deal with probabilistic events. One of the later contenders is Nancy Cartwright's model, that proposes to conceive of regularities as so-called "nomological machines"—"fixed (enough) arrangements] ofcomponents, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of regular behavior that we represent in our scientific laws" (Cartwright 50). Cartwright claims that scientific models work on the ceteris paribus (everything else being equal) basis. That is to say they "hold only in circumscribed conditions or so long as no factors relevant to the effect besides those specified occur" (Cartwright 28), such as inside a battery, refrigerator, or a rocket—insofar as special shielding conditions are in effect (as a counterexample, imagine usingNewton's laws ofmechanics for describing the motion of a dropped crumpled dollar). Cartwright's explanatory framework is thus pragmatic and local, avoiding the twin pitfalls of the metaphysical assumptions underlying the two older models—the totalizing and unifying idea of a covering law, on the one hand, and the influence of causal "force," on the other. What is instead emphasized by her is a procedural aspect of scientific knowledge that has been historically prevalent and reflects the practical exigencies of human existence—the experience of dangers and natural obstacles to human 126Explaining 'Anna Karenina ' desires. When ill-defined impediments are analyzed and reduced to concrete difficulties, specific... (shrink)

Using as case studies two early diagrams that represent mechanisms of the cell division cycle, we aim to extend prior philosophical analyses of the roles of diagrams in scientific reasoning, and specifically their role in biological reasoning. The diagrams we discuss are, in practice, integral and indispensible elements of reasoning from experimental data about the cell division cycle to mathematical models of the cycle’s molecular mechanisms. In accordance with prior analyses, the diagrams provide functional explanations of the cell cycle and (...) facilitate the construction of mathematical models of the cell cycle. But, extending beyond those analyses, we show how diagrams facilitate the construction of mathematical models, and we argue that the diagrams permit nomological explanations of the cell cycle. We further argue that what makes diagrams integral and indispensible for explanation and model construction is their nature as locality aids: they group together information that is to be used together in a way that sentential representations do not. (shrink)

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true (...) of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi. (shrink)

ABSTRACT This work presents efficient algorithms to update knowledge bases in the presence of integrity constraints. The algorithms ensure that the changes to the knowledge bases are minimal. We use the deductive database paradigm to represent knowledge. Minimality is defined as a natural partial order over possible models of the database and expresses a preference for data explicity stored in the database over the data deduced by default. This requirement seems rational for many applications and yet it is hard (...) to be expressed with a set of integrity constraints. We handle safe negation and local variables in the bodies of the rules. We tackle the issue of modifying the active domain (i.e. adding new constants or removing some of the constants which are present in the knowledge base), and we demonstrate how interaction with the user can be used to handle recursive definitions. (shrink)

We investigate how the perceived uncertainty of a conditional affects a person's choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne. These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given (...) a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results. (shrink)

The Quantified Argument Calculus (Quarc) is a formal logic system, first developed by Hanoch Ben-Yami in (Ben-Yami 2014), and since then extended and applied by several authors. The aim of this paper is to further these contributions by, first, providing a philosophical motivation for the truth-valuational, substitutional approach of (Ben-Yami 2014) and defending it against a common objection, a topic also of interest beyond its specific application to Quarc. Second, we fill the formal lacunae left in the original presentation, which (...) did not incorporate identity systematically into Quarc, and although it proved the soundness of the system did not prove its completeness. (shrink)

Dynamic topological logic combines topological and temporal modalities to express asymptotic properties of dynamic systems on topological spaces. A dynamic topological model is a triple 〈X ,f , V 〉, where X is a topological space, f : X → X a continuous function and V a truth valuation assigning subsets of X to propositional variables. Valid formulas are those that are true in every model, independently of X or f. A natural problem that arises is to identify (...) the logics obtained on familiar spaces, such as . It [9] it was shown that any satisfiable formula could be satisfied in some for n large enough, but the question of how the logic varies with n remained open.In this paper we prove that any fragment of DTL that is complete for locally finite Kripke frames is complete for . This includes DTL○; it also includes some larger fragments, such as DTL1, where “henceforth” may not appear in the scope of a topological operator. We show that satisfiability of any formula of our language in a locally finite Kripke frame implies satisfiability in by constructing continuous, open maps from the plane into arbitrary locally finite Kripke frames, which give us a type of bisimulation. We also show that the results cannot be extended to arbitrary formulas of DTL by exhibiting a formula which is valid in but not in arbitrary topological spaces. (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)

We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules are all valid in fixed-point models; the second by ‘\-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that (...) to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth. (shrink)