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Journal of Applied Logic
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ISSN (Print) 1570-8683
Published by Elsevier

[2563 journals] [SJR: 1.079] [H-I: 17]

An application of Carnapian inductive logic to an argument in the
philosophy of statistics
- Abstract: Publication date: Available online 21 May 2014
  Source:Journal of Applied Logic
  Author(s): Teddy Groves
  I claim that an argument from the philosophy of statistics can be improved by using Carnapian inductive logic. Gelman and Shalizi [9] criticise a philosophical account of how statisticians ought to choose statistical models which they call ‘the received view of Bayesian inference’ and propose a different account inspired by falsificationist philosophy of science. I introduce another philosophical account inspired by Carnapian inductive logic and argue that it is even better than Gelman and Shalizi's falsificationist account.
  
  PubDate: 2014-06-14T16:15:46Z
Temporal logics for concurrent recursive programs: Satisfiability and
model checking
- Abstract: Publication date: Available online 14 May 2014
  Source:Journal of Applied Logic
  Author(s): Benedikt Bollig , Aiswarya Cyriac , Paul Gastin , Marc Zeitoun
  We develop a general framework for the design of temporal logics for concurrent recursive programs. A program execution is modeled as a partial order with multiple nesting relations. To specify properties of executions, we consider any temporal logic whose modalities are definable in monadic second-order logic and which, in addition, allows PDL-like path expressions. This captures, in a unifying framework, a wide range of logics defined for ranked and unranked trees, nested words, and Mazurkiewicz traces that have been studied separately. We show that satisfiability and model checking are decidable in EXPTIME and 2EXPTIME, depending on the precise path modalities.
  
  PubDate: 2014-06-14T16:15:46Z
Editor's note: Special issue on Combining Probability and Logic to Solve
Philosophical Problems
- Abstract: Publication date: Available online 26 May 2014
  Source:Journal of Applied Logic
  Author(s): Niki Pfeifer
  Editor's note: Special Issue on Combining Probability and Logic to Solve Philosophical Problems (Progic 2013).
  
  PubDate: 2014-06-14T16:15:46Z
Information, confirmation, and conditionals
- Abstract: Publication date: Available online 26 May 2014
  Source:Journal of Applied Logic
  Author(s): Peter Milne
  Loosely speaking, a proposition adds the more information to corpus b, the greater the proportion of possibilities left open by b that it rules out. Plausible qualitative constraints lead to the result that any measure of information-added is a strictly decreasing rescaling of a conditional probability function sending 1 to 0. The two commonest rescalings are − log P and 1 − P . In a similar vein, e is favourable evidence for hypothesis h relative to background b if h rules out a smaller proportion of the possibilities left open by b and e jointly, than left open by b alone. In terms of the underlying probability measure, this secures the familiar positive relevance conception of confirmation and that f is more favourable evidence for h than e iff h rules out a smaller proportion of the possibilities left open by b and f jointly than left open by b and e jointly. In these terms, a measure of confirmation should be a function of the information added by h to b ∧ e and to b, decreasing with the first and increasing with the second. When e = h , the possibilities that drop out as we narrow the focus with e are exactly the possibilities left open by b but excluded by h. Thus the extent to which h confirms h relative to b is a measure of the information h adds to b. Given a measure I of information added, we can think of I ( a c , b ) − I ( a , b ) as a measure of the “deductive gap”, relative to b, between a and a ∧ c . When I ( a , b ) = − log P ( a b ) , I ( a c , b ) − I ( a , b ) = − log P ( c a b ) , the amount of information the indicative conditional ‘if a then c’ adds to b on Ernest Adams' account of that conditional. When I ( a , b ) = 1 − P ( a b ) , I ( a c , b ) − I ( a , b ) = I ( a ⊃ c , b ) where a ⊃ c is a material conditional. What, if anything, can be said in general about “information theoretic” conditionals obtained from measures of information-added in this way' We find that, granted a couple of provisos, all satisfy modus ponens and that the conditionals fall victim to Lewis-style triviality results if, and only if, I ( a ∧ ¬ a , b ) = ∞ (as happens with −
  PubDate: 2014-06-14T16:15:46Z
S7
- Abstract: Publication date: December 2013
  Source:Journal of Applied Logic, Volume 11, Issue 4
  Author(s): Andrew David Irvine
  Following Halldén, we define S7 as the system generated by the addition of ◊ ◊ p to S3. Initial motivation for the extension comes from Halldénʼs paradox. In addition to resolving the paradox, the resulting system generates a helpful framework for comparing classical propositional logic (CPL) with otherwise incommensurable logics, including multi-valued logics such as L3 and paraconsistent logics such as LP. S7, although non-regular and non-normal, thus turns out to be preferable to systems such as S4 and S5 as an account of alethic modality.
  
  PubDate: 2014-04-29T07:30:36Z
Towards classifying propositional probabilistic logics
- Abstract: Publication date: Available online 6 February 2014
  Source:Journal of Applied Logic
  Author(s): Glauber De Bona , Fabio Gagliardi Cozman , Marcelo Finger
  This paper examines two aspects of propositional probabilistic logics: the nesting of probabilistic operators, and the expressivity of probabilistic assessments. We show that nesting can be eliminated when the semantics is based on a single probability measure over valuations; we then introduce a classification for probabilistic assessments, and present novel results on their expressivity. Logics in the literature are categorized using our results on nesting and on probabilistic expressivity.
  
  PubDate: 2014-04-29T07:30:36Z
Probabilities of counterfactuals and counterfactual probabilities
- Abstract: Publication date: Available online 7 February 2014
  Source:Journal of Applied Logic
  Author(s): Alan Hájek
  Probabilities figure centrally in much of the literature on the semantics of conditionals. I find this surprising: it accords a special status to conditionals that other parts of language apparently do not share. I critically discuss two notable ‘probabilities first’ accounts of counterfactuals, due to Edgington and Leitgeb. According to Edgington, counterfactuals lack truth values but have probabilities. I argue that this combination gives rise to a number of problems. According to Leitgeb, counterfactuals have truth conditions-roughly, a counterfactual is true when the corresponding conditional chance is sufficiently high. I argue that problems arise from the disparity between truth and high chance, between approximate truth and high chance, and from counterfactuals for which the corresponding conditional chances are undefined. However, Edgington, Leitgeb and I can unite in opposition to Stalnaker and Lewis-style ‘similarity’ accounts of counterfactuals.
  
  PubDate: 2014-04-29T07:30:36Z
Reasoning about evidence
- Abstract: Publication date: Available online 7 February 2014
  Source:Journal of Applied Logic
  Author(s): Igor Douven
  Bayesians understand the notion of evidential support in terms of probability raising. Little is known about the logic of the evidential support relation, thus understood. We investigate a number of prima facie plausible candidate logical principles for the evidential support relation and show which of these principles the Bayesian evidential support relation does and which it does not obey. We also consider the question which of these principles hold for a stronger notion of evidential support.
  
  PubDate: 2014-04-29T07:30:36Z
Stochastic λ-calculi: An extended abstract
- Abstract: Publication date: Available online 1 April 2014
  Source:Journal of Applied Logic
  Author(s): Dana S. Scott
  It is shown how the operators in the “graph model” for λ-calculus (which can function as a programming language for Recursive Function Theory) can be expanded to allow for “random combinators”. The result then is a model for a new language for random algorithms.
  
  PubDate: 2014-04-29T07:30:36Z
Editorial Board
- Abstract: Publication date: March 2014
  Source:Journal of Applied Logic, Volume 12, Issue 1
  
  PubDate: 2014-04-29T07:30:36Z
Category theory, logic and formal linguistics: Some connections, old and
new
- Abstract: Publication date: March 2014
  Source:Journal of Applied Logic, Volume 12, Issue 1
  Author(s): Jean Gillibert , Christian Retoré
  We seize the opportunity of the publication of selected papers from the Logic, categories, semantics workshop to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.
  
  PubDate: 2014-04-29T07:30:36Z
On the logical structure of de Finetti's notion of event
- Abstract: Publication date: Available online 12 April 2014
  Source:Journal of Applied Logic
  Author(s): Tommaso Flaminio , Lluis Godo , Hykel Hosni
  This paper sheds new light on the subtle relation between probability and logic by (i) providing a logical development of Bruno de Finetti's conception of events and (ii) suggesting that the subjective nature of de Finetti's interpretation of probability emerges in a clearer form against such a logical background. By making explicit the epistemic structure which underlies what we call Choice-based probability we show that whilst all rational degrees of belief must be probabilities, the converse doesn't hold: some probability values don't represent decision-relevant quantifications of uncertainty.
  
  PubDate: 2014-04-29T07:30:36Z
Erratum to “Confirmation as partial entailment” [Journal of
Applied Logic 11 (2013) 364–372]
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  Author(s): Vincenzo Crupi , Katya Tentori
  We provide a correction to the proof of the main result in Crupi and Tentori (2013).
  
  PubDate: 2014-04-29T07:30:36Z
Propositional dynamic logic for searching games with errors
- Abstract: Publication date: Available online 13 April 2014
  Source:Journal of Applied Logic
  Author(s): Bruno Teheux
  We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the n + 1 -valued Kripke models and a corresponding language based on a modal extension of Łukasiewicz many-valued logic. We illustrate the definitions by providing a framework for an analysis of the Rényi–Ulam searching game with errors. Our main result is the axiomatization of the theory of the n + 1 -valued Kripke models. This result is obtained through filtration of the canonical model of the smallest n + 1 -valued propositional dynamic logic.
  
  PubDate: 2014-04-29T07:30:36Z
Capturing equilibrium models in modal logic
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  Author(s): Luis Fariñas del Cerro , Andreas Herzig , Ezgi Iraz Su
  Here-and-there models and equilibrium models were investigated as a semantical framework for answer-set programming by Pearce, Valverde, Cabalar, Lifschitz, Ferraris and others. The semantics of equilibrium logic is given in an indirect way: the notion of an equilibrium model is defined in terms of quantification over here-and-there models. We here give a direct semantics of equilibrium logic, stated for a modal language embedding the language of equilibrium logic.
  
  PubDate: 2014-04-29T07:30:36Z
A propositional linear time logic with time flow isomorphic to ω2
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  Author(s): Bojan Marinković , Zoran Ognjanović , Dragan Doder , Aleksandar Perović
  Primarily guided with the idea to express zero-time transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal ω 2 (concatenation of ω copies of ω). If we think of ω 2 as lexicographically ordered ω × ω , then any particular zero-time transition can be represented by states whose indices are all elements of some { n } × ω . In order to express non-infinitesimal transitions, we have introduced a new unary temporal operator [ ω ] (ω-jump), whose effect on the time flow is the same as the effect of α ↦ α + ω in ω 2 . In terms of lexicographically ordered ω × ω , [ ω ] ϕ is satisfied in 〈 i , j 〉 -th time instant iff ϕ is satisfied in 〈 i + 1 , 0 〉 -th time instant. Moreover, in order to formally capture the natural semantics of the until operator U, we have introduced a local variant u of the until operator. More precisely, ϕ u ψ is satisfied in 〈 i , j 〉 -th time instant iff ψ is satisfied in 〈 i , j + k 〉 -th time instant for some nonnegative integer k, and ϕ is satisfied in 〈 i , j + l 〉 -th time instant for all 0 ≤ l < k . As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.
  
  PubDate: 2014-04-29T07:30:36Z
Editorial Board
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  
  PubDate: 2014-04-29T07:30:36Z
JohnWoodsErrors of Reasoning: Naturalizing the Logic of
Inference2013College PublicationsLondon978-1-84890-114-8
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  Author(s): Matthieu Fontaine
  
  PubDate: 2014-04-29T07:30:36Z
The paradoxes of permission an action based solution
- Abstract: Publication date: June 2014
  Source:Journal of Applied Logic, Volume 12, Issue 2
  Author(s): Dov Gabbay , Loïc Gammaitoni , Xin Sun
  The aim of this article is to construct a deontic logic in which the free choice postulate allow (Ross, 1941) [11] would be consistent and all the implausible result mentioned in (Hanson, in press) [5] will be blocked. To achieve this we first developed a new theory of action. Then we build a new deontic logic in which the deontic action operator and the deontic proposition operator are explicitly distinguished.
  
  PubDate: 2014-04-29T07:30:36Z
A logical framework for privacy-preserving social network publication
- Abstract: Publication date: Available online 31 December 2013
  Source:Journal of Applied Logic
  Author(s): Tsan-sheng Hsu , Churn-Jung Liau , Da-Wei Wang
  Social network analysis is an important methodology in sociological research. Although social network data are valuable resources for data analysis, releasing the data to the public may cause an invasion of privacy. In this paper, we consider privacy preservation in the context of publishing social network data. To address privacy concerns, information about a social network can be released in two ways. Either the global structure of the network can be released in an anonymized way; or non-sensitive information about the actors in the network can be accessed via a query-answering process. However, an attacker could re-identify the actors in the network by combining information obtained in these two ways. The resulting privacy risk depends on the amount of detail in the released network structure and expressiveness of the admissible queries. In particular, different sets of admissible queries correspond to different types of attacks. In this paper, we propose a logical framework that can represent different attack models uniformly. Specifically, in the framework, individuals that satisfy the same subset of admissible queries are considered indiscernible by the attacker. By partitioning a social network into equivalence classes (i.e., information granules) based on the indiscernibility relation, we can generalize the privacy criteria developed for tabulated data to social network data. To exemplify the usability of the framework, we consider two instances of the framework, where the sets of admissible queries are the ALCI and ALCQI concept terms respectively; and we exploit social position analysis techniques to compute their indiscernibility relations. We also show how the framework can be extended to deal with the privacy-preserving publication of weighted social network data. The uniformity of the framework provides us with a common ground to compare existing attack models; while its generality could extend the scope of research to meet privacy concerns in the era of social semantic computing.
  
  PubDate: 2014-01-04T04:23:59Z
Empiricism, probability, and knowledge of arithmetic: A preliminary
defense
- Abstract: Publication date: Available online 31 December 2013
  Source:Journal of Applied Logic
  Author(s): Sean Walsh
  The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Fregeʼs concern about indiscernibility and circularity (§ 2.1), while other are more recent, such as Bakerʼs concern about the unreliability of small samplings in the setting of arithmetic (§ 2.2). Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the ω-rule (§ 3.1) or to the non-computability and non-continuity of probability assignments (§ 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments (Appendix A–Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4 Appendix B). In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar.
  
  PubDate: 2014-01-04T04:23:59Z
Editorial Board
- Abstract: Publication date: December 2013
  Source:Journal of Applied Logic, Volume 11, Issue 4
  
  PubDate: 2013-10-28T11:10:21Z
SLAP: Specification Logic of Actions with Probability
- Abstract: Publication date: Available online 23 October 2013
  Source:Journal of Applied Logic
  Author(s): Gavin Rens , Thomas Meyer , Gerhard Lakemeyer
  A logic for specifying probabilistic transition systems is presented. Our perspective is that of agents performing actions. A procedure for deciding whether sentences in this logic are valid is provided. One of the main contributions of the paper is the formulation of the decision procedure: a tableau system which appeals to solving systems of linear equations. The tableau rules eliminate propositional connectives, then, for all open branches of the tableau tree, systems of linear equations are generated and checked for feasibility. Proofs of soundness, completeness and termination of the decision procedure are provided.
  
  PubDate: 2013-10-24T03:34:35Z
A neural cognitive model of argumentation with application to legal
inference and decision making
- Abstract: Publication date: Available online 23 October 2013
  Source:Journal of Applied Logic
  Author(s): Artur S. dʼAvila Garcez , Dov M. Gabbay , Luis C. Lamb
  Formal models of argumentation have been investigated in several areas, from multi-agent systems and artificial intelligence (AI) to decision making, philosophy and law. In artificial intelligence, logic-based models have been the standard for the representation of argumentative reasoning. More recently, the standard logic-based models have been shown equivalent to standard connectionist models. This has created a new line of research where (i) neural networks can be used as a parallel computational model for argumentation and (ii) neural networks can be used to combine argumentation, quantitative reasoning and statistical learning. At the same time, non-standard logic models of argumentation started to emerge. In this paper, we propose a connectionist cognitive model of argumentation that accounts for both standard and non-standard forms of argumentation. The model is shown to be an adequate framework for dealing with standard and non-standard argumentation, including joint-attacks, argument support, ordered attacks, disjunctive attacks, metalevel attacks, self-defeating attacks, argument accrual and uncertainty. We show that the neural cognitive approach offers an adequate way of modelling all of these different aspects of argumentation. We have applied the framework to the modelling of a public prosecution charging decision as part of a real legal decision making case study containing many of the above aspects of argumentation. The results show that the model can be a useful tool in the analysis of legal decision making, including the analysis of what-if questions and the analysis of alternative conclusions. The approach opens up two new perspectives in the short-term: the use of neural networks for computing prevailing arguments efficiently through the propagation in parallel of neuronal activations, and the use of the same networks to evolve the structure of the argumentation network through learning (e.g. to learn the strength of arguments from data).
  
  PubDate: 2013-10-24T03:34:35Z
Modelling Martin Löf Type Theory in Categories
- Abstract: Publication date: Available online 4 September 2013
  Source:Journal of Applied Logic
  Author(s): François Lamarche
  We present a model of Martin-Löf type theory that includes both dependent products and the identity type. It is based on the category of small categories, with cloven Grothendieck bifibrations used to model dependent types. The identity type is modelled by a path functor that seems to have independent interest from the point of view of homotopy theory. We briefly describe this modelʼs strengths and limitations.
  
  PubDate: 2013-09-08T02:05:58Z
Selectional restrictions, types and categories
- Abstract: Publication date: Available online 23 August 2013
  Source:Journal of Applied Logic
  Author(s): Nicholas Asher
  
  PubDate: 2013-08-27T03:32:03Z
Natural language semantics in biproduct dagger categories
- Abstract: Publication date: Available online 23 August 2013
  Source:Journal of Applied Logic
  Author(s): Anne Preller
  Biproduct dagger categories serve as models for natural language. In particular, the biproduct dagger category of finite dimensional vector spaces over the field of real numbers accommodates both the extensional models of predicate calculus and the intensional models of quantum logic. The morphisms representing the extensional meanings of a grammatical string are translated to morphisms representing the intensional meanings such that truth is preserved. Pregroup grammars serve as the tool that transforms a grammatical string into a morphism. The chosen linguistic examples concern negation, relative noun phrases, comprehension and quantifiers.
  
  PubDate: 2013-08-27T03:32:03Z
Infinity and verifiability in Carnapʼs inductive logic
- Abstract: Publication date: Available online 20 August 2013
  Source:Journal of Applied Logic
  Author(s): Ruurik Holm
  Truth of sentences in infinity is discussed in the framework of Rudolf Carnapʼs inductive logic, which uses finite state descriptions and an asymptotic limit approach for defining probabilities in infinity. This means that Carnapʼs approach suits well for a semantics which is based on finite observability. However, a proper link between asymptotic probability and truth in infinity is missing from Carnapʼs treatment. A novel notion of truth in infinity is introduced and referred to as the extended truth. The idea is that the truth of the sentence S is extended by a particular sequence of state descriptions (where the larger one contains all of the smaller ones) iff S is true in each state description of the sequence. The corresponding notion of extended probability is introduced. Some important results are proved for extended truth and extended probability.
  
  PubDate: 2013-08-23T12:31:00Z
The sure thing principle, dilations, and objective probabilities
- Abstract: Publication date: Available online 13 August 2013
  Source:Journal of Applied Logic
  Author(s): Haim Gaifman
  The common theme that unites the four sections is STP, the sure thing principle. But the paper can be divided neatly into two parts. The first, consisting of the first two sections, contains an analysis of STP as it figures in Savageʼs system and proposals of changes to that system. Also possibilities for partially ordered acts are considered. The second, consisting of the last two sections, is about imprecise probabilities, dilations and objective probabilities. Variants of STP are considered but this part is self-contained and can be read separately. The main claim there is that dilations, which can have extremely counterintuitive consequences, can be eliminated by a more careful analysis of the phenomenon. It outlines a proposal of how to do it.
  
  PubDate: 2013-08-15T14:08:31Z
Relational semantics for full linear logic
- Abstract: Publication date: Available online 7 August 2013
  Source:Journal of Applied Logic
  Author(s): Dion Coumans , Mai Gehrke , Lorijn van Rooijen
  Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In [4] it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [5] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form. In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [4,5] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms. Traditionally, so-called phase semantics are used as models for (provability in) linear logic [7]. These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.
  
  PubDate: 2013-08-11T02:06:43Z
Editorial Board
- Abstract: Publication date: September 2013
  Source:Journal of Applied Logic, Volume 11, Issue 3
  
  PubDate: 2013-08-03T08:06:39Z
Formal Ontologies and Coherent Spaces
- Abstract: Publication date: Available online 25 July 2013
  Source:Journal of Applied Logic
  Author(s): V. Michele Abrusci , Christophe Fouqueré , Marco Romano
  The paper contains a short summary – oriented by a logical point of view – of a joint work on Formal Ontolgies. We shall show how Formal Ontologies correspond to Coherent Spaces, and operations on Formal Ontologies correspond to operations on corresponding Coherent Spaces. So, we are offering a new way to establish the semantics of Formal Ontologies. Surely, we are giving a contribution towards a geometrical treatment of Formal Ontologies (as decidable organizations of digital data).
  
  PubDate: 2013-07-26T02:07:39Z
An epistemic and dynamic approach to abductive reasoning: Abductive
problem and abductive solution
- Abstract: Publication date: Available online 25 July 2013
  Source:Journal of Applied Logic
  Author(s): Fernando R. Velázquez-Quesada , Fernando Soler-Toscano , Ángel Nepomuceno-Fernández
  We propose a study of abductive reasoning addressing it as an epistemic process that involves both an agentʼs information and the actions that modify this information. More precisely, we present and discuss definitions of an abductive problem and an abductive solution in terms of an agentʼs information, that is, in terms of knowledge and beliefs. The discussion is then formalised by ‘implementing’ our definitions in a dynamic epistemic logic framework, where the properties of these definitions are studied, an epistemic action that represents the application of an abductive step is introduced, and an illustrative example is provided. A number of the most interesting properties of abductive reasoning (those highlighted by Peirce) are shown to be better modelled within this approach.
  
  PubDate: 2013-07-26T02:07:39Z
Continuity and geometric logic
- Abstract: Publication date: Available online 25 July 2013
  Source:Journal of Applied Logic
  Author(s): Steven Vickers
  This paper is largely a review of known results about various aspects of geometric logic. Following Grothendieckʼs view of toposes as generalized spaces, one can take geometric morphisms as generalized continuous maps. The constructivist constraints of geometric logic guarantee the continuity of maps constructed, and can do so from two different points of view: for maps as point transformers and maps as bundles.
  
  PubDate: 2013-07-26T02:07:39Z
On the a priori and a posteriori assessment of probabilities
- Abstract: Publication date: Available online 11 July 2013
  Source:Journal of Applied Logic
  Author(s): A. Vasudevan
  We argue that in spite of their apparent dissimilarity, the methodologies employed in the a priori and a posteriori assessment of probabilities can both be justified by appeal to a single principle of inductive reasoning, viz., the principle of symmetry. The difference between these two methodologies consists in the way in which information about the single-trial probabilities in a repeatable chance process is extracted from the constraints imposed by this principle. In the case of a posteriori reasoning, these constraints inform the analysis by fixing an a posteriori determinant of the probabilities, whereas, in the case of a priori reasoning, they imply certain claims which then serve as the basis for subsequent probabilistic deductions. In a given context of inquiry, the particular form which a priori or a posteriori reason may take depends, in large part, on the strength of the underlying symmetry assumed: the stronger the symmetry, the more information can be acquired a priori and the less information about the long-run behavior of the process is needed for an a posteriori assessment of the probabilities. In the context of this framework, frequency-based reasoning emerges as a limiting case of a posteriori reasoning, and reasoning about simple games of chance, as a limiting case of a priori reasoning. Between these two extremes, both a priori and a posteriori reasoning can take a variety of intermediate forms.
  
  PubDate: 2013-07-14T03:56:19Z
Abducted by Bayesians?
- Abstract: Publication date: Available online 3 July 2013
  Source:Journal of Applied Logic
  Author(s): Jan-Willem Romeijn
  This paper discusses the role of theoretical notions in making predictions and evaluating statistical models. The core idea of the paper is that such theoretical notions can be spelled out in terms of priors over statistical models, and that such priors can themselves be assigned probabilities. The discussion substantiates the claim that the use of theoretical notions may offer specific empirical advantages. Moreover, I argue that this use of theoretical notions explicates a particular kind of abductive inference. The paper thus contributes to the discussion over Bayesian models of abductive inference.
  
  PubDate: 2013-07-06T03:36:03Z
Combining Probability and Logic: Papers from Progic 2011
- Abstract: Publication date: Available online 1 July 2013
  Source:Journal of Applied Logic
  Author(s): Jeffrey Helzner
  
  PubDate: 2013-07-02T02:09:59Z
Preservation of Craig interpolation by the product of matrix logics
- Abstract: Publication date: Available online 18 June 2013
  Source:Journal of Applied Logic
  Author(s): C. Sernadas , J. Rasga , A. Sernadas
  The product of matrix logics, possibly with additional interaction axioms, is shown to preserve a slightly relaxed notion of Craig interpolation. The result is established symbolically, capitalizing on the complete axiomatization of the product of matrix logics provided by their meet-combination. Along the way preservation of the metatheorem of deduction is also proved. The computation of the interpolant in the resulting logic is proved to be polynomially reducible to the computation of the interpolants in the two given logics. Illustrations are provided for classical, intuitionistic and modal propositional logics.
  
  PubDate: 2013-06-20T02:07:17Z
A sequent calculus for a logic of contingencies
- Abstract: Publication date: Available online 4 June 2013
  Source:Journal of Applied Logic
  Author(s): Michael Tiomkin
  We introduce a sequent calculus that is sound and complete with respect to propositional contingencies, i.e., formulas which are neither provable nor refutable. Like many other sequent and natural deduction proof systems, this calculus possesses the subformula property and has a simple proof search mechanism.
  
  PubDate: 2013-06-08T02:07:04Z
A general first-order solution to the ramification problem with cycles
- Abstract: Publication date: Available online 4 June 2013
  Source:Journal of Applied Logic
  Author(s): Hannes Strass , Michael Thielscher
  We provide a solution to the ramification problem that integrates findings of different axiomatic approaches to ramification from the last ten to fifteen years. For the first time, we present a solution that: (1) is independent of a particular time structure, (2) is formulated in classical first-order logic, (3) treats cycles – a notoriously difficult aspect – properly, and (4) is assessed against a state-transition semantics via a formal correctness proof. This is achieved as follows: We introduce indirect effect laws that enable us to specify ramifications that are triggered by activation of a formula rather than just an atomic effect. We characterise the intended models of these indirect effect laws by a state-transition semantics. Afterwards, we show how to compile a class of indirect effect laws into first-order effect axioms that then solve the ramification and frame problems. We finally prove the resulting effect axioms sound and complete with respect to the semantics defined earlier.
  
  PubDate: 2013-06-08T02:07:04Z
Sound Approximate Reasoning about Saturated Conditional Probabilistic
Independence under Controlled Uncertainty
- Abstract: Publication date: Available online 3 June 2013
  Source:Journal of Applied Logic
  Author(s): Sebastian Link
  Knowledge about complex events is usually incomplete in practice. We distinguish between random variables that can be assigned a designated marker to model missing data values, and certain random variables to which the designated marker cannot be assigned. The ability to specify an arbitrary set of certain random variables provides an effective mechanism to control the uncertainty in form of missing data values. A finite axiomatization for the implication problem of saturated conditional independence statements is established under controlled uncertainty, relative to discrete probability measures. The completeness proof utilizes special probability models where two assignments have probability one half. The special probability models enable us to establish an equivalence between the implication problem and that of a propositional fragment in Cadoli and Schaerfʼs S -3 logic. Here, the propositional variables in S correspond to the random variables specified to be certain. The duality leads to an almost linear time algorithm to decide implication. It is shown that this duality cannot be extended to cover general conditional independence statements. All results subsume classical reasoning about saturated conditional independence statements as the idealized special case where every random variable is certain. Under controlled uncertainty, certain random variables allow us to soundly approximate classical reasoning about saturated conditional independence statements.
  
  PubDate: 2013-06-04T03:38:01Z
The expressibility of fragments of Hybrid Graph Logic on finite digraphs
- Abstract: Publication date: Available online 24 May 2013
  Source:Journal of Applied Logic
  Author(s): James Gate , Iain A. Stewart
  Hybrid Graph Logic is a logic designed for reasoning about graphs and is built from a basic modal logic, augmented with the use of nominals and a facility to verify the existence of paths in graphs. We study the finite model theory of Hybrid Graph Logic. In particular, we develop pebble games for Hybrid Graph Logic and use these games to exhibit strict infinite hierarchies involving fragments of Hybrid Graph Logic when the logic is used to define problems involving finite digraphs. These fragments are parameterized by the quantifier-rank of formulae along with the numbers of propositional symbols and nominals that are available. We ascertain exactly the relative definability of these parameterized fragments of the logic.
  
  PubDate: 2013-05-27T02:07:28Z
Ordered domain algebras
- Abstract: Publication date: Available online 2 May 2013
  Source:Journal of Applied Logic
  Author(s): Robin Hirsch , Szabolcs Mikulás
  We give a finite axiomatisation to representable ordered domain algebras and show that finite algebras are representable on finite bases.
  
  PubDate: 2013-05-03T02:08:16Z
A model of type theory in simplicial sets
- Abstract: Publication date: Available online 24 April 2013
  Source:Journal of Applied Logic
  Author(s): T. Streicher
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  PubDate: 2013-04-29T02:07:39Z
Skew Lattices and Binary Operations on Functions
- Abstract: Available online 30 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  A recent study of the override and update operations defined on sets of partial functions placed both operations within the algebraic context of a certain variety of algebras. We show the latter to be term equivalent to the variety of right-handed skew Boolean algebras. Both operations are then studied within the broader context of skew lattices with an eye towards achieving greater insight into their joint algebraic behavior. A decision procedure is given to determine whether an equation in both operations holds for all sets of partial functions.
  
  PubDate: 2013-04-01T02:09:36Z
An observation on Carnapʼs Continuum and Stochastic Independencies
- Abstract: Available online 18 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnsonʼs Sufficientness Principle holds, thus giving Carnapʼs Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnapʼs Continuum for the language with just a single predicate.
  
  PubDate: 2013-03-20T03:07:19Z
From Bayesian epistemology to inductive logic
- Abstract: Available online 18 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  Inductive logic admits a variety of semantics [7, Part 1]. This paper develops semantics based on the norms of Bayesian epistemology [16, Chapter7]. §1 introduces the semantics and then, in $ 2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris [2]. §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising.
  
  PubDate: 2013-03-20T03:07:19Z
Confirmation as partial entailment: A representation theorem in inductive
logic
- Abstract: Available online 18 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  The most prominent research program in inductive logic – here just labelled The Program, for simplicity – relies on probability theory as its main building block and aims at a proper generalization of deductive-logical relations by a theory of partial entailment. We prove a representation theorem by which a class of ordinally equivalent measures of inductive support or confirmation is singled out as providing a uniquely coherent way to work out these two major sources of inspiration of The Program.
  
  PubDate: 2013-03-20T03:07:19Z
Probabilities on Sentences in an Expressive Logic
- Abstract: Available online 18 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.
  
  PubDate: 2013-03-20T03:07:19Z
Ultralarge lotteries: Analyzing the Lottery Paradox using non-standard
analysis
- Abstract: Available online 15 March 2013
  Publication year: 2013
  Source:Journal of Applied Logic
  
  A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that is intrinsically vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness.
  
  PubDate: 2013-03-16T03:06:55Z