A propositional logic of explicit proofs, LP, was introduced in [S. Artemov, Explicit provability and constructive semantics, The Bulletin for Symbolic Logic 7 1–36], completing a project begun long ago by Gödel, [K. Gödel, Vortrag bei Zilsel, translated as Lecture at Zilsel’s in: S. Feferman , Kurt Gödel Collected Works III, 1938, pp. 62–113]. In fact, LP can be looked at in a more general way, as a logic of explicit evidence, and there have been several papers along these (...) lines. A major result about LP is the Realization Theorem, that says any theorem of S4 can be converted into a theorem of LP by some replacement of necessitation symbols with explicit proof terms. Thus the necessitation operator of S4 can be seen as a kind of implicit existential quantifier: there exists a proof term such that…. In this paper, quantification over evidence is introduced into LP, and it is shown that the connection between S4 necessitation and the existential quantifier becomes an explicit one. The extension of LP with quantifiers is called QLP. A semantics and an axiom system for QLP are given, soundness and completeness are established, and several results are proved relating QLP to LP and to S4. (shrink)

One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while (...) making the mathematical machinery simpler and more perspicuous. In addition, work such as that of Yablo fits naturally into the bilattice setting. What I do here is present the general background of bilattices, discuss why they are natural, and show how fixpoint approaches to truth in languages that allow self-reference can be applied. This is not new work, but rather is a summary of research I have done over many years. (shrink)

Hybrid logics internalize their own semantics. Members of the newer family of justification logics internalize their own proof methodology. It is an appealing goal to combine these two ideas into a single system, and in this paper we make a start. We present a hybrid/justification version of the modal logic T. We give a semantics, a proof theory, and prove a completeness theorem. In addition, we prove a Realization Theorem, something that plays a central role for justification logics generally. Since (...) justification logics are newer and less well-known than hybrid logics, we sketch their background, and give pointers to their range of applicability. We conclude with suggestions for future research. Indeed, the main goal of this paper is to encourage others to continue the investigation begun here. (shrink)

According to Richard Swinburne, the principle of simplicity is of great importance to theory choice scenarios and theoretical changes in the sciences. In particular, he holds that the theory choice criterion of fit with background evidence can be reduced to the criteria of simplicity and of yielding the data. We will, however, rebut this reduction thesis and show that three central aspects of theoretical change (confirming power of empirical data, reliability of experimental methods, and truth of new (...) theoretical proposals) cannot be adequately reconstructed if simplicity is regarded as a key criterion of theory choice. (shrink)

In a short 1997 book entitled Simplicity as Evidence of Truth, the Oxford philosopher Richard Swinburne has put forward the following thesis summarily: ‘… for theories (of equal scope) rendering equally probable our observational data (which, for brevity I shall call equally good at “predicting”), fitting equally well with background knowledge, the simplest is most probably true’.

Background: Limited evidence exists comprehensively assessing physical fitness attributes, physical activity behaviors, nutritional habits, and nutritional knowledge according to sex in basketball players during early adolescence. Insight of this nature could be used to optimize the training process and lifestyles in young basketball players.Objective: To compare physical fitness attributes, physical activity levels, nutritional habits, and nutritional knowledge between elite male and female basketball players under 14 years of age.Methods: Twenty-three U-14 basketball players from the same elite basketball academy (...) participated in this study. Physical fitness attributes were assessed using a basketball-specific test battery, while physical activity levels, nutritional habits, and nutritional knowledge were assessed using questionnaires.Results: Male players exhibited better physical fitness in all tests. Of note, a high proportion of players declared never or only sometimes eating fruit and vegetables. In addition, relatively poor nutritional knowledge was evident in all players with the group correctly answering <50% of nutritional questions overall and according to sex.Conclusion: These findings emphasize the necessity to perform individualized prescription of training stimuli across sexes to optimize the physical preparedness and development of youth basketball players. Additionally, strategies such as nutrition-focused education interventions may be necessary in this population given the low consumption of fruits and vegetables, as well as the poor nutritional knowledge observed in players. (shrink)

This paper examined the existence of gender differences in the degree to which leader’s perceptions of successor potential is influenced by interpersonal fit. In Study 1 (n = 97 leaders and n = 280 followers) multi-source field data revealed that for male leaders, ratings of followers’ potential as successors were positively related to interpersonal fit, measured by the degree to which followers’ saw their leadership as being close and interpersonal (i.e., being coaching, transformational and leading by example). For female leaders, (...) these relationships were absent, suggesting that they are less driven by interpersonal fit. In Study 2 (n = 311 leaders) a scenario study provided causal evidence that male leaders rated potential successors more positively when they perceived greater interpersonal fit with followers, whereas female leaders’ successor ratings where not informed by fit perceptions. We discuss the theoretical and practical implications for gendered leadership successor perceptions in organizations. (shrink)

Some prominent accounts of scientific evidence treat evidence as an unrelativized concept. But whether belief in a hypothesis is justified seems relative to the epistemic situation of the believer. The issue becomes yet more complicated in the context of group epistemic agents, for then one confronts the problem of relativizing to an epistemic situation that may include conflicting beliefs. As a step toward resolution of these difficulties, an ideal of justification is here proposed that incorporates both an unrelativized (...) evidence requirement and the requirement of the security of the evidence on which a conclusion from data is based. The latter requirement incorporates the consideration of epistemic modal statements. (shrink)

A closer examination of scientific practice has cast doubt recently on the thesis that observation necessarily fails to determine theory. In some cases scientists derive fundamental hypotheses from phenomena and general background knowledge by means of demonstrative induction. This note argues that it is wrong to interpret such an argument as providing inductive support for the conclusion, e.g. by eliminating rival hypotheses. The examination of the deduction of the inverse square law of gravitation due to J. Bertrand, and R. (...) Fowler's deduction of the quantization of the linear harmonic oscillator's energy spectrum from Planck's radiation law illustrates this point. It is suggested that demonstrative induction is a computational step in fitting a theoretical model and a set of phenomena, with little direct confirmational impact. The thesis of underdetermination, whatever one may think of it, is not threatened by demonstrative induction. (shrink)

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \ can be (...) read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense. (shrink)

Background: No studies have examined the underlying structure or predictive validity of the Cognitive Therapy Adherence and Competence Scale. Examining the structure of the CTACS is of great relevance because it could provide information on what constitutes competence in CBT, and whether some underlying factors are more important for predicting treatment outcomes than others. This study investigates the psychometric properties of the Norwegian version of CTACS and its associations with treatment outcomes in a sample of primary care clients (...) who received CBT for anxiety and/or depression.Method: Independent assessors rated audiotaped therapy sessions in a sample of 132 primary care clients, participating in the Prompt Mental Health Care trial. Outcomes were symptoms of anxiety and depression assessed by patient self-report questionnaires. Structural validity was examined by means of confirmatory and exploratory factor analyses, whereas longitudinal associations with treatment outcome were explored by adopting multilevel modeling.Results: No evidence was found for the divergent validity of the constructs competence and adherence as indicated by a very high correlation between these two subscales in CTACS. Regarding reliability, ICCs for the mean score of the full competence scale and its associated subscales were generally good to excellent, although the subscale measuring the quality of the therapeutic relationship was relatively low. Internal consistency was overall acceptable, but our CFA models did not provide an acceptable fit for the pre-specified one-factor and four-factor solutions. EFA results were difficult to interpret, with a sub-optimal three-factor solution providing best model fit and only two meaningful factors [CBT specific skills and session structure ]. Overall, the results indicated no evidence for the scales' predictive validity.Conclusion: Our findings point to several psychometric problems of the CTACS that may limit both its research and clinical utility. The importance of providing empirical evidence for both reliability and validity aspects of scales are discussed and suggestions for future research are provided. (shrink)

Many individuals with disabling conditions have difficulty with gait and balance control that may result in a fall. Exoskeletons are becoming an increasingly popular technology to aid in walking. Despite being a significant aid in increasing mobility, little attention has been paid to exoskeleton features to mitigate falls. To develop improved exoskeleton stability, quantitative information regarding how a user reacts to postural challenges while wearing the exoskeleton is needed. Assessing the unique responses of individuals to postural perturbations while (...) wearing an exoskeleton provides critical information necessary to effectively accommodate a variety of individual response patterns. This report provides kinematic and neuromuscular data obtained from seven healthy, college-aged individuals during posterior support surface translations with and without wearing a lower limb exoskeleton. A 2-min, static baseline standing trial was also obtained. Outcome measures included a variety of 0 dimensional measures such as center of pressure RMS, peak amplitude, velocities, pathlength, and electromyographic RMS, and peak amplitudes. These measures were obtained during epochs associated with the response to the perturbations: baseline, response, and recovery. T-tests were used to explore potential statistical differences between the exoskeleton and no exoskeleton conditions. Time series waveforms of the COP and EMG data were also analyzed. Statistical parametric mapping was used to evaluate the 1D COP and EMG waveforms obtained during the epochs with and without wearing the exoskeleton. The results indicated that during quiet stance, COP velocity was increased while wearing the exoskeleton, but the magnitude of sway was unchanged. The OD COP measures revealed that wearing the exoskeleton significantly reduced the sway magnitude and velocity in response to the perturbations. There were no systematic effects of wearing the exoskeleton on EMG. SPM analysis revealed that there was a range of individual responses; both behaviorally and among neuromuscular activation patterns. Using both the OD and 1D measures provided a more comprehensive representation of how wearing the exoskeleton impacts the responses to posterior perturbations. This study supports a growing body of evidence that exoskeletons must be personalized to meet the specific capabilities and needs of each individual end-user. (shrink)

This is an expository paper in which the basic ideas of a family of Justification Logics are presented. Justification Logics evolved from a logic called LP, introduced by Sergei Artemov [1, 3], which formed the central part of a project to provide an arithmetic semantics for propositional intuitionistic logic. The project was successful, but there was a considerable bonus: LP came to be understood as a logic of knowledge with explicit justifications and, as such, was capable of addressing in (...) a natural way long-standing problems of logical omniscience. Since then, LP has become one member of a family of related logics, all logics of knowledge with explicit knowledge terms. In this paper the original problem of intuitionistic foundations is discussed only briefly. We concentrate entirely on issues of reasoning about knowledge. (shrink)

A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.

Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiple-valued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the paper is relatively self-contained.

Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common (...) modal language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness. (shrink)

Kleene’s well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.

Strict/tolerant logic is a formally defined logic that has the same consequence relation as classical logic, though it differs from classical logic at the metaconsequence level. Specifically, it does not satisfy a cut rule. It has been proposed for use in work on theories of truth because it avoids some objectionable features arising from the use of classical logic. Here we are not interested in applications, but in the formal details themselves. We show that a wide range of logics have (...) strict/tolerant counterparts, with the same consequence relations but differing at the metaconsequence level. Among these logics are Kleene’s K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {K_3}$$\end{document}, Priest’s LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {LP}}$$\end{document}, and first-degree entailment, FDE. The primary tool we use is the bilattice. But it is more than a tool, it seems to be the natural home for this kind of investigation. (shrink)

Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences are treated more tolerantly. More specifically, ST is a three-valued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does. A version of this result has been extended to meta, (...) metameta, … consequence levels in Barrio et al.. In my earlier paper Fitting I showed that the original ideas behind ST are, in fact, much more general than first appeared, and an infinite family of many valued logics have Strict/Tolerant counterparts. This family includes both Kleene’s and Priest’s logic individually, as well as first degree entailment. For instance, for both the Kleene and the Priest logic, the corresponding strict/tolerant logic is six-valued, but with differing sets of strictly and tolerantly designated truth values. The present paper extends that generalization in two directions. We examine a reverse notion, of Tolerant/Strict logics, which exist for the same structures that were investigated in Fitting. And we show that the generalization extends through the meta, metameta, … consequence levels for the same infinite family of many valued logics. Finally we close with remarks on the status of cut and related rules, which can actually be rather nuanced. Throughout, the aim is not the philosophical applications of the Strict/Tolerant idea, but the determination of how general a phenomenon it is. (shrink)

The use of the Law of Likelihood (LL) as a general tool for assessing rival hypotheses has been criticized for its ambiguous treatment of background information. The LL endorses radically different answers depending on what information is designated as background versus evidence. I argue that once one distinguishes between two questions about evidentiary support, the ambiguity vanishes. I demonstrate this resolution by applying it to a debate over the status of the ‘fine-tuning argument’.

This paper continues my work of [9], which showed there was a broad family of many valued logics that have a strict/tolerant counterpart. Here we consider a generalization of weak Kleene three valued logic, instead of the strong version that was background for that earlier work. We explain the intuition behind that generalization, then determine a subclass of strict/tolerant structures in which a generalization of weak Kleene logic produces the same results that the strong Kleene generalization did. This paper (...) provides much background, but is not self-contained. Some results from [9] are called on, and are not reproved here. [9] Melvin C. Fitting. “A Family of Strict/Tolerant Logics”. In: Journal of Philosophical Logic. Online. Print publication forthcoming. (shrink)

While Kripke's original paper on the theory of truth used a three-valued logic, we believe a four-valued version is more natural. Its use allows for possible inconsistencies in information about the world, yet contains Kripke's development within it. Moreover, using a four-valued logic makes it possible to work with complete lattices rather than complete semi-lattices, and thus the mathematics is somewhat simplified. But more strikingly, the four-valued version has a wide, natural generalization to the family of interlaced bilattices. Thus, (...) with little more work, the theory is extended to a broad class of settings. Indeed, a result like Theorem 6.2 would not even be possible to state without the interlaced bilattice machinery. We hope the notion of interlaced bilattice will make apparent further such connections. (shrink)

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational (...) variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators. (shrink)

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with (...) relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies. (shrink)

First - order modal logic is very much under current development, with many different semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently, several semantics based on counterparts have been examined, in a development that goes back to David Lewis. There is yet another line of research, using intensional objects, that traces back to Richard Montague. I have been involved with this line of development for some time. In the present paper, I briefly (...) sketch several of the approaches to first - order modal logic. Then I present one that I call FOIL in the Montague tradition that, I believe, is both expressive and natural. I briefly discuss in what sense it can be made to encompass the other approaches. Finally, I provide tableau rules to go with the FOIL semantics. (shrink)

There is an obvious difference between what a term designates and what it means. At least it is obvious that there is a difference. In some way, meaning determines designation, but is not synonymous with it. After all, “the morning star” and “the evening star” both designate the planet Venus, but don't have the same meaning. Intensional logic attempts to study both designation and meaning and investigate the relationships between them.

In the tech report Artemov and Yavorskaya [4] an elegant formulation of the first-order logic of proofs was given, FOLP. This logic plays a fundamental role in providing an arithmetic semantics for first-order intuitionistic logic, as was shown. In particular, the tech report proved an arithmetic completeness theorem, and a realization theorem for FOLP. In this paper we provide a possible-world semantics for FOLP, based on the propositional semantics of Fitting [5]. We also give an Mkrtychev semantics. Motivation and intuition (...) for FOLP can be found in Artemov and Yavorskaya [4], and are not fully discussed here.This paper is dedicated to Sergei Artemov, an honored colleague and friend, who has made wonderful things for the rest of us to play with. (shrink)

Several justification logics have been created, starting with the logic LP, [1]. These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding (...) justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open. (shrink)

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal (...) operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. (shrink)

LP can be seen as a logic of knowledge with justifications. See [S. Artemov, The logic of justification, The Review of Symbolic Logic 1 477–513] for a recent comprehensive survey of justification logics generally. Artemov’s Realization Theorem says justifications can be extracted from validities in the more conventional Hintikka-style logic of knowledge S4, in which they are not explicitly present. Justifications, however, are far from unique. There are many ways of realizing each theorem of S4 in the logic LP. (...) If the machinery of justifications is to be applied to artificial intelligence, or better yet, to everyday reasoning, we will need to work with whatever justifications we may have at hand—one version may not be interchangeable with another, even though they realize the same S4 formula. In this paper we begin the process of providing tools for reasoning about justifications directly. The tools are somewhat complex, but in retrospect this should not be surprising. Among other things, we provide machinery for combining two realizations of the same formula, and for replacing subformulas by equivalent subformulas. The results are algorithmic in nature—semantics for LP plays no role. We apply our results to provide a new algorithmic proof of Artemov’s Realization Theorem itself. This paper is a much extended version of [M.C. Fitting, Realizations and LP, in: S. Artemov, A. Nerode , Logical Foundations of Computer Science—New York ’07, in: Lecture Notes in Computer Science, vol. 4514, Springer-Verlag, 2007, pp. 212–223]. (shrink)

First-order modal logics, as traditionally formulated, are not expressive enough. It is this that is behind the difficulties in formulating a good analog of Herbrand’s Theorem, as well as the well-known problems with equality, non-rigid designators, definite descriptions, and nondesignating terms. We show how all these problems disappear when modal language is made more expressive in a simple, natural way. We present a semantic tableaux system for the enhanced logic, and (very) briefly discuss implementation issues.

A well-known problem with Hintikka-style logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpliciter, to knowledge for a reason. Then, as consequences become ‘further away’ from one’s basic knowledge, reasons for them become more complex, thus providing a kind of resource measurement. One kind of reason (...) is a formal proof. Sergei Artemov has introduced a logic of explicit proofs, LP. I present a semantics for this, based on the idea that it is a logic of knowledge with explicit reasons. A number of fundamental facts about LP can be established using this semantics. But it is equally important to realize that it provides a natural logic of more general applicability than its original provenance, arithmetic provability. (shrink)

Classical first-order logic can be extended in two different ways to serve as a foundation for mathematics: introduce higher orders, type theory, or introduce sets. As it happens, both approaches have natural analogs for quantified modal logics, both approaches date from the 1960’s, one is not very well-known, and the other is well-known as something else. I will present the basic semantic ideas of both higher order intensional logic, and intensional set theory. Before doing so, I’ll quickly sketch some necessary (...) background material from quantified modal logic. Except for standard material concerning propositional modal logics, the paper is essentially self-contained. (shrink)

Propositional modal logic is a standard tool in many disciplines, but first-order modal logic is not. There are several reasons for this, including multiplicity of versions and inadequate syntax. In this paper we sketch a syntax and semantics for a natural, well-behaved version of first-order modal logic, and show it copes easily with several familiar difficulties. And we provide tableau proof rules to go with the semantics, rules that are, at least in principle, automatable.

In an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.In this (...) paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view. (shrink)

We present non-clausal resolution systems for propositional modal logics whose Kripke models do not involve symmetry, and for first order versions whose Kripke models do not involve constant domains. We give systems for K, T , K4 and S4; other logics are also possible. Our systems do not require preliminary reduction to a normal form and, in the first order case, intermingle resolution steps with Skolemization steps.

In this paper we argue that it is often adaptive to use one's background beliefs when interpreting information that, from a normative point of view, is incomplete. In both of the experiments reported here participants were presented with an item possessing two features and were asked to judge, in the light of some evidence concerning the features, to which of two categories it was more likely that the item belonged. It was found that when participants received (...) class='Hi'>evidence relevant to just one of these hypothesised categories they used their background beliefs to interpret this information. In Experiment 2, on the other hand, participants behaved in a broadly Bayesian manner when the evidence they received constituted a completed likelihood ratio. We discuss the circumstances under which participants, when making their judgements, consider the alternative hypothesis. We conclude with a discussion of the implications of our results for an understanding of hypothesis testing, belief revision, and categorisation. (shrink)

ABSTRACTBeing able to resist temptation at a young age is crucial for successful functioning yet it can be challenging. According to the Selection, Optimization, and Compensation with Emotion Regulation framework, one central element of successful functioning is selection which involves choosing among regulatory options whose resource requirements fits with the amount of available resources an individual possesses. Although conceptually important, direct empirical evidence is lacking. Accordingly, the present study utilised performance based measures to examine the interactive effect (...) of regulatory selection to resist temptation, and individual differences in executive resources, on functioning in young children. Specifically, 39 first grade children that varied in executive resources, selected between two major regulatory strategies to resist temptation, that varied in their resource demands, and were evaluated on successful functioning... (shrink)

Several justification logics have evolved, starting with the logicLP, (Artemov 2001). These can be thought of as explicit versions of modal logics, or logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, (...) various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open. (shrink)

First we have individual variables, as usual in first-order logics. (We do not have individual constants, but this is a minor point.) The propositional logic LP has justification constants, but in FOLP these are generalized to allow individual variables as arguments. Thus we have as justification constants c, c(x), c(x, y), . . . . Similarly LP has justification variables, but in FOLP these can be parametrized with individual variables p, p(x), p(x, y), . . . . To keep (...) terminology in line with past papers, we will still refer to things as justification constants and justification variables, even though they have structure to them. As in LP, justification terms are built up from justification constants and justification variables using ·, +, ! as usual. In addition there is a new constructor, genx, introduced by Artemov, and there is one further new constructor, exsx, introduced in this paper. If t is a justification term and x is an individual variable, genxt and exsxt are justification terms. An individual variable x is free in a justification term unless it is bound by genx or exsx. More specifically, the free variables of p(x, y, . . .) and of c(x, y, . . .) are {x, y, . . .}, the free variables of s · t and of s + t are the free variables of s together with the free variables of t, the free variables of !s are the free variables of s, and the free variables of genxt and of exsxt are the free variables of t except for x. Formulas are built up from atomic formulas, including ⊥, in the way standard in first-order logic, together with the additional formation rule: t:X is a formula provided t is a justification term, X is a formula, and all free variables of X occur in t. We assume ⊃, ⊥, and ∀ are basic, with other connectives and quantifier defined. The axiomatization used here is a combination of an LP axiomatization and a standard axiomatization of first-order logic, together with a version of the Barcan formula, and one additional axiom that corresponds to the converse Barcan formula.. (shrink)

Participants maneuvered a rat image through a circular region on the computer screen to find a hidden target platform, blending aspects of two well-known spatial tasks. Like the Morris water maze task, participants first experienced a series of learning trials before having to navigate to the hidden target platform from different locations and orientations. Like the dot-location task, they determined the location of a position within a two-dimensional circular region. This procedure provided a way to examine how the number of (...) surrounding cues (1, 2, or 3) affects the memory for spatial location in navigation. Memory performance was better when there were more cues and when targets were close to cues, consistent with the idea that cues bolster fine-grain memory, especially in proximal regions. Early and late measures of bias in memory reflected biases in a direction toward the nearest cue, implicating a cue-based category structure of the navigational space. Collectively, results suggest cue-based spatial memory representations that have been inferred from the dot-location task generalize to a navigation task within a simple, computer-based environment, as demonstrated by the good fits of the spatial model developed for the dot-location task (Fitting, Wedell, & Allen, 2005, 2007). (shrink)

As used by professional logicians today, is the name of their chosen subject singular or plural, "logic" or "logics"? This is a special case of a more general question. For instance, an algebraist might write a book entitled "Algebra", which is about algebras. Though many mathematicians are not aware of it, logic today most decidedly has its plural aspect. Indeed, it always did. Classical logic, which mathematicians often tend to identify with the entirety of logic, was in place roughly (...) by the beginning of the twentieth century. Since then a wide range of so-called non-classical logics have been developed. But indeed, before the creation of classical logic, there were multiple versions of logic, some of them more-or-less formalized. The current growing interest in medieval and ancient European and Asian logics has brought much of this back to modern awareness. Perhaps a later volume in this series will look at the history from a contemporary viewpoint. But that is not our task here. This is the second volume in a series called Landscapes in Logic. The intention of the series is to present reports illustrating the interplay between contemporary work in logic and mainstream mathematics. Of course this is both vague and overly ambitious, and must result in heterogeneous collections. The first volume in the series, Contemporary Logic and Computing, appeared in 2020. The contents divided plausibly into topics from contemporary logic, and from contemporary computing. The present volume is more diverse, and includes articles about both classical and non-classical logics, sometimes from the semantic side and sometimes from the proof-theoretic side. Some articles are primarily technical, often algebraic, while others are more philosophical in nature. Many fit into multiple categories. This multiplicity should not be seen as a defect. The papers here do not just explore logics house by house, but say something about their general neighborhoods as well. (shrink)

A generalization of conventional Horn clause logic programming is proposed in which the space of truth values is a pseudo-Boolean or Heyting algebra, whose members may be thought of as evidences for propositions. A minimal model and an operational semantics is presented, and their equivalence is proved, thus generalizing the classic work of Van Emden and Kowalski.

odel-L¨ ob computability logic. In order to make things relatively self-contained, I sketch the essential ideas of GL, and discuss the significance of its fixpoint theorem. Then I give the algorithm embodied in the program in a little more detail. It should be emphasized that nothing new is presented here — all the theory and methodology are due to others. The main interest is, in a sense, psychological. The approach taken here has been declared in the literature, more than once, (...) to be of theoretical interest only, being too unwieldly in practice. Experimentation shows this is not so provided formulas are not too complicated. A copy of the program can be obtained by ftp from venus.gc.cuny.edu. It is in the subdirectory “/pub/fitting,” under the name “glfixpt.pro.” Log on as “anonymous” and use your full e-mail address as password. If there are difficulties with ftp, send e-mail to the author at mlfl[email protected]. The Prolog code is standard, and should run under any implementation, though minor modifications may be necessary for some systems. These modifications are described in the last section, and also in the program itself. (shrink)

There is an error in the completeness proof for the {λ, =} part of FOIL-K. The error occurs in Section 4, in the text following the proof of Corollary 4.7, and concerns the definition of the interpretation I on relation symbols. Before this point in the paper, for each object variable v an equivalence class v has been defined, and for each intension variable f a function f has been defined. Then the following definition is given for a relation symbol (...) P : v1, v2, . . . , f1, f2, . . . ∈ I(P )(Γ) just in case there are w1, w2, . . . in d(Γ) with wi ∈ vi such that P (w1, w2, . . . , f1, f2, . . .) ∈ Γ. It was pointed out by Torben Brauner that we could have f1 and g1 being the same function, but also have P (w1, w2, . . . , f1, f2, . . .) ∈ Γ without having P (w1, w2, . . . , g1, f2, . . .) ∈ Γ. Our solution is to modify the definition of the model, rather artificially, so that if f and g are the same function, then f and g are syntactically the same intension variable. This is done as follows. First, arbitrarily choose some object variable w, and its corresponding equivalence class w. For each intension variable f we define a disambiguation world ˆ. (shrink)

First-order modal logic, in the usual formulations, is not suf- ficiently expressive, and as a consequence problems like Frege’s morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure (...) using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the “true” semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in G¨ odel’s ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation. (shrink)

Herbrand’s theorem is a central fact about classical logic, [9, 10]. It provides a constructive method for associating, with each first-order formula X, a sequence of formulas X1, X2, X3, . . . , so that X has a first-order proof if and only if some Xi is a tautology. Herbrand’s theorem serves as a constructive alternative to..

This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the (...) proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true. (shrink)