his is a book to broaden the mind. It brings the reader into a world only vaguely imaginable and richly enlightens it with extraordinary attention to interesting historical details. It is beautifully written and endlessly interesting.

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

It is often suggested that we are equipped with a set of cognitive tools that help us to filter out unreliable testimony. But are these tools effective? I answer this question in two steps. Firstly, I argue that they are not real-time effective. The process of filtering, which takes place simultaneously with or right after language comprehension, does not prevent a particular hearer on a particular occasion from forming beliefs based on false testimony. Secondly, I argue that they are (...) long-term effective. Some hearers sometimes detect false testimony, which increases speakers’ incentives for honesty and stabilizes the practice of human communication in which deception is risky and costly. In short, filtering prevents us from forming a large number of beliefs based on false testimony, not by turning each of us into a high-functioning polygraph but by turning the social environment of human communication into one in which such polygraphs are not required. Finally, I argue that these considerations support strong anti-reductionism about testimonial entitlement. (shrink)

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of (...) the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. (shrink)

In this paper we introduce the notion of filter on the neutrosophic crisp set, then we consider a generalization of the filter’s studies. Afterwards, we present the important neutrosophic crisp filters. We also study several relations between different neutrosophic crisp filters and neutrosophic topologies. Possible applications to database systems are touched upon.

We propose a model of uncertain belief. This models coherent beliefs by a filter, ????, on the set of probabilities. That is, it is given by a collection of sets of probabilities which are closed under supersets and finite intersections. This can naturally capture your probabilistic judgements. When you think that it is more likely to be sunny than rainy, we have{????|????(????????????????????)>????(????????????????????)}∈????. When you think that a gamble ???? is desirable, we have {????|Exp????[????]>0}∈????. It naturally extends the model of credal (...) sets; and we will show it captures all the expressive power of the desirable gambles model. It also captures the expressive power of sets of desirable gamble sets (with a mixing axiom, but no Archimadean axiom). (shrink)

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...) of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}$ on subsets of the Cantor space. (shrink)

In Neal Stephenson’s fictional novel, Diamond Age (1995), the protagonist Nell acquires a prototype of what we might today recognise as a highly sophisticated e-reader with a voice-assistant. This e-reader, the “Young Lady’s Illustrated Primer”, uses artificial intelli- gence to serve as Nell’s personal teacher. What is key to the Primer is how it is designed to respond to Nell. The Primer has a theory of Nell – her needs, her real-world situation, her abilities – and it tailors its lessons (...) for her. The Primer is a highly personalised artificially intelligent device. For many readers – myself included – the Primer was a utopian vision of the bespoke digital tutors of the future. The appearance of new media and technologies in the early 2000s all pointed to the near-term reality of devices like Primers. The real-life experience of personalised digital media, though, is so far from Stephen- son’s vision that it is hard to imagine we will ever get there. Nell’s time with the Primer develops her intellectually, emotionally, and physically. Our “bespoke” media experience, by contrast, is more likely to turn us in on ourselves rather than guide us to live well in our societies. Personalised media is making us more like Narcissus: our consumption is driven by personal choices and automated algorithms that reflect what we already want to hear, see, and consume, and we are captivated. Central to our understanding of the problems with the personalisation of digital media are filter bubbles and echo chambers. These phenomena are used interchangeably by some, but in distinct ways by others. I take the position that filter bubbles and echo chambers are distinct phenomena, but ones that raise similar epistemic worries. I begin with a general discussion of filter bubbles and echo chambers, and I introduce the concept of “selective exposure” as central to both phenomena. There is minimal uniformity across uses of the terms “filter bubble” and “echo chamber”. Consequently, I divide the later discussion into two more precise topics: homophily and polarisation. I conclude with some future-looking comments about responsibility and the epistemic risks posed by digital personalisation. (shrink)

Social media have undoubtedly changed our ways of living. Their presence concerns an increasing number of users (over 4,74 billion) and pervasively expands in the most diverse areas of human life. Marketing, education, news, data, and sociality are just a few of the many areas in which social media play now a central role. Recently, some attention toward the link between social media and political participation has emerged. Works in the field of artificial intelligence have already pointed out that there (...) is a close link between the use of machine learning algorithms in social media and possible epistemic isolation, which could lead to political radicalization. The idea supporting this paper is that artificial intelligence for social media can actively put users’ deliberative capacity at risk and foster political extremism. To prove these claims, I proceed along two lines of inquiry. First, I focus on filter bubbles, namely the result of selections made by algorithms that recommend contents that meet users’ expectations and opinions. To analyze this phenomenon, I refer to the Deweyan model of experience. Second, I connect the filter bubbles problem to the Deweyan idea of deliberative and participatory democracy and Nussbaum’s concept of political compassion. The purpose of this paper is to provide a philosophical foundation that can both (1) effectively serve as a method for analyzing machine learning algorithms and their potential problems in relation to political extremism, and (2) be adopted as a standard to counter the danger of extremism associated with social media experience. (shrink)

This article explores what John Dewey’s political philosophy can offer in regard to the current crisis in digital democracy. It focuses on two digital mechanisms, the “filter bubble” and the “echo chamber”. While there is a prominent, Dewey-inspired debate on “digital publics” in the literature, a reconstruction of the Deweyan concepts of the public and of shared experience shows that it does not adequately reflect the aspect of situated and embodied experience. Based on this, it is shown that digital media (...) offerings must also be rooted in local contexts of experience in order to answer the challenge of those two problematic mechanisms. (shrink)

Two countable filters on $\omega$ are incompatible if they have no common infinite pseudointersection. Letting $\alpha(P_f)$ denote the minimal size of a maximal uncountable family of pairwise incompatible countable filters on $\omega$, we prove the consistency of t $< \alpha(P_f)$.

It is commonly assumed that personalization technologies used by Google for the purpose of tailoring search results for individual users create filter bubbles, which reinforce users’ political views. Surprisingly, empirical evidence for a personalization-induced filter bubble has not been forthcoming. Here, we investigate whether filter bubbles may result instead from a searcher’s choice of search queries. In the first experiment, participants rated the left-right leaning of 48 queries (search strings), 6 for each of 8 topics (abortion, benefits, climate change, sex (...) equality, immigration, nuclear family, Islam, and taxation). An independent sample of participants were then asked to select one of these queries for each of the 8 topics. With the exception of the topic of Islam, participants were significantly more likely to select a query corresponding to their own political leaning, compared to other queries, explaining between 12% and 39% of the variance. A second experiment investigated the effect of the political leaning of the same queries on the overall political leaning of Search Engine Result Pages (SERPs) in Google Search. The top six results of each SERP were rated collectively by a third group of participants, explaining 36.3% of the variance across all 48 search terms (p < .00001). That is, (1) participants in our experiments tended to select own-side search queries, and (2) using those queries tended to yield own-side search results when using the Google search engine. Our results are consistent with the notion of a self-imposed filter bubble in which query selection plays a salient role. (shrink)

To the French critics who originally labelled certain films noir it seemed that a class of Hollywood products had gone darker during the war years – as though a dark filter had been placed over the lens. Films were not designed or marketed as noir, and retrospectively noir's status as a genre is still unsettled. Yet there is widespread interest today in experiencing diverse films as noir, and even in using a Noir Filter in Instagram and video games. Pursuing the (...) filter clue, the noir experience can be thought of as subjection to a dark filtering of narrative. Image filtering styles can help to resolve not only the puzzle of noir's quasi-genre status but also an issue of general interest in aesthetic experience. The use of image filters makes a distinctively powerful contribution to the experience of a work or genre, or to the cultural dominance of an aesthetic regime, due to the unobtrusive formative role it plays in the economy of experience. (shrink)

An adequate moral theory must – or so many philosophers have argued – be compatible with the attitudes and practical requirements of deep friendship. Bernard Williams suggested that the decision procedure required by both deontology and consequentialism inserts a fetishising filter between the natural moral motivation of any normal person to prioritise friends and the decision to act on it. But this injects ‘one thought too many’ into the moral reaction mechanism. It is standardly assumed that virtue ethics is somehow (...) immune to this objection. The present article explores this assumption and finds it wanting in various respects. Virtue ethics filters friendship throughphronesisand thus inserts an extra thought into the mechanism in question. To escape Williams's curse, the only way is to argue that the extra thought required by virtue ethics is not ‘one thought too many’. The article closes with an attempt to show that, contra deontology, the friendship motivation in virtue ethics is derived from the moral virtue, not the intellectual filter, and, contra consequentialism,phronesisdoes not require the maximisation of value. The presumed advantage of virtue ethics must lie in the content of its filter rather than the filter's non-existence. (shrink)

I argue that classical counterfactual semantics in the style of Stalnaker, Lewis, and Kratzer validates an inference pattern that is disconfirmed in natural language. The solution is to alter the algorithm we use to handle inconsistency in premise sets: rather than checking all maximally consistent fragments of a premise sets, as in Krazter’s semantics, we selectively remove some of the premises. The proposed implementation starts from standard premise semantics and involves a new ‘filtering’ operation that achieves just this removal. (...) The resulting semantics is interestingly related to the semantics for counterfactuals emerging from Judea Pearl’s causal models framework in computer science: in particular, filtering is a possible worlds semantics counterpart of Pearl’s interventions. (shrink)

Advances in artificial intelligence, as well as its increased presence in everyday life, have brought the emergence of many new phenomena, including an intriguing appearance of what seems to be a variant of body dysmorphic disorder, coined “Snapchat dysmorphia”. Body dysmorphic disorder is a DSM-5 psychiatric disorder defined as a preoccupation with one or more perceived defects or flaws in physical appearance that are not observable or appear slight to others. Snapchat dysmorphia is fueled by automated selfie filters that reflect (...) unrealistic sociocultural standard. In this paper, we discuss how body dysmorphic disorder and related body image distortions could arise, using the conceptual resources provided by the active inference framework. We suggest that these disorders involve dysfunctional self-modelling which entails maladaptive internalization of sociocultural preferences during adolescent identity formation. Identity formation is hereby described as cycles of interpersonal active inference that arbitrate between identity exploration and commitment. We propose that impaired self-modelling is unable to reduce interpersonal uncertainty during identity exploration, which, over time, degenerates into uncontrollable epistemic habits that isolate the body image from corrective sensory evidence. In light of these insights, we subsequently explore some of the consequences of image-centered social media platforms on the identity formation process. We conclude that heightened interpersonal uncertainty in this novel context could precipitate the onset of body dysmorphic disorder and related body image distortions, particularly when selfie filters are involved. (shrink)

Schemas contribute to adaptation, filtering novelty though knowledge-expectancy structures, the residue of past contingencies and their consequences. Adaptation requires a balance between flexible, dynamic context-sensitivity and the cognitive efficiency that schemas afford in promoting persistent goal pursuit despite distraction. Affects can form and disrupt schemas. Transient affective experiences systematically alter selectivity of attentiveness to the directly experienced present environment, the internal environment, and to the stored experiences of memory. Enduring personal stylistic predispositions, like implicit motives and affective schemas, influence (...) how experience is perceived, responded to, and integrated; they shape memory and influence present experiential patterns, individually and intersubjectively. Such systematic influences are potential sources of error in the study of memory if not mapped; so far, individual personality differences have just been a source of complication in the literature on emotion-congruent perception and memory. I synthesize what findings there are about how personality differences, emotions, and affects contribute to the structuring and integration of perceptions and memories both directly and by way of hot, affectively-anchored schemas. Case studies from experimental and personality psychology highlight a conception of personality and affective experience relevant to memory research and cognitive science. (shrink)

Advances in artificial intelligence, as well as its increased presence in everyday life, have brought the emergence of many new phenomena, including an intriguing appearance of what seems to be a variant of body dysmorphic disorder, coined “Snapchat dysmorphia”. Body dysmorphic disorder is a DSM-5 psychiatric disorder defined as a preoccupation with one or more perceived defects or flaws in physical appearance that are not observable or appear slight to others. Snapchat dysmorphia is fueled by automated selfie filters that reflect (...) unrealistic sociocultural standard. In this paper, we discuss how body dysmorphic disorder and related body image distortions could arise, using the conceptual resources provided by the active inference framework. We suggest that these disorders involve dysfunctional self-modelling which entails maladaptive internalization of sociocultural preferences during adolescent identity formation. Identity formation is hereby described as cycles of interpersonal active inference that arbitrate between identity exploration and commitment. We propose that impaired self-modelling is unable to reduce interpersonal uncertainty during identity exploration, which, over time, degenerates into uncontrollable epistemic habits that isolate the body image from corrective sensory evidence. In light of these insights, we subsequently explore some of the consequences of image-centered social media platforms on the identity formation process. We conclude that heightened interpersonal uncertainty in this novel context could precipitate the onset of body dysmorphic disorder and related body image distortions, particularly when selfie filters are involved. (shrink)

The Filter Dichotomy says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A medial limit is a universally measurable function from [Formula: see text] to the unit interval [0, 1] which is finitely additive for disjoint sets, and maps singletons to 0 and ω to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. (...) We show that the Filter Dichotomy implies that there are no medial limits. (shrink)

With the use of a suitable assumption about the structure of the class of experimental filters, it is shown that the sequence of alternating replicas of two filters is their greatest lower bound, as Jauch suggests. A generalization of his suggestion yields the greatest lower bound of a denumerable set of filters. The criteria of admissibility of filters are briefly discussed.

Leibniz filters play a prominent role in the theory of protoalgebraic logics. In [3] the problem of the definability of Leibniz filters is considered. Here we study the definability of Leibniz filters with parameters. The main result of the paper says that a protoalgebraic logic S has its strong version weakly algebraizable iff it has its Leibniz filters explicitly definable with parameters.

Assuming the consistency of the theory “ZFC + there exists a measurable cardinal”, we construct 1. a model in which the first cardinal κ, such that 2κ > κ+, bears a normal filter F whose associated boolean algebra is κ+-distributive ,2. a model where there is a measurable cardinal κ such that, for every regular cardinal ρ < κ, 2ρ = ρ++ holds,3. a model of “ZFC + GCH” where there exists a non-measurable cardinal κ bearing a normal filter F (...) whose associated boolean algebra is κ+-distributive. (shrink)

With the troubles of core losses and all-state confined to certain limitations which are the innate traits of permanent magnet synchronous motors, this article develops a command filtered adaptive backstepping approach to follow the track of PMSM’s desired rotor position. To begin with, the RBF neural network technique is utilized to get close to the uncharted nonlinear terms which existed in PMSM’s mathematical model. Meanwhile, an advanced adaptive command filter control methodology is constructed to avoid the computing explosion during the (...) process of backstepping design. Furthermore, to make sure that all the state variables are confined into certain ranges, we employed the barrier Lyapunov function at every step of the controllers construction. In addition, an error compensating mechanism is proposed to neutralize filtering errors and only one adaptive law is required. At last, simulation results bear out the superiority of the aforementioned control scheme. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those filter distributive (...) logics for which the class Matr (C) prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2]. (shrink)

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle (...) "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. (shrink)

In view of the poor recommendation performance of traditional resource collaborative filtering recommendation algorithms, this article proposes a collaborative filtering recommendation model based on deep learning for art and MOOC resources. This model first uses embedding vectors based on the context of metapaths for learning. Embedding vectors based on the context of metapaths aggregate different metapath information and different MOOCs may have different preferences for different metapaths. Secondly, to capture this preference drift, the model introduces an attention mechanism, (...) which can improve the interpretability of the recommendation results. Then, by introducing the Laplacian matrix into the prior distribution of the hidden factor feature matrix, the relational network information is effectively integrated into the model. Finally, compared with the traditional model using the scoring matrix, the model in this article using text word vectors effectively alleviates the impact of data sparsity and greatly improves the accuracy of prediction. After analyzing the experimental results, compared with other algorithms, the resource collaborative filtering recommendation model proposed in this article has achieved better recommendation results, with good stability and scalability. (shrink)

Narrow Band-pass filtering techniques have been a challenging task since the inception of audio and telecommunication applications. The challenge involves keeping quality factor, gain and mid-frequency of the filter independent of each other. The critical applications require a design that ensures mid-frequency immune to the circuit component tolerances. It becomes increasingly difficult for low-frequency applications where the shift in few Hz in mid-frequency would cause desired frequencies to fall outside the filter’s bandwidth and go undetected. The selection of right (...) topology of the filter for the best performance is the key to successful design. This book objectively compares the relative performance of none popular narrow band-pass filter topologies. The filter topologies are evaluated using a real-world practical example of designing an extremely narrow band-pass filter. The book provides guidelines for selecting the right topology for the low-frequency narrow band-pass filter. (shrink)

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a stationary (...) subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.

We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection. We prove the following results:Many classical examples of nice tall filters contain no towers.It is consistent that tall analytic P-filters contain towers of arbitrary regular height.It is consistent that all towers generate nonmeager filters, in particular Borel filters do not contain towers.The statement “Every ultrafilter contains towers.” is independent of ZFC.Furthermore, we study many possible logical implications between the existence of towers in filters, (...) inequalities between cardinal invariants of filters $,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left$), and the existence of Luzin type families, that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$. (shrink)

When confronting the unexplained, it is helpful to consider it from many different points of view. In an essay published in 2004, entitled "Sensors, Filters, and the Source of Reality," Robert Jahn and Brenda Dunne of Princeton University's PEAR laboratory proposed that consciousness constructs its reality by ordering the information it derives from the external world through an array of physiological, psychological, and cultural filters. This thesis has now been considered by nineteen distinguished scholars who here present their commentaries from (...) a broad spectrum of professional and personal perspectives. Drawn from such diverse backgrounds as art, Buddhism, evolutionary biology, fantasy, out-of-body experiences, philosophy, physics, psychology, semiotics, and systems engineering, among others, these contributions offer an assortment of unique and fascinating glimpses of how our experiences and their styles of representation are reflected through these filters of consciousness. (shrink)

Let I be a quasimaximal subset of a computable basis of the fully efective vector space V ∞ . We give a necessary and sufficient condition for the existence of an isomorphism between the principal filter respectivelly. We construct both quasimaximal sets that satisfy and quasimaximal sets that do not satisfy this condition. With the latter we obtain a negative answer to Question 5.4 posed by Downey and Remmel in [3].

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the (...) matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into 10 different values. (shrink)

Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (M) depends (...) only on a few basic properties of F. Similar characterizations are given of the class of filters F for which L F (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (U, q), where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

Logics $L^F(M)$ are considered, in which $M$ ("most") is a new first-order quantifier whose interpretation depends on a given filter $F$ of subsets of $\omega$. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that $F$ is not of the form $\{(\bigcap F) \cup X: |\omega - X| < \omega\}$ with $|\omega - \bigcap F| = \omega$. Moreover the set of validities of $L^F(M)$ and even of $L^F_{\omega_1\omega}(M)$ depends only on a few basic properties of (...) F. Similar characterizations are given of the class of filters $F$ for which $L^F(M)$ has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models $(\mathfrak{U}, q)$, where the collection $q$ is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter $F$). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)