Predicate Logic Research Articles

ON DATABASE QUERIES OVER A WEAKLY O-MINIMAL DOMAIN WITH A SMALL COUNTABLE SPECTRUM

In the relational database model introduced by E.F. Codd, the database state is understood as a finite set of relationships between elements. The names of the relationships and their arnts (locations) are fixed and called the database schema. The individual information stored in the relationships of this schema is called the state of the database. Although relational databases were devised for finite sets of data, it is often convenient to assume that there is an infinite domain. We consider relational databases organized over an ordered domain with some additional relations – a typical example is the set of rational numbers together with the relation of linear order and binary operation of addition. If the first-order predicate logic language is used as the query language, then queries can use both database relationships and domain relationships, with variables changing throughout the domain. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving permutations – such queries are called order-generic. It was discovered that for some domains order-generic FO queries fail to express more than pure order queries. Here we prove the collapse result theorem over a weakly o-minimal domain having convexity rank 1 and a small countable spectrum.

A completeness theorem for intuitionistic predicate logic. An intuitionistic proof

A completeness theorem for intuitionistic predicate logic. An intuitionistic proof

Embedding causal team languages into predicate logic

Causal team semantics ([2]) supports causal-observational languages, which enrich the languages for deterministic causation ([11,18]) with dependencies and other team-specific operators. Handling the causal aspects of these languages requires a richer semantics than propositional team semantics; nonetheless, in this paper we show that the causal-observational languages considered in [2] can be embedded into first-order dependence logic by means of a translation and a careful choice of models. We show that, in some significant cases, the translation can be refined to an embedding into the Bernays-Schönfinkel-Ramsey fragment of dependence logic or, in the restricted case of recursive causal models, into the existential fragment. As an application, we use the embeddings to show the decidability of a satisfiability problem for the causal-observational languages. Along the way, we question the correctness of the semantics for interventionist counterfactuals proposed by Halpern ([18]) and propose an alternative one which behaves as usual in the uncontroversial recursive case.

On the Universality of Atomic and Molecular Logics via Protologics

After observing that the truth conditions of connectives of non–classical logics are generally defined in terms of formulas of first–order logic, we introduce ‘protologics’, a class of logics whose connectives are defined by arbitrary first–order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equi-expressivity between two logics based on different classes of models. We prove that, according to that notion, every pure predicate logic with \(k\ge 0\) variables and constants is as expressive as a predicate atomic logic, some sort of atomic logic. Then, we prove that the class of protologics is equally expressive as the class of molecular logics. That formally supports our claim that atomic and molecular logics are somehow ‘universal’. Finally, we identify a subclass of molecular logics that we call predicate molecular logics and which constitutes its representative core: every molecular logic is as expressive as a predicate molecular logic.

Grothendieck’s theory of schemes and the algebra–geometry duality

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \(A \rightarrow B\) into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and (what we shall call) the structure-semiotics duality (of which the syntax-semantics duality for propositional and predicate logic are particular cases). Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general (commutative) rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics (e.g. Stone duality, Gelfand duality, Pontryagin duality, Galois-Grothendieck duality, etc.) can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” (Friedman) that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding (co)domains.

Fast parametric curve matching (FPCM) for automatic spike detection

Objective. Epilepsy is a widely spread neurological disease, whose treatment often requires resection of the pathological cortical tissue. Interictal spike analysis observed in the non-invasively collected EEG or MEG data offers an attractive way to localize epileptogenic cortical structures for surgery planning purposes. Interictal spike detection in lengthy multichannel data is a daunting task that is still often performed manually. This frequently limits such an analysis to a small portion of the data which renders the appropriate risks of missing the potentially epileptogenic region. While a plethora of automatic spike detection techniques have been developed each with its own assumptions and limitations, none of them is ideal and the best results are achieved when the output of several automatic spike detectors are combined. This is especially true in the low signal-to-noise ratio conditions. To this end we propose a novel biomimetic approach for automatic spike detection based on a constrained mixed spline machinery that we dub as fast parametric curve matching (FPCM). Approach. Using the peak-wave shape parametrization, the constrained parametric morphological model is constructed and convolved with the observed multichannel data to efficiently determine mixed spline parameters corresponding to each time-point in the dataset. Then the logical predicates that directly map to verbalized text-book like descriptions of the expected interictal event morphology allow us to accomplish the spike detection task. Main results. The results of simulations mimicking typical low SNR scenarios show the robustness and high receiver operating characteristic AUC values of the FPCM method as compared to the spike detection performed using more conventional approaches such as wavelet decomposition, template matching or simple amplitude thresholding. Applied to the real MEG and EEG data from the human patients and to rat ECoG data, the FPCM technique demonstrates reliable detection of the interictal events and localization of epileptogenic zones concordant with independent conclusions made by the epileptologist. Significance. Since the FPCM is computationally light, tolerant to high amplitude artifacts and flexible to accommodate verbalized descriptions of an arbitrary target morphology, it is likely to complement the existing arsenal of means for analysis of noisy interictal datasets.

On Three Levels of Abstractness in Peirce’s Beta Graphs

Peirce’s beta graphs are roughly equivalent to our first-order predicate logic. However, Bellucci and Pietarinen have recently argued that the beta graphs are not well-equipped to handle asymmetric relative terms. I survey four proposed solutions to the problem and find them all wanting. I offer a fifth solution according to which Peirce’s beta graphs function at three different levels of abstractness from natural language. I diagnose the problem of asymmetric relative terms as arising when we transition from the first to the second level of abstractness. Making grammatical information encoded in natural language explicit at the first level of abstractness and interpreting graphs at the second level of abstractness as shorn of grammatical information resolves the problem. The solution is both well-motivated by Peirce’s own commitments and increases the expressiveness of the beta graphs.

The Expressive Power of the N-Operator and the Decidability of Logic in Wittgenstein’s Tractatus

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results.

Rule Fusion of Privacy Protection Strategies for Co-Ownership Data Sharing

With the rapid development of social networks, personal privacy leakage has become more and more serious. A social network is a shared platform. Resources in a social network may be shared by multiple owners. In order to prevent privacy leakage, each owner assigns a corresponding privacy protection strategy. For the same shared contents, integrating the privacy protection strategies of all owners is the key problem for sharing. This paper proposes a rule fusion method of privacy protection for the co-ownership of data shared in social networks. First, the content of the protection is defined according to different privacy requirements. Second, this paper uses predicate logic formulas to abstract the natural language-based description of privacy protection and further provides a logical model of privacy protection rules. Third, this paper gives the definition of privacy protection heterogeneous rules and provides a rule fusion algorithm to ensure no conflict exists among these rules. The experimental results show that the proposed rule-based fusion method of privacy protection strategy performs at a higher level than the privacy protection strategy fusion.

Philosophical and methodological bases of cut and sentence interpretation in formal linguistics

Philosophical and methodological bases of cut and sentence interpretation in formal linguistics. Now it remains to answer the question of which of these units - logical units or linguistic units - is more scientifically older. Science has already answered this question - linguistics (broader philology) was separated from the structure of philosophy only in the XVIII century, that is, linguistic definitions were copied from philosophy - from logic. This can be proved by another incident. This section of our work can be concluded with the following general conclusion: In Uzbek formal linguistics, the interpretation of the position of the cut in the sentence structure was given not based on the internal (native, ontological) features of the Uzbek language, but based on the understanding of the cut in Russian formal linguistics. In formal linguistics itself, the interpretation of the cut is based on the understanding of the logical predicate.

An extension of Jónsson‐Tarski representation and model existence in predicate non‐normal modal logics

AbstractWe give an extension of the Jónsson‐Tarski representation theorem for both normal and non‐normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson‐Tarski representation to non‐normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q‐filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the extended representation theorem, we show that every predicate modal logic, whether it is normal or non‐normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas.

Policy-driven Data Sharing over Attribute-Based Encryption supporting Dual Membership

Attribute-Based Encryption (ABE) plays an important role in current secure data sharing through fine-grained customizable policies. However, the existing ABE schemes only support simple predicates, = and ≠, but cannot express a more general membership predicates, ∈ and ∉, in policies. The low expressivity of ABE will enlarge the ciphertext storage and reduce the communication efficiency. To overcome this problem, we propose an ABE supporting Dual Membership (DM-ABE). The core problem for implementing this scheme is how to use cryptographic methods to decide the membership between the verified element and the given set. In order to solve this problem, we design a cryptographic algorithm, called Secure Decision of Membership (SDM), based on aggregation functions. In this algorithm, any set can be aggregated into one cryptographic element, and the verified element and the given set can be converted into another cryptographic element in decision process. The membership between them can be decided by the above two cryptographic elements. Furthermore, we construct the DM-ABE by using SDM. Because of the good expressivity of our DM-ABE, we further propose a novel cryptographic data sharing framework by integrating DM-ABE and attribute-based access control to provide fine-grained access control and security protection for private data. In the security proof of DM-ABE, we prove that the DM-ABE satisfies the semantic security against chosen-plaintext attacks under the DBDHE assumption in the standard model through a unified way, considering both two encryption methods for ∈ and ∉ at the same time. Finally, we analyze our scheme in terms of time and space complexity, and compare it with some existing schemes. The results show that our DM-ABE has a better expressive ability on the boolean logic of general membership predicates, ∈ and ∉.

Sense, reference, and contemporary “predicativism”

Abstract Contributing to the debate between referentialist and predicativist accounts of the semantics of proper names, this paper partly endorses a recent trend to reject unitary accounts of their semantics. It does so by restoring a Fregean version of the variety of use account. It criticizes alternative variety of use accounts for not clearly distinguishing pragmatic, syntactic, and semantic issues and argues that, once these are distinguished, the necessity of accepting that names have a variety of uses, and are sometimes logical singular terms and at others logical predicates shows that Frege’s claim, that we should recognize that names have both reference and sense, is vindicated.

Unifying Physical Interaction, Linguistic Communication, and Language Acquisition of Cognitive Agents by Minimalist Grammars

Cognitive agents that act independently and solve problems in their environment on behalf of a user are referred to as autonomous. In order to increase the degree of autonomy, advanced cognitive architectures also contain higher-level psychological modules with which needs and motives of the agent are also taken into account and with which the behavior of the agent can be controlled. Regardless of the level of autonomy, successful behavior is based on interacting with the environment and being able to communicate with other agents or users. The agent can use these skills to learn a truthful knowledge model of the environment and thus predict the consequences of its own actions. For this purpose, the symbolic information received during the interaction and communication must be converted into representational data structures so that they can be stored in the knowledge model, processed logically and retrieved from there. Here, we firstly outline a grammar-based transformation mechanism that unifies the description of physical interaction and linguistic communication and on which the language acquisition is based. Specifically, we use minimalist grammar (MG) for this aim, which is a recent computational implementation of generative linguistics. In order to develop proper cognitive information and communication technologies, we are using utterance meaning transducers (UMT) that are based on semantic parsers and a mental lexicon, comprising syntactic and semantic features of the language under consideration. This lexicon must be acquired by a cognitive agent during interaction with its users. To this aim we outline a reinforcement learning algorithm for the acquisition of syntax and semantics of English utterances. English declarative sentences are presented to the agent by a teacher in form of utterance meaning pairs (UMP) where the meanings are encoded as formulas of predicate logic. Since MG codifies universal linguistic competence through inference rules, thereby separating innate linguistic knowledge from the contingently acquired lexicon, our approach unifies generative grammar and reinforcement learning, hence potentially resolving the still pending Chomsky-Skinner controversy.

Logic of Sense: 2021 Version

Logic of sense is presented as a theoretical approach which avoids the “hard problem of consciousness” pitfall. The logic of sense theory objective is to follow the consciousness development “from inside”, as an answer to the question “how can we explain the consciousness and its content on the basis of itself with no dogmatic presupposition”. Nothing is accepted, however evident or complying with common sense it may seem, if not justified from that point of view. The basic categories of the logic of sense are introduced: smysl (sense), smyslopolaganiye (sense positing), tselostnost’ (integrity, wholeness), svyaznost’ (coherence, linkage) and others, and its main theses are argued for. The pure principle of consciousness could not have been disclosed without a close scrutiny of the millennia-old non-Western big cultures which unfold, in the course of their history, the non-Western versions of consciousness, thus testifying to plurality, and not universality, of reason. The philosophy of mind, as well as philosophy on the whole, remained so far limited by the initial set of dogmatically accepted presuppositions which they could not get rid of, erroneously taking them as having no alternative. Hence the importance of an attentive study of big cultures as a precious experience of unfolding the reason alternative to the version which lies at the basis of the philosopher’s native big culture. The main lines of a logic-and-meaning study of the big Arab-Islamic culture are discussed and its results are analyzed. The principal one is the discovery of process logic (P-logic, not to be confused with what is meant by this expression in contemporary Western philosophy) as an alternative to substance logic (S-logic) which lies at the basis of the big European culture. Any of the variety of the logics of sense is first and foremost the logic of predication, i.e., the way of defining the epistemic chain unfolding itself as the technology of the subject-predicate glue-together, which explains the coherence of human consciousness at the traditionally defined levels of sense perception, speech ability and reason. The results of implementing the logic-and-meaning approach in psychology are described. The objectives of further development of the logic-and-meaning project are sketched.

A General Glivenko–Gödel Theorem for Nuclei

Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an inductively generated abstract consequence relation, and from propositional logic to any set of objects whatsoever. The resulting conservation theorem comes with precise criteria for its validity, which allow us to instantly include G\"odel's counterpart for first-order predicate logic of Glivenko's theorem. The open nucleus gives us a form of the deduction theorem for positive logic, and the closed nucleus prompts a variant of the reduction from intuitionistic to minimal logic going back to Johansson.

EVOLUTION OF ABSOLUTE PARTICIPIAL CONSTRUCTION IN LANGUAGES OF ROMANCE, GERMANIC, AND SLAVIC GROUPS

The study was carried out within the framework of the current problems associated with the evolution of the absolute participial construction in several living and extinct languages of the Romance, Germanic and Slavic groups. The controversial issues, versions of the origin and development of the structure in these languages ("Latin", "Greek" and "autochthonous") are discussed. The structural and semantic features of the absolute participial construction are compared. It has been established that in the languages under study, the construction has a binary structure that includes a name (noun or pronoun) playing the role of a logical subject, and a participle in the role of a logical predicate. Together with the main sentence, the construction forms a paratactic syntactic complex, the constituents of which are not connected with each other by means of service words. Similarities include the ability to express definitively or syncretically temporary meaning; as for the differences, they are the expression in some languages of a causal, conditional, concessive, target, connecting meaning. Depending on the peculiarities of the development of grammatical systems of languages, the structure may include participles of different types, prepositions may be present, the structure may take both the general case form and another case fixed by the language for this type of structures. The words order, which can be either direct or inverse or depend on the transmitted meaning or part of speech of the subject, also differs in the languages. In conclusion, the necessity of further comprehensive analysis of this type of structures is substantiated.

Identity in Martin‐Löf type theory

AbstractThe logic of identity contains riches not seen through the coarse lens of predicate logic. This is one of several lessons to draw from the subtle treatment of identity in Martin‐Löf type theory, to which the reader will be introduced in this article. After a brief general introduction we shall mainly be concerned with the distinction between identity propositions and identity judgements. These differ from each other both in logical form and in logical strength. Along the way, connections to philosophical debates concerning identity are noted. Some use of logical symbolism is inevitable in any serious discussion of type theory, but the emphasis here is on basic ideas rather than technicalities.

New Remarks on the Concept in Logical Use

The main thesis of this paper is directed against the traditional (cognitivetheoretical) definition of the concept which claims that the concept is the '' thought about the essence of the object being thought'', i.e. that it is “a set of essential features or essential characteristics of an object''. But the '' set of essential features or essential characteristics of an object of thought'' is a '' content’’ of the thought. The thought about the essence of an object is definition and the concept is not definition but the part of definition! Besides as the part of formal structure of thought, the concept possesses calculative logical properties that in formal logic (be it syllogistics, or the logic of propositions, or the logic of predicates) come to the front place of formal logical computation. Without the calculative properties of the concept, there would be no calculative properties of propositions which express the thought (thought structures). The calculative properties of a concept include the (1) degree of its logical generality (degree of variability), the (2) logical relations it can establish within the whole of the conceptual content, the (3) operability of the concept in structure of affirmation and negation, the (4) deducibility of either axiomatic or probabilistic systems. Therefore, I believe that, from the logical point of view, the definition of a concept should be applied in favor of its calculative properties that it possesses.

An efficient automatic accident detection system using inertial measurement through machine learning techniques for powered two wheelers

The two-wheeler accidents in most populated and developing countries have become vulnerable and six accidents happen every hour on average. This paper proposes an efficient automatic accident detection system that attempts to detect the occurrences of the accidents in powered two-wheelers (PTW) automatically using vehicle-dependent parameters and the physiological parameters of the rider in real-time. The proposed system builds an accident detection system in PTW using three steps namely, critical event detection system, accident detection system, and severity assessment system. The critical event detection system reads the accelerometer sensor values from the On-Board Diagnostic (OBD) unit mounted on the PTW and classifies the state of the vehicle as normal, fall-like, and fall through the enhanced decision tree algorithm. The enhanced decision tree algorithm uses a tanh function to calculate entropy values. The rules are extracted to fix the threshold by pruning the decision tree to identify the fall of the vehicle and the rider. Due to the unstable nature of PTW and the rider, a novel Adaptive Sequence Window algorithm (ASW) is proposed to substantiate and validate the occurrence of accidents based on the sequence of states identified. Once the accident is detected, the Decision Support System (DSS) running on the OBD mounted on the PTW decides the severity of the accident by combining the three parameters namely fall of the vehicle, fall of the rider, and pulse rate of the rider using the first-order predicate logic rules. The enhanced decision tree algorithm outperforms the other classifiers such as naïve Bayes, artificial neural network, and recurrent neural network with an accuracy of 99.8%. The OBD unit mounted on the PTW and the rider’s helmet is used to detect the occurrence of accidents automatically along with its severity with less time. The ASW algorithm enables the system to detect the fall of the vehicle and rider within five minutes and prevents false positives. Further, the information can be communicated based on the severity of the accident to the emergency medical service for quick response.