$${{{\mathcal {F}}}}$$ -systems are useful digraphs to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and the one of Yablo can be analyzed with that tool to find graph-theoretic patterns. In this paper we studied this general model consisting of a set of sentences and the binary relation ‘ $$\ldots $$ affirms the falsity of $$\ldots $$ ’ among them. The possible existence of non-referential sentences was also considered. To model the sets of all (...) the sentences that can jointly be valued as true we introduced the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enabled us to characterize referential contradictions, i.e., sentences that can only be false under a classical valuation due to the interactions with other sentences in the model. A Kripke-style fixed-point characterization of groundedness was offered, and complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) fixed points were put in correspondence with conglomerates. Furthermore, argumentation frameworks are special cases of $$\mathcal{F}$$ -systems. We showed the relation between local conglomerates and admissible sets of arguments and argued about the usefulness of the concept for the argumentation theory. (shrink)

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with “Since” and “Until”) in a special way. We refer to this method as “adding a temporal dimension to L” or just “temporalising L”. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence. Charles Sanders Peirce was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean (...) semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic. (shrink)

Although formal system verification has been around for many years, little attention was given to the case where the specification of the system has to be changed. This may occur due to a failure in capturing the clients’ requirements or due to some change in the domain (think for example of banking systems that have to adapt to different taxes being imposed). We are interested in having methods not only to verify properties, but also to suggest how the system (...) model should be changed so that a property would be satisfied. For this purpose, we will use techniques from the area of Belief Revision, that deals with the problem of changing a knowledge base in view of new information. In the last thirty years, several authors have contributed with change operations and ways of characterizing them. However, most of the work concentrates on knowledge bases represented using classical propositional logic. In the last decade, there have been efforts to apply belief revision theory to description and modal logics. In this work, we analyze what is needed for a theory of belief revision which can be applied to the temporal logic, such as the Computation Tree Logic (CTL). In particular, we illustrate different alternatives for formalizing the concept of revision of CTL. Our interest in this particular logic comes both from practical issues, since it is used for software specification, as from theoretical issues, as it is a non-compact logic and most existing results rely on compactness. We focus here on the revision of CTL models and present a characterization result for the revision of partial models. (shrink)

Recent work has shown that the input-output behavior of some common machine learning classifiers can be captured in symbolic form, allowing one to reason about the behavior of these classifiers using symbolic techniques. This includes explaining decisions, measuring robustness, and proving formal properties of machine learning classifiers by reasoning about the corresponding symbolic classifiers. In this work, we present a theory for unveiling the _reasons_ behind the decisions made by Boolean classifiers and study some of its theoretical and practical implications. (...) At the core of our theory is the notion of a _complete reason,_ which can be viewed as a necessary and sufficient condition for why a decision was made. We show how the complete reason can be used for computing notions such as sufficient reasons (also known as PI-explanations and abductive explanations), how it can be used for determining decision and classifier bias and how it can be used for evaluating counterfactual statements such as “a decision will stick even if...because....” We present a linear-time algorithm for computing the complete reasoning behind a decision, assuming the classifier is represented by a Boolean circuit of appropriate form. We then show how the computed complete reason can be used to answer many queries about a decision in linear or polynomial time. We finally conclude with a case study that illustrates the various notions and techniques we introduced. (shrink)

The paper aims at developing the idea that the standard operator of noncontingency, usually symbolized by Δ, is a special case of a more general operator of dyadic noncontingency Δ(−, −). Such a notion may be modally defined in different ways. The one examined in the paper is __Δ__(B, A) = df ◊B ∧ (A ⥽ B ∨ A ⥽ ¬B), where ⥽ stands for strict implication. The operator of dyadic contingency __∇__(B, A) is defined as the negation of __Δ__(B, (...) A). Possibility (◊A) may be then defined as __Δ__(A, A), necessity (□A) as __∇__(¬A, ¬A) and standard monadic noncontingency (__Δ__A) as __Δ__( \({\textsf{T}}\), A). In the second section it is proved that the deontic system KD is translationally equivalent to an axiomatic system of dyadic noncontingency named KDΔ 2, and that the minimal system KΔ of monadic contingency is a fragment of KDΔ 2. The last section suggests lines for further inquiries. (shrink)

In this paper we design a new logical system to explicitly model the different deductive reasoning steps of a boundedly rational agent. We present an adequate system in line with experimental findings about an agent’s reasoning limitations and the cognitive effort that is involved. Inspired by Dynamic Epistemic Logic, we work with dynamic operators denoting explicit applications of inference rules in our logical language. Our models are supplemented by (a) impossible worlds (not closed under logical consequence), suitably structured according (...) to the effect of inference rules, and (b) quantitative components capturing the agent’s cognitive capacity and the cognitive costs of rules with respect to certain resources (e.g. memory, time). These ingredients allow us to avoid problematic logical closure principles, while at the same time deductive reasoning is reflected in our dynamic truth clauses. We further show that our models can be reduced to awareness-like plausibility structures that validate the same formulas and we give a sound and complete axiomatization with respect to them. This approach to the agent’s internal deductive reasoning is finally combined with actions of external information. (shrink)

Weak ontic necessity is the ontic necessity expressed by “should/ought to” in English. An example of it is “I should be dead by now”. A feature of this necessity is that whether it holds at the present world is irrelevant to whether its prejacent holds at the present world. In this paper, by combining premise semantics and update semantics for conditionals, we present a logical theory for conditional weak ontic necessity based on context update. A context is a set of (...) ordered defaults, determining expected possible states of the present world. Sentences are evaluated with respect to contexts. When evaluating the conditional weak ontic necessity with respect to a context, we first update the context with the antecedent, then check whether the consequent holds with respect to the updated context. We compare this theory to some related works. The logic’s expressivity is studied. (shrink)

We describe the foundations and the systematization of natural logic-like monotonic inference using unscoped episodic logical forms (ULFs) that as reported by Kim et al. (Proceedings of the 1st and 2nd Workshops on Natural Logic Meets Machine Learning (NALOMA), Groningen, 2021a, b) introduced and first evaluated. In addition to providing a more detailed explanation of the theory and system, we present results from extending the inference manager to address a few of the limitations that as reported by Kim et al. (...) (Proceedings of the 1st and 2nd Workshops on Natural Logic Meets Machine Learning (NALOMA), Groningen, 2021b) naive system has. Namely, we add mechanisms to incorporate lexical information from the hypothesis (or goal) sentence, enable the inference manager to consider multiple possible scopings for a single sentence, and match against the goal using English rather than the ULF. (shrink)

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with (...) converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF2 include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed-point operators. (shrink)

This paper aims at comparing and relating belief revision and argumentation as approaches to model reasoning processes. Referring to some prominent literature references in both fields, we will discuss their (implicit or explicit) assumptions on the modeled processes and hence commonalities and differences in the forms of reasoning they are suitable to deal with. The intended contribution is on one hand assessing the (not fully explored yet) relationships between two lively research fields in the broad area of defeasible reasoning and (...) on the other hand pointing out open issues and potential directions for future research. (shrink)

In this paper, we propose a framework capable of dealing with anaphora and ellipsis which is both general and algorithmic. This generality is ensured by the compination of two general ideas. First, we use a dynamic semantics which reperent effects using a monad structure. Second we treat scopes flexibly, extending them as needed. We additionally implement this framework as an algorithm which translates abstract syntax to logical formulas. We argue that this framework can provide a unified account of a large (...) number of anaphoric phenomena. Specifically, we show its effectiveness in dealing with pronominal and VP-anaphora, strict and lazy pronouns, lazy identity, bound variable anaphora, e-type pronouns, and cataphora. This means that in particular we can handle complex cases like Bach–Peters sentences, which require an account dealing simultaneously with several phenomena. We use Haskell as a meta-language to present the theory, which also consitutes an implementation of all the phenomena discussed in the paper. To demonstrate coverage, we propose a test suite that can be used to evaluate computational approaches to anaphora. (shrink)

The Feferman–Vaught theorem provides a way of evaluating a first order sentence \(\varphi \) on a disjoint union of structures by producing a decomposition of \(\varphi \) into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than \(\varphi \). We introduce a “tree” generalization of the prenex normal form (PNF) for first order sentences, and show that for an input sentence (...) in this form having a fixed number of quantifier alternations, a Feferman–Vaught decomposition can be obtained in time elementary in the size of the sentence. The sentences in the decomposition are also in tree PNF, and further have the same number of quantifier alternations and the same quantifier rank as the input sentence. We extend this result by considering binary operations other than disjoint union, in particular sum-like operations such as join, ordered sum and NLC-sum, that are definable using quantifier-free translation schemes. (shrink)

The transformers that drive chatbots and other AI systems constitute large language models (LLMs). These are currently the focus of a lively discussion in both the scientific literature and the popular media. This discussion ranges from hyperbolic claims that attribute general intelligence and sentience to LLMs, to the skeptical view that these devices are no more than “stochastic parrots”. I present an overview of some of the weak arguments that have been presented against LLMs, and I consider several of (...) the more compelling criticisms of these devices. The former significantly underestimate the capacity of transformers to achieve subtle inductive inferences required for high levels of performance on complex, cognitively significant tasks. In some instances, these arguments misconstrue the nature of deep learning. The latter criticisms identify significant limitations in the way in which transformers learn and represent patterns in data. They also point out important differences between the procedures through which deep neural networks and humans acquire knowledge of natural language. It is necessary to look carefully at both sets of arguments in order to achieve a balanced assessment of the potential and the limitations of LLMs. (shrink)

Putting individual constants and quantifiers into the same syntactic category within first-order language promises to have far-reaching consequences: a syntax of this kind can reveal the potential of any such language, allowing us to realize that a vast class of noun phrases, including non-denoting terms, can be accommodated in the new syntax as expressions suited to being subjects of sentences. In the light of this, a formal system that is an extension of classical first-order logic is developed here, and is (...) equipped with an appropriate semantics. An ontological interpretation of the new logic is then also provided, with several categories of object and notions of existence being distinguished. Last but not least, a modal version of the logic, with some interesting formal features, is proposed. (shrink)

In an earlier paper [Rational choice and AGM belief revision, _Artificial Intelligence_, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, 2020) where information may (...) be either (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of _filtered belief revision_ and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic _partial_ belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information. (shrink)

In a recent work, Pizzi proposes a notion of dyadic non-contingency, and then gives an axiomatic system of dyadic non-contingency named \(\text {KD}\Delta ^2\), which is shown to be translationally equivalent to the deontic system KD and has the minimal system \(\text {K}\Delta \) of monadic contingency as a fragment. However, the reason why he defines dyadic non-contingency like that is unclear. In this article, inspired by the notion of relativized knowing-value in the literature, we give a plausible explanation for (...) this. We also show that \(\text {KD}\Delta ^2\) is _not_ sound with respect to the class of serial frames, and then correct a mistake in the previous-mentioned Pizzi’s work. (shrink)

In this paper we present an extension of (bunched) separation logic, Boolean BI, with epistemic and dynamic epistemic modalities. This logic, called action model separation logic ( \(\mathrm {AMSL}\) ), can be seen as a generalization of public announcement separation logic in which we replace public announcements with action models. Then we not only model public information change (public announcements) but also non-public forms of information change, such as private announcements. In this context the semantics for the connectives (...) \(*\) and \(\mathrel {-*}\) from separation logic are epistemic versions of their usual semantics. This is because formulas are interpreted in states, not in resources, and agents may be uncertain between different states representing the same resource. We present the logic \(\mathrm {AMSL}\) and its semantics, with a detailed case study that highlights its interest for modeling. We also prove the elimination of the dynamics modalities and discuss some alternative epistemic semantics for the separation connectives. (shrink)

In formal semantics based on modern type theories, some sentences may be interpreted as judgements and some as logical propositions. When interpreting composite sentences, one may want to turn a judgemental interpretation or an ill-typed semantic interpretation into a proposition in order to obtain an intended semantics. For instance, an incorrect judgement $$a:A$$ may be turned into its propositional form $$\textsc {is}(A,a)$$ and an ill-typed application p(a) into $$\textsc {do}(p,a)$$, so that the propositional forms can take part in logical compositions (...) that interpret composite sentences, especially those that involve negations and conditionals.In this paper, we propose an operator not that facilitates such a transformation. Introducing not axiomatically, with five axiomatic laws to govern its behaviour, we shall use it to define $$\textsc {is}$$ and $$\textsc {do}$$ and give examples to illustrate its use in semantic interpretation. The introduction of not into type theories is logically consistent – this is justified by showing that not can be defined by means of the heterogeneous equality $$\textrm{JMeq}$$ so that all of the axiomatic laws for not become provable. Therefore, since the extension with $$\textrm{JMeq}$$ preserves logical consistency, so does the extension with not. We shall also study conditions under which $$\textsc {is}$$ and $$\textsc {do}$$ operators can be used safely without the risk of over-generation. (shrink)

The DAPRECO knowledge base is the main outcome of the interdisciplinary project bearing the same name. It is a repository of rules written in LegalRuleML, an XML formalism designed to be a standard for representing the semantic and logical content of legal documents. The rules represent the provisions of the General Data Protection Regulation, the new Regulation that is significantly affecting the digital market in the European Union and beyond. The DAPRECO knowledge base builds upon the Privacy Ontology, which provides (...) a model for the legal concepts involved in the GDPR, by adding a further layer of constraints in the form of if-then rules, referring either to standard first order logic implications or to deontic statements. If-then rules are formalized in reified Input/Output logic and then codified in LegalRuleML. Reified Input/Output logic is an application of standard Input/Output logic for legal reasoning, in which Input/Output logic is combined with the reification-based approach in Hobbs and Gordon. The DAPRECO knowledge base is then a case study for reified Input/Output logic, and it shows that the formalism indeed appears to be a good candidate to effectively formalize, via uniform and simple representations, complex linguistic/deontic phenomena that may be found in legal texts. To date, the DAPRECO knowledge base is the biggest knowledge base in LegalRuleML and Input/Output logic freely available online. (shrink)

‘Case’ and ‘grammatical relations’ are central to syntactic theory, but rigorous treatments of these concepts in surface-oriented grammars such as Dynamic Syntax are pending. In this respect, Japanese is worthy of mention; in this language, the nominative case particle ga, which typically marks a subject, may mark an object in certain syntactic contexts, and more than one instance of ga may be present within a single clause. These patterns cannot be captured if we simply assume that ga marks a subject. (...) In the present article, we aim to advance formal aspects of the framework, especially the mechanism of ‘structural underspecification,’ by proposing that the parse of a case particle maximally excludes potential landing sites of an unfixed node at the time of parsing the case particle, delaying the resolution of the unfixed node until a subsequent stage of structure building. This maximal exclusion approach to structural underspecification accounts for a range of case marking patterns and their connections with grammatical relations. (shrink)

‘Case’ and ‘grammatical relations’ are central to syntactic theory, but rigorous treatments of these concepts in surface-oriented grammars such as Dynamic Syntax are pending. In this respect, Japanese is worthy of mention; in this language, the nominative case particle ga, which typically marks a subject, may mark an object in certain syntactic contexts, and more than one instance of ga may be present within a single clause. These patterns cannot be captured if we simply assume that ga marks a subject. (...) In the present article, we aim to advance formal aspects of the framework, especially the mechanism of ‘structural underspecification,’ by proposing that the parse of a case particle maximally excludes potential landing sites of an unfixed node at the time of parsing the case particle, delaying the resolution of the unfixed node until a subsequent stage of structure building. This maximal exclusion approach to structural underspecification accounts for a range of case marking patterns and their connections with grammatical relations. (shrink)

‘Case’ and ‘grammatical relations’ are central to syntactic theory, but rigorous treatments of these concepts in surface-oriented grammars such as Dynamic Syntax are pending. In this respect, Japanese is worthy of mention; in this language, the nominative case particle ga, which typically marks a subject, may mark an object in certain syntactic contexts, and more than one instance of ga may be present within a single clause. These patterns cannot be captured if we simply assume that ga marks a subject. (...) In the present article, we aim to advance formal aspects of the framework, especially the mechanism of ‘structural underspecification,’ by proposing that the parse of a case particle maximally excludes potential landing sites of an unfixed node at the time of parsing the case particle, delaying the resolution of the unfixed node until a subsequent stage of structure building. This maximal exclusion approach to structural underspecification accounts for a range of case marking patterns and their connections with grammatical relations. (shrink)

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully (...) advanced by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science. (shrink)

Centered around the analysis of the prescriptive portion of the Vedas, the Sanskrit philosophical school of Mīmāṃsā provides a treasure trove of normative investigations. We focus on the leading Mīmāṃsā authors Prabhākara, Kumārila and Maṇḍana, and discuss three modal logics that formalize their deontic theories. In the first part of this paper, we use logic to analyze, compare and clarify the various solutions to the _śyena_ controversy, a two-thousand-year-old problem arising from seemingly conflicting commands in the Vedas. In the second (...) part, the formalized Mīmāṃsā theories are analyzed and employed to provide alternative perspectives on well-known paradoxes from the contemporary field of deontic logic. Thus, we go from logic to Mīmāṃsā and back again. (shrink)

The paper gives a formal analysis of public lies, explains how public lying is related to public announcement, and describes the process of recoveries from false beliefs engendered by public lying. The framework treats two kinds of public lies: simple lying update and two-step lying, which consists of suggesting that the lie may be true followed by announcing the lie. It turns out that agents’ convictions of what is true are immune to the first kind, but can be shattered by (...) the second kind. Next, recovery from public lying is analyzed. Public lies that are accepted by an audience cannot be undone simply by announcing their negation. The paper proposes a recovery process that works well for restoring beliefs about facts but cannot be extended to beliefs about beliefs. The formal machinery of the paper consists of KD45 models and conditional neighbourhood models, with various update procedures on them. Completeness proofs for a number of reasoning systems (converse belief logic, public lies logic, lying and recovery logic, conditional neighbourhood logic, plus its dynamic version) are included. (shrink)

R. Suszko’s Sentential Calculus with Identity \( SCI \) results from classical propositional calculus \( CPC \) by adding a new connective \(\equiv \) and axioms for identity \(\varphi \equiv \psi \) (which we interpret here as ‘propositional identity’). We reformulate the original semantics of \( SCI \) using Boolean prealgebras which, introduced in different ways, are known in the literature as structures for the modeling of (hyper-) intensional semantics. We regard intensionality here as a measure for the discernibility of (...) propositions (and hyperintensionality as a high degree of intensionality). As concrete examples of \( SCI \) -based intensional modeling, we review and study algebraic semantics of some Lewis-style modal logics in the vicinity of \( S3 \) and present conditions under which those modal systems can be restored, in a precise sense, as certain axiomatic extensions of \( SCI \). This generalizes work of Suszko which is focused on the modal systems \( S4 \) and \( S5 \). Our approach is particularly intended as a proposal to consider and to further study \( SCI \) (and its extensions) as a general framework for the modeling of (hyper-) intensional semantics. (shrink)

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. (...) On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics. (shrink)

We present a Revised Projectivity Calculus that extends the scope of inclusion and exclusion inferences derivable under the Projectivity Calculus developed by Icard :705–725, 2012). After pointing out the inadequacies of C, we introduce four opposition properties which have been studied by Chow Proceedings of the 18th Amsterdam Colloquium, Springer, Berlin, 2012; Beziau, Georgiorgakis New dimensions of the square of opposition, Philosophia Verlag GmbH, München, 2017) and are more appropriate for the study of exclusion reasoning. Together with the monotonicity properties, (...) the OPs will form the basis of RC instead of the additive/multiplicative properties used in C. We also prove some important results of the OPs and their relation with the MPs. We then introduce a set of projectivity signatures together with the associated operations and conditions for valid inferences, and develop RC by inheriting the key features of C. We then show that under RC, we can derive some inferences that are not derivable under C. We finally discuss some properties of RC and point to possible directions of further studies. (shrink)

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent (...) logic with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (shrink)

Transparent intensional logic (TIL) is a well-explored type-theoretical framework for semantics of natural language. However, its treatment of polymorphic functions, which are essential for the analysis of various natural language phenomena, is still underdeveloped. In this paper, we address this issue and propose an extension of TIL that introduces polymorphism via type variables ranging over types and generalized variables ranging over constructions and types. Furthermore, we offer an analysis of sentences involving non-specific notional attitudes of the general form ‘_A_ considers (...) (believes, desires, wants, seeks,...) something’. (shrink)

Idempotency requires any phonotactically licit forms to be faithfully realized. Output-drivenness requires any discrepancies between underlying and output forms to be driven exclusively by phonotactics. These formal notions are relevant for phonological theory and play a crucial role in learnability. Tesar and Magri provide tight guarantees for OT output-drivenness and idempotency through conditions on the faithfulness constraints. This paper derives analogous faithfulness conditions for HG idempotency and output-drivenness and develops an intuitive interpretation of the various OT and HG faithfulness conditions (...) thus obtained. The intuition is that faithfulness constraints measure the phonological distance between underlying and output forms. They should thus comply with a crucial axiom of the definition of distance, namely that any side of a triangle is shorter than the sum of the other two sides. This intuition leads to a faithfulness triangle inequality which is shown to be equivalent to the faithfulness conditions for idempotency and output-drivenness. These equivalences hold under various assumptions, crucially including McCarthy’s :75–138, 2003b) generalization that constraints are all categorical. (shrink)

This paper proposes new semantics for propositional dynamic logic (PDL), replacing the standard relational semantics. Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) (Kremer and Mints in Ann Pure Appl Logic (...) 131:133–158, 2005). We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic (Dabrowski et al. in Ann Pure Appl Logic 78:73–110, 1996), and show that the topological interpretation of public announcements as given in Bjorndahl (in: van Ditmarsch and Sandu (eds) Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to Logic, vol 12, Springer, 2018) coincides exactly with a natural interpretation of test programs. (shrink)

We present a number of puzzles arising for the interpretation of modified numerals. Following Büring and others we assume that the main difference between comparative and superlative modifiers is that only the latter convey disjunctive meanings. We further argue that the inference patterns triggered by disjunction and superlative modifiers are hard to capture in existing semantic and pragmatic analyses of these phenomena (neo-Gricean or grammatical alike), and we propose a novel account of these inferences in the framework of bilateral state-based (...) modal logic defining a first order extension of Aloni (Semant Pragmat 15:5-EA, 2022. https://doi.org/10.3765/sp.15.5)’s. (shrink)

In his pioneering work in the field of inductive inference, Gold (Inf Control 10:447–474, 1967) proved that a set containing all finite languages and at least one infinite language over the same fixed alphabet is not identifiable in the limit (learnable in the exact sense) from complete texts. Gold’s work paved the way for computational learning theories of language and has implications for two linguistically relevant classes in the Chomsky hierarchy (cf. Chomsky in Inf Control 2:137–167, 1959, Chomsky in Knowledge (...) of language: its nature, origin, and use, Praeger, 1986; Shieber in Linguist Philos 8:333–343, 1985; Heinz in Linguist Inq 41:623–661, 2010). Within that same framework, Angluin (Inf Control 45:117–135, 1980) provided a complete characterization for the learnability of language families. Mathematically, the concept of identification in the limit from that classical setting can be seen as the use of a particular type of metric for learning in the limit (Wharton in Inf Control 26:236–255, 1974). In this research note, I use Niyogi’s extended version of a theorem by Blum and Blum (Inf Control 28:125–155, 1975) on the existence of locking data sets to prove a necessary condition for learnability in the limit of any family of languages with the Gold property in any given metric. This recovers the most general version of Gold’s Theorem as a particular case. Moreover, when the language family is further assumed to contain all finite languages, the same condition also becomes sufficient for learnability in the limit. Finally, we discuss questions that are left open and outline a research program regarding language learnability in the limit from other input types and cognitive feasibility. (shrink)

We investigate an approach to improving statistical text classification by combining machine learners with an ontology-based identification of domain-specific topic categories. We apply this approach to ad hoc disclosures by public companies. This form of obligatory publicity concerns all information that might affect the stock price; relevant topic categories are governed by stringent regulations. Our goal is to classify disclosures according to their effect on stock prices (negative, neutral, positive). In the study reported here, we combine natural language parsing (...) with a formal background ontology to recognize disclosures concerning particular topics from a prescribed list. The semantic analysis identifies some of these topics with reasonable accuracy. We then demonstrate that machine learners benefit from the additional ontology-based information when predicting the cumulative abnormal return attributed to the disclosure at hand. (shrink)

Minimizers are widely acknowledged cross-linguistically to denote a minimal quantity, extent or degree. With respect to minimizers in Mandarin Chinese, Shyu claims that their so-called negative polarity is purely syntactically determined and is facilitated by the lian…dou EVEN construction. Within the framework of Dynamic Syntax which allows for interaction between syntactic, semantic and pragmatic information, we demonstrate that the total negation is actually derived from the interaction between syntax, semantics and pragmatics, rather than being determined by purely syntactic means.

Jc Beall’s off-topic interpretation of Weak Kleene logic offers a logic of ‘true-and-topic’ preservation. However, Nissim Francez has recently argued that being ‘off-topic’ is a relational and not an _absolute_ semantic property; as such, it fails to satisfy the conditions of truth-functionality. For Francez, this means that it ‘cannot serve as an interpretation of a truth-value’. In this paper, I propose a two-layered _re_interpretation of Beall’s off-topic semantics. This two-layered framework has two crucial features: a sentential topic-tagging device and a (...) two-layered evaluation tool. I show that this framework results in a logic that preserves Beall’s key insight and addresses Francez’s concerns. (shrink)

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

In this paper, we build on earlier work by Standefer (Logic J IGPL 27(4):543–569, 2019) in investigating extensions of substructural logics, particularly relevant logics, with the machinery of justification logics. We strengthen a negative result from the earlier work showing a limitation with the canonical model method of proving completeness. We then show how to enrich the language with an additional operator for implicit commitment to circumvent these problems. We then extend the logics with axioms for D, 4, and 5, (...) which requires additional justification term operators, following the work of Pacuit (in proceedings of the fifth panhellenic logic symposium, 2005) and Rubtsova (in Grigoriev D, Harrison J (eds) Computer science—theory and applications, CSR 2006; J Logic Comput 16(5):671–684, 2006), and present the required modifications to the frame semantics. We present a simplification of the neighborhood frames from the earlier work and we close by investigating the distinctive contribution of the + operator to the logic. (shrink)

This paper presents a new logical analysis of quantity comparatives (i.e. More linguists than philosophers came to the party.) within the Delineation Semantics approach to gradability and comparison (McConnell-Ginet in Comparison constructions in English. PhD thesis, University of Rochester, Rochester, 1973; Kamp in Formal semantics of natural language. Cambridge University Press, Cambridge, 1975; Klein in Linguist Philos 4:1–45, 1980) among many others. Along with the Degree Semantics framework (Cresswell in Montague grammar. Academic Press, New York, 1976; von Stechow in J (...) Semant 3:1–77, 1984; Kennedy in Projecting the adjective. PhD thesis. University of California, Santa Cruz, 1997, among many others) Delineation Semantics is one of the dominant logical frameworks for analyzing the meaning of gradable constituents of the adjectival syntactic category; however, there has been very little work done investigating the application of this framework to the analysis of gradability outside the adjectival domain. This state of affairs distinguishes the Delineation Semantics framework from its Degree Semantics counterpart, where such questions have been investigated in great deal since the beginning of the twenty-first century. Nevertheless, it has been observed [for example, by Doetjes (Seventeenth amsterdam colloquium. University of Amsterdam, Amsterdam, 2011); van Rooij (The vagueness handbook. Springer, Dordrecht, 2011c)] that there is nothing inherently adjectival about the way that the interpretations of scalar predicates are calculated in Delineation Semantics, and therefore that there is enormous potential for this approach to shed light on the nature of gradability and comparison in the nominal and verbal domains. This paper is a first contribution to realizing this potential within a Mereological extension of a simple version of the DelS system. (shrink)

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. (...) The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change. (shrink)

In this paper we present a study of adjectival/adverbial modification using modern type theories, i.e. type theories within the tradition of Martin-Löf. We present an account of various issues concerning adjectival/adverbial modification and argue that MTTs can be used as an adequate language for interpreting NL semantics. MTTs are not only expressive enough to deal with a range of modification phenomena, but are furthermore well-suited to perform reasoning tasks that can be easily implemented given their proof-theoretic nature. In MTT-semantics, common (...) nouns are interpreted as types rather than predicates. Therefore, in order to capture the semantics of adjectives adequately, one needs to meet the challenge of modeling CNs modified by adjectives as types. To explicate that this can be done successfully, we first look at the mainstream classification of adjectives, i.e. intersective, subsective and non-subsective adjectives. There, we show that the rich type structure available in MTTs, along with a suitable subtyping framework, offers an adequate mechanism to model these cases. In particular, this modelling naturally takes care of the characterising inferences associated with each class of adjectives. Then, more advanced issues on adjectival modification are discussed: degree adjectives, comparatives and multidimensional adjectives. There, it is shown that the use of indexed types can be usefully applied in order to deal with these cases. In the same vein, the issue of adverbial modification is discussed. We study two general typings for sentence and VP adverbs respectively. It is shown that the rich type structure in MTTs further provides useful organisational mechanisms in giving formal semantics for adverbs. In particular, we discuss the use of \-types to capture the veridicality/non-veridicality distinction and further discuss cases of intensional adverbs using the type theoretic notion of context. We also look at manner, subject and speech act adverbials and propose solutions using MTTs. Finally, we show that the current proof technology can help mechanically check the associated inferences. A number of our proposals concerning adjectival and adverbial modification have been formalised in the proof assistant Coq and many of the associated inference patterns are checked to be correctly captured. (shrink)

Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information, which can be incomplete or conflicting, but not explicitly about facts. This layer (...) of available information or evidence, which is the object of the agents’ knowledge, can be seen as a database. By adopting this sceptical posture in our semantics, we prepare the ground for logics where the notion of knowledge—or more appropriately, belief—is entirely based on evidence. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system. In summary, our contributions are twofold: on the one hand we present an intuition and possible application for many-valued modal logics, and on the other hand we develop a logic that models the dynamics of evidence in a simple and intuitively clear fashion. (shrink)

In Gärdenfors and Makinson :197–245, 1994) and Gärdenfors it was shown that it is possible to model nonmonotonic inference using a classical consequence relation plus an expectation-based ordering of formulas. In this article, we argue that this framework can be significantly enriched by adopting a conceptual spaces-based analysis of the role of expectations in reasoning. In particular, we show that this can solve various epistemological issues that surround nonmonotonic and default logics. We propose some formal criteria for constructing and updating (...) expectation orderings based on conceptual spaces, and we explain how to apply them to nonmonotonic reasoning about objects and properties. (shrink)