Coalgebraic Automata Theory

We propose a productivity-based account of vagueness. Predicates remain bivalent over the intended universe, but their extensions may be productive, explaining the persistence of borderline cases as a structural failure of effective decidability rather than a semantic gap. We distinguish contrary (term) negation from complementary negation and prove that, for sharp predicates, term negation coincides with logical/complementary negation. We further show an effective obstruction: no effectively enumerable family of precisifications, each providing a c.e. approximation, can capture a productive extension; a uniform escape operator always produces counterexamples. The model yields a dynamic, one‑sided account of vagueness that improves on supervaluationist, fuzzy, and rough‑set approaches by explaining the perpetual emergence of new borderline cases without appealing to a predicate–complement partition.

We motivate and present a novel semantic theory for universal generalizations ('every 𝐴 is a 𝐵'), contributing to a growing theoretical line that gives equal prominence to subject matter and truth conditions when modelling propositional content. Our proposed theory aligns with the presently dominant linguistic programme for capturing natural language quantifiers (the generalized quantifier framework) and demonstrably respects a package of influential theses on subject matter (e.g. that a subject matter corresponds to a set of basic questions), many of which extend in interesting and natural ways to the quantified setting. Along the way, we take a principled stand on some divisive issues: most notably, we follow Nelson Goodman in denying that universal generalizations are about individuals, relative to the domain of quantification.

Fragmentation is a widely discussed thesis on the architecture of mental content, saying, roughly, that the content of an agent's belief state is best understood as a set of information islands that are individually coherent and logically closed, but need not be jointly coherent and logically closed, nor uniformly accessible for guiding the agent's actions across different deliberative contexts. Expressivism is a widely discussed thesis on the mental states conventionally expressed by certain categories of declarative discourse, saying, roughly, that prominent forms of declarative utterance should be taken to express something other than the speaker's outright acceptance of a representational content. In this paper, I argue that specific versions of these views-Topical Fragmentation and Semantic Expressivism-present a mutually beneficial combination. In particular, I argue that combining Topical Fragmentation with Semantic Expressivism fortifies the former against (what I call) the Connective Problem, a pressing objection that lays low more familiar forms of Fragmentation. This motivates a novel semantic framework: Fragmented Semantic Expressivism, a bilateral state-based system that (i) prioritizes fragmentationist acceptance conditions over truth conditions, (ii) treats representational content as hyperintensional, and (iii) gives expressivistic acceptance conditions for the standard connectives. Finally, we discuss the distinctive advantages of this system in answering the problem of logical omniscience and Karttunen's problem for epistemic 'must'.

Quine's famous assertion that "to be is to be the value of a bound variable" translates, in syntactic form, the way in which first-order formal logic correctly expresses existence, which can no longer be taken as a real predicate since Kant. However, this operation of quantification does not exhaust the metaphysical breadth of the theme of Being, involved by dragging through the use of the verb "to be". In particular, when it comes to mathematical objects, it is verified that quantification depends on the reality of non-empirical and non-mental domains, which support the truth of mathematical propositions. This work seeks to demonstrate that such domains must have an autonomous, independent, abstract and necessary way of being, which implies the adoption of mathematical Platonism in a strong sense, a position similar to that adopted by G. Frege in the proposition of his thesis of the "Third Kingdom" ("drittes Reich") in his famous article Der Gedanke.

This paper addresses the problem of bridging the gap between the fields of Knowledge Renresentation OCR) and Uncertain Reasoning (UR). The prohosed solution consists of a framework for representing uncertain knowledge in which two components, one dealing with (categorical) knowledge and one dealing with uncertainty about this knowledge, are singled out. In this sense, the framework is "hybrid". This framework is characterized in both modeltheoretic and proof-theoretic terms. State of belief is represented by "belief sets", defined in terms of the "functional approach to knowledge representation" suggested by Levesque. Examples are given, using first order logic and (a minimal subset of) M-Krypton for the KR side, and a yes/no trivial case and Dempster-Shafer theory for the UR side. * This research has been partially supported by the ARCHON project, funded by grants from the Commission of the European Communities under the ESPRIT-II Program, P-2256. The partners in the ARCHON project are: Krupp Atlas Elektronik, Amber,

We take a look at the “forbidden route” backward from the Wilsonian/Feynmanian paradigm in (effective) quantum field theory and emergent geometry in quantum gravity.
Our rear-ward exploration of the forgotten origins of quantum (field) theory will reveal crucial missing ingredients in order to recover a relation between “quantum” and “geometry”.

The text “The Ontology of Knowledge, Logic, Arithmetic, Set Theory, and Geometry” by Jean-Louis Boucon explores a deeply philosophical interpretation of knowledge, its logical structure, and the foundational elements of mathematical and scientific reasoning.
Here’s an overview condensed by an AI of the key themes and ideas, summarized into a quite general conceptual structure.
These two pages are instructive on their own, but their main purpose is to facilitate the reading of the entire article, allowing the reader to situate in the coherence of the whole both the meaning and the usefulness of each proposed idea.

The paper analyses the argument proposed by Milne (2005) against truthmaker maximalism and shows that the objections raised to this argument by de Sa and Zardini, and Rodriguez-Pereyra are misguided because the first one misuses the vagueness of some terms; and the second one is based on a fallacy of petitio principii (this is exactly the same type of objection as it was raised by Rodriguez-Pereyra against the Milne's argument).

The Rule of Necessitation allows us to conclude Necessary B (Box B) if we have a derivation  of the formula B. A typical justification of the rule of necessitation runs as follows: if through pure reasoning we conclude the formula B, then this formula B holds valid in virtue of pure logic and hence should hold valid in every possible worlds whence (Box B). A tacit assumption here is that pure logic is constant in every possible world. In this talk we sketch the beginnings of a framework that allows for different possible worlds to have different logics.

The notion of definiteness has played a fundamental role in the early developments of set
theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different
forms of definiteness. First, a condition can be definite in the sense that, given any object, either the
condition applies to that object or it does not. We call this intensional definiteness. Second, a condition
or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has
been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns
whether an intension applies to objects considered one by one, extensional definiteness concerns the
totality of objects to which the intension applies. We discuss how these two forms of definiteness
admit of precise mathematical analyses. We argue that two main types of explication of extensional
definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based
on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation.
We submit that the two notions of extensional definiteness that emerges from our investigation enable
us to identify and understand some of the most important fault lines in the philosophy and foundations
of mathematics.

We study imagination as reality-oriented mental simulation (ROMS): the activity of simulating nonactual scenarios in one's mind, to investigate what would happen if they were realized. Three connected questions concerning ROMS are: What is the logic, if there is one, of such an activity? How can we gain new knowledge via it? What is voluntary in it and what is not? We address them by building a list of core features of imagination as ROMS, drawing on research in cognitive psychology and the philosophy of mind. We then provide a logic of imagination as ROMS which models such features, combining techniques from epistemic logic, action logic, and subject matter semantics. Our logic comprises a modal propositional language with non-monotonic imagination operators, a formal semantics, and an axiomatization. §1. Introduction. Imagination is "the faculty or action of forming new ideas, or images or concepts of external objects not present to the senses" (OED). This allows for the notion to cover a range of cognitive activities, from daydreaming, to freely following a train of thought, to hallucinating. We focus on one, labeled, for reasons that will become clear soon, reality-oriented mental simulation (ROMS): the episodic activity of simulating alternatives to reality in our mind, to investigate what would or is likely to happen, or what would have happened, if they were, or had been, realized. In cognitive psychology, ROMS is widely agreed to play important roles (Markman, Klein, & Surh, 2009): it allows us to improve performance, to make contingency plans by anticipating future outcomes, and to learn from mistakes by considering how we could have done better by acting differently. It is used to account for counterfactual thought (

This work is devoted to incorporating into QFT the notion that particles and hence the particle states should be localizable in space. It focuses on the case of the Dirac field in 1+1 dimensional flat spacetime, generalizing a recently developed formalism for scalar fields. This is achieved exploiting again the non-uniqueness of quantization process. Instead of elementary excitations carrying definite amounts of energy and momentum, we construct the elementary excitations of the field localized (at some instant) in a definite region of space. This construction not only leads to a natural notion of localized quanta, but also provides a local algebra of operators. Once constructed, the new representation is confronted to the standard global (Fock) construction. In spite of being unitarily inequivalent representations, the localized operators are well defined in the conventional Fock space. By using them we dig up the issues of localization in QFT showing how the globality of the vacuu...

We present a logic of evidence that reduces agents' epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine's semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine's semantics. Finally, we prove that evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic.

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Resorting to suitable representations of q-algebras, we give a new proof of the theorem stating that the Gaussian polynomial defined by the q-binomial coefficient is the weight (polynomial generating function) of the restricted (i.e. limited in number and size of summands) k-rowed plane partitions.

This paper deals with a famous problem of epistemic logic – logical omniscience. Logical omniscience occurs in the logical systems where the axiomatics is complete and consequently an agent using inference rules knows everything about the system. Logical omniscience is a major problem due to complexity problems and the inability for adequate human reasoning modeling It is studied both informal logic and philosophy of psychology (bounded rationality). It is important for bounded rationality because it reflects the problem of formal characterization of purely psychological mechanisms. Paper proposes to solve it using the ideas from the philosophical bounded rationality and intuitionistic logic. Special regions of deductible formulas developed according to psychologistic criterion should guide the deductive model. The method is compared to other ones presented in the literature on logical omniscience such as Hintikka’s and Vinkov and Fominuh. Views from different perspectives such as computer science and artificial intelligence are also provided. Keywords: bounded rationality, epistemic logic, intuitionism, omniscience.

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T(op), respectively. The q-deformation parameter is rela...

Bayes' theorem is shown to be implicit in the quantum formalism describing the entanglement between an open system X and its thermal bath Y. The formalism adopted here is the Thermo Field Dynamics in quantum field theory at finite temperature. Bayes' theorem turns out to be naturally related to the minimization of free energy of the system. A discussion of possible consequences of our result is also presented with reference to some neuronal functional activities described in the dissipative quantum model of the brain. Bayes' theorem appears to play a crucial role in the brain action-perception cycle.

The quantum model of the brain proposed by Ricciardi and Umezawa is extended to dissipative dynamics in order to study the problem of memory capacity. It is shown that infinitely many vacua are accessible to memory printing in a way that in sequential information recording the storage of a new information does not destroy the previously stored ones, thus allowing a huge memory capacity. The mechanism of information printing is shown to induce breakdown of time-reversal symmetry. Thermal properties of the memory states as well as their relation with squeezed coherent states are finally discussed.

The received view says that possibility is the dual of necessity: a proposition is (metaphysically, logically, epistemically etc.) possible iff it is not the case that its negation is (metaphysically, logically, epistemically etc., respectively) necessary. This reading is usually taken for granted by modal logicians and indeed seems plausible when dealing with logical or metaphysical possibility. But what about epistemic possibility? We argue that the dual definition of epistemic possibility in terms of epistemic necessity generates tension when reasoning about non-idealized agents and is a problem of concern for most hyperintensional epistemic logics that alleviate the problem of logical omniscience. The tension is particularly evident when knowledge is taken as a primitive to define other epistemic concepts, such as justification and belief, as done in the knowledge-first tradition. We propose a non-dual interpretation of epistemic possibility, employing a hyperintensionality filt...

Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain superfluous restrictions are removed'. The aim of this paper is to develop a clearer understanding of Gödel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.

What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett's, that verification-truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong's that modality can reside in non-modal truthmakers.

Language equivalence and inclusion can be checked coinductively by establishing a (bi)simulation on suitable deterministic automata. In this paper we present an enhancement of this technique called (bi)simulation-up-to. We give general conditions on language operations for which bisimulation-up-to is sound. These results are illustrated by a large number of examples, giving new proofs of classical results such as Arden's rule, and involving the regular operations of union, concatenation and Kleene star as well as language equations with complement and intersection, and shuffle (closure).

We present a study of the notion of coalgebraic simulation introduced by Hughes and Jacobs. Although in their original paper they allow any functorial order in their definition of coalgebraic simulation, for the simulation relations to have good properties they focus their attention on functors with orders which are strongly stable. This guarantees a so-called "composition-preserving" property from which all the desired good properties follow. We have noticed that the notion of strong stability not only ensures such good properties but also "distinguishes the direction" of the simulation. For example, the classic notion of simulation for labeled transition systems, the relation "p is simulated by q", can be defined as a coalgebraic simulation relation by means of a strongly stable order, whereas the opposite relation, "p simulates q", cannot. Our study was motivated by some interesting classes of simulations that illustrate the application of these results: covariant-contravariant simulations and conformance simulations.

It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but analyze the interesting idea on which it is based, namely that an acceptable abstraction has to 'generate' the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.

In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ωlogic and the Ω-conjecture.

According to Weyl, " 'inexhaustibility' is essential to the infinite". However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are "extensionally determinate" and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.

Language equivalence and inclusion can be checked coinductively by establishing a (bi)simulation on suitable deterministic automata. In this paper we present an enhancement of this technique called (bi)simulation-up-to. We give general conditions on language operations for which bisimulation-up-to is sound. These results are illustrated by a large number of examples, giving new proofs of classical results such as Arden's rule, and involving the regular operations of union, concatenation and Kleene star as well as language equations with complement and intersection, and shuffle (closure).

Much work has been done with'the Elementary Theory of the Category of Sets (ETCS) introduced by Lawvere in (21, particu!ariy since its reformulation by lawvere and Tierney I1.31. This work has been done under the assumption that a model of 'this theory is actually. in some sense. 3 category of sets. Our work is based on a subsystem of ETCS, the axioms for 3 Baolcan topos (BT). In Section 1, we define precisely the systenrs involved and give some basic constructions. Then in Section Z we define, for an arbitrary Boolean topos E, a Boolean model A{(E) and show, assuming that E satistItcs a weak axiom of choice (that the subobjects of I form a set of generators). that M(E) is a model da weak systetn %O of set theory. A stronger system 2, of set theory is also defined, and we show that if E satisfies brought to you by CORE View metadata, citation and similar papers at core.ac.uk

An important problem of the modal logic is the interpretation of modality symbols. The most used program of the interpretation of modalities is the "possible world's semantics". In this study, it is demonstrated that the semantics of possible worlds doesn't offer a correct interpretation for modalities because it fails in the attempt to give a solution to the strict implication paradoxes and, more than that, the possible worlds cannot exist. This study shows that the modalities can be adequately interpreted only if we take the time into account. For instance, a proposition is necessary true at a given moment only if it is a consequence of the past at that moment. In this way, the modal value of a proposition can be variable in time.

up to isomorphism by its place in some pattern of maps. In research, structures are studied by these patterns of maps. Foundations can be given entirely in terms of these patterns. In ontology, mathematical objects need have no properties beyond the structural relations re- vealed by the maps. Category theory unied

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T(op), respectively. The q-deformation parameter is rela...

This paper provides a rebuttal to the argument in Elohim (2018) in `Synthese'. Elohim provides a novel hyperintensional, ground-theoretic regimentation of the proposals in the metaphysics of consciousness. He then argues that Chalmers' (2010) intensional two-dimensional conceivability argument against physicalism is unsound, in light of the hyperintensional metaphysics of consciousness. Thus, intensional conceivability cannot be a guide to hyperintensional metaphysics. This paper demonstrates that a multi-hyperintensional version of epistemic two-dimensional semantics can be countenanced, and is sufficient for conceivability to be a guide to metaphysics in the hyperintensional setting such that Chalmers' argument, hyperintensionally construed, is in fact sound.

becoming much easier if we can express them in, should we say, suitable framework. For instance, S 3 can be treated as model of Euclidian space, so we could use apparatus of analysis, algebra and model theory for solving particular problem in Euclidian geometry. On the other hand, some analytical, or algebraical problems became much easier if we can find their geometrical interpretation. Logical background of this method can be found in two different approaches: interpretation method, i.e. formal representation of one first order theory in another (in general case one formal system in another), and category theory, where we exploit equivalence and existence of adjoint functors between certain categories. Our particular interest is application of these methods in automated reasoning. 1

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T(op), respectively. The q-deformation parameter is rela...

The notion of constraint system (cs) is central to declarative formalisms from concurrency theory such as process calculi for concurrent constraint programming (ccp). Constraint systems are often represented as lattices: their elements, called constraints, represent partial information and their order corresponds to entailment. Recently a notion of n-agent spatial cs was introduced to represent information in concurrent constraint programs for spatially distributed multi-agent systems. From a computational point of view a spatial constraint system can be used to specify partial information holding in a given agent's space (local information). From an epistemic point of view a spatial cs can be used to specify information that a given agent considers true (beliefs). Spatial constraint systems, however, do not provide a mechanism for specifying the mobility of information/processes from one space to another. Information mobility is a fundamental aspect of concurrent systems. In th...

We critically review two extant paradigms for understanding the systematic interaction between modality and tense, as well as their respective modifications designed to do justice to the contingency of time's structure and composition. We show that on either type of theory, as well as their respective modifications, some principles prove logically valid whose truth might sensibly be questioned on metaphysical grounds. These considerations lead us to devise a more general logical framework that allows accommodation of those metaphysical views that its predecessors rule out by fiat.

Some systems of modal logic, such as S5, which are often used as epistemic logics with the 'necessity' operator read as 'the agent knows that', are problematic as general epistemic logics for agents whose computational capacity does not exceed that of a Turing machine because they impose unwarranted constraints on the agent's theory of non-epistemic aspects of the world, for example by requiring the theory to be decidable rather than merely recursively axiomatizable. To generalize this idea, two constraints on an epistemic logic are formulated: r.e. conservativeness, that any recursively enumerable theory R in the sublanguage without the epistemic operator is conservatively extended by some recursively enumerable theory in the language with the epistemic operator which is permitted by the logic to be the agent's overall theory; the weaker requirement of r.e. quasi-conservativeness

Every definable forcing class Γ gives rise to a corresponding forcing modality, for which Γ ϕ means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of ω 1-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise.

The paper presents an epistemic logic with quantification over agents of knowledge and with a syntactical distinction between de re and de dicto occurrences of terms. Knowledge de dicto is characterized as 'knowledge that', and knowlegde de re as 'knowledge of'. Transition semantics turns out to be an adequate tool to account for the distinctions introduced.

Theories of scientic representation, following Chakravartty's catego-rization, are divided into two groups. Whereas cognitive-functional views emphasize agents ' intentions, informational theories stress the objective relation between represented and representing. In the rst part, a modied structuralist theory is introduced that takes into ac-count agents ' intentions. The second part is devoted to dismissing a criticism against the structural account of representation on which similarity as the backbone of representation raises serious problems, since it has denite logical features, i.e. re exivity, symmetry and transitivity, which representation lacks. Drawing on the representa-tional relation between quantum and statistical eld theories, I argue that scientic representation displays these logical features, although depending on the context they may be used or not. 1. Scientic Representation and Its Theories