This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

A world is trivial if it makes every proposition true all at once. Such a world is impossible, an absurdity. Our world, we hope, is not an absurdity. It is important, nevertheless, for semantic and metaphysical theories that we be able to reason cogently about absurdities—if only to see that they are absurd. In this note we describe methods for ‘observing’ absurd objects like the trivial world without falling in to incoherence, using some basic techniques from modal logic. The goal (...) is to begin to locate the trivial world’s relative position in modal space; the outcome is that the less we assume about relative possibility, the more detail we can discern at the edge of reason. (shrink)

According to certain normative theories in epistemology, rationality requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omniscience requirement in Bayesian epistemology. Building on a key insight by Hacking :311–325, 1967), I develop a version of Bayesianism that permits logical ignorance. This includes: an account of the synchronic norms that govern a logically ignorant individual at any given time; (...) an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence; and an account of when logical ignorance is irrational and when it isn’t. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations. (shrink)

Several authors maintain that we are capable of having inconsistent intentional states, either in cases of illusion, in certain cases of imagination, or because the observable world is (partly) inconsistent and we perceive it as such. These views are all premised on the assumption that inconsistent intentional states—even if acknowledged—are peculiar and have nothing essential to do with our perceptual capacities. In the present article, I would like to present, and argue for, a much stronger thesis: that inconsistent intentional states (...) underlie the possibility of having intentional content in mind. I argue for this thesis based on a Husserlian phenomenological analysis of our grasp of objects, which I formulate in terms of incompatibility semantics. (shrink)

Modal Meinongianism is a form of Meinongianism whose main supporters are Graham Priest and Francesco Berto. The main idea of modal Meinongianism is to restrict the logical deviance of Meinongian non-existent objects to impossible worlds and thus prevent it from “contaminating” the actual world: the round square is round and not round, but not in the actual world, only in an impossible world. In the actual world, supposedly, no contradiction is true. I will show that Priest’s semantics, as originally formulated (...) in Towards Non-being, tell us something different. According to certain models, there are true contradictions in the actual world. Berto and Priest have noticed this unexpected consequence and have suggested a solution, but I will show that their solution is highly questionable. In the last section of this paper, I will introduce a new and simpler version of modal Meinongianism that avoids the problem. (shrink)

This paper defends a rational account of conceivability according to which conceiving is a kind of modal thinking that is distinct from imagining effectively allowing us to think beyond what we can imagine, and that we are subject to rational rather than experiential constraints when we do so. Defending this view involves appealing to the perspective of an idealized agent and I’ll argue that this appeal is not worrisome given an “objective” view of propositional justification.

Imagination has received a great deal of attention in different fields such as psychology, philosophy and the cognitive sciences, in which some works provide a detailed account of the mechanisms involved in the creation and elaboration of imaginary worlds. Although imagination has also been formalized using different logical systems, none of them captures those dynamic mechanisms. In this work, we take inspiration from the Common Frame for Imagination Acts, that identifies the different processes involved in the creation of imaginary worlds, (...) and we use it to define a dynamic formal system called the Logic of Imagination Acts. We build our logic by using a possible-worlds semantics, together with a new set of static and dynamic modal operators. The role of the new dynamic operators is to call different algorithms that encode how the formal model is expanded in order to capture the different mechanisms involved in the creation and development of imaginary worlds. We provide the definitions of the language, the semantics and the algorithms, together with an example that shows how the model is expanded. By the end, we discuss some interesting features of our system, and we point out to possible lines of future work. (shrink)

Imagining is something we use everyday in our lives and in a wide variety of ways. In spite of the amount of works devoted to its study from both psychology and philosophy, there are only a few formal systems capable of modelling it; besides, almost all of those systems are static, in the sense that their models are initially predefined, and they fail to capture the dynamic process behind the creation of new imaginary scenarios. In this work, we review some (...) influential theories of imagination and use their insights to distil an algorithm describing such process. Then, we use this algorithm to define a dynamic logical system built upon on a single-agent epistemic logic that provides the necessary tools to capture how the agent voluntarily creates new imaginary worlds; in other words, our system allows the model to be expanded dynamically at any time as a result of the agent performing an act of imagination. Furthermore, we provide an axiomatization and prove that the system is sound and complete. (shrink)

Imagination has received a great deal of attention in different fields such as psychology, philosophy and the cognitive sciences, in which some works provide a detailed account of the mechanisms involved in the creation and elaboration of imaginary worlds. Although imagination has also been formalized using different logical systems, none of them captures those dynamic mechanisms. In this work, we take inspiration from the Common Frame for Imagination Acts, that identifies the different processes involved in the creation of imaginary worlds, (...) and we use it to define a dynamic formal system called the Logic of Imagination Acts. We build our logic by using a possible-worlds semantics, together with a new set of static and dynamic modal operators. The role of the new dynamic operators is to call different algorithms that encode how the formal model is expanded in order to capture the different mechanisms involved in the creation and development of imaginary worlds. We provide the definitions of the language, the semantics and the algorithms, together with an example that shows how the model is expanded. By the end, we discuss some interesting features of our system, and we point out to possible lines of future work. (shrink)

In this paper, we review three influential theories of imagination in order to understand how the dynamics of imagination acts could be modeled using formal languages. While reviewing them, we notice that they are not detailed enough to account for all the mechanisms involved in creating and developing imaginary worlds. We claim those theories could be further refined into what we call the Common Frame for Imagination Acts, which defines a framework that can be used to study the dynamics of (...) imagination using formal languages, and we support our claim by showing how the framework allows to model certain imagination acts step by step. Then, we introduce the Rhombus of Imagination, which is a visual tool that can be used in conjunction with our framework to study the procedural structure of different kinds of imagination acts and identify their dynamic properties. (shrink)

In this paper, we review three influential theories of imagination in order to understand how the dynamics of imagination acts could be modeled using formal languages. While reviewing them, we notice that they are not detailed enough to account for all the mechanisms involved in creating and developing imaginary worlds. We claim those theories could be further refined into what we call the Common Frame for Imagination Acts, which defines a framework that can be used to study the dynamics of (...) imagination using formal languages, and we support our claim by showing how the framework allows to model certain imagination acts step by step. Then, we introduce the Rhombus of Imagination, which is a visual tool that can be used in conjunction with our framework to study the procedural structure of different kinds of imagination acts and identify their dynamic properties. (shrink)

Dilip Ninan has raised a puzzle for centered world accounts of de re attitude reports extended to accommodate what he calls “counterfactual attitudes.” As a solution, Ninan introduces multiple centers to the standard centered world framework, resulting in a more robust semantics for de re attitude reports. However, while the so-called multi-centered world proposal solves Ninan’s counterfactual puzzle, this additional machinery is not without problems. In Section 1, I present the centered world account of attitude reports, followed by the extension (...) to counterfactual attitudes which Ninan targets with his puzzle. In Section 2, I pose the counterfactual puzzle and present Ninan’s multi-centered world solution, emphasizing similarities and differences between multi-centered and centered world accounts of attitude reports. In Section 3, I argue the counterfactual attitude wishing falls under the purview of the multi-centered proposal, but that the proposal generates false predictions for wish reports concerning unsatisfiable content. I canvass responses, ultimately concluding Ninan’s proposal requires substantial revisions. (shrink)

I present a possible worlds semantics for a hyperintensional belief revision operator, which reduces the logical idealization of cognitive agents affecting similar operators in doxastic and epistemic logics, as well as in standard AGM belief revision theory. (Revised) belief states are not closed under classical logical consequence; revising by inconsistent information does not perforce lead to trivialization; and revision can be subject to ‘framing effects’: logically or necessarily equivalent contents can lead to different revisions. Such results are obtained without resorting (...) to non-classical logics, or to non-normal or impossible worlds semantics. The framework combines, instead, a standard semantics for propositional S5 with a simple mereology of contents. (shrink)

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the imagined (...) non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics. (shrink)

It is a venerable slogan due to David Hume, and inherited by the empiricist tradition, that the impossible cannot be believed, or even conceived. In Positivismus und Realismus, Moritz Schlick claimed that, while the merely practically impossible is still conceivable, the logically impossible, such as an explicit inconsistency, is simply unthinkable. -/- An opposite philosophical tradition, however, maintains that inconsistencies and logical impossibilities are thinkable, and sometimes believable, too. In the Science of Logic, Hegel already complained against “one of the (...) fundamental prejudices of logic as hitherto understood”, namely that “the contradictory cannot be imagined or thought” (Hegel 1931: 430). Our representational capabilities are not limited to the possible, for we appear to be able to imagine and describe also impossibilities — perhaps without being aware that they are impossible. -/- Such impossibilities and inconsistencies are what this entry is about... (shrink)

This paper aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of the equivalence relations countenanced in Homotopy Type Theory, in order to specify an abstraction principle for epistemic intensions. The homotopic abstraction principle for epistemic intensions provides an epistemic conduit into our knowledge of intensions as abstract objects. I examine, then, how intensional functions in Epistemic Modal Algebra are deployed as core models in the philosophy of mind, Bayesian perceptual psychology, (...) and the program of natural language semantics in linguistics, and I argue that this provides abductive support for the truth of homotopic abstraction. Epistemic modality can thus be shown to be both a compelling and a materially adequate candidate for the fundamental structure of mental representational states, comprising a fragment of the Language of Thought. (shrink)

It is commonly believed that the role of context cannot be ignored in the analysis of conditionals, and counterfactuals in particular. On truth conditional accounts involving possible worlds semantics, conditionals have been analysed as expressions of relative necessity: “If A, then B” is true at some world w if B is true at all the A-worlds deemed relevant to the evaluation of the conditional at w. A drawback of this approach is that for the evaluation of conditionals with the same (...) antecedents at some world, the same worlds are deemed as relevant for all occasions of utterance. But surely this is inadequate, if shifts of contexts between occasions are to be accounted for. Both the linguistic and logical implications of this defect are discussed, and in order to overcome it a modification of David Lewis’ ordering semantics for counterfactuals is developed for a modified language. I follow Lewis by letting contexts determine comparative similarity assignments, and show that the addition of syntactic context parameters (context indices) to the language gives the freedom required to switch between sets of relevant antecedent worlds from occasion to occasion by choosing the corresponding similarity assignment accordingly. Thus an account that extends Lewis’ analysis of a language containing a single counterfactual connective > to a language containing infinitely many counterfactual connectives >_c, each indexed by a different context name c, overcomes the limitations of traditional analyses. Finally it is also shown that these traditional accounts can be recovered from the modified account if certain contextual restrictions are in place. -/- . (shrink)

The two main features of this thesis are (i) an account of contextualized (context indexed) counterfactuals, and (ii) a non-vacuist account of counterpossibles. Experience tells us that the truth of the counterfactual is contingent on what is meant by the antecedent, which in turn rests on what context is assumed to underlie its reading (intended meaning). On most conditional analyses, only the world of evaluation and the antecedent determine which worlds are relevant to determining the truth of a conditional, and (...) consequently what its truth value is. But that results in the underlying context being fixed, when evaluating distinct counterfactuals with the same antecedent on any single occasion, even when the context underlying the evaluation of each counterfactual may vary. Alternative approaches go some of the way toward resolving this inadequacy by appealing to a difference in the consequents associated with counterfactuals with the same antecedent. That is, in addition to the world of evaluation and the antecedent, the consequent contributes to the counterfactual’s evaluation. But these alternative approaches nevertheless give a single, determinate truth value to any single conditional (same antecedent and consequent), despite the possibility that this value may vary with context. My reply to these shortcomings (chapter 4) takes the form of an analysis of a language that makes appropriate explicit access to the intended context available. That is, I give an account of a contextualized counterfactual of the form ‘In context C: If it were the case that … , then it would be the case that …’. Although my proposal is largely based on Lewis’ (1973, 1981) analyses of counterfactuals (the logic VW and its ordering semantics), it does not require that any particular logic of counterfactuals should serve as its basis – rather, it is a general prescription for contextualizing a conditional language. The advantage of working with ordering semantics stems from existing results (which I apply and develop) concerning the properties of ordering frames that facilitate fashioning and implementing a notion of contextual information preservation. Analyses of counterfactuals, such as Lewis’ (1973), that cash out the truth of counterfactuals in terms of the corresponding material conditional’s truth at possible worlds result in all counterpossibles being evaluated as vacuously true. This is because antecedents of counterpossibles are not true at any possible world, by definition. Such vacuist analyses have already been identified and challenged by a number of authors. I join this critical front, and drawing on existing proposals, I develop an impossible world semantics for a non-vacuist account of counterpossibles (chapter 5), by modifying the same system and semantics that serve the basis of the contextualized account offered in chapter 4, i.e. Lewis’ (1986) ordering semantics for the logic VW. I critically evaluate the advantages and disadvantages of key conditions on the ordering of worlds on the extended domain and show that there is a sense in which all of Lewis’ analysis of mere counterfactuals can be preserved, whilst offering an analysis of counterpossibles that meets our intuitions. The first part of chapter 1 consists of an outline of the usefulness of impossible worlds across philosophical analyses and logic. That outline in conjunction with a critical evaluation of Lewis’ logical arguments in favour of vacuism in chapter 2, and his marvellous mountain argument against impossible worlds in chapter 3, serves to motivate and justify the impossible world semantics for counterpossibles proposed in chapter 5. The second part of chapter 1 discusses the limitations that various conditional logics face when tasked to give an adequate treatment of the influence of context. That introductory discussion in conjunction with an overview of conditional logics and their various semantics in chapter 2 – which includes an in-depth exposition of Stalnaker-Lewis similarity semantics for counterfactuals – serves as the motivation and conceptual basis for the contextualized account of counterfactuals proposed in chapter 4. (shrink)

In my thesis, I aim to develop a systematic and philosophically coherent thesis of ultimate reality for Ibn Arabi. In this pursuit, I adopt the style of analytic philosophy, seeking to employ and utilise some of its methods and theories. The philosophical aspects of Ibn Arabi’s doctrine are in dire need of conceptual clarification and systematic analysis with a closer focus on argumentation. The analytic tradition will prove most helpful in this regard. In my thesis, I begin by tracing Ibn (...) Arabi’s related views and concepts as they are dispersed throughout his writings. I then clarify, sharpen and, in many cases, develop these views and concepts into fully constructed forms. Finally, I weave the developed concepts and views into a systematic thesis or set of sub-theses. Where necessary, I provide my own arguments and concepts to help substantiate and strengthen the structure of Ibn Arabi’s thesis. I propose that Ibn Arabi’s various, and sometimes apparently inconsistent, views are best presented in terms of three main concepts: the concept of ultimate reality as existence, which has long been known as Ibn Arabi’s thesis of the oneness of existence ; the concept of ultimate reality as the Divine Self ; and the concept of ultimate reality as God. The three concepts neither represent different ultimate realities nor different perspectives of the same ultimate reality. Instead, they represent the same ultimate reality but each one has its own unique scope and encompassment. The three concepts are aimed at encompassing Ibn Arabi’s various views and concepts of ultimate reality in a consistent and systematic manner. (shrink)