Recent years witnessed a growing interest in non-standard epistemic logics of knowing whether, knowing how, knowing what, knowing why and so on. The new epistemic modalities introduced in those logics all share, in their semantics, the general schema of ∃x◻φ, e.g., knowing how to achieve φ roughly means that there exists a way such that you know that it is a way to ensure that φ. Moreover, the resulting logics are decidable. Inspired by those particular logics, in this work, we (...) propose a very general and powerful framework based on quantifier-free predicate language extended by a new modality, which packs exactly ∃x◻ together. We show that the resulting language, though much more expressive, shares many good properties of the basic propositional modal logic over arbitrary models, such as finite-tree-model property and van Benthem-like characterization w.r.t.\ first-order modal logic. We axiomatize the logic over S5 frames with intuitive axioms to capture the interaction between the new modality and know-that operator in an epistemic setting. (shrink)

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 \textbf{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 two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-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 $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-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 \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{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 \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{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 \textbf{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 $\Omega$-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 \textbf{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 \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{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 \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{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)

ABSTRACT Quine insisted that the satisfaction of an open modalised formula by an object depends on how that object is described. Kripke's ‘objectual’ interpretation of quantified modal logic, whereby variables are rigid, is commonly thought to avoid these Quinean worries. Yet there remain residual Quinean worries for epistemic modality. Theorists have recently been toying with assignment-shifting treatments of epistemic contexts. On such views an epistemic operator ends up binding all the variables in its scope. One might worry that this yields (...) the undesirable result that any attempt to ‘quantify in’ to an epistemic environment is blocked. If quantifying into the relevant constructions is vacuous, then such views would seem hopelessly misguided and empirically inadequate. But a famous alternative to Kripke's semantics, namely Lewis' counterpart semantics, also faces this worry since it also treats the boxes and diamonds as assignment-shifting devices. As I'll demonstrate, the mere fact that a variable is bound is no obstacle to binding it. This provides a helpful lesson for those modelling de re epistemic contexts with assignment sensitivity, and perhaps leads the way toward the proper treatment of binding in both metaphysical and epistemic contexts: Kripke for metaphysical modality, Lewis for epistemic modality. (shrink)

Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (shrink)

Information exchange can be seen as a dynamic process of raising and resolving issues. The goal of this paper is to provide a logical framework to model and reason about this process. We develop an inquisitive dynamic epistemic logic , which enriches the standard framework of dynamic epistemic logic , incorporating insights from recent work on inquisitive semantics. At a static level, IDEL does not only allow us to model the information available to a set of agents, like standard epistemic (...) logic, but also the issues that the agents entertain. At a dynamic level, IDEL does not only allow us to model the effects of communicative actions that provide new information, like standard DEL, but also the effects of actions that raise new issues. Thus, IDEL provides the fundamental tools needed to analyze information exchange as a dynamic process of raising and resolving issues. (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)

Argument evaluation, one of the central problems in argumentation theory, consists in studying what makes an argument a good one. This paper proposes a formal approach to argument evaluation from the perspective of justification logic. We adopt a multi-agent setting, accepting the intuitive idea that arguments are always evaluated by someone. Two general restrictions are imposed on our analysis: non-deductive arguments are left out and the goal of argument evaluation is fixed: supporting a given proposition. Methodologically, our approach uses several (...) existing tools borrowed from justification logic, awareness logic, doxastic logic and logics for belief dependence. We start by introducing a basic logic for argument evaluation, where a list of argumentative and doxastic notions can be expressed. Later on, we discuss how to capture the mentioned form of argument evaluation by defining a preference operator in the object language. The intuitive picture behind this definition is that, when assessing a couple of arguments, the agent puts them to a test consisting of several criteria. As a result of this process, a preference relation among the evaluated arguments is established by the agent. After showing that this operator suffers a special form of logical omniscience, called preferential omniscience, we discuss how to define an explicit version of it, more suitable to deal with non-ideal agents. The present work exploits the formal notion of awareness in order to model several informal phenomena: awareness of sentences, availability of arguments and communication between agents and external sources. We discuss several extensions of the basic logic and offer completeness and decidability results for all of them. (shrink)

It is well known that the convex hull of the classical truth value functions contains all and only the probability functions. Work by Paris and Williams has shown that this also holds for various kinds of nonclassical logics too. This note summarises the formal details of this topic and extends the results slightly.

We show that subspaces of the space ${\mathbb{Q}}$ of rational numbers give rise to uncountably many d-logics over K4 without the finite model property.

Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: after which , does it hold that Kφ? We give various semantic results and show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.

We give a definition of bisimulation for conditional modalities interpreted on selection functions and prove the correspondence between bisimilarity and modal equivalence, generalizing the Hennessy–Milner Theorem to a wide class of conditional operators. We further investigate the operators and semantics to which these results apply. First, we show how to derive a solid notion of bisimulation for conditional belief, behaving as desired both on plausibility models and on evidence models. These novel definitions of bisimulations are exploited in a series of (...) undefinability results. Second, we treat relativized common knowledge, underlining how the same results still hold for a different modality in a different semantics. Third, we show the flexibility of the approach by generalizing it to multi-agent systems, encompassing the case of multi-agent plausibility models. (shrink)

We introduce a new understanding of deontic modals that we callobligations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory.

We need to understand the impossible. Francesco Berto and Mark Jago start by considering what the concepts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition have in common. They are all concepts which divide the world up more finely than logic does. Logically equivalent sentences may carry different meanings and information and may differ in how they're believed. Fictions can be inconsistent yet meaningful. We can suppose impossible things without collapsing into total incoherence. Yet for the leading philosophical (...) theories of meaning, these phenomena are an unfathomable mystery. To understand these concepts, we need a metaphysical, logical, and conceptual grasp of situations that could not possibly exist: Impossible Worlds. This book discusses the metaphysics of impossible worlds and applies the concept to a range of central topics and open issues in logic, semantics, and philosophy. It considers problems in the logic of knowledge, the meaning of alternative logics, models of imagination and mental simulation, the theory of information, truth in fiction, the meaning of conditional statements, and reasoning about the impossible. In all these cases, impossible worlds have an essential role to play. (shrink)

The aim of this paper, is to provide a logical framework for reasoning about actions, agency, and powers of agents and coalitions in game-like multi-agent systems. First we define our basic Dynamic Logic of Agency ( ). Differently from other logics of individual and coalitional capability such as Alternating-time Temporal Logic (ATL) and Coalition Logic, in cooperation modalities for expressing powers of agents and coalitions are not primitive, but are defined from more basic dynamic logic operators of action and (historic) (...) necessity. We show that STIT logic can be reconstructed in . We then extend with epistemic operators, which allows us to distinguish capability and power. We finally characterize the conditions under which agents are aware of their capabilities and powers. (shrink)

We introduce a three-tiered framework for modelling and reasoning about agents who (i) can use possibly complete reasoning systems without any restrictions but who nevertheless are (ii) bounded in the sense that they never reach infinitely many results and, finally, who (iii) perform their reasoning in time. This last aspect does not concern so much the time it takes for agents to actually carry out their reasoning, as the time which can bring about external changes in the agents’ states such (...) as arriving of new information or discarding previously available information due to bounds of the agent’s resources. These three aspects are treated with the maximal possible degree of independence from each other. The treatment of layer (iii) can be combined with arbitrary logic at level (ii) which, in turn, can be combined with arbitrary agent logic at level (i). At the level (iii), we discuss briefly the duality (or rather, complementarity) of system descriptions based on actions and transitions, on the one hand, and states and their changes, on the other. We settle for the latter and present a simple language, for describing state changes, which is parameterized by an arbitrary language for describing properties of the states. The language can be viewed as a simple fragment of step logic, admitting however various extensions by appropriate choices of the underlying logic. Alternatively, it can be seen as a very specific fragment of temporal logic (with a variant of ‘until’ or ‘chop’ operator), and is interpreted over dense linear time. The reasoning system presented here is sound, as well as strongly complete and decidable (provided that so is the parameter logic for reasoning about single states). We give the main idea of the completeness proof and suggest a wide range of possible applications, which is a simple consequence of the parametric character of both the language and the reasoning system. We address then in more detail the case of non-omniscient rational agents, (ii). Their models are syntactic structures (sets of available formulae) similar in spirit, though not in the technical formulation, to the models used in other syntax oriented approaches and in active logic. We give a new sound, complete and decidable system for reasoning about such agents. Finally, we illustrate its extensions with the internal reasoning of agents, (i), by equipping them with some example logics. (shrink)

McCall (1984) offered a semantics of counterfactual conditionals based on “real possible worlds” that avoids using the vague notion of similarity between possible worlds. I will propose an interpretation of McCall’s counterfactuals in a formal framework based on Baltag-Moss-Solecki events and protocols. Moreover, I will argue that using this interpretation one can avoid an objection raised by Otte (1987).

Scientists collect evidence in order to confirm or falsify scientific theories. Unfortunately, scientific evidence may sometimes be false or deceiving and as a consequence lead individuals to believe in a false theory. By interaction between scientists, such false beliefs may spread through the entire community. There is currently a debate about the effect of various network configurations on the epistemic reliability of scientific communities. To contribute to this debate from a logical perspective, this paper introduces an epistemic logical framework of (...) observation, interaction and belief revision in scientific communities. The presented sound and complete system provides the formal tools for qualitative analysis of the social dynamics of scientific inquiry. Furthermore, this paper includes detailed suggestions for future applications of the framework. (shrink)

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop short of a (...) formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics. (shrink)

The effects of public announcements, private communications, deceptive messages to groups, and so on, can all be captured by a general mechanism of updating multi-agent models with update action models, now in widespread use. There is a natural extension of the definition of a bisimulation to action models. Surely enough, updating with bisimilar action models gives the same result. But the converse turns out to be false: update models may have the same update effects without being bisimilar. We propose action (...) emulation as a notion of equivalence more appropriate for action models, and generalizing standard bisimulation. It is proved that action emulation provides a full characterization of update effect. We first concentrate on the general case, and next focus on the important case of action models with propositional preconditions. Our notion of action emulation yields a simplification procedure for action models, and it gives designers of multi-agent systems a useful tool for comparing different ways of representing a particular communicative action. (shrink)

In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.

We present an extension of the basic revision theory of circular definitions with a unary operator, □. We present a Fitch-style proof system that is sound and complete with respect to the extended semantics. The logic of the box gives rise to a simple modal logic, and we relate provability in the extended proof system to this modal logic via a completeness theorem, using interpretations over circular definitions, analogous to Solovay’s completeness theorem forGLusing arithmetical interpretations. We adapt our proof to (...) a special class of circular definitions as well as to the first-order case. (shrink)

In the previous paper with a similar title :311–344, 2018), we presented a family of propositional epistemic logics whose languages are extended by two ingredients: by quantification over modal operators or over agents of knowledge and by predicate symbols that take modal operators as arguments. We denoted this family by \}\). The family \}\) is defined on the basis of a decidable higher-order generalization of the loosely guarded fragment of first-order logic. And since HO-LGF is decidable, we obtain the decidability (...) of logics of \}\). In this paper we construct an alternative family of decidable propositional epistemic logics whose languages include ingredients and. Denote this family by \}\). Now we will use another decidable fragment of first-order logic: the two variable fragment of first-order logic with two equivalence relations +2E) [the decidability of FO\+2E was proved in Kieroński and Otto :729–765, 2012)]. The families \}\) and \}\) differ in the expressive power. In particular, we exhibit classes of epistemic sentences considered in works on first-order modal logic demonstrating this difference. (shrink)

We study set algebras with an operator (SAO) that satisfy the axioms of S5 knowledge. A necessary and sufficient condition is given for such SAOs that the knowledge operator is defined by a partition of the state space. SAOs are constructed for which the condition fails to hold. We conclude that no logic singles out the partitional SAOs among all SAOs.

Two kinds of theoretical framework for syntax are encountered in current linguistics. One emerged from the mathematization of proof theory, and is referred to here as generative-enumerative syntax (GES). A less explored alternative stems from the semantic side of logic, and is here called model-theoretic syntax (MTS). I sketch the outlines of each, and give a capsule summary of some mathematical results pertaining to the latter. I then briefly survey some diverse types of evidence suggesting that in some ways MTS (...) seems better suited to theorizing about the relevant linguistic phenomena. (shrink)

This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of (...) the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs. (shrink)

We introduce and study a PDL-style logic for reasoning about protocols, or plans, under imperfect information. Our paper touches on a number of issues surrounding the relationship between an agent’s abilities, available choices, and information in an interactive situation. The main question we address is under what circumstances can the agent commit to a protocol or plan, and what can she achieve by doing so?

We propose a general framework for dynamic epistemic logics. It consists of a generic language for DELs and a class of structures, called model transition systems, that describe model transformations in a static way. An MTS can be viewed as a two-layered Kripke model and consequently inherits standard concepts such as bisimulation and bounded morphism from the ordinary Kripke models. In the second half of this article we add the global operator to the language, which enables us to define the (...) notions of a canonical MTS and canonicity of a DEL formula for a property of MTSs. Using these notions, we clarify correspondences between axioms of DELs and properties of MTSs. (shrink)

We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract (...) properties and are preserved and reflected by translations under suitable conditions. (shrink)

The global consequence relation of a normal modal logic \ is formulated as a global sequent calculus which extends the local sequent theory of \ with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over \. The preservation of Craig interpolation property from local to global sequent theories of any normal modal (...) logic is shown by proof-theoretic method. (shrink)

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.

We present a modal logic called (logic of intention and attempt) in which we can reason about intention dynamics and intentional action execution. By exploiting the expressive power of , we provide a formal analysis of the relation between intention and action and highlight the pivotal role of attempt in action execution. Besides, we deal with the problems of instrumental reasoning and intention persistence.

We continue the work initiated in Herzig and Lorini (J Logic Lang Inform, in press) whose aim is to provide a minimalistic logical framework combining the expressiveness of dynamic logic in which actions are first-class citizens in the object language, with the expressiveness of logics of agency such as STIT and logics of group capabilities such as CL and ATL. We present a logic called ( Deterministic Dynamic logic of Agency ) which supports reasoning about actions and joint actions of (...) agents and coalitions, and agentive and coalitional capabilities. In it is supposed that, once all agents have selected a joint action, the effect of this joint action is deterministic. In order to assess we prove that it embeds Coalition Logic. We then extend with modal operators for agents’ preferences, and show that the resulting logic is sufficiently expressive to capture the game-theoretic concepts of best response and Nash equilibrium. (shrink)

We study a generalization of the standard syntax and game-theoretic semantics of logic, which is based on a duality between two players, to a multiplayer setting. We define propositional and modal languages of multiplayer formulas, and provide them with a semantics involving a multiplayer game. Our focus is on the notion of equivalence between two formulas, which is defined by saying that two formulas are equivalent if under each valuation, the set of players with a winning strategy is the same (...) in the two respective associated games. We provide a derivation system which enumerates the pairs of equivalent formulas, both in the propositional case and in the modal case. Our approach is algebraic: We introduce multiplayer algebras as the analogue of Boolean algebras, and show, as the corresponding analog to Stone’s theorem, that these abstract multiplayer algebras can be represented as concrete ones which capture the game-theoretic semantics. For the modal case we prove a similar result. We also address the computational complexity of the problem whether two given multiplayer formulas are equivalent. In the propositional case, we show that this problem is co-NP-complete, whereas in the modal case, it is PSPACE-hard. (shrink)

In this paper, we first propose a simple formal language to specify types of agents in terms of necessary conditions for their announcements. Based on this language, types of agents are treated as ‘first-class citizens’ and studied extensively in various dynamic epistemic frameworks which are suitable for reasoning about knowledge and agent types via announcements and questions. To demonstrate our approach, we discuss various versions of Smullyan’s Knights and Knaves puzzles, including the Hardest Logic Puzzle Ever (HLPE) proposed by Boolos (...) (in Harv Rev Philos 6:62–65, 1996). In particular, we formalize HLPE and verify a classic solution to it. Moreover, we propose a spectrum of new puzzles based on HLPE by considering subjective (knowledge-based) agent types and relaxing the implicit epistemic assumptions in the original puzzle. The new puzzles are harder than the previously proposed ones in the literature, in the sense that they require deeper epistemic reasoning. Surprisingly, we also show that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. Our formalism paves the way for studying these new puzzles using automatic model checking techniques. (shrink)