Modal Logic

We introduce the Ordinal Folding Index (OFI), a computable, countable ordinal assigned to every well-formed formula of a reflective language by a monotone-with-delay evaluation operator. This operator is (i) continuous on countable chains, (ii) layer-aware for probabilistic truth values, and (iii) parameterized by a tunable evidence functor capturing empirical updates. The OFI of a formula is defined as the first stage at which the fold-back of the operator into a syntactic normal form becomes idempotent (i.e. further unfolding yields no new (...) information). Intuitively, OFI measures the "depth" of self-reference needed before a sentence's truth value stabilizes. We show that OFI strictly refines classical closure ordinals from fixed-point logics while remaining recursively enumerable, admits polynomial-time approximations on finite models, and coincides with the length of the shortest parity-fold winning strategy in the associated evaluation game. This furnishes a single transfinite scale that unifies transfinite fixed-point depth, ordinal game values, and practical convergence diagnostics for large language models. We situate OFI in relation to the modal????-calculus alternation hierarchy, coalgebraic modal logic ranks, and proof-theoretic ordinals from formal arithmetic. An empirical section demonstrates how OFI can be estimated for transformer-based language models by iteratively feeding model outputs back into the model (a form of self-consistency probing), with the measured stabilization ordinals correlating with model perplexity and chain-of-thought complexity. Finally, we catalog five open problems in this new area-ranging from the completeness of the spectrum of OFI (can every computable ordinal arise as an OFI?) to the design of a self-bounding reflective operator-and we outline possible research pathways toward their resolution. (shrink)

A brief overview of the system S5 in modal logic as defined by Brian F. Chellas, author of "Modal Logic: An Introduction." The history and usage of modal logic are given mention, along with some applications. Very much a draft. Written for PhileInSophia on July 5, 2021.

This paper defends a thesis stronger than epistemic humility about modality and weaker than metaphysical eliminativism: the concept of framework-transcendent impossibility is theoretically idle. It does no explanatory or inferential work that framework-relative impossibility cannot do, it has no independent route of epistemic access, and the appearance that it is required arises from a hypostatisation of framework-internal achievements. I call this the Idleness Thesis. It is logically stronger than the claim that we cannot demonstrate absolute impossibility (since something undemonstrable might (...) still play indispensable theoretical roles) and weaker than the claim that there are no absolute impossibilities (since the absence of theoretical work is compatible with mind-independent modal facts). The Idleness Thesis is the only version of framework-relativism that engages necessitism and absolutism on terms they cannot evade by retreating to pure metaphysics. I argue that meaning is itself framework-determined, that this dissolves the standard "change of subject" objection to logical pluralism, and that absolutism cannot give content to its central modal claims without conscripting framework-internal resources. The conclusion is not that nothing is impossible, but that nothing is impossible "absolutely" in any sense that earns its theoretical keep. (shrink)

Possibility semantics offers an elegant framework for a semantic analysis of modal logic that does not recruit fully determinate entities such as possible worlds. The present papers considers the application of possibility semantics to the modeling of the indeterminacy of the future. Interesting theoretical problems arise in connection to the addition of object-language determinacy operator. We argue that adding a two-dimensional layer to possibility semantics can help solve these problems. The resulting system assigns to the two-dimensional determinacy operator a well-known (...) logic (coinciding with the logic of universal modalities under global consequence). The paper concludes with preliminary investigations of how to integrate the system with a temporalist conception of propositions and how to compare to more established frameworks. (shrink)

Quantificationalism is the view that some true propositions are false at or relative to some domains of quantification, in much the same sense in which propositions can be true or false at times or possible worlds. This paper seeks to give solid formal foundations to quantificationalism, comparable to what modal and tense logics are to the notions of contingent and temporary truth. It has the twofold purpose of understanding what sort of problems arise when we try and make precise sense (...) of Quantificationalism, and of solving these problems by outlining a comprehensive higher-order logical framework in which Quantificationalism can be precisely formulated and shown to be consistent. (shrink)

This essay aims to provide a modal logic and hyperintensional semantics for Charles Parsons (1980)'s treatment of rational intuition as a mathematical modality. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of `intuition-of', i.e., (...) the cognitive phenomenal properties of thoughts, and the modal operators regimenting the notion of `intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via a topic-sensitive epistemic two-dimensional truthmaker semantics. I relate the modal logic and hyperintensional semantics for the modality to self-knowledge, because of Gödel's footnotes and fragment in which (I) Gödel relates rational intuition to (i) the realistic aspect of idealism, and to (ii) self-knowledge, and (II) I countenance fixed points in automata for the epistemic modal mu-calculus (iii) to characterize the iteration of epistemic states, and (iv) to specify how Gödel's claim to Wang that intuition might be computational might be characterized. (shrink)

Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a number of results such as compactness, a semantic characterisa- tion of elementary equivalence, the existence of a quadratic-time decision pro- cedure, and Brandom’s incompatibility semantics property. We demonstrate the usefulness of the logic as a language for knowledge representation.

This paper revisits and extends the proof first presented in Epistemological Observations about Mind–Machine Equivalence by Farzad Didehvar and Mohammad Saleh Zarepour (2007). The central claim is that no agent can ever attain certainty of being a machine — even in the case that it truly is one. The argument is based on two principles: fairness, which forbids privileging self-testimony, and comparability, which requires that self- and adversary testimony be treated as epistemically equal. By embedding the original proof into modal (...) logic systems (K, T, D, S4, S5), we show that fairness alone suffices to block certainty in weaker systems, while comparability becomes essential in S5. This meta-theorem clarifies the logical foundations of self-certainty and highlights the philosophical significance of accessibility conditions in epistemic logic. (THIS PAPER IS UNDER CONSTRUCTION, SEEMINGLY IT NEEDS TO MUCH WORKS IN LOGICAL PART). (shrink)

It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.

This paper develops formal semantics and tableau-based proof procedures for several modal logics using Resolution Matrix Semantics (RMS), a non-relational approach that employs determinate truth values (tn, tc, fc, fn) alongside indeterminate values (true t, false f, both t/f). Unlike Kripke's relational models, which excel in relational contexts, RMS offers a productive alternative for modeling elusive precision, akin to quantum indeterminacy, while preserving modal distinctions. A tableau method is introduced, extending classical decomposition rules to RMS truth values. Branching rules for (...) connectives and □ handle determinate and indeterminate cases, with dual algorithms (FA for f, TFA for t/f) for KD and K. Examples demonstrate validity (e.g., □p → p in KT, □p → ¬□¬p in KD) and non-validity (e.g., p → □◇p in KT), closing all branches for theorems. These contributions enhance RMS's applicability, providing efficient, non-relational tools for modal reasoning without worlds. (shrink)

Human thinking does not proceed within a single logic. It stabilizes meaning at the intersection of multiple, partially incompatible logics while tolerating indeterminacy. This paper develops quantum-inspired polylogical systems - formal framework in which this cognitive fact becomes a principle of inference. Building on Resolution Matrix Semantics, indeterminate truth values are interpreted as semantic superpositions, and logical systems themselves form a space of interacting constraints. Inference is reconceived not as derivation within a fixed logic, but as the emergence of stable (...) semantic configurations through interference and suppression. Unlike conventional quantum AI, which accelerates classical reasoning, the proposed model alters the semantics of inference itself. Poly-Logic Quantum AI thus offers a structural model of human reasoning—one that explains coherence without certainty, context sensitivity without logic switching, and the emergence of new ideas without randomness. The result is a theory of artificial reasoning closer to human thinking than to calculation. (shrink)

The text explores the moral law as a super-natural phenomenon that can be demonstrated rationally. Starting from the observation that, in nature, every action is governed by the instinct for survival and personal interest, it shows how purely altruistic behavior seems apparently impossible. Yet, the moral law emerges as a coherent and universal principle: although it can generate personal advantages, its essence does not depend on self-interest, thus distinguishing it from natural causality. The novelty of this work lies in the (...) logical and conceptual demonstration of the moral law, without relying on traditional religious, cultural, or empirical assumptions. Unlike Kant, who postulated it a priori, or Thomas Aquinas, who grounded it in theology, here morality derives from ontological coherence and the concrete possibility of application. In this way, the text proposes an original paradigm of moral philosophy: the moral law is not merely a postulate, but a rationally necessary emergence, conceptually super-natural and applicable. The six appendices build the argumentative framework: Appendix A defines the implications of the principle of conservation. Appendix B distinguishes between freedom of choice, which is subordinated to personal interest, and the choice of freedom, attainable only through adherence to the moral law, highlighting how morality represents the only coherent choice capable of emancipating the individual from servitude. Appendix C explores the dialectic between the Self and the Ego: left to instinct alone, the individual degenerates; the moral method, instead, enables authentic freedom, coherence, and objective understanding. Appendix D introduces scientific analogies, such as the wave-particle duality, to show the symbolic limits of human knowledge, while emphasizing that the moral law can be understood through concrete and universal examples. Appendix E, through a logical-formal demonstration, makes explicit the conceptual validity of the moral law, clearly distinguishing absolute moral freedom from illusory freedom of choice. The lemmas and theorem show how a coherent moral action, guided by rational will and immune to self-interest, is necessarily possible and applicable, providing a rigorous and rational foundation for the idea of super-natural moral behavior. Appendix F presents a logical-formal example. In summary, the text builds a bridge between philosophical narrative and logical formalization, proposing an innovative, coherent, and deeply original framework for moral philosophy. (shrink)

This paper argues that the contrast between “something” and “nothing” fails before explanation begins: absolute nothingness is not a failed rival to actuality, but never qualifies as a rival at all. The argument requires no necessary being, no privileged inventory of existents, and no particular modal logic. Nor does it begin from intuitions about empty worlds or from the claim that we cannot imagine “nothing.” It turns instead on a prior question: what must be in place for a candidate to (...) count as an alternative? That condition is the Structure-Base Principle (SBP): determinacy cannot outrun being. Whenever a claim or scenario is treated as a genuine alternative—a candidate that could be true rather than false, or obtain rather than fail to obtain—some determinacy-enabling structure already obtains. This is not about minds, language, or representation; it concerns objective conditions of determinacy that would obtain even in a mindless, language-less cosmos. Ordinary absence poses no problem: a room can contain no chairs, a region no electrons, a model no objects in its domain. But such absences are determinate only relative to a contrast-field. Absolute nothingness is different: it is the alleged cancellation of every such field. The proposed alternative therefore fails in either reading: read as a state, absolute nothingness makes something the case while requiring that nothing be the case; read as a bare denial, it leaves no candidate alternative to actuality. Attempts to place nothingness at a world, region, or void reintroduce the backdrop they deny. Because the point concerns determinacy itself, not proof-system choice, the paper argues that the result is invariant across classical, free, intuitionistic, relevant, and paraconsistent logics. More radically than a square circle or married bachelor, absolute nothingness is not even an impossible configuration but rather the attempted cancellation of configuration itself. (shrink)

Ontological arguments like those of Gödel (1995) and Pruss (2009; 2012) rely on premises that initially seem plausible, but on closer scrutiny are not. The premises have modal import that is required for the arguments but is not immediately grasped on inspection, and which ultimately undermines the simpler logical intuitions that make the premises seem plausible. Furthermore, the notion of necessity that they involve goes unspecified, and yet must go beyond standard varieties of logical necessity. This leaves us little reason (...) to believe the premises, while their implausible existential import gives us good reason not to. (shrink)

This paper presents a deductive argument for the structural necessity of a unique model—referred to as the M Model—that underlies the very possibility of rationality, meaning, and evaluative thought. Beginning with the minimal epistemic premise that a rational mind is possible, the argument identifies a set of necessary conditions (distinction, coherent relationality, normative evaluability, and exemplifiability) that any grounding structure must satisfy. Through five axioms and a modal progression from system K to a refined notion of structural necessity (denoted ◻ᴹ), (...) it is demonstrated that only the M Model fulfills all requirements without contradiction. The conclusion is that reason is not contingently emergent, but structurally grounded in a necessary framework. This foundation is not postulated but deduced: rationality itself presupposes it. The result is a metaphysically neutral, formally rigorous account of the structural preconditions of thought—a philosophical foundation for intelligibility that any worldview must reckon with. (shrink)

The theory of determinism and deterministic causation has persuaded many for millennia that unbending laws of physics produce inevitable outcomes from given causes. I argue that claims supporting determinism, both logical and empirical, lack rational and factual foundation. I show that Laplace, often seen as determinism’s foremost proponent, in fact supports my argument; that the claim “everything has a cause” leads to a regress and relies on an unproven assumption about the uniformity of nature that, even if granted, does not (...) support inevitability or necessity; that brain scan studies sometimes cited for determinism are irrelevant; and that deterministic causation at lower levels does not necessarily dictate outcomes at higher levels. Modal logic, which at best establishes only the possibility, not the necessity, of determinism across all possible logical worlds, further undermines its claims. Using a test for knowledge that I recently proposed, which rejects certainty while emphasizing careful analysis of evidence by the most reliable means, I conclude that determinism is mistaken and can be regarded as shown to be false. (shrink)

The dogma that the propositional logic of metaphysical modality is S5 is rebutted in related installments (previously published and unpublished essays).

Desarzens proved that a formula in one variable and modal degree one is unifiable in the basic modal logic K if and only if it is unified by one of the four substitutions p → ⊤, ⊥, □⊥, ♢⊤, equivalently if and only if it is true everywhere on a certain four-point Kripke frame. We sharpen this in two directions. First, we show that Desarzens’ frame is minimal and unique: among finite test frames for Fm ({p}, 1), four points are (...) necessary, and every four-point test frame is isomorphic to Desarzens’ frame. Second, we classify degree-one ground substitutions in arbitrary finite languages by pairs of sets A, B recording truth at leaves and nonleaves. Using this classification, we exhibit a fixed formula Ψ ∈ Fm ({p, q}, 1) which is true everywhere on every Kripke frame under a suitable valuation, but is not unified by any ground substitution of modal degree at most one. Consequently, no class of Kripke frames characterises degree-one ground-unifiability on Fm({p, q}, 1). (shrink)

Different definitions of deduction are offered. The first is that deduction is an inference type that is both possibly valid and possibly invalid. No inference can satisfy this definition, because valid inferences are not possibly invalid and invalid inferences are not possibly valid. In the second definition, deductive inferences are defined as being possibly valid or possibly invalid, but not both. This hypothesis does not work, since it implies that inductive inferences are deductive. To avoid this objection, deduction can be (...) understood as an inference that aims for validity. This definition also has unwanted consequences, including the fact that invalid inferences are only deductive when they are thought to be possibly valid. If the only genuine deductions are the valid ones, then our talk about deduction is an indirect and thoughtless way of referring to valid inferences rather than to an inference type. There is simply no deduction to speak of. Only valid inferences. (shrink)

Strong moral absolutism is sometimes understood as a modal thesis: that at least some moral prohibitions hold necessarily across all possible worlds. This paper argues that, under a minimal deontic constraint, such necessity cannot be maintained for prohibitions whose targets are actions whose possibility varies with the structural features of a world. The argument proceeds in three steps. First, it specifies a strong necessitarian reading of moral prohibition as the claim that for some action A it is necessary that A (...) is forbidden. Second, it adopts a minimal “ought implies can” principle: obligations cannot apply to actions that are impossible. Third, it constructs a counterexample world W* in which A is impossible. In W*, the prohibition of A cannot apply without violating the minimal deontic constraint. It follows that the prohibition is not necessary across all possible worlds. The result is limited: it does not establish moral relativism, nor does it deny the possibility of objective moral assessment within a given world. It establishes only that moral prohibitions, insofar as they concern the space of possible action, cannot be modal absolutes. The paper closes with a minimal consequence: moral norms are structurally dependent on the relational configuration of a world that determines which actions are possible. (shrink)

Possible worlds are commonly seen as an interpretation of modal operators such as "possible" and "necessary". Here, we develop possible world semantics (PWS) which can be expressed in basic set theory and first-order logic, thus offering a reductionist account of modality. Specifically, worlds are understood as complete sets of statements and possible worlds are sets whose statements are consistent with a set of conceptual laws. We introduce the construction calculus (CC), a set of axioms and rules for truth, possibility, worldness (...) and consistency. We show that CC allows to prove fundamental theorems about necessity, possibility, impossibility and contigency, thus demonstrating prima-facie plausibility of PWS. Finally, we discuss the explanatory power of our approach and draw connections between our account and established philosophical conceptions of possible worlds. (shrink)

El sistema de demostración KE es una variante de los tableaux analíticos pero es computacionalmente más eficiente que estos. Este sistema fue extendido a la lógica intuicionista proposicional recientemente. Además de su eficiencia y a diferencia de los resolvedores SAT, el sistema KE intuicionista no requiere transformaciones a formas normales y es modular en tanto que se puede extender, simple y naturalmente, a una familia amplia de lógicas que admiten semánticas relacionales. En este trabajo exploramos parte de esta modularidad, extendiendo (...) KE a una familia de lógicas modales intuicionistas, las cuales tienen aplicaciones relevantes en razonamiento automatizado y verificación formal. (shrink)

We develop general strategies for transferring soundness and completeness from more expressive modal languages to less expressive ones, unifying several existing notions of operator definability along the way. For soundness, we exploit semantic insensitivity: if a less expressive language is insensitive to a frame operation, soundness extends to the operation's closure of the original frame class. For completeness, restricting to relational semantics and languages with a single operator, we present strategies for relating the target logic's canonical model to that of (...) a normal modal logic via a truth-preserving translation. Three of those dispense entirely with specifying an accessibility condition for the target logic, inheriting it from a normal modal logic instead. (shrink)

Existential Realism (ER) proposes a two-tier ontology: only the present exists, yet the past and the future are real. This thesis arises from the diagnosis that in debates on the ontology of time, “existence” and “reality” are often implicitly identified, thereby making presentism and eternalism appear as competing positions that merely disagree about the extension of a single ontological predicate. ER instead separates a narrow, present-bound predicate of existence from a broader notion of reality that allows non-present entities to count (...) as real when, for instance, they existed in the past or will exist in the future, allowing these to participate in time-directed deductions. This paper develops the core system ERL (Existential-Realism Logic): a classical first-order linear temporal logic, in the Prior tradition, enriched with two distinguished unary predicates E (“exists now”) and R (“is real”). ERL is (i) syntactically defined with precision, (ii) given a model-theoretic semantics via Kripke-style ER-frames, (iii) axiomatized through a Hilbert-style system, and (iv) supported by metalogical results establishing soundness and outlining a canonical completeness program. Where strong completeness requires additional standard assumptions (such as Henkinization or generalized models), this is made explicit. Further central technical results include: conservativity over classical first-order linear temporal logic without E/R; an explicit Barcan profile (Barcan schemas hold for unrestricted quantifiers under constant domains, but may fail for existence-restricted quantifiers ∀ᴱ/∃ᴱ); and non-reducibility results (E is not definable, over the class of ER-models, using purely temporal operators; R cannot be forced as a definitional derivative of E together with temporal operators). The overall aim is to demonstrate that ER constitutes a logically stable and metalogically respectable position within established modal-temporal traditions, without unnecessarily over-engineering the formal system. Soundness is proven; completeness is outlined via a canonical-model strategy but left for full technical development in subsequent work. (shrink)

Most theories of time conflate ontological existence with structural reality. They then pose the familiar question: whether the past or the future is existent. Existential Realism (ER) rejects this conflation by distinguishing existence from temporal reality: only the present exists, while past and future remain real as structured temporal states within the framework. This paper extends the ER logical program by developing a branching-time semantics that represents: (i) an open future as a structured space of real possibilities, (ii) the transition (...) from possibility to actuality as history resolution within a branching-time structure, (iii) the persistence and irreversibility of past reality, and (iv) agency in branching time via standard STIT (“seeing to it that”) machinery. We define ER–BT frames that preserve the Prior–Ockhamist tradition of evaluation at moment–history pairs while adding an ER layer that distinguishes three time-indexed modal statuses for events: modal (future-real), existent (present), and past-real. We then integrate STIT choice functions and independence constraints and prove model-theoretic non-collapse results showing that ER–BT preserves openness. These results support a non-eternalist interpretation without commitment to modal realism or block instantiation, at the level of the formal model. This paper extends the two-tier framework introduced in Formal Logic for Existential Realism by providing an axiomatized system, ER–BT frame semantics, and selected meta-theoretic results (including soundness and partial completeness). (shrink)

Debates in temporal ontology—most prominently between presentism, eternalism, and the growing block theory—are often conducted via intuitive and linguistic contrasts that leave the underlying structural commitments underspecified. Existential Realism (ER) proposes a two-tier distinction: existence is present-bound, while reality extends beyond the present to include (at least) causal traces of the past and structured anticipations of the future. This paper offers a model-theoretic comparison of ER with its major rivals within a single, shared fixed-domain semantic environment. We introduce a unified (...) first-order tense language with an ER-style existence predicate and a reality predicate; define a neutral class of temporal frames; characterize presentism, eternalism, growing-block theory, and ER as subclasses determined by explicit frame conditions; and prove representation results. We then establish separation results showing that ER is frame-theoretically distinct from presentist, eternalist, and growing-block rivals, and prove that the reality predicate R cannot be conservatively eliminated over ER frames. A key technical component is a non-definability result (with an explicit countermodel) showing that ER’s reality predicate is not eliminable in favor of tense and existence alone, together with a bisimulation-based robustness discussion. -/- Series note. This installment extends the two-tier framework and logical core of Formal Logic for Existential Realism by treating temporal ontologies as frame classes and proving comparative model-theoretic results, while taking soundness/completeness/conservativity results from the core system as background. (shrink)

In his 1981 paper “From Worlds to Possibilities”, Lloyd Humberstone developed an approach to modal logic using possibilities rather than possible worlds. Possibilities, unlike worlds, may be incomplete. This paper sets out the possibility frame approach to modal logic, proves some results about its logic (including that some logics definable on Humberstone frames are not definable on Kripke frames), and surveys several applications, including to conditionals, vagueness, and fiction.

Some central epistemological notions are expressed by sentential operators O that entail the possibility of knowledge in the sense that 'Op' entails 'It is possible to know that p'. We call these modal-epistemological notions. Using apriority and being in a position to know as case studies, we argue that the logics of modal epistemological notions are extremely weak. In particular, their logics are not normal and do not include any closure principles.

These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text influenced me the most, though the order of (...) presentation is inspired more by Goldblatt.2 My goal was to write a text for dedicated undergraduates with no previous experience in modal logic. The text had to meet the following desiderata: (1) the level of difficulty should depend on how much the student tries to prove on his or her own—it should be an easy text for those who look up all the proofs in the appendix, yet more difficult for those who try to prove everything themselves; (2) philosophers (i.e., colleagues) with a basic training in logic should be able to work through the text on their own; (3) graduate students should find it useful in preparing for a graduate course in modal logic; (4) the text should prepare people for reading advanced texts in modal logic, such as Goldblatt, Chellas, Hughes and Cresswell, and van Benthem, and in particular, it should help the student to see what motivated the choices in these texts; (5) it should link the two conceptions of logic, namely, the conception of a logic as an axiom system (in which the set of theorems is constructed from the bottom up through proof sequences) and the conception of a logic as a set containing initial ‘axioms’ and closed under ‘rules of inference’ (in which the set of theorems is constructed from the top down, by carving out the logic from the set of all formulas as the smallest set closed under the rules); finally, (6) the pace for the presentation of the completeness theorems should be moderate—the text should be intermediate between Goldblatt and Chellas in this regard (in Goldblatt, the completeness proofs come too quickly for the undergraduate, whereas in Chellas, too many unrelated.... (shrink)

This paper is a study of a distinctively chemical notion of possibility. This is the notion of possibility that occurs in chemical discourses when chemists speak of the possibility or impossibility of achieving a given result through chemical means. This notion pertains to the possibility of processes, not of compounds, so it differs from the kind of chemical possibility mentioned in Wittgenstein's Philosophical Investigations or the kinds discussed in the literature on Putnam's Twin Earth argument. I argue that this process-oriented (...) notion of possibility cannot be reduced in a simple way to physical possibility, and that standard possible worlds semantics does not allow a natural analysis of this notion. I suggest an extension of possible worlds semantics that may overcome this limitation. I finish by pointing out some open questions about chemical possibility. (shrink)

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

In this article, I investigate the modal logic of exact equivalence (i.e., sameness of exact verifiers). In particular, by building on Kim's exact truthmaker semantics for modal logic, I provide an answer to the following question: which sentences of the language of propositional modal logic are exactly equivalent by virtue of their logical form?

We introduce pre-filtrations and pre-stable canonical rules for the Kuznetsov–Muravitsky system of intuitionistic modal logic and provide a new proof of the Kuznetsov–Muravitsky isomorphism, along with several preservation results. The proofs employ these rules and a duality between modal (Heyting) algebras and their corresponding order-topological spaces.

The paper “Tautology elimination, cut elimination, and S5” published in this journal presents a novel method for establishing by proof analysis the admissibility of the rule of tautology elimination for certain sequent calculi. Since tautology elimination will typically imply the admissibility of cut, the method promises a new path to show the admissibility of cut for cut-free calculi on which the standard techniques within structural proof theory seem inapplicable. This paper shows that the method as presented involves an error.

In a paper published in this journal, Golan argues for the thesis that there is no tenable notion of global metainferential validity. The conclusion rests on the premise that metainferential validity for propositional languages should be closed under uniform substitution of arbitrary formulas for atoms. This paper looks into Golan's justification for this premise in terms of the formality of logic. It is argued that neither formality as schematicity nor formality as topic-neutrality implies closure under uniform substitution of arbitrary formulas (...) for atoms as requirement. It is also argued that the justification of such a requirement in terms of the formality of logic makes Golan's conclusion irrelevant for logical research. (shrink)

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

Fragmentalism allows incompatible facts to constitute reality in an absolute manner, provided that they fail to obtain together. In recent years, the view has been extensively discussed, with a focus on its formalisation in model-theoretic terms. This paper focuses on three formalisations: Lipman’s approach, the subvaluationist interpretation, and a novel view that has been so far overlooked. The aim of the paper is to explore the application of these formalisations to the alethic modal case. This logical exploration will allow us (...) to study (i) cases of metaphysical incompatibility between modal facts and (ii) cases of modal dialetheias. In turn, this will enrich our understanding of the role of impossibility in the fragmentalist framework. (shrink)

Arguments challenging the existence of free will frequently share a common structure, relying on variants of a principle we call Closure, according to which having no choice about a truth is preserved under entailment. We show that, under plausible assumptions, Closure is valid if and only if the `no choice' operator is intensional. By framing the debate in terms of the intensionality of this operator, this paper illuminates previously underappreciated constraints on defenses of Closure-based arguments against the existence of free (...) will. (shrink)

Modal Interpretations of quantum mechanics distinguish between the realm of possibility, where the dynamical state determines what may be the case, and the realm of actuality, where the value state represents what actually is the case. In particular, the Modal-Hamiltonian Interpretation proposes an ontology in which there are no individual objects: quantum systems are characterized as non-individual bundles of properties. The aim of the present paper is to propose some rst steps towards the development of a new modal logic adapted (...) to describe a possibilist modality in an ontology of properties. The main idea consists in conceiving possibility as fundamental and introducing actuality as an operator acting on possible facts. On this basis, propositions referring to possible facts and propositions referring to actual facts are distinguished, and some relations between the realm of possibility and the realm of actuality are introduced. (shrink)

I develop a modal extension of the QUantified ARgument Calculus (QUARC) - a novel logical system introduced by Hanoch Ben-Yami. QUARC is meant to better capture the logic of natural language. The purpose of this paper is to evaluate this claim by considering how modal QUARC (M-QUARC) handles the Barcan and Converse Barcan Formulas and how this correlates to surrounding debates. To do so, I develop a variable domain semantics for M-QUARC and show that even if the usual domain conditions (...) are imposed on models with variable domains, simple M-QUARC-analogues of the Barcan and Converse Barcan Formulas are not valid. I introduce new conditions and show that they validate the formulas. I go on to extend the language of M-QUARC to simulate unrestricted quantification and show that in this setting the domain conditions validate the Barcan and Converse Barcan Formulas. Based on these results, I evaluate the relationship of the formal systems M-QUARC and standard quantified modal logic with respect to one another and to natural language. I argue that if M-QUARC does capture the logic of natural language, then natural language is capable to express neither the Barcan and Converse Barcan Formulas nor counterexamples to them. Given that, I raise doubts that M-QUARC captures quantification as it occurs in natural language. (shrink)

In artificial intelligence (AI), responses generated by machine-learning models (most often large language models) may be unfactual information presented as a fact. For example, a chatbot might state that the Mona Lisa was painted in 1815. Such phenomenon is called AI hallucinations, seeking inspiration from human psychology, with a great difference of AI ones being connected to unjustified beliefs (that is, AI “beliefs”) rather than perceptual failures). -/- AI hallucinations may have their source in the data itself, that is, the (...) source content, or in the training procedure, i.e. the way the knowledge was encoded in the model’s parameters, so that errors in encoding and decoding textual and non-textual representations can cause hallucinations. In this paper, we will observe how such errors come to life and how they might be mitigated. For this purpose, we will analyze the usability of justification logics, to behave as a proof checker for validating the correctness of large language models’ (LLM) responses. Justification logic was developed by S. Artemov, and later on mostly by Artemov and M. Fitting, deriving its main idea from the logic of proofs (LP): knowledge and belief modalities are seen as justification terms, i.e. t:X stands for t is a (proper) justification for X. Justification logic originated from attempts to create semantics for intuitionistic logic where proofs were the most proper justifications, but in further development, justification logic could be applied to different kinds of justifications). With the recent attempts to mitigate incorrect LLM responses, we will analyze various guardrails that are currently used for LLM responses, and see how the logic of justification may provide its benefits as an AI safety layer against false data. (shrink)

We present a construction of nondeterministic semantics for some deontic logics based on the class of paraconsistent logics known as Logics of Formal Inconsistency (LFIs), for the first time combining swap structures and Kripke models through the novel notion of swap Kripke models. We start by making use of Nmatrices to characterize systems based on LFIs that do not satisfy axiom (cl), while turning to RNmatrices when the latter is considered in the underlying LFIs. This paper also presents, for the (...) first time, a full axiomatization and a semantics for the CDn hierarchy, utilizing the aforementioned mixed semantics with RN matrices. It includes the historical system CD1 of da Costa and Carnielli (1986), the first deontic paraconsistent system proposed in the literature. (shrink)

Kant famously carves out space for transcendental freedom by dint of the distinction between appearances and things in themselves. In doing so, he denies establishing the actuality of freedom, as this would transcend the legitimate use of theoretical reason. Strikingly, he also denies proving the possibility of freedom (KrV, A 558/B 586). There is widespread agreement that here Kant denies having demonstrated the real possibility of freedom. However, scholars are divided on the kind of possibility that does pertain to it. (...) This paper explores the modal status of transcendental freedom from the standpoint of theoretical reason. I argue that Kant establishes both the logical possibility of freedom and a more robust status – what I call real non-impossibility. Short of proving transcendental freedom to be really possible, he shows that its real impossibility cannot be demonstrated, thereby leaving room for a deduction of its reality on practical grounds. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for (...) bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic KM and modal rule systems extending the provability logic GL. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases. (shrink)

This paper sets out to extend the epistemic semantics presented in Dahl (2023) to modal and conditional logics. To do so, I extend the notion of belief expansion systems, inspired by the AGM-model of belief change, to include belief revision, and uses the resulting structures as models for both conditional and modal logic. In the first case, this applies the well-known approach to conditionals initiated by Gärdenfors (1978), but in a weaker setting which doesn't assume an underlying logic. As such, (...) it results in semantics for both classical conditional logic as well as the intuitionistic conditionals studied by Weiss (2019). For modal logics, it uses the condition that □φ is accepted if and only if φ is accepted under every belief revision. As a result of this system of semantics, we get soundness and completeness theorems for both classical and non-classical systems of modal and conditional logic on the basis of a single type of structure. Finally, I briefly discuss how models based on belief change can still be thought to explain the semantics of objective modal claims. (shrink)

A major issue for the trustworthiness of modern AI-models is their lack of robustness. A notorious example is that putting a small sticker on a stop sign can cause AI-models to classify it as a speed limit sign. This is not just an engineering challenge, but also a philosophical one: we need to better understand the concepts of robustness and trustworthiness. Here, we contribute to this using methods from (formal) epistemology and prove a no-go result: No matter how these concepts (...) are understood exactly, they cannot have four prima facie desirable properties without trivializing. To do so, we describe a modal logic to reason about the robustness of an AI-model, and then we prove that the four properties imply triviality via a novel interpretation of Fitch’s lemma. We then discuss the consequences for explicating a viable notion of robustness for AI. A broader theme of the paper is to build bridges between AI and epistemology: Not only does epistemology provide novel methods for AI, but modern AI also provides many new questions and perspectives for epistemology. (shrink)