In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal (...) conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be the truth in all possible worlds. (shrink)

Hardcore actualism (HA) grounds all modal truths in the concrete constituents of the actual world (see, e.g., Borghini and Williams (2008), Jacobs (2010), Vetter (2015)). I bolster HA, and elucidate the very nature of possibility (and necessity) according to HA, by considering if it can validate S5 modal logic. Interestingly, different considerations pull in different directions on this issue. To resolve the tension, we are forced to think hard about the nature of the hardcore actualist's modal reality and how radically (...) this departs from possible worlds orthodoxy. Once we achieve this departure, the prospects of a hardcore actualist validation of S5 look considerably brighter. This paper thus strengthens hardcore actualism by arguing that it can indeed validate S5–arguably the most popular logic of metaphysical modality–and, in the process, it elucidates the very nature of modality according to this revisionary, but very attractive, modal metaphysics. (shrink)

In this paper I introduce a sequent system for the propositional modal logic S5. Derivations of valid sequents in the system are shown to correspond to proofs in a novel natural deduction system of circuit proofs (reminiscient of proofnets in linear logic, or multiple-conclusion calculi for classical logic). -/- The sequent derivations and proofnets are both simple extensions of sequents and proofnets for classical propositional logic, in which the new machinery—to take account of the modal vocabulary—is directly motivated in terms (...) of the simple, universal Kripke semantics for S5. The sequent system is cut-free and the circuit proofs are normalising. (shrink)

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

This paper examines the contemporary theistic use of the modal system S5 and argues that it is methodologically unjustified. By treating S5 as a neutral logical framework, such arguments import strong metaphysical assumptions that allow necessity to be derived from possibility. The paper shows that this results in framework-dependent conclusions and modal symmetry problems, undermining claims of genuine epistemic warrant.

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.

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , (...) K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover. The proof formalized is close to that of Hughes and Cresswell, but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical (...) connectives and necessity as the only primitive modal operator. The full source code is available online and has been typechecked with Lean 3.4.2. (shrink)

In this paper I analyse the relationship between the modal ontological argument (MOA) and its popular “reverse” version in modal logic S5. In the existing literature these arguments are often treated as formally symmetric, since they employ analogous modal constructions. I argue, however, that this symmetry disappears once the discussion is shifted to the metalogical level and the modal operator of possibility is interpreted epistemically, as a postulate of consistency of the corresponding extension of a theory. Under this interpretation, the (...) MOA premise (“it is possible that God exists necessarily”) corresponds to assuming that the theory of the world remains consistent after adding the thesis of God’s necessary existence. By contrast, the proper negation of this premise (“it is not possible that God exists necessarily”) corresponds to the claim that this extension of the theory is inconsistent. I show that the commonly used counterargument formulated as “it is possible that God does not exist necessarily” is not the logical negation of the MOA premise and is therefore not epistemically equivalent to it. In the second part of the paper, drawing on Gödel’s second incompleteness theorem, I show that the consistency of a sufficiently rich theory cannot be established within that theory itself, leading to an iterated, fractal structure of epistemic limitation. As a result, the claim that God as a necessary being is impossible turns out to be epistemically much stronger than the claim that such a being is possible. This establishes a structural epistemic asymmetry at the core of the debate between modal theism and its critics. (shrink)

What is the relation between metaphysical necessity and essence? This paper defends the view that the relation is one of identity: metaphysical necessity is a special case of essence. My argument consists in showing that the best joint theory of essence and metaphysical necessity is one in which metaphysical necessity is just a special case of essence. The argument is made against the backdrop of a novel, higher-order logic of essence, whose core features are introduced in the first part of (...) the paper. The second part investigates the relation between metaphysical necessity and essence in the context of HLE. Reductive hypotheses are among the most natural hypotheses to be explored in the context of HLE. But they also have to be weighed against their non-reductive rivals. I investigate three different reductive hypotheses and argue that two of them fare better than their non-reductive rivals: they are simpler, more natural, and more systematic. Specifically, I argue that one candidate reduction, according to which metaphysical necessity is truth in virtue of the nature of all propositions, is superior to the others, including one proposed by Kit Fine, according to which metaphysical necessity is truth in virtue of the nature of all objects. The paper concludes by offering some reasons to think that the best joint theory of essence and metaphysical necessity is one in which the logic of metaphysical necessity includes S4, but not S5. (shrink)

When it comes to evaluating reductive hypotheses in metaphysics, supervenience arguments are the tools of the trade. Jaegwon Kim and Frank Jackson have argued, respectively, that strong and global supervenience are sufficient for reduction, and others have argued that supervenience theses stand in need of the kind of explanation that reductive hypotheses are particularly suited to provide. Simon Blackburn's arguments about what he claims are the specifically problematic features of the supervenience of the moral on the natural have also been (...) influential. But most discussions of these arguments have proceeded under the strong and restrictive assumptions of the S5 modal logic. In this paper we aim to remedy that defect, by illustrating in an accessible way what happens to these arguments under relaxed assumptions and why. The occasion is recent work by Ralph Wedgwood, who seeks to defend non-reductive accounts of moral and mental properties together with strong supervenience, but to evade both the arguments of Kim and Jackson and the explanatory challenge by accepting only the weaker, B, modal logic. In addition to drawing general lessons about what happens to supervenience arguments under relaxed assumptions, our goal is therefore to shed some light on both the virtues and costs of Wedgwood's proposal. (shrink)

Can we find propositions that cannot rationally be denied in any possible world without assuming the existence of that same proposition, and so involving ourselves in a contradiction? In other words, can we find transworld propositions needing no further foundation or justification? Basically, three differing positions can be imagined: firstly, a relativist position, according to which ultimately founded propositions are impossible; secondly, a meta-relativist position, according to which ultimately founded propositions are possible but unnecessary; and thirdly, an absolute position, according (...) to which such propositions are necessary. In this short essay I show that under the premise of modal logic S5 with constant domain there are ultimately founded propositions and that their existence is even necessary, and I will give some reasons for the superiority of S5 over other logics. (shrink)

This paper develops a novel modal-semantic ontological argument that challenges the semantic coherence of atheistic discourse itself. By introducing the Reasonability Principle (RP), it is argued that any truth — including atheism — must be supported by at least one analytic or synthetic reason. However, synthetic arguments against God’s existence, such as those based on evil, divine hiddenness, or explanatory parsimony, rely on semantic structures that presuppose the metaphysical possibility of a being whose existence they deny. If the concept of (...) God functions as a rigid designator referring to a necessarily existing being, then — under modal logic S5 — mere possibility entails necessity. Thus, atheistic conditionals implicitly commit to God's necessary existence. The paper concludes that atheism entails a semantic self-undermining, unless it can provide a purely analytic proof of God’s incoherence — a standard not met by existing arguments. Hence, theism emerges as the semantic price of atheism’s intelligibility. (shrink)

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)

This paper presents a multi-faceted critique of the Modal Ontological Argument (MOA), focusing on its semantic and epistemic weaknesses. The analysis identifies a dual failure within the argument's structure. First, logically, the MOA fails because it treats "necessary existence" as a first-order predicate. This results in an ill-typed application within standard Kripke semantics and constitutes a misuse of S5 modal operators. Second, epistemologically, the argument engages in circular reasoning by embedding the conclusion (necessary existence) within the initial definition of a (...) "Maximally Great Being" (MGB) and assuming the coherence of MGB's properties without independent justification. The argument is also vulnerable to the logical symmetry problem, where the counter-argument ("Possibly an MGB does not exist") is structurally equipollent. In conclusion, the MOA is not a pure logical proof; rather, it is a formalization of premise-dependent metaphysical assumptions and thus fails to uniquely establish God's existence as a necessary truth. (shrink)

This paper develops a metatheoretical account of the limits of epistemic evaluation in modal arguments with strong ontological consequences. Focusing on the modal ontological argument (MOA) and its formally symmetric reverse version, both reconstructed in the standard S5 modal system, the paper shows that formal validity alone cannot determine epistemic legitimacy. Although the two arguments are identical in their logical structure, their key premises have radically different epistemic statuses. The paper proves that this asymmetry cannot be captured within the language (...) or proof theory of the modal system itself. Epistemic admissibility depends on normative and metalevel conditions that are not representable as formal properties of modal formulas. The result is established by a general meta-theorem: there exists no criterion definable within a modal system that maps formal validity to epistemic justification. This limitation is not specific to S5 but follows from the structure of reflective formal systems and is an epistemological counterpart of classical incompleteness and undefinability results. The paper concludes that any adequate evaluation of modal arguments must be two-level: formal analysis is necessary, but epistemic legitimacy requires an irreducible metatheoretical perspective. This has general consequences for the methodology of arguments that aim to establish substantive metaphysical or ontological conclusions. (shrink)

Resolution Matrix Semantics (RMS) introduces the alternative truth-value-based framework for modal logic, providing a substantive alternative to Kripke’s relational semantics of possible worlds. Drawing inspiration from Y. Ivlev’s substantive semantics, RMS utilizes a 4-valued structure—necessary truth (tn), contingent truth (tc), contingent false (fc), and necessary false (fn)—augmented by indeterminate values (t, f, t/f) to define modal systems Km, KDm, KTm, S4m, and S5m, analogous to Kripke’s K, KD, T, S4, and S5. By directly assigning determined and indeterminate truth values via (...) an interpretation function, RMS validates formulas without relying on accessibility relations, as demonstrated through soundness and completeness proofs and a tailored tableau method. The framework’s versatility extends to applications in deontic logic, artificial intelligence, and quantum computing, enabling context-dependent reasoning with computational efficiency. RMS thus bridges philosophical logic and practical domains, offering a truth-centric perspective on modal reasoning’s complexities. (shrink)

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions (...) arguably include some necessitist principles. -/- Too often, philosophers have discussed ‘metaphysical’ modality — possibility, contingency, necessity — in isolation. Yet metaphysical modality is just a special case of a broad range of modalities, which we may call ‘objective’ by contrast with epistemic and doxastic modalities, and indeed deontic and teleological ones (compare the distinction between objective probabilities and epistemic or subjective probabilities). Thus metaphysical possibility, physical possibility and immediate practical possibility are all types of objective possibility. We should study the metaphysics and epistemology of metaphysical modality as part of a broader study of the metaphysics and epistemology of the objective modalities, on pain of radical misunderstanding. Since objective modalities are in general open to, and receive, natural scientific investigation, we should not treat the metaphysics and epistemology of metaphysical modality in isolation from the metaphysics and epistemology of the natural sciences. -/- In what follows, Section 1 gives a preliminary sketch of metaphysical modality and its place in the general category of objective modality. Section 2 reviews some familiar forms of scepticism about metaphysical modality in that light. Later sections explore a few of the many ways in which natural science deals with questions of objective modality, including questions of quantified modal logic. (shrink)

This paper presents the "Enhanced Ten-Step Model of Judgemental Philosophy," a comprehensive framework detailing the human judgement process from sensory input to social norm formation. Building on the original 10-step model (Kim, 2025), this enhanced version integrates five core parallel/modulatory systems (Affective Processing, Value Assessment & Motivation, Prediction & Prediction Error, Executive Functions / Cognitive Control, and Unconscious Memory Consolidation) with the main sequential pathway (S1-S10). The model elucidates the structural conditions for judgemental possibility, centered on the Judgemental Triad (Constructivity, (...) Coherence, Resonance) and the Pre-Judgemental Field (PJF). It particularly emphasizes the multi-layered role of Resonance – the "return of meaning" – in conscious experience (S5: Explicit Resonance), inter-subjective interaction (S9: Inter-subjective Resonance, including human-animal and mediated communication), and normative processes (S10). The model connects its components to neurocognitive indicators and discusses pathological implications, offering an integrative neurophilosophical perspective on judgement, consciousness, self-awareness, and agency, and provides a basis for empirical validation and future research. (shrink)

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...) to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given. (shrink)

This paper first argues that we can bring out a tension between the following three popular doctrines: (i) the canonical reduction of metaphysical modality to essence, due to Fine, (ii) contingentism, which says that possibly something could have failed to be something, and (iii) the doctrine that metaphysical modality obeys the modal logic S5. After presenting two such arguments (one from the theorems of S4 and another from the theorems of B), I turn to exploring various conclusions we might draw (...) in light of these results, and argue that none comes cost free. In the course of laying out possible responses to my arguments, we'll have a chance to evaluate various doctrines about the interplay between contingency and essence, as well as develop some alternative reductions of metaphysical modality to essence. I don't decisively come down in favor of one response over the others, though I say some things that point towards the conclusion that essence has no role to play in reducing metaphysical modality. (shrink)

The dogma that the propositional logic of metaphysical modality is S5 is rebutted. The author exposes fallacies in standard arguments supporting S5, arguing that propositional metaphysical modal logic is weaker even than both S4 and B, and is instead the minimal and weak metaphysical-modal logic T.

We present epistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati and Schlöder 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic S5 modulo an appropriate translation. The logical framework developed provides the (...) basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve. (shrink)

In "Essence and Modality", Kit Fine proposes that for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all objects whatsoever. Call this view Fine's Thesis. This paper is a study of Fine's Thesis in the context of Fine's logic of essence (LE). Fine himself has offered his most elaborate defense of the thesis in the context of LE. His defense rests on the widely shared assumption that metaphysical necessity obeys the (...) laws of the modal logic S5. In order to get S5 for metaphysical necessity, he assumes a controversial principle about the nature of all objects. I will show that the addition of this principle to his original system E5 leads to inconsistency with an independently plausible principle about essence. In response, I develop a theory that avoids this inconsistency while allowing us to maintain S5 for metaphysical necessity. However, I conclude that our investigation of Fine's Thesis in the context of LE motivates the revisionary conclusion that metaphysical necessity obeys the principles of the modal logic S4, but not those of S5. I argue that this constitutes a distinctively essentialist challenge to the received view that the logic of metaphysical necessity is S5. (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)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

This work introduces the Topological Unified Field Theory (TUFT), a unified framework formulated on a nine-dimensional spacetime defined by the complex Hopf fibration S1 -> S9 -> CP4. In TUFT, gravity and gauge interactions emerge simultaneously from topology rather than from postulated dynamics. Gravity appears as a metric-independent topological field theory whose dimensional reduction uniquely reproduces four-dimensional General Relativity. We prove that the Milnor universal bundle for nontrivial U(1) structure is uniquely realized by the complex Hopf fibration, implying that any (...) consistent unification based on nontrivial U(1) topology is homeomorphic to TUFT. The Standard Model gauge group SU(3) x SU(2) x U(1) arises inevitably from the allowed bundle connections; no alternative fibration reproduces the observed particle multiplets or charge assignments. Compactness of the total space enforces ultraviolet finiteness, eliminates the vacuum landscape, and selects a unique physical vacuum. All physical worldlines are globally nonsingular, resolving both cosmological and black-hole singularities. -/- All particle masses are derived from first principles, directly from topological and spectral invariants of the bundle, with no fitted parameters. Charged leptons arise as Beltrami modes on S3, quarks as S5 modes with color enforced by holonomy, gluons on S7, and neutrinos on S9. TUFT yields absolute neutrino mass predictions consistent with oscillation data and beyond current experimental reach. The photon and graviton emerge massless as required by topology. The muon anomalous magnetic moment is resolved within experimental uncertainty, and TUFT further provides parameter-free predictions for the tau anomalous magnetic moment. -/- A transparent working Python verification code is provided, demonstrating that all predictions follow exclusively from topological invariants of the bundle. A blinded comparison against the latest experimental mass-spectral data yields a p-value of < 0.001, indicating a greater than 99.9% likelihood that the observed mass spectrum is a direct manifestation of the TUFT topological invariants rather than a stochastic alignment. Finally, an appendix presents direct experimental confirmation of TUFT holonomy via quantum transport measurements of Sala et al. (2025) and torsion via the anomalous cavitation data of Barger (1964), which constitute precise empirical realizations of the unique theory. -/- This paper has completed a rigorous, stringent peer review upon invitation of the International Journal of Topology, an applied topology / mathematics journal with Editor-in-Chief Michel Planet (a quantum gravity theorist), and is awaiting final editor decision. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical justification, have attracted discrete attention (...) among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

The paper aims to show that, if S5 is the logic of metaphysical necessity, then no state of affairs in any possible world constitutes any non-trivial evidence for or against the existence of the traditional God. There might well be states of affairs in some worlds describing extraordinary goods and extraordinary evils, but it is false that these states of affairs constitute any (non-trivial) evidence for or against the existence of God. The epistemological and metaphysical consequences for philosophical theology of (...) assuming that S4 or Kσρ is the logic of metaphysical necessity are equally untenable. S4 guarantees that God does not exist if there is the slightest evidence against the existence of God. And Kσρ guarantees that God might survive the loss or acquisition of any essential property at all. (shrink)

This paper studies proof systems for the logics of super-strict implication \(\textsf{ST2}\) – \(\textsf{ST5}\), which correspond to C.I. Lewis’ systems \(\textsf{S2}\) – \(\textsf{S5}\) freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating \(\textsf{STn}\) in \(\textsf{Sn}\) and backsimulating \(\textsf{Sn}\) in \(\textsf{STn}\), respectively (for \({\textsf{n}} =2, \ldots, 5\) ). Next, \(\textsf{G3}\) -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive (...) of \(\textsf{G3}\) -style calculi, that they are sound and complete, and it is shown that the proof search for \(\mathsf {G3.ST2}\) is terminating and therefore the logic is decidable. (shrink)

The minimal God exemplifies essential omnipotence, omniscience, and moral perfection, but none of the other properties of the traditional God. I examine the consequences of the minimal God in augmented S5, S4, and Kρσ. The metaphysical consequences for the minimal God in S5 include the impossibility that God—or any other object—might acquire, lose, or exchange an essential property. It is impossible that an essentially divine being might become essentially human, for instance. The epistemological consequences include the impossibility of agnosticism—it is (...) impossible that P(◻︎FG |E) = P(~◻︎FG| E). Indeed, every weaker forms of agnosticism are also impossible—it is impossible that P(◻︎FG |E) = n, n(0 < n < 1). The metaphysical and epistemological consequences for the minimal God in S4 are equally untenable. I show that we can avoid all of these unwelcome epistemological consequences for the minimal God in Kρσ. But Kρσ has some extraordinary metaphysical consequences for the minimal God. All of the problems for the minimal God generalize to the maximal God. (shrink)

This paper proposes a theory on the necessity of consciousness from the perspective of Judgemental Philosophy (JP). It argues that the 'Resonance Drive' (RD)—the fundamental human impetus to expand meaningful connections with the world—inevitably leads to a confrontation with profound existential and cognitive challenges. The process of transcending established meaning-structures (Constructivity-Coherence, C1-C2) requires a re-experiencing of fundamental 'Indeterminacy,' which in turn generates not only anxiety but also acute cognitive dissonance and a potent temptation to delegate judgement as a means of (...) evading this discomfort. This paper posits that consciousness, particularly the functions of Metacognition (S6) and Explicit Resonance (S5) within the JP framework, is a structurally necessary adaptation to overcome these specific challenges. Metacognition (S6) provides the self-reflective capacity to navigate dissonance and consciously resist the urge to delegate. Subsequently, Explicit Resonance (S5)—specifically the affective pleasure experienced upon the autonomous resolution of this dissonance—serves as a powerful motivational reward, reinforcing the value of sovereign judgement. Consciousness is, therefore, not merely an optimal state for growth, but a necessary structure for maintaining judgemental integrity and autonomy in the face of the inherent psychological costs of meaning expansion. (shrink)

Dynamic epistemic logic (DEL) extends purely modal epistemic logic (S5) by adding dynamic operators that change the model structure. Propositional dynamic logic (PDL) extends basic modal logic with programs that allow the de nition of complex modalities. We provide a common generalisation: a logic that is dynamic in both senses, and one that is not limited to S5 as its modal base. It also incorporates, and signi cantly generalises, all the features of existing extensions of DEL such as BMS [3] (...) and LCC [21]. Our dynamic operators work in two steps. First, they provide a multiplicity of transformations of the original model, one for each action in a purely syntactic action structure (in the style of BMS). Second, they specify how to combine these multiple copies to produce a new model. In each step, we use the generality of PDL to specify the transformations. The main technical contribution of the paper is to provide an axiomatisation of this general dynamic dynamic logic (GDDL). This is done by providing a computable translation of GDDL formulas to equivalent PDL formulas, thus reducing the logic to PDL, which is decidable. The proof involves switching between representing programs as terms and as automata. We also show that both BMS and LCC are special cases of GDDL, and that there are interesting applications that require the additional generality of GDDL, namely the modelling of private belief update. More recent extensions and variations of BMS and LCC are also discussed. (shrink)

This paper presents a new theoretical framework for resolving the long-standing Gettier problem in epistemology by applying Judgemental Philosophy (JP) and its "Enhanced Ten-Step Model." While traditional approaches have attempted to amend the "Justified True Belief" (JTB) definition of knowledge, this paper argues that Gettier cases represent a fundamental 'structural failure' within the judgement process itself. Specifically, it posits that the belief in all Gettier-style problems is the product of a 'Resonance(R) Illusion,' which arises from a 'misled resonance pathway'. The (...) analysis is systematically applied to Gettier's original counterexamples, classic variations such as the 'stopped clock' case, and more refined challenges like those proposed by Zagzebski. Furthermore, through an advanced counterexample (the 'Kind Operator'), the paper explores the deep connection between this structural failure, Hume's problem of induction, and the self-correcting nature of the scientific method. In each case, it is shown that a flawed premise within the subject's 'Global Self-Model Buffer' leads to a state of 'pseudo-coherence' (S3: Constructivity-Coherence), which in turn generates a strong but illusory subjective certainty (S5: Explicit Resonance). The causal link between the belief and the actual state of affairs that makes it true is severed. Consequently, JP proposes that genuine justification requires not just internal coherence but a 'veridical pathway of Resonance,' ensuring the belief is structurally and causally connected to reality. This framework provides a unified structure for understanding both the Gettier problem and the 'Brain in a Vat' hypothesis, rooting both epistemological challenges in the Kantian limit of our inability to directly perceive the 'thing-in-itself'. Furthermore, the paper explores the role of Metacognition (S6) as the faculty that allows for doubt regarding this limit, and interprets this very doubt as a positive manifestation of the 'Resonance Drive' (RD) striving for a more complete and veridical understanding. Ultimately, this Judgemental Philosophical account reframes knowledge not as a static state (JTB), but as the successful outcome of a dynamic process: an un-misled, structurally sound resonance between the judging subject and the world. (shrink)

This paper presents a comprehensive proof-theoretic analysis of Jaśkowski’s discussive (or discursive) logic, working with a set of connectives including classical negation and disjunction, as well as so-called (right-)discussive conjunction and discussive implication. By employing established techniques two labelled frameworks are introduced: sequent and natural deduction systems. The paper explores the ability of the proposed calculi to accurately represent Jaśkowski’s discussive logic, particularly in light of its paraconsistent nature, and establishes cut- admissibility and normalization theorems. Additionally, the introduced sequent calculus (...) – shown to allow terminating proof search – is employed to prove the embedding of discussive logic within modal logic S5. Finally, it is proved that the natural deduction calculus translates into the corresponding sequent system, with soundness and completeness established for both calculi. Concluding remarks highlight the potential for expanding this study and suggest directions for future research. (shrink)

Imagination is a source of evidence for objective modality. It is through this epistemic connection that the idea of modality first gains traction in our intellectual life. A proper theory of modality should be able to explain our imagination’s modal epistemic behaviors. This chapter highlights a peculiar asymmetry regarding epistemic defeat for imagination-based modal justification. Whereas imagination-based evidence for possibility cannot be undermined by information about the causal origin of our imaginings, unimaginability-based evidence for impossibility can be undermined by information (...) about the causal origin of the unimaginability. It is argued that an acceptance of S4 over S5 as the true logic for objective modality best explains this epistemic asymmetry. (shrink)

Michael Almeida has recently tried to show that if S5 correctly represents metaphysical necessity, there can be no non-trivial evidence for or against the existence of the traditional God. Evidence would thus be irrelevant to the reasonability of traditional theistic belief. Almeida's argument has implications beyond its announced target: it amounts to a new argument for sweeping scepticism. Almeida's argument for the irrelevance of evidence to the existence of God would apply to any state of affairs that entails some metaphysical (...) necessity. In S5, every state of affairs entails some metaphysical necessity, so Almeida should conclude that non-trivial evidence for or against any state of affairs is impossible. I argue that the problem is not with S5 but with inadequately motivated assumptions about evidential support. Avoiding those assumptions disables the argument for sweeping scepticism and without foreclosing the possibility of non-trivial evidence for or against the existence of the traditional God. (shrink)

Michael Dummett argues, against Saul Kripke, that there could have been unicorns. He then claims that this possibility shows that the logic of metaphysical modality is not S5, and, in particular, that the B axiom is false. Dummett’s argument against B, however, is invalid. I show that although there are number of ways to repair Dummett’s argument against B, each requires a controversial metaphysical or semantic commitment, and that, regardless of this, the case against B is undermotivated. Dummett’s case is (...) still of interest, however, as if his assumptions are correct, S5 has to go, with the natural culprit being S4. (shrink)

We present a sound and complete Fitch-style natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is two-dimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...) on the diagonal. Thus, they are just like the epistemic operators for apriority and its dual. We take this extension of Fitch’s familiar derivation system to be a very natural one, since the new rules and labeled lines hereby introduced preserve the structure of Fitch’s own rules for the modal case. (shrink)

In this paper, I evaluate the Modal Ontological Argument and whether the argument has sufficient grounds to be an effective one to prove Theism. In this paper I explain the Modal Ontological Argument, explain S5 modality and then finally list objections and general rebuttals to the arguments given against the Modal Ontological Argument.

A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate D, and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are D we in Section 2 derive the thesis (40) that all properties are D or some individual is G. In Section 3 theorems 1 to 3 suggest (...) a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of D suggests weakened Gaunilo-like objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts `God’ allow thorough indifference as to whether God exists or not. (shrink)

Could the truths of mathematics have been different than they in fact are? If so, which truths could have been different? Do the contingent mathematical facts supervene on physical facts, or are they free floating? I investigate these questions within a framework of higher-order modal logic, drawing sometimes surprising connections between the necessity of arithmetic and analysis and other theses of modal metaphysics: the thesis that possibility in the broadest sense is governed by a logic of S5, that what is (...) possible holds in some maximally specific possibility, and that every property can be rigidified. The investigation will distinguish sharply between platonic contingency---contingency about whether particular abstract ``platonic'' mathematical objects are arranged in a certain way (e.g. in a natural number or real number structure)---from a deeper variety of structural contingency concerning what holds of objects whenever they *are* arranged in that way. (shrink)

It is commonly believed that intrinsically bad possible worlds are inconsistent with the perfect goodness of God. A perfectly good being could not exist in possible worlds that are intrinsically bad. Indeed it is widely believed that possible worlds that are insufficiently good are inconsistent with a perfectly good God. Modal atheological arguments aim to show that, since the pluriverse includes intrinsically bad worlds and insufficiently good worlds, there necessarily does not exist a perfectly good God. I show that modal (...) atheological arguments are in general invalid. Modal atheological arguments fail to show that there exists no perfectly good God even on the assumption that there are extremely bad worlds in the pluriverse. Modal atheological arguments depend entirely on the logic of modality in S5. But the logic of S5 ensures that a perfectly good God is consistent with extremely bad possible worlds and so guarantees that a perfectly good God is consistent with a plenitudinous pluriverse. (shrink)

This paper distinguishes functional reproducibility from lawful determinism. A system may repeat its errors perfectly and still remain lawless. Using the CODES framework and the Resonance Intelligence Core (RIC), the work defines determinism as coherence law: emission occurs only when phase alignment (PASₛ ≥ θ_L), drift stability (|ΔPAS_ζ| ≤ ε_drift), symbolic legality (GLYPHLOCK = 1), and temporal lock (TEMPOLOCK = 1) all hold simultaneously. Five verification tests (S1–S5) demonstrate how pseudo-deterministic systems replay incoherence while lawful systems remain phase-stable under replay. (...) The paper establishes falsifiable criteria for deterministic inference, introduces the Lawful Emission Check (LEC-1), and outlines the path to measurable coherence in computation. (shrink)

Few commentators (if any) would question Schrader's poignant obser­vation that 'the doctrine of the thing in itself presents the single greatest stumbling block in the Kantian philosophy' [S5:49]. Understanding what Kant meant by the doctrine i.e., the role it plays both in his overall System and in his transcendental idealism can help prevent it from being discarded 'as a per­versity' [49], inasmuch as it can be interpreted in such a way that it makes quite good sense [see VI.2]. Yet even (...) the most coherent interpretation could not prevent the philosopher who demands knowledge from 'stumbling' over it; for, accord­ing to Kant, the thing in itself is by definition unknow­able. In V.3 we saw, however, that there is one alternative to faith as the ultimate jus­tification for its employment which, if successful, would satisfy even the most persistent skeptic: viz., to justify the thing in itself by constructing a valid transcendental argument for the necessity of its existence. Since any ap­peal to faith would thereby be rendered superflu­ous, we must now examine more carefully the possibility of realizing this goal. For, although Kant him­self did not believe he required such a transcendental argu­ment, it may be pos­sible to reconstruct his System on the basis of a slightly different presup­posi­tion, such as that the thing in itself can, in fact, be proved to exist and to have certain knowable characteristics. Hence, in this Appendix I shall analyze the logical consistency of an affirmative answer to the meta­critical question: Is the thing in itself knowable? Our answer to this question will inevitably determine to a large extent how we should approach the task of interpreting the elements of Kant's Critical philosophy (as in Part Three), so it is important to deal with it seriously. (shrink)

We develop a deontic ontological schema that derives the existence and unity of the Good from a uniquely self–realizing “ought.” The core move is typed: two apex reflections do the heavy lifting; the role premise “the Good ought to exist” and the unity premise “the Good ought to be one” are inputs to these reflections. Guardrails (essentiality, stability, unity) block overgeneration and keep the system conservative over ordinary deontic reasoning. The route is distinct from modal ontological arguments: no possibility premise, (...) no global S5; the optional categorical mirror recasts the same pattern as an initiality/representability question and clarifies principled failure modes (e.g. Pyrrhonian non–representability) and a hybrid middle ground (apex–visible Pyrrhonism). The schema is explicitly conditional and modest in ontic load: it does not fix a mode of subsistence, nor promise constructive or algorithmic witnesses absent further structure. Finally, we sketch how the ideal case functions diagnostically—exposing clean fracture lines for disagreement and informing minimal metaphysical commitments for implementation in ordinary, defeasible normativity. (shrink)

Most theists do not put a (subjective) probability of 1 (certainty) on God's existence. Most atheists do not put a probability of 0 on God's existence. I argue that these familiar positions are incoherent. On the assumption of S5 and the probability calculus it can be shown that the only coherent (subjective) probabilities an agent can assign to God's existence/non-existence are 0 or 1. Believers must be completely committed believers and non-believers must be completely committed non-believers. Agnosticism is not a (...) coherent position. (shrink)

This paper develops the Necessity Dilemma, a structural challenge to classical theism formulated in minimal modal logic. Classical theistic arguments across ontological, cosmological, teleological, moral, and epistemic domains rely on Essential Necessity—the principle that if God exists, then God exists necessarily. We demonstrate that combining this commitment with the minimal concession that God’s non-existence is possible (◇¬G) entails atheism. This “knife-edge” result shows that both horns of the theistic dichotomy collapse: a contingent God ceases to be divine, while a necessary (...) God is either refuted by minimal modal reasoning or must be asserted as a brute fact, ending philosophical argumentation. Our analysis highlights a fundamental asymmetry: atheistic reasoning requires only the weakest modal system (K), while theistic reasoning requires increasingly strong commitments such as S5. We conclude that atheism is not merely the absence of belief but the theoretically more economical stance, justified by the modal instability of classical theism. (shrink)