Lots of things are usefully modelled in science as dynamical systems: growing populations, flocking birds, engineering apparatus, cognitive agents, distant galaxies, Turing machines, neural networks. We argue that relevant logic is ideal for reasoning about dynamical systems, including interactions with the system through perturbations. Thus, dynamical systems provide a new applied interpretation of the abstract Routley-Meyer semantics for relevant logic: the worlds in the model are the states of the system, while the (in)famous ternary relation is a combination (...) of perturbation and evolution in the system. Conversely, the logic of the relevant conditional provides sound and complete laws of dynamical systems. (shrink)

We have to gain from recognizing a relation between epistemic agents and the parts of subject matters that play a role in their cognitive lives. I call this relation “grasping”. Namely, I zone in on one notion of having a partial grasp of a subject matter—that of agents grasping part of the subject matter that they are attending to—and characterize it. I propose that giving up the idealization that we fully grasp the subject matters we attend to allows one to (...) more realistically characterize the epistemic life of agents. To show this, I propose an epistemic logic with partial grasp that has in mind considerations from first-order aboutness theory with the aim of avoiding certain forms of logical omniscience, and which provides an alternative to immanent closure (Yablo Aboutness, Princeton University Press, 2014). (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental (...) result in semantics. Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

Most readers of the Tractatus take Wittgenstein's logic to be contentless, since tautologies do not picture reality. However, this fails to explain its transcendental status, especially given that Kant introduced transcendental logic through the notion of content. In this essay, I propose a transcendental interpretation of Wittgenstein’s logic, as "logic of depiction” endowed with a transcendental content: the formal rules governing the subject’s speech about objects. Tautologies turn out to be dispensable by-products of the transcendental (i.e., (...) content-endowed) logic of depiction. The result is a reconception of Wittgenstein's logic, grounded in a systematic comparison with Kant. (shrink)

In spite of its significance for everyday and philosophical discourse, the explanatory connective has not received much treatment in the philosophy of logic. The present paper develops a logic for based on systematic connections between and the truth-functional connectives.

This essay explores the cosmological argument through the lens of Aristotelian logic and the Principle of Sufficient Reason, arguing that the existence of contingent beings requires a necessary being as the ultimate ground of reality.

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.

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning (...) showing by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the (...) idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

When agents face the most consequential decisions in their lives, they engage in a form of reasoning which is categorically distinct from the reasoning that occupies nearly all of practical philosophy and nearly all of psychotherapy. I call this between-world reasoning, and I argue that the distinction between it and ordinary within-world reasoning corresponds to the distinction between modal logic and propositional logic as applied to lived experience. Within-world reasoning asks what to do given fixed constraints; it is (...) the domain of cognitive-behavioural reframing, standard decision theory, and Aristotelian practical wisdom. Between-world reasoning asks whether to change the constraints themselves by transitioning to a different set of circumstances where the structure of possibility is altogether different; it is the domain of modal logic, where the operators of possibility and necessity range over different sets of circumstances rather than over different propositions within a single set. Drawing on Kripke semantics, Schopenhauer's metaphysics of representation, and clinical evidence from Modal Integrative Psychotherapy (Fatić 2025), I develop a formal characterisation of practical modal worlds, show that the accessibility relation governing transitions between them is non-transitive and non-Euclidean, introduce the geometry of motivation which governs structural decisions, and present three clinical cases as evidence that between-world reasoning is a real pattern of practical cognition. The framework provides resources for understanding why propositional psychotherapeutic interventions (reframing, insight, cognitive restructuring) fail when the problem is structural, why modal interventions (changing the client's objective circumstances) succeed where propositional approaches cannot, and why much of what clinical psychology classifies as impulsive or avoidant behaviour may in fact be between-world reasoning operating under constraint. (shrink)

Logic appears indispensable to every coherent description of reality, yet admits no external grounding: any attempt to justify logic presupposes it. This paper argues that the circularity is diagnostic rather than vicious. It reveals _constitutive closure_: a structure whose components mutually enable each other such that no component's absence can be coherently questioned. Logic is the worked example, but the structure is general. Using Gödel-Löb provability logic to characterize the boundary of self-referential demonstration, the paper establishes (...) a parity result: within demonstrable reach, constitutive closure and any proposed external grounding have equivalent explanatory force. The result is dissolution, not stalemate. Grounding questions dissolve when posing them requires what they ask about. (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion (...) of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

In both philosophy and science, standard logics struggle to model a key insight of Existential Realism (ER): that existence is a strictly present, empirical notion, whereas reality spans beyond the present. Classical first-order logic and even temporal modal logics tend to conflate being real with existing simpliciter, making it difficult to formally distinguish something that exists now from something that is real but not present. This paper introduces a formal two-tier logical framework tailored to ER’s ontology. We motivate this (...) framework by showing how orthodox systems—from classical predicate logic to temporal modal logic—fail to capture ER’s core distinction between present existence and temporally extended reality. We then develop a layered logic with explicit operators or quantifiers for “existence-now” versus “reality-over-time,” including formal axioms that preserve ER’s asymmetries: only present entities exist (in the fullest sense), yet past and future entities can be real (ontologically significant) without existing. The logic adopts a modal-style treatment of time to model the collapse of potential into actual (present-becoming), and explicitly encodes causal asymmetry — where the past influences the present, but not the reverse. We illustrate the framework with symbolic representations, including an iconic “ER equation” capturing the relationship between reality and existence. To demonstrate its versatility, we apply the logic to examples from quantum physics (e.g. wavefunction collapse and delayed-choice experiments), cognitive neuroscience (memory traces and anticipatory processing), ethics (moral duties to future persons and the reality of future outcomes), and AI (agents modeling counterfactual or temporally extended scenarios). Finally, we show how the formalism sheds new light on classic metaphysical debates such as the truthmaker problem for past truths, persistence and identity over time, and the ontology of merely possible or future entities. Throughout, our presentation maintains a rigorous yet accessible tone, consistent with prior ER works, with technical clarity and footnoted references. The result is a self-contained preprint that extends T.C. Trepp’s Existential Realism into a formal logical system, demonstrating its broad applicability across science, philosophy, and practical reasoning. (shrink)

Sentences about logic are often used to show that certain embedding expressions are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that (...) determines the interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on (...) in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

Theories of epistemic justification are commonly assessed by exploring their predictions about particular hypothetical cases – predictions as to whether justification is present or absent in this or that case. With a few exceptions, it is much less common for theories of epistemic justification to be assessed by exploring their predictions about logical principles. The exceptions are a handful of ‘closure’ principles, which have received a lot of attention, and which certain theories of justification are well known to invalidate. But (...) these closure principles are only a small sample of the logical principles that we might consider. In this paper, I will outline four further logical principles that plausibly hold for justification and two which plausibly do not. While my primary aim is just to put these principles forward, I will use them to evaluate some different approaches to justification and (tentatively) conclude that a ‘normic’ theory of justification best captures its logic. (shrink)

Is logic normative for reasoning? In the wake of work by Gilbert Harman and John MacFarlane, this question has been reduced to: are there any adequate bridge principles which link logical facts to normative constraints on reasoning? Hitherto, defenders of the normativity of logic have exclusively focussed on identifying adequate validity bridge principles: principles linking validity facts—facts of the form 'gamma entails phi'—to normative constraints on reasoning. This paper argues for two claims. First, for the time being at (...) least, Harman’s challenge cannot be surmounted by articulating validity bridge principles. Second, Harman’s challenge can be met by articulating invalidity bridge principles: principles linking invalidity facts of the form 'gamma does not entail phi' to normative constraints on reasoning. In doing so, I provide a novel defence of the normativity of logic. (shrink)

Those who conceive logic as a science have generally favoured one of two alternative conceptions as to what the subject-matter of this science ought to be. On the one hand is the nowadays somewhat old-fashioned-seeming view of logic as the science of judgment, or of thinking or reasoning activities in general. On the other hand is the view of logic as a science of ideal meanings, 'thoughts', or 'propositions in themselves'. There is, however, a third alternative conception, (...) which enjoyed only a brief flowering in the years leading up to the first World War, but whose lingering presence can be detected in the background of more recent ontologising trends in logic, as for example in the 'situation semantics' of Barwise and Perry. This third conception sees logic as a science of special objects called 'Sachverhalte' or 'states of affairs'. A view of this sort is present in simplified form in the works of Meinong, but it received its definitive formulation in the writings of Adolf Reinach, a student of Husserl who is otherwise noteworthy for having anticipated, in a monograph of 1913, large chunks of what later became known as the theory of speech acts.(1). (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)

In this work, we present the first general logic of attention. Attention is a powerful cognitive ability that allows agents to focus on potentially complex information, such as logically structured propositions, higher-order beliefs, or what other agents pay attention to. This ability is a strength, as it helps to ignore what is irrelevant, but it can also introduce biases when some types of information or agents are systematically ignored. Existing dynamic epistemic logics for attention cannot model such complex attention (...) scenarios, as they only model attention to atomic formulas. Additionally, such logics quickly become cumbersome, as their size grows exponentially in the number of agents and announced literals. Here, we introduce a logic that overcomes both limitations. First, we generalize edge-conditioned event models, which we show to be as expressive as standard event models yet exponentially more succinct (generalizing both standard event models and generalized arrow updates). Second, we extend attention to arbitrary formulas, allowing agents to also attend to other agents' beliefs or attention. Our work treats attention as a modality, like belief or awareness. We introduce attention principles that impose closure properties on that modality and that can be used in its axiomatization. Throughout, we illustrate our framework with examples of AI agents reasoning about human attention, demonstrating how such agents can discover attentional biases. (shrink)

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages (...) when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness. (shrink)

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.

A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, recent decades have (...) shown remarkable comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨. (shrink)

The criticism of Immanuel Kant’s logic commenced with the advent of the so-called ‘new logic’ in the 20th century. One particular passage from the second preface to the Critique of Pure Reason has been a source of contention, where Kant asserted that logic has not taken a step forward or backward since Aristotle (B VIII). In Kant scholarship, one current strategy to avoid this criticism is to relocate Kant within the domain of philosophy of logic or (...) by segregating his general logic from modern formal systems. In this paper, it will be contended that this strategy is too weak, given that the B-preface has currently been analyzed in a markedly divergent manner by the so-called ‘methodological interpretation’. In his methodology and history of science of the B-preface, Kant means something different by progress and regression than what his 20th-century critics assume he meant. By examining what Kant and his critics considered to be progress and regress in science and logic, good reasons can be put forward for the argument that Kant was correct in his assertion that logic has not gone a step forward or backward. (shrink)

This note presents, motivates and details, a logic —model and proof theory— for a first order language with contextually restricted quantification.

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)

This paper provides a reconstruction of the argumentative logic behind the view of ideal agency in the Zhuangzi 莊子. I begin by articulating two principles that describe the reality in which the text understands agents to be acting: (1) Global Transformation—roughly, the view that reality is constantly transforming and (2) Inscrutability of Change—roughly, the view that agents cannot know how reality will change (§1). Then, I analyze the conditions that one’s agency must satisfy to count as an instance of (...) ideal agency. I extract from the text an argument for the principle of Ideal Adaptive Agency—roughly, the claim that one is an ideal agent if and only if they are maximally adaptive (§2). Then, I consider the question of what capacities are involved in the exercise of ideal agency. First, I identify the nonideal agential stance “Deeming Stance” and extract from the text an argument to the conclusion that those who adopt it are not ideal agents (§3). Second, I identify the agential stance “Adaptive Stance” and extract from the text an argument to the conclusion that agents who adopt it are ideal agents (§4). (shrink)

This paper analyzes the concept of ineffability and outlines a theory of the logic and semantics of ineffability claims. I begin by identifying three core problems for any theory of ineffability: (a) the problem of consistency: how is it possible to say that some entity x is ineffable without contradicting oneself? (b) The problem of reference: how is it possible to refer to some ineffable identity x if reference presupposes the ability to identify and thus to describe the entity (...) in question? (c) The problem of meaning: what is the meaning of negative statements about an ineffable entity x if x being ineffable means that nothing can be said about x? I then present and discuss three possible solutions to these problems that have been advanced in the recent literature: the theory of illuminating falsehoods (Lebens 2014), the theory of non-fundamental truths (Jacobs 2015), and the theory of apo-minimalism (van Elswyk forthcoming). I reject all three and instead develop an epistemic capacities-based account of ineffability that relativizes it to types of epistemic systems and their linguistic capabilities. (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)

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)

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)

In this paper, I ask whether we should see different logical systems as appropriate for different domains (or perhaps in different contexts) and whether this would amount to a form of logical pluralism. One, though not the only, route to this type of position, is via pluralism about truth. Given that truth is central to validity, the commitment the typical truth pluralist has to different notions of truth for different domains may suggest differences regarding validity in those different domains. Indeed, (...) as we’ll see, the differences between the proposed multiple notions of truth are often of a type that is clearly significant in relation to logical features, such as whether or not a constructive notion of truth is at issue. I criticise domain-based logical pluralism. Having done so I introduce a context-based framework that operates with a context-relative notion of validity. I show that this context-based framework can be employed by the domain-specific logical pluralist, but that framework also allows for logical pluralism that does not involve several domains. Different contexts may demand rules of classical logic, where others only justify intuitionistic rules, even when the same domain (e.g. mathematics) is at issue. (shrink)

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of (...) partitions. Or, putting it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives). (shrink)

Many have claimed that legitimate constitutional democracy is either conceptually or practically impossible, given infinite regress paradoxes deriving from the requirement of simultaneously democratic and constitutional origins for legitimate government. This paper first critically investigates prominent conceptual and practical bootstrapping objections advanced by Barnett and Michelman. It then argues that the real conceptual root of such bootstrapping objections is not any specific substantive account of legitimacy makers, such as consent or democratic endorsement, but a particular conception of the logic (...) of normative standards—the determinate threshold conception—that the critic attributes to the putatively undermined account of legitimacy. The paper further claims that when we abandon threshold conceptions of the logic of legitimacy in favor of regulative-ideal conceptions, then the objections, from bootstrapping paradoxes to the very idea of constitutional democracy, disappear. It concludes with considerations in favor of adopting a more demanding conception of the regulative ideal of constitutional democracy, advanced by Habermas, focusing on potentials for developmental learning. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the (...) perspective of abstract algebraic logic. We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú. We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

In this chapter, we begin by sketching in the broadest possible strokes the ideas behind two formal systems that have been introduced with to goal of explicating the ways in which reasons interact to support the actions and conclusions they do. The first of these is the theory of defeasible reasoning developed in the seminal work of Pollock; the second is a more recent theory due to Horty, which adapts and develops the default logic introduced by Reiter to provide (...) an account of reasons. However, the implementations are complex enough, in both cases, to prevent anything more than this sketch. And we would not want to give the impression that we think that work on the logic of reasons must follow the path mapped out in either of these theories—indeed, we feel that the field is wide open. In the remainder of the chapter, therefore, will concentrate on a number of issues bearing on the logic of reasons that are either not treated in the work of Pollock and Horty, or whose treatment there is, we feel, either inadequate or incomplete. These are: first, the question of whether it is necessary to understand logical interactions among reasons themselves, rather than simply between reasons and the actions or conclusions they support, and if so, what principles might govern these interactions; second, priority relations among reasons and the notion of reason accrual; and third, some problems posed by undercutting defeat. (shrink)

This is a working-paper version of the manuscript which presents a quantification of The NMC Theory (Nguyen 2025). Abstract. This article reconceives global order as a self-reproducing architecture of closure rather than an open field of power. Against the illusion of multipolarity, it advances the New Multipolar Convergence (NMC) Theory, which defines bipolarity as the structural law of the nuclear–digital system. Power is not exercised through expansion but through insulation—the recursive consolidation of chokepoints across Security, Finance, Technology, Human Capital, and (...) Norms. The NMC-5 Index operationalises this logic, tracing how closure density, not aggregate capability, determines systemic hierarchy. Through comparative analysis of pseudo-poles and gray zones, the paper demonstrates that bipolarity persists as the only stable configuration of world order. The result is both theoretical and epistemic: NMC reveals global structure as a cognitive law—a system that thinks through closure. This manuscript further operationalises that structural law by introducing the NMC-5 Index as the first proto-operational architecture for quantifying systemic closure in real time. Unlike legacy theories that collapse when removed from their local or temporal context, NMC’s status as law necessitates a real-time measurement framework capable of sensing chokepoint dynamics at the speed of structural formation. By demonstrating that bipolar reality is not merely interpretable but computationally measurable, the article completes the transition of structural geopolitics from a descriptive discipline into a computational science. The NMC-5 Index is thus the indispensable instrument that allows NMC to read the world at the velocity and complexity of the system it describes. Keywords. Epistemology; Bipolarity; Global Order; Methodology; Structural law; NMC Theory. (shrink)

Logic in Buddhist Philosophy concerns the systematic study of anumāna (often translated as inference) as developed by Dignāga (480-540 c.e.) and Dharmakīti (600-660 c.e.). Buddhist logicians think of inference as an instrument of knowledge (pramāṇa) and, thus, logic is considered to constitute part of epistemology in the Buddhist tradition. According to the prevalent 20th and early 21st century ‘Western’ conception of logic, however, logical study is the formal study of arguments. If we understand the nature of (...) class='Hi'>logic to be formal, it is difficult to see what bearing logic has on knowledge. In this paper, by weaving together the main threads of thought that are salient in Dignāga’s and Dharmakīti’s texts, I shall re-conceive the nature of logic in the context of epistemology and demarcate the logical part of epistemology which can be recognised as logic. I shall demonstrate that we can recognise the logical significance of inference as understood by Buddhist logicians despite the fact that its logical significance lies within the context of knowledge. (shrink)

Leibniz argues that there must be a fundamental level of simple substances because composites borrow their reality from their constituents and not all reality can be borrowed. I contend that the underlying logic of this ‘borrowed reality argument’ has been misunderstood, particularly the rationale for the key premise that not all reality can be borrowed. Contrary to what has been suggested, the rationale turns neither on the alleged viciousness of an unending regress of reality borrowers nor on the Principle (...) of Sufficient Reason, but on the idea that composites are phenomena and thus can be real only insofar as they have a foundation in substances, from which they directly ‘borrow’ their reality. The claim that composites are phenomena rests in turn on Leibniz's conceptualism about relations. So understood, what initially looked like a disappointingly simple argument for simples turns out to be a rather rich and sophisticated one. (shrink)

Biomedical research is increasingly a matter of the navigation through large computerized information resources deriving from functional genomics or from the biochemistry of disease pathways. To make such navigation possible, controlled vocabularies are needed in terms of which data from different sources can be unified. One of the most influential developments in this regard is the so-called Gene Ontology, which consists of controlled vocabularies of terms used by biologists to describe cellular constituents, biological processes and molecular functions, organized into hierarchies (...) via the relation of class subsumption. Here we seek to provide a rigorous account of the logic of classification that underlies GO and similar biomedical ontologies. Drawing on Aristotle, we develop a system of axioms and definitions for the treatment of biological classes and instances. (shrink)

In his article “Issues of Pragmaticism” published in 1905, in The Monist, Charles S. Peirce complains that “Logicians have been at fault in giving Vagueness the go-by, so far as not even to analyze it.” That same year, occupying himself with the consequences of “Critical commonsensism,” he affirmed, “I have worked out the logic of vagueness with something like completeness,” a statement that causes the majority of the commentators on his work, including the editors of the Collected Papers to (...) ask where this logic is to be found. The fever for finding Peirce's manuscripts is fed by the hope of some researchers of discovering the logic of vagueness, a hope that has grown since Carolyn Eisele's publication of his mathematical works. Others - and I count myself among them - believe that in reality this is a matter of something already known. That is, they interpret the affirmation ending the paragraph of reproach addressed to logicians, “The present writer has done his best to work out the Stechiology, Critic, and Methodeutik of the subject,” as a tripartite semiotic of the vague, still limited, according to Peirce's older works, to symbols, that is, to the signs of natural language examined from the perspective of logic. (shrink)

Any logic is represented as a certain collection of well-orderings admitting or not some algebraic structure such as a generalized lattice. Then universal logic should refer to the class of all subclasses of all well-orderings. One can construct a mapping between Hilbert space and the class of all logics. Thus there exists a correspondence between universal logic and the world if the latter is considered a collection of wave functions, as which the points in Hilbert space can (...) be interpreted. The correspondence can be further extended to the foundation of mathematics by set theory and arithmetic, and thus to all mathematics. (shrink)

Fuzzy logic extends deductive methods to situations in which the information available may be only partly or approximately true. Fuzzy logic has often been championed as a logic of vague terms, and it does indeed provide an intuitive analysis of what goes wrong in Sorites reasoning. Here a fuzzy semantics is given for a language containing the quasi-modal operators “Determinately” (Delta) and “Indeterminately” (Nabla) and the identity predicate (=). The semantics is sensitive to higher-order vagueness. For example, (...) the semantics distinguishes between Herbert’s being a clear borderline case of a bald man and his being a borderline borderline case of a bald man. I show that a famous reductio ad absurdum of the statement “Indeterminately (a=b)”, due to Gareth Evans, is not valid when the background logic is fuzzy logic. Moreover, an improved form of Evans’s reductio due to Harold Noonan is also not valid. The proposed fuzzy semantics appears to provide a promising route to resolving Sorites-type paradoxes. (shrink)

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions (...) to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (shrink)

I defend logical realism. I begin by motivating the realist approach by underlining the difficulties for its main rival: inferentialism. I then focus on AND and OR, and delineate a realist view of these two logical constants. The realist view is developed in terms of Alexander’s Principleshowing that AND and OR have distinctive determining roles. After that, I say what logic is not. We should not take logic to be essentially about the mind, or language, or exclusively about (...) an abstract realm, or about reasoning, truth, truth-tables, truth-functions, topic-neutrality or form. Lastly, I turn to consider NEGATION and argue that we cannot escape negative facts, and facts conjoining and disjoining negative facts with positive facts. I then give NEGATION a distinctive role, one that contrasts with AND and OR. I reflect on the notion of logic in the coda. (shrink)

This essay takes logic and ethics in broad senses: logic as the science of evidence; ethics as the science justice. One of its main conclusions is that neither science can be fruitfully pursued without the virtues fostered by the other: logic is pointless without fairness and compassion; ethics is pointless without rigor and objectivity. The logician urging us to be dispassionate is in resonance and harmony with the ethicist urging us to be compassionate.

This article introduces a modal-tensed logic of actualization, grounded in a processual and non-metrical ontology of the real. Classical modal logics evaluate truth across pre-constituted possible worlds, treating modality as extensional and symmetric. In contrast, we propose that modality arises from the internal dynamics of stabilization within a field of tensions: configurations become possible, effective, or actual according to their capacity to resolve constraints. The logic defines five operators, present actualization, past trace, future projection, structural possibility, and necessity, (...) governed by axioms encoding irreversibility, fragility, and asymmetrical transition. We provide a semantic model based on dynamically accessible configurations with intensity-based stabilization. A case study in quantum measurement illustrates how actualization captures the transition from superposition to definite outcome without invoking collapse. This framework reconceptualizes modality, temporality, and ontology as emergent from internal dynamics rather than imposed from external metaphysical structures. It offers new perspectives on metaphysical logic and the foundations of physics. (shrink)