I focus on three mereological principles: the Extensionality of Parthood (EP), the Uniqueness of Composition (UC), and the Extensionality of Composition (EC). These principles are not equivalent. Nonetheless, they are closely related (and often equated) as they all reflect the basic nominalistic dictum, No difference without a difference maker. And each one of them—individually or collectively—has been challenged on philosophical grounds. In the first part I argue that such challenges do not quite threaten EP insofar as they are either self-defeating (...) or unsupported. In the second part I argue that they hardly undermine the tenability of EC and UC as well. (shrink)

Extensional scientific realism is the view that each believable scientific theory is supported by the unique first-order evidence for it and that if we want to believe that it is true, we should rely on its unique first-order evidence. In contrast, intensional scientific realism is the view that all believable scientific theories have a common feature and that we should rely on it to determine whether a theory is believable or not. Fitzpatrick argues that extensional realism is immune, while intensional (...) realism is not, to the pessimistic induction. I reply that if extensional realism overcomes the pessimistic induction at all, that is because it implicitly relies on the theoretical resource of intensional realism. I also argue that extensional realism, by nature, cannot embed a criterion for distinguishing between believable and unbelievable theories. (shrink)

Extensionalism is, roughly, the view that perception occurs in episodes that are temporally extended (and thus capable of accomodating in their entirety phenomena taking a nonzero lapse of time to occur). This view is widely acknowledged to be incompatible with thin presentism, the second most popular position in temporal ontology. In this paper, I argue that extensionalism is also incompatible with several other positions in temporal ontology, namely those positing the existence of non-present times that host sentience—positions I (...) collectively refer to as the sentient non-present view. Most notably, extensionalism proves to be incompatible with B-theoretical eternalism, which is the most popular position in temporal ontology. (shrink)

Continuum physical theories model states as real- or complex-valued fields and dynamics as linear operators on infinite dimensional spaces. Under explanatory realism, an ontic interpretation incurs two semantic commitments: (i) real-valued physical magnitudes must denote relative to the theory’s admissible state interface, and (ii) the theory must be semantically closed under its own evolution and readout rules. Denotation is interface-relative: it requires the existence of a total continuous witness on names. Equivalently, it requires bounded input dependence at each fixed finite (...) precision. Failure of bounded input dependence is captured by forced extensional totalization (FET), in which even coarse output precision cannot be stabilized by any finite prefix of a state name. We give a physics-facing construction of FET for the three dimensional wave equation with Kirchhoff point readout. We construct a countable family of smooth, compactly supported, finite-energy initial states localized on pairwise disjoint shrinking latitude belts on the Huygens sphere, and restrict the admissible domain to states containing at most one localized contingency. Each nonzero belt perturbation is normalized to contribute the same order-one amount to the readout, while the zero state contributes none. Under any non-oracular state interface satisfying a minimal locality-of-access condition, no finite prefix can rule out activation of some sufficiently deep belt. The induced point readout therefore behaves like the existential predicate ``some belt is active'', exhibits FET at the zero state, and fails to denote. Semantic closure fails: the evolution--readout pipeline yields a localized point magnitude that is non-denoting relative to the fixed state interface. The result isolates a semantic obstruction to ontic readings of continuum dynamics. It is independent of computability, predictability, or measurement feasibility, and it holds despite smooth compact support, finite energy, and classical well-posedness. (shrink)

Continuum physical theories represent states as real- or complex-valued fields and dynamics as linear operators on infinite dimensional spaces. When interpreted ontically under explanatory realism, such theories are committed to a semantic closure requirement: admissible physical magnitudes must be denoted, and physical evolution must preserve denotation. The work of Pour-El and Richards shows that even well-posed linear continuum dynamics can violate effective uniformity and computability. In this paper we identify a more elementary semantic obstruction. We present a self-contained construction showing (...) that linear continuum evolution, when coupled to an entirely standard localized averaging readout, can force the aggregation of countably many independent contingencies into a single demanded real-valued magnitude. The resulting scalar is extensionally equivalent to the halting real. Under any admissible semantics, its specification is non denoting, since approximating it to arbitrary precision requires forced extensional resolution of a non-total predicate. The construction uses only smooth, compactly supported, finite-energy initial data for the three-dimensional wave equation and does not rely on function-space semantics, regional statehood assumptions, or modulus-of-continuity arguments. It therefore establishes a minimal semantic obstruction for ontic readings of continuum dynamics, isolating forced extensional totalization as the operative mechanism behind failures of semantic closure. (shrink)

Modern deep generative models display a fundamental paradox: while bounded strictly by the limits of Turing computability (T), they routinely provide highly accurate, computationally tractable solutions to physical and biological problems known to be NP-hard or non-computable from first principles. This paper presents a unified two-tier mathematical framework that resolves this paradox, spanning from foundational computability theory to engineering-ready intelligent system paradigms. At the foundational tier, we model physical reality and biological neural systems as an infinite-dimensional continuous smooth manifold M (...) governed by a non-computable evolution operator Φt / ∈T. Through physical measurement and symbolic recording, continuous states undergo a topological collapse via a dissipative projection operator P drop onto a discrete symbolic space D, leaking fossilized solutions of the continuous operator into empirical datasets. We prove that deep learning architectures do not execute intrinsic hyper-computation; instead, they achieve extensional alignment, minimizing empirical risk over leaked oracle pairs to yield an ϵ-approximation of Φt’s image over a compact domain. We further formulate and prove the Semiotic Collapse Theorem, showing that recursive training on isolated synthetic data loops leads to monotonic decay of mutual information I(M; Dn) and zero-measure support collapse. Building on this foundation, we present Operator-Centric Intelligence (OCI), a post-reinforcement learning paradigm that transcends the limitations of traditional RL via two core insights: (1) neural operators provide a dimensionality reduction advantage by directly learning intrinsic physical dynamics, and (2) the success of intelligent systems stems from computational delegation to external physical systems that solve intractable problems in O(1) physical time. We formalize OCI as a four-layer architecture centered on the Continuous Flow Operator (CFO), with three core mechanisms: symplectic Hamiltonian operator alignment, Pareto-optimal active querying for computational delegation, and reward demotion to a prior hint. We prove that OCI unifies all existing control and intelligence paradigms as special boundary cases, achieving unprecedented sample efficiency and engineering scalability. This framework provides a first-principles explanation for AI4Science success, establishes strict algorithmic boundaries for synthetic data scalability, offers a rigorous foundation for embodied cognition, and opens a new era of post-reinforcement learning artificial intelligence. (shrink)

Simulation hypotheses are often challenged by an energy objection: a universe as large, structured, and observationally coherent as ours would require impossible or effectively unbounded energy and computation to simulate. This objection usually assumes two strong architectural premises. First, it assumes that simulated reality must be instantiated as an externally expanding, full-resolution map whose cost scales with apparent internal size. Second, it assumes that global coherence must be maintained through brute-force storage of an ever-growing table of facts across observers, measurements, (...) histories, and correlations. This paper argues that both assumptions are stronger than necessary. Apparent scale may arise through recursive internal differentiation of a fixed totality rather than external expansion, and coherence itself may be generated from compact lawful structure rather than stored extensionally as a total fact-table. On this view, the relevant question is not simply whether apparent size can be compressed, but whether coherence can be law-compressed. The paper does not defend the truth of the simulation hypothesis. It advances a narrower claim: the standard energy objection is weaker than it appears if both scale and coherence can, in principle, be generated from compact structure rather than requiring proportional external substrate. (shrink)

Ethical extensionism is a common argument pattern in environmental and animal ethics, which takes a morally valuable trait already recognized in us and argues that we should recognize that value in other entities such as nonhuman animals. I exposit ethical extensionism’s core argument, argue for its validity and soundness, and trace its history to 18th century progressivist calls to expand the moral community and legal franchise. However, ethical extensionism has its critics. The bulk of the paper responds to recent criticisms, (...) including (1) environmental ethicists’ objection against its austere conception of moral value (2), feminist ethicists’ claim that extensionism fails to account for the moral significance of difference, (3) disability theorists’ concern that ethical extensionist arguments are offensive, and (4) animal rights theorists’ lament that extensionism is a practical failure. While something is to be gained from each criticism, I argue that they ultimately fail and that extensionism remains compelling. (shrink)

This paper defends extensionality, a principle of normative rationality according to which the value of an option should not depend on the way it is represented or framed. It considers two accounts of decision-theoretic representation challenging the principle, Schick’s theory of understandings (1992) and Bermúdez’s theory of frames (2020). Both contend that in certain situations of inner conflict, one may knowingly and rationally see or grasp the same option under several perspectives leading to different evaluations of the option. I argue (...) that situations of conflicts do not constitute a convincing case against extensionality and offer an alternative model of conflict where decision-theoretic consequences are intensional properties of the option. This second model is more appealing as it treats inner conflict as an inability to weigh the consequences of the same action and is consistent with extensionality. I end the paper with a discussion of what Bermúdez calls factual propositions and defend the view that the specification of factual propositions is not evaluatively neutral for the decision-maker. This suggests that all specifications of decision problems and their consequences are value-driven. (shrink)

I argue that Universalism (the thesis that mereological composition is unrestricted) entails Extensionalism (the thesis that sameness of composition is sufficient for identity) as long as the parthood relation is transitive and satisfies the Weak Supplementation principle (to the effect that whenever a thing has a proper part, it has another part disjoint from the first).

The mereological account of social groups based on Classical Extensional Mereology (CEM) is notoriously associated with the reductionist stance that social groups are regular fusions identified by mereological conditions alone. This approach has many limitations and has suffered knock-down objections due to its transitive and coextensional features. The reaction to this impasse has been twofold. Philosophers have either moved away from CEM-based accounts altogether or attempted to resist these objections by appealing to extra-ontological explanations while retaining the reductive qualification of (...) social groups. In this paper, I develop a non-reductive yet CEM-based framework that employs the notion of φ-parthood to restrict the interpretation of parthood simpliciter through extra-mereological, socially relevant conditions. Social groups turn out to be specified fusions and, depending on how the extra-mereological conditions are interpreted, the resulting account can be extensional or intensional. I show that both accounts successfully handle transitivity and coextensionality objections, and argue that long-standing objections from modal and temporal inflexibility are misguided for any CEM-based account. (shrink)

I discuss Frege's argument - later called the slingshot - that if a construction is extensional and preserves logical equivalence then it is truth-functional. I consider some simple apparent counterexamples and conclude that they are not sentence-embedding in the required way.

Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a second-order entity $X$, there may be an object $\varepsilon X$, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities. This paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is (...) more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second-order logic. I do so by showing how the $\varepsilon$ operator can have the effect of “extensionalizing” intensional second-order entities. (shrink)

Teague (2024) argues that leading theories of good guessing (Holguín 2022; Dorst and Mandelkern 2021) generate incoherent guesses: on one body of evidence it can be rational to guess Yale to Where will Latif go to law school? yet rational to guess not-Yale to Will Latif go to Yale?, seemingly committing one agent to both p and ¬p. I argue that the puzzle rests on a single conation that can be stated exactly. A guess is the selection of an optimal (...) answer from the answer- space a question makes available in the simplest case an argmax over the question's partition. Two facts about coarsening then settle the matter. First, truth distributes over the merging of cells, so belief, which tracks truth, transmits across the part-relation; this is why Hoek's (2025) coherence principles hold for belief. Second, optimal selection does not distribute over the merging of cells: the most probable atom of a ne partition need not lie on the most probable side of a coarsening of it. Selection therefore does not transmit. The puzzle treats a guess as though it transmitted as a belief would; once selection and truth are separated, the two contradictory guesses are seen to be optimal answers to two questions that share a cell extensionally but not the inquiry that adjudicates them. I distinguish extensional from inferential parthood the latter the relation a sub- question bears to a whole when its settled answer is carried forward and contracts the whole's live partition show that whether-Yale is an extensional but not an inferential part of where, and argue that the resulting picture withstands Teague's inquiry-theoretic pressures and accounts for them within a single mechanism. The same asymmetry, seen under conjunction rather than coarsening, resolves Fang's (2026) puzzle about closure across independent questions. (shrink)

The original interest of this article lies in existential import. It provides a broader view on the problem by reference to modern, symbolic logic (ch.1). Gradually, however, our interest will change into the amalgamated expressions often used in logic; that is, why are such expressions as “x is a round triangle” applied in logic? We critically discuss this question from an extensional viewpoint, namely model theoretic semantics (ch.2). We also touch on Church’s λ-calculus in the appendix (app.2).

Many believe that, if true, reason-statements of the form ‘that X is F is a reason to φ’ describe a ‘favouring-relation’ between the fact that X is F and the act of φing. This favouring-relation has been assumed to share many features of other, more concrete relations. This combination of views leads to immediate problems. Firstly, unlike statements about many other relations, reason-statements can be true even when the relata do not exist, i.e., when the relevant facts do not obtain (...) and the relevant acts are not done. Secondly, the previous combination of views also makes it very difficult to draw the distinction between agent-relative and agent-neutral reasons. I argue that we should think that the predicate ‘is a reason to’ creates non-extensional contexts in the statements in which it is used. This would both solve the previous problems and avoid the awkward consequences of the so-called slingshot argument. (shrink)

There are two main ways in which the notion of mereological fusion is usually defined in the current literature in mereology which have been labelled ‘Leśniewski fusion’ and ‘Goodman fusion’. It is well-known that, with Minimal Mereology as the background theory, every Leśniewski fusion also qualifies as a Goodman fusion. However, the converse does not hold unless stronger mereological principles are assumed. In this paper I will discuss how the gap between the two notions can be filled, focussing in particular (...) on two specific sets of principles that appear to be of particular philosophical interest. The first way to make the two notions equivalent can be used to shed some interesting light on the kind of intuition both notions seem to articulate. The second shows the importance of a little-known mereological principle which I will call ‘Mild Supplementation’. As I will show, the mereology obtained by adding Mild Supplementation to Minimal Mereology occupies an interesting position in the landscape of theories that are stronger than Minimal Mereology but weaker than what Achille Varzi and Roberto Casati have labelled ‘Extensional Mereology’. (shrink)

This paper develops a novel philosophical paradigm addressing the metaphysical and epistemological implications of generative artificial intelligence. For over half a century, the philosophy of mind has been divided between computationalists, who argue that all mental processes are algorithmic, and anti-computationalists, who maintain that human cognition involves non-algorithmic processes that cannot be simulated by Turing machines. We propose a resolution to this long-standing debate by introducing the concepts of the hypercomputational ontological manifold M and the dissipative projection operator Pdrop. We (...) demonstrate that human brains and physical reality form a unified continuous dynamical system whose evolution is fundamentally non-computable (Φt /∈ T). However, through symbolic representation and data recording, humans project the outputs of this hypercomputational system onto a discrete countable symbolic space D, a process we call oracle leakage. Generative AI systems, which are strictly Turing-computable, do not possess intrinsic hypercomputational capabilities. Instead, they achieve their remarkable performance through extensional alignment: minimizing empirical risk over the oracle solutions encoded in human-generated data. We prove mathematically that when AI systems are trained exclusively on synthetic data generated by other AI systems, they enter a state of semiotic convolution that inevitably leads to epistemological collapse. This result provides a rigorous philosophical foundation for embodied cognition and reaffirms the irreplaceable role of human subjectivity as the ultimate source of epistemic value in the digital age. (shrink)

If one takes seriously the idea that a scientific language must be extensional, and accepts Quine’s notion of truth-value-related extensionality, and also recognizes that a scientific language must allow for singular terms that do not refer to existing objects, then there is a problem, since this combination of assumptions must be inconsistent. I will argue for a particular solution to the problem, namely, changing what is meant by the word ‘extensionality’, so that it would not be the truth-value that had (...) to be preserved under the substitution of co-extensional expressions, but the state of affairs that the sentence described. The question is whether or not elementary sentences containing empty singular terms, such as ‘Vulcan rotates’, are extensional in the substitutivity sense. Five conditions are specified under which extensionality in the substitutivity sense of such sentences can be secured. It is demonstrated that such sentences are state-of-affairs-as-extension-related extensional. This implies that such sentences are also truth-value-related extensional in Quine’s sense, but not truth-value-as-extension-related extensional. (shrink)

Much of the ontology made in the analytic tradition of philosophy nowadays is founded on some of Quine’s proposals. His naturalism and the binding between existence and quantification are respectively two of his very influential metaphilosophical and methodological theses. Nevertheless, many of his specific claims are quite controversial and contemporaneously have few followers. Some of them are: (a) his rejection of higher-order logic; (b) his resistance in accepting the intensionality of ontological commitments; (c) his rejection of first-order modal logic; and (...) (d) his rejection of the distinction between analytic and synthetic statements. I intend to argue that these controversial negative claims are just interconnected consequences of those much more accepted and apparently less harmful metaphilosophical and methodological theses, and that the glue linking all these consequences to its causes is the notion of extensionality. (shrink)

Bermúdez argues for rational framing effects in the form of quasi-cyclical preferences. This is supposed to refute the extensionality principle in standard decision theory. In response, I argue that it is better to analyze seemingly quasi-cyclical preferences as ceteris paribus preferences. Furthermore, if frames are included as objects of choice, we can acknowledge rational framing effects without rejecting extensionality.

Hylomorphically complex objects are things that change their parts or matter or that might have, or have had, different parts or matter. Often ontologists analyze such objects in terms of sets (or functions, understood set-theoretically) or other extensional entities such as mereological fusions or quantities of matter. I urge two reasons for being wary of any such analyses. First, being extensional, such things as sets are ill-suited to capture the characteristic modal and temporal flexibility of hylomorphically complex objects. Secondly, sets (...) are often appealed to because they seem to contain their members. But the idea that sets do contain their members, in the ordinary sense of containment, is a substantive metaphysical position that makes analyses that rely on that idea for their plausibility much more metaphysically committing than is generally thought. (shrink)

Object x being a member of itself or not and x being a member of R or not constitute two vastly different concepts. This paper attempts to locate the reflection of such an utter difference within the formal structure of the axiom of extensionality. -/- .

Intensional evidence is any reason to accept a proposition that are not the truth values of the proposition accepted or, if it is a complex proposition, are not the truth values of its propositional contents. Extensional evidence is non-intensional evidence. Someone can accept a complex proposition, but deny its logical consequence when her acceptance is based on intensional evidence, while the logical consequences of the proposition presuppose the acceptance of extensional evidence, e.g., she can refuse the logical consequence of a (...) proposition she accepts because she does not know what are the truth-values of its propositional contents. This tension motivates counterexamples to the negation of conditionals, the propositional analysis of conditionals, hypothetical syllogism, contraposition, and or-to-if. It is argued that these counterexamples are non-starters because they rely on a mix of intensionally based premises and extensionally based conclusions. Instead, a genuine counterexample to classical argumentative forms should present circumstances where an intuitively true and extensionally based premise leads to an intuitively false conclusion that is also extensionally based. The other point is that evidentiary concerns about intensionally based beliefs should be constrained by the truth conditions of propositions presented by classical logic, which are nothing more than requirements of coherence in distributions of truth value. It is argued that this restriction also dissolves some known puzzles such as conditional stand-offs, the Adams’ pair, the opt-out property, and the burglar’s puzzle. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...) to some speculative research programs. (shrink)

In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is (...) identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. (shrink)

In “Psychopower and Ordinary Madness” my ambition, as it relates to Bernard Stiegler’s recent literature, was twofold: 1) critiquing Stiegler’s work on exosomatization and artefactual posthumanism—or, more specifically, nonhumanism—to problematize approaches to media archaeology that rely upon technical exteriorization; 2) challenging how Stiegler engages with Giuseppe Longo and Francis Bailly’s conception of negative entropy. These efforts were directed by a prevalent techno-cultural qualifier: the rise of Synthetic Intelligence (including neural nets, deep learning, predictive processing and Bayesian models of cognition). This (...) paper continues this project but first directs a critical analytic lens at the Derridean practice of the ontologization of grammatization from which Stiegler emerges while also distinguishing how metalanguages operate in relation to object-oriented environmental interaction by way of inferentialism. Stalking continental (Kapp, Simondon, Leroi-Gourhan, etc.) and analytic traditions (e.g., Carnap, Chalmers, Clark, Sutton, Novaes, etc.), we move from artefacts to AI and Predictive Processing so as to link theories related to technicity with philosophy of mind. Simultaneously drawing forth Robert Brandom’s conceptualization of the roles that commitments play in retrospectively reconstructing the social experiences that lead to our endorsement(s) of norms, we compliment this account with Reza Negarestani’s deprivatized account of intelligence while analyzing the equipollent role between language and media (both digital and analog). (shrink)

This continues from Part 1. It is shown how an intensional interpretation of physics object languages can be formalised, and how a syntactic compositional time reversal operator can subsequently be defined. This is applied to solve the problems used as examples in Part 1. A proof of a general theorem that such an operator must be defineable is sketched. A number of related issues about the interpretation of theories of physics, including classical and quantum mechanics and classical EM theory are (...) discussed. (shrink)

In this paper, I argue that current attempts at classifying life–mind continuity (LMC) feature several important ambiguities. We can resolve these ambiguities by distinguishing between the extensional, categorical, and systematic relationships that LMC might encompass. In Sect. 1, I begin by introducing the notion of LMC and the theory behind it. In Sect. 2, I show how different ideas of mind shape different approaches to continuity and how to achieve its aim. In Sect. 3, I canvas various canonical formulations and (...) classifications of LMC; I then demonstrate that they retain important ambiguities. Section 4 builds on this by arguing that we must conceive of the extensional and categorical aspects of continuity independently. In Sect. 5, I show further that current literature has underexplored multiple systematic aspects of continuity. I then take a constructive approach in Sect. 6 by providing a classification model for LMC based on extensional and categorical commitments. Here, I comment on aspects of the thesis omitted from the model but essential for a full classification and thorough comparison between various approaches to LMC. All of these arguments lay the foundation for more exhaustively classifying accounts of LMC. (shrink)

Future Logic is an original, and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities. Traditional and Modern logic have covered in detail only formal deduction from actual categoricals, or from logical conditionals (conjunctives, hypotheticals, and disjunctives). Deduction from modal categoricals has also been considered, though very vaguely and roughly; whereas deduction from natural, temporal and extensional forms of conditioning has been all but totally ignored. (...) As for induction, apart from the elucidation of adductive processes (the scientific method), almost no formal work has been done. This is the first work ever to strictly formalize the inductive processes of generalization and particularization, through the novel methods of factorial analysis, factor selection and formula revision. This is the first work ever to develop a formal logic of the natural, temporal and extensional types of conditioning (as distinct from logical conditioning), including their production from modal categorical premises. Future Logic contains a great many other new discoveries, organized into a unified, consistent and empirical system, with precise definitions of the various categories and types of modality (including logical modality), and full awareness of the epistemological and ontological issues involved. Though strictly formal, it uses ordinary language, wherever symbols can be avoided. Among its other contributions: a full list of the valid modal syllogisms (which is more restrictive than previous lists); the main formalities of the logic of change (which introduces a dynamic instead of merely static approach to classification); the first formal definitions of the modal types of causality; a new theory of class logic, free of the Russell Paradox; as well as a critical review of modern metalogic. But it is impossible to list briefly all the innovations in logical science — and therefore, epistemology and ontology — this book presents; it has to be read for its scope to be appreciated. (shrink)

Quine argues that if sentences that are set theoretically equivalent are interchangeable salva veritate, then all transparent operators are truth-functional. Criticisms of this argument fail to take into account the conditional character of the conclusion. Quine also argues that, for any person P with minimal logical acuity, if ‘belief’ has a sense in which it is a transparent operator, then, in that sense of the word, P believes everything if P believes anything. The suggestion is made that he intends that (...) result to show us that ‘believes’ has no transparent sense. Criticisms of this argument are either based on unwarranted assertions or on definitions of key terms that depart from Quine’s usage of those terms. (shrink)

Real-world phenomena frequently involve vagueness, partial truth, and incomplete information. To capture such uncertainty in a mathematically rigorous manner, numerous generalized set-theoretic frameworks have been introduced, including Fuzzy Sets [1], Intuitionistic Fuzzy Sets [2], Neutrosophic Sets [3, 4], Vague Sets [5], Hesitant Fuzzy Sets [6], Picture Fuzzy Sets [7], Quadripartitioned Neutro-sophic Sets [8], PentaPartitioned Neutrosophic Sets [9], Plithogenic Sets [10], HyperFuzzy Sets [11], and HyperNeutrosophic Sets [12]. Within these frameworks, a vast number of concepts have been proposed and studied, especially (...) in the contexts of Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Sets. This extensive body of work highlights both the importance of these theories and the breadth of their application domains. Consequently, many ideas, constructions, and structural patterns recur across these four families of uncertainty-oriented models. In this book, we present a comprehensive, large-scale survey of Fuzzy, Intuitionistic Fuzzy, Neutro-sophic, and Plithogenic Sets. Our aim is to offer r eaders a s ystematic overview of existing develop-ments and, through this unified exposition, to foster new insights, further conceptual extensions, and additional applications across a wide range of disciplines. (shrink)

x and y are at least weakly indiscernible if they are distinct with respect to two predicates F and G (FxGy or FyGx and not both) but for all z except x and y, Fz if and only if Gz.

According to the Bad Company objection, the fact that Frege’s infamous Basic Law V instantiates the general definitional pattern of higher-order abstraction principles is a good reason to doubt the soundness of this sort of definitions. In this paper I argue against this objection by showing that the definitional pattern of abstraction principles – as extrapolated from §64 of Frege’s Grundlagen– includes an additional requirement (which I call the specificity condition) that is not satisfied by the Basic Law V while (...) is satisfied by other higher-order abstractions such as Hume’s Principle. I also show that the failure of this additional requirement in the case of Basic Law V is engendered by an essential feature of Frege’s conception of logic and thus that Frege himself should not have regarded the Basic Law V as a definition by abstraction. (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In (...) this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

At present, there are at least two set theories motivated by quantum ontology: Décio Krause’s quasi-set theory (Q) and Maria Dalla Chiara and Giuliano Toraldo di Francia’s quasi-set theory (QST). Recent work [Jorge-Holik-Krause, 2023] has established certain links between QST and Pawlak’s rough set theory (RST), showing that both are strong candidates for providing a non-deterministic semantics of N matrices that generalizes those based on ZF. In this work, we show that the new atomless quasi-set theory Q^− , recently introduced (...) to account for a quantum property ontology [Krause-Jorge, 2024], has strong structural similarities with QST and RST. We study the level of extensionality that each theory presents, its relation to the Leibniz principle and the rigidity property. We believe that developing common features among these three theories can motivate common fields of research. By revealing shared structures, the developments of each theory can have a positive impact on the others. (shrink)

This paper introduces the Co-Equal Structure Thesis (CEST), the proposal that the identity of any infinite set S is given by the ordered pair Φ(S) = ⟨card(S), ord(S)⟩. After showing that Φ is definable within standard ZFC and that ZFC + Φ constitutes a conservative definitional extension, the paper argues that cardinality alone is operationally insufficient for distinguishing structurally divergent sets. Ordinal-sensitive operations—such as ordinal multiplication—produce distinct order-types that cardinal equivalence collapses. CEST is offered as a conceptual remedy: a dual-invariant (...) identity criterion integrating structural adequacy with extensional economy. The paper concludes with implications for foundations, suggesting the possibility of an Axiom of Structural Identity (ASI), and outlines potential directions for further logical and philosophical development. (shrink)

This paper is about both an application of virtue ethics, and about virtue ethics itself. A popular application of neo-Aristotelian virtue ethics to environmental issues is called interpersonal extensionism. It argues that we should view the normative range of traditional interpersonal virtues, such as compassion and humility, as extending beyond our interactions with people to also include our interactions with non-human environments. This paper uncovers an unaddressed problem for this view, then proposes a solution by revising how we understand neo-Aristotelian (...) virtue ethics itself. The tenet in question, defended by virtue ethicists such as Hursthouse and Foot, claims that a character trait counts as a virtue only if it is strictly necessary for human flourishing. This tenet threatens interpersonal extensionism because, as I show, the extended or environment-regarding aspects of virtues such as compassion and humility are not strictly speaking necessary for human flourishing: people can in principle flourish without those. But I then argue that diversities across both people and environments give us good, seldom acknowledged, reasons to revise neo-Aristotelian virtue ethics so that it no longer imposes the tenet that led to this problem. Neither a trait nor an aspect of it need be necessary for human flourishing in order to count as a neo-Aristotelian virtue. Indeed, marked variation between people and between environments can be complementary with respect to flourishing. Sometimes, for instance, part of the reason an environment-regarding aspect of compassion is virtuous is because of its distinct role in creating conditions that increase the chance that others will flourish. To show these things, I draw from work in philosophy of science on cluster approaches to categories and constitution, and from experiences working with diverse people in Canada's diverse national and provincial parks. (shrink)

The identity theory’s rise to prominence in analytic philosophy of mind during the late 1950s and early 1960s is widely seen as a watershed in the development of physicalism, in the sense that whereas logical behaviourism proposed analytic and a priori ascertainable identities between the meanings of mental and physical-behavioural concepts, the identity theory proposed synthetic and a posteriori knowable identities between mental and physical properties. While this watershed does exist, the standard account of it is misleading, as it is (...) founded in erroneous intensional misreadings of the logical positivists’—especially Carnap’s—extensional notions of translation and meaning, as well as misinterpretations of the positivists’ shift from the strong thesis of translation-physicalism to the weaker and more liberal notion of reduction-physicalism that occurred in the Unity of Science programme. After setting the historical record straight, the essay traces the first truly modern identity theory to Schlick’s pre-positivist views circa 1920 and goes on to explore its further development in Feigl, arguing that the fundamental difference between the Schlick-Feigl identity theory and the more familiar and influential Place-Smart-Armstrong identity theory has resurfaced in the deep and seemingly unbridgeable gulf in contemporary philosophy of consciousness between inflationary mentalism and deflationary physicalism. (shrink)

The categorical syllogism, as the core of traditional formal logic, has long been criticized for its inability to handle relational reasoning, leading many to argue that it should be replaced by first-order logic. This paper argues that this criticism misunderstands the nature of traditional logic. Traditional logic is grounded in "content consistency" rather than "truth preservation," and its core principle—Dictum de Omni et Nullo—was never inherently restricted to property-based reasoning; its perceived limitations arose from a historical contraction in its mode (...) of expression. The central contribution of this paper is the proposal of a "dual-mechanism model" for natural language logic. First, an extensional reasoning mechanism, based on the subordination relations of concept extensions, is developed. This mechanism extends the predicate in the Dictum de Omni et Nullo from a single concept to a "compound predicate," thereby enabling it to handle reasoning involving properties, relations, and their combinations. Second, an intensional reasoning mechanism, based on the satisfaction of a concept's defining conditions, is introduced. This mechanism systematically addresses relational reasoning, complex property judgments, and object reasoning by verifying whether objects or groups of objects meet the intensional conditions of a concept. Both mechanisms adopt the form of "two premises leading to one conclusion" and are collectively termed the generalized syllogism. Through this model, traditional formal logic is refined and extended, allowing it to comprehensively cover both property-based and relational reasoning in natural language, thereby establishing its independent and irreplaceable status in this domain. (shrink)

Tichý’s Transparent Intensional Logic (TIL) is an overarching logical framework apt for the analysis of all sorts of discourse, whether colloquial, scientific, mathematical or logical. The theory is a procedural (as opposed to denotational) one, according to which the meaning of an expression is an abstract, extra-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure that is the object denoted by the expression. Such procedures are rigorously defined as (...) TIL constructions. Though TIL analytical potential is very large, deduction in TIL has been rather neglected. Tichý defined a sequent calculus for pre-1988 TIL, that is TIL based on the simple theory of types. Since then no other attempt to define a proof calculus for TIL has been presented. The goal of this paper is to propose a generalization and adjustment of Tichý’s calculus to TIL 2010. First, I briefly recapitulate the rules of simple-typed calculus as presented by Tichý. Then, I propose the adjustments of the calculus so that it be applicable to hyperintensions within the ramified hierarchy of types. TIL operates with a single procedural semantics for all kinds of logical-semantic context, be it extensional, intensional or hyperintensional. I show that operating in a hyperintensional context is far from being technically trivial. Yet, it is feasible. To this end, we introduce a substitution method that operates on hyperintensions. It makes use of a four-place substitution function (called Sub) defined over hyperintensions. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

This paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behavior of the conditional in HYPE) and show (...) its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behavior of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories. (shrink)

The paper concentrates on the problem of adequate reflection of fragments of reality via expressions of language and inter-subjective knowledge about these fragments, called here, in brief, language adequacy. This problem is formulated in several aspects, the most being: the compatibility of language syntax with its bi-level semantics: intensional and extensional. In this paper, various aspects of language adequacy find their logical explication on the ground of the formal-logical theory T of any categorial language L generated by the so-called classical (...) categorial grammar, and also on the ground of its extension to the bi-level, intensional and extensional semantic-pragmatic theory ST for L. In T, according to the token-type distinction of Ch.S. Peirce, L is characterized first as a language of well-formed expression-tokens (wfe-tokens) - material, concrete objects - and then as a language of wfe-types - abstract objects, classes of wfe-tokens. In ST the semantic-pragmatic notions of meaning and interpretation for wfe-types of L of intensional semantics and the notion of denotation of extensional semanics for wfe-types and constituents of knowledge are formalized. These notions allow formulating a postulate (an axiom of categorial adequacy) from which follow all the most important conditions of the language adequacy, including the above, and a structural one connected with three principles of compositionality. (shrink)

This paper demonstrates that externalism, the claim that entities exist independently of all possible experience, is not merely unjustified or implausible but logically impossible. The argument proceeds through conceptual analysis of what "existence" means, revealing that the concept is necessarily and entirely grounded in experiential access. Any attempt to apply "existence" to what lies beyond all possible experience commits what we term category fraud: using experientially-derived and validated concepts to assert the reality of a domain where those concepts, by definition, (...) cannot be validated or meaningfully applied. This is worse than simple performative contradiction. It is applying semantic tools beyond the scope where they have meaning, like using a centimeter-ruler to measure distances in a realm where spatial extension doesn't exist. Through examination of the complete set of all possible entities, removing only those with any experiential impact whatsoever, we demonstrate that the distinction between "does not exist" and "exists with zero experiential impact ever" collapses via the Principle of Extensionality: sets with identical members are identical sets. The shelf problem crystallizes this: both ontological space (shelf) and the posited entities (trophy) require experiential grounding. You cannot build a real shelf from dream-materials. What remains is not a narrow edge case but recognition that externalism applies experientially-grounded concepts beyond their coherent domain, rendering the proposition semantically meaningless and therefore logically impossible to be true. Building on Experiential Empiricism's foundational axioms (Sergent, n.d.-a), we show that materialism, theism, Platonism, and all variants of externalism share this identical logical flaw. The conclusion is not that we cannot know whether external entities exist, but that the claim itself lacks coherent semantic content. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

There are at least two different concepts that philosophers might target when analyzing causation: a pre-selective notion and a selective notion. This paper argues that these two distinct conceptions have been conflated to date, citing the puzzles of overdetermination, extensionality, and transitivity as evidence. The primary aim of the paper is to help reset the methodological scene concerning analyses of causation.

This paper develops a unified theoretical framework for AI economics integrating computability theory, non-equilibrium thermodynamics, endogenous growth theory, and the philosophy of mind. We resolve the half-century debate between computationalism and anti-computationalism by showing that human brains and physical reality are inherently hypercomputational systems with continuous nonlinear dynamics, capable of solving problems undecidable or intractable for any Turing machine. We introduce the hypercomputational ontological manifold M and dissipative projection operator Pdrop, demonstrating that data is not neutral bits but a topological (...) collapse of continuous hypercomputational dynamics onto discrete symbolic spaces. Modern AI’s success stems not from a breakthrough in Turing computation, but from its ability to compress and approximate hypercomputational solutions encoded in low-entropy data from human minds and physical environments—a process we term extensional alignment. We show that deep reinforcement learning’s strong performance on NP-hard and continuous control problems constitutes a computational outsourcing mechanism: rather than solving problems internally, AI agents delegate computation to the physical environment as an online hypercomputational oracle. We then build a three-tier nested endogenous growth model where growth is driven by AI’s absorption of hypercomputational data, not pure computational power alone. Our model yields three counterintuitive, testable predictions: (1) Pure Turing synthetic data has diminishing marginal value and leads to semiotic collapse via entropy growth in isolated Markov chains; (2) Long-run AI economy growth is bounded by new hypercomputational information from human innovation and physical exploration, not Moore’s Law; (3) Non-algorithmic creative and experimental human labor remains irreplaceable as the ultimate source of economic value. These findings carry profound implications for AI policy, labor economics, and the future of economic growth. (shrink)

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and (...) semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)