A formal model of metaphor is introduced. It models metaphor, first, as an interaction of “frames” according to the frame semantics, and then, as a wave function in Hilbert space. The practical way for a probability distribution and a corresponding wave function to be assigned to a given metaphor in a given language is considered. A series of formal definitions is deduced from this for: “representation”, “reality”, “language”, “ontology”, etc. All are based on Hilbert space. A few statements about a (...) quantum computer are implied: The sodefined reality is inherent and internal to it. It can report a result only “metaphorically”. It will demolish transmitting the result “literally”, i.e. absolutely exactly. A new and different formal definition of metaphor is introduced as a few entangled wave functions corresponding to different “signs” in different language formally defined as above. The change of frames as the change from the one to the other formal definition of metaphor is interpreted as a formal definition of thought. Four areas of cognition are unified as different but isomorphic interpretations of the mathematical model based on Hilbert space. These are: quantum mechanics, frame semantics, formal semantics by means of quantum computer, and the theory of metaphor in linguistics. (shrink)

A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson and Benci et al. have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s “isomorphic” events are not in fact isomorphic, but Howson (...) is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

Alonzo Church famously provided three principal competing criteria for “strict synonymy,” i.e., sameness of semantic content. These are his Alternatives (0), (1), and (2)—numbered in order of increasing course-grainedness of content. On Alternative (2), expressions are deemed strictly synonymous iff they are logically equivalent. This criterion seems hopeless as an account of the objects of propositional attitude. On Alternative (1), expressions are deemed synonymous iff they are λ-convertible. Alternative (1) also evidently conflicts with discourse about the attitudes. On Alternative (0), (...) expressions are deemed strictly synonymous iff they are “synonymously isomorphic” in Church’s sense. On Alternative (0), semantic content is so fine-grained that not even “Romeo loves Juliet” and “Juliet is loved by Romeo” qualify as strictly synonymous. A fourth alternative, here called “Alternative (3),” deems expressions strictly synonymous iff they are co-intensional, i.e., they have the same semantic extension with respect to the same possible worlds. This criterion’s notion of content is even more course grained than that of Alternative (2). Several objections to Alternative (3) are considered. (This article is contained within the author’s more definitive essay, “Singular Concepts”.). (shrink)

This paper develops co-ordinated multiple-domain supervenience relations to model determination and dependence relations between complex entities and their constituents by appealing to R-related pairs and by making use of associated isomorphisms. Supervenience relations are devised for order-sensitive and repetition-sensitive mereologies, for mereological systems that make room for many-many composition relations, as well as for hierarchical mereologies that incorporate compositional and hylomorphic structure. Finally, mappings are provided for theories that consider wholes to be prior to their parts.

What is the proper metaphysics of quantum mechanics? In this dissertation, I approach the question from three different but related angles. First, I suggest that the quantum state can be understood intrinsically as relations holding among regions in ordinary space-time, from which we can recover the wave function uniquely up to an equivalence class (by representation and uniqueness theorems). The intrinsic account eliminates certain conventional elements (e.g. overall phase) in the representation of the quantum state. It also dispenses with first-order (...) quantification over mathematical objects, which goes some way towards making the quantum world safe for a nominalistic metaphysics suggested in Field (1980, 2016). Second, I argue that the fundamental space of the quantum world is the low-dimensional physical space and not the high-dimensional space isomorphic to the ``configuration space.'' My arguments are based on considerations about dynamics, empirical adequacy, and symmetries of the quantum mechanics. Third, I show that, when we consider quantum mechanics in a time-asymmetric universe (with a large entropy gradient), we obtain new theoretical and conceptual possibilities. In such a model, we can use the low-entropy boundary condition known as the Past Hypothesis (Albert, 2000) to pin down a natural initial quantum state of the universe. However, the universal quantum state is not a pure state but a mixed state, represented by a density matrix that is the normalized projection onto the Past Hypothesis subspace. This particular choice has interesting consequences for Humean supervenience, statistical mechanical probabilities, and theoretical unity. (shrink)

In this paper, we propose a regularity theory of causation. The theory aims to be reductive and to align with our pre‐theoretic understanding of the causal relation. We show that our theory can account for a wide range of causal scenarios, including isomorphic scenarios, omissions, and scenarios which suggest that causation is not transitive.

The hole argument purportedly shows that spacetime substantivalism implies a pernicious form of indeterminism. We show that the argument is seductive only because it mistakes a trivial claim (viz. there are isomorphic models) for a significant claim (viz. there are hole isomorphisms). We prove that the latter claim is false -- thereby closing the debate about whether substantivalism implies indeterminism.

This paper introduces and defends a new principle for when a structural equation model is apt for analyzing actual causation. Any such analysis in terms of these models has two components: a recipe for reading claims of actual causation off an apt model, and an articulation of what makes a model apt. The primary focus in the literature has been on the first component. But the problem of structural isomorphs has made the second especially pressing (Hall 2007; Hitchcock 2007a). Those (...) with realist sympathies have reason to resist the standard response to this problem, which introduces a normative parameter into the metaphysics (Hall 2007; Halpern and Hitchcock 2010, 2015; Halpern 2016a; Menzies 2017; Gallow 2021). However, the only alternative solution in the literature leaves central questions unanswered (Blanchard and Schaffer 2017). I propose an independently motivated aptness requirement, Evident Mediation, that provides the missing details and resolves the structural isomorph problem without need for a normative parameter. (shrink)

The first-order temporal logics with □ and ○ of time structures isomorphic to ω (discrete linear time) and trees of ω-segments (linear time with branching gaps) and some of its fragments are compared: the first is not recursively axiomatizable. For the second, a cut-free complete sequent calculus is given, and from this, a resolution system is derived by the method of Maslov.

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s (...) quasi-categoricity theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...) to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction (...) instead simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much (...) more, I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

In this paper, I contest increasingly common "realist" interpretations of Hegel's theory of "the concept" (der Begriff), offering instead a "isomorphic" conception of the relation of concepts and the world. The isomorphism recommended, however, is metaphysically deflationary, for I show how Hegel's conception of conceptual form creates a conceptually internal standard for the adequacy of concepts. No "sideways-on" theory of the concept-world relationship is envisioned. This standard of conceptual adequacy is also "graduated" in that it allows for a lack (...) of fit between concept and world. The possibility for a "maximally isomorphic" fit between concept and world obtains through the teleological realization of concepts, which marks especially the "artificial" world of human culture (law, art, religion, etc.). Some of the most seemingly exaggerated claims Hegel makes about the concept, I contend, can be understood when we consider the significance Hegel ascribes to human making, which is provided for in his conceptual theory. But my framework provides an interpretive key for the way Hegel sees concepts imperfectly realized in the natural world as well. (shrink)

In a previous volume of Ethical Theory & Moral Practice, Melis Erdur defends the provocative claim that postulating a stance-independent ground for morality constitutes a substantive moral mistake that is isomorphic to the substantive moral mistake that many realists attribute to antirealists. In this discussion paper I reconstruct Erdur’s argument and raise two objections to the general framework in which it arises. I close by explaining why rejecting Erdur’s approach doesn’t preclude normative criticism of metaethical theories.

In this paper, I suggest an outline of a new interpretation of core issues in Spinoza’s metaphysics and philosophy of mind. I argue for three major theses. (1) In the first part of the paper I show that the celebrated Spinozistic doctrine commonly termed “the doctrine of parallelism” is in fact a confusion of two separate and independent doctrines of parallelism. Hence, I argue that our current understanding of Spinoza’s metaphysics and philosophy of mind is fundamentally flawed. (2) The clarification (...) and setting apart of the two doctrines will also put us in a position to present my second major thesis and address one of the more interesting and enduring problems in Spinoza’s metaphysics: how can the attribute of thought be, on the one hand, isomorphic with any other attribute, and yet, on the other hand, be isomorphic with God himself, who has infinitely many attributes? In the second part of the paper, I present Spinoza’s solution to this problem. I argue that the number and order of modes is the same in all attributes. Yet, modes of Thought, unlike modes of any other attribute, have an infinitely-faceted internal structure so that one and the same idea represents infinitely many modes by having infinitely many facets (or aspects). (3) This new understanding of the inner structure of ideas in Spinoza will lead us to my third thesis in which I explain and solve another old riddle in Spinoza’s metaphysics: his insistence on the impossibility of the human mind knowing any of God’s infinite attributes other than Thought and Extension. In the third part, I show some of the major ramifications of my new interpretation and respond to some important objections. In my conclusion I discuss the philosophical importance of my interpretation. I explain why Spinoza could not embrace reductive idealism in spite of the preeminence he grants to the attribute of Thought. I argue that Spinoza is a dualist -- not a mind-body dualist, as he is commonly conceived to be, but rather a dualist of Thought and Being. Finally, I suggest that Spinoza’s position on the mind-body issue breaks with the traditional categories and ways of addressing the subject by suggesting a view which grants clear primacy to Thought without accepting any idealist reduction of bodies to thought. (shrink)

A common way to understand memory structures in the cognitive sciences is as a cognitive map​. Cognitive maps are representational systems organized by dimensions shared with physical space. The appeal to these maps begins literally: as an account of how spatial information is represented and used to inform spatial navigation. Invocations of cognitive maps, however, are often more ambitious; cognitive maps are meant to scale up and provide the basis for our more sophisticated memory capacities. The extension is not meant (...) to be metaphorical, but the way in which these richer mental structures are supposed to remain map-like is rarely made explicit. Here we investigate this missing link, asking: how do cognitive maps represent non-spatial information?​ We begin with a survey of foundational work on spatial cognitive maps and then provide a comparative review of alternative, non-spatial representational structures. We then turn to several cutting-edge projects that are engaged in the task of scaling up cognitive maps so as to accommodate non-spatial information: first, on the spatial-isometric approach​ , encoding content that is non-spatial but in some sense isomorphic to spatial content; second, on the ​ abstraction approach​ , encoding content that is an abstraction over first-order spatial information; and third, on the ​ embedding approach​ , embedding non-spatial information within a spatial context, a prominent example being the Method-of-Loci. Putting these cases alongside one another reveals the variety of options available for building cognitive maps, and the distinctive limitations of each. We conclude by reflecting on where these results take us in terms of understanding the place of cognitive maps in memory. (shrink)

I defend a theory of mental representation that satisfies naturalistic constraints. Briefly, we begin by distinguishing (i) what makes something a representation from (ii) given that a thing is a representation, what determines what it represents. Representations are states of biological organisms, so we should expect a unified theoretical framework for explaining both what it is to be a representation as well as what it is to be a heart or a kidney. I follow Millikan in explaining (i) in terms (...) of teleofunction, explicated in terms of natural selection. -/- To explain (ii), we begin by recognizing that representational states do not have content, that is, they are neither true nor false except insofar as they both “point to” or “refer” to something, as well as “say” something regarding whatever it is they are about. To distinguish veridical from false representations, there must be a way for these separate aspects to come apart; hence, we explain (ii) by providing independent theories of what I call f-reference and f-predication (the ‘f’ simply connotes ‘fundamental’, to distinguish these things from their natural language counterparts). -/- Causal theories of representation typically founder on error, or on what Fodor has called the disjunction problem. Resemblance or isomorphism theories typically founder on what I’ve called the non-uniqueness problem, which is that isomorphisms and resemblance are practically unconstrained and so representational content cannot be uniquely determined. These traditional problems provide the motivation for my theory, the structural preservation theory, as follows. F-reference, like reference, is a specific, asymmetric relation, as is causation. F-predication, like predication, is a non-specific relation, as predicates typically apply to many things, just as many relational systems can be isomorphic to any given relational system. Putting these observations together, a promising strategy is to explain f-reference via causal history and f-predication via something like isomorphism between relational systems. -/- This dissertation should be conceptualized as having three parts. After motivating and characterizing the problem in chapter 1, the first part is the negative project, where I review and critique Dretske’s, Fodor’s, and Millikan’s theories in chapters 2-4. Second, I construct my theory about the nature of representation in chapter 5 and defend it from objections in chapter 6. In chapters 7-8, which constitute the third and final part, I address the question of how representation is implemented in biological systems. In chapter 7 I argue that single-cell intracortical recordings taken from awake Macaque monkeys performing a cognitive task provide empirical evidence for structural preservation theory, and in chapter 8 I use the empirical results to illustrate, clarify, and refine the theory. (shrink)

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, (...) neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (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)

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and re-presentable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, (...) and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen. (shrink)

Standard compositionality is the doctrine that the semantic content of a compound expression is a function of the semantic contents of the contentful component expressions. In 1954 Hilary Putnam proposed that standard compositionality be replaced by a stricter version according to which even sentences that are synonymously isomorphic (in the sense of Alonzo Church) are not strictly synonymous unless they have the same logical form. On Putnam’s proposal, the semantic content of a compound expression is a function of: (i) (...) the contentful component expressions; and (ii) the expression’s logical form. Kit Fine recently expanded and modified Putnam’s idea into a sweeping theory in philosophy of language and philosophy of mind. The present paper is a detailed critique of Fine’s “semantic relationism.” Fine’s notion of coordination is explained in terms of the familiar pragmatic phenomenon of recognition. A serious error in Fine’s formal disproof of standard Millianism is exposed. It is demonstrated furthermore that Church’s original criticism of Putnam’s proposal can be extended to Fine’s semantic relationism. Finally, it is also demonstrated that the positive position Fine proffers to supplant standard Millianism is in fact exactly equivalent to standard Millianism, so that Fine’s overall position not only does not displace standard Millianism but is in fact inconsistent. (shrink)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

It is astonishing that we humans are able to have, act on and express moral beliefs. This dissertation aims to provide a better philosophical understanding of why and how this is possible especially when we assume metaethical expressivism. Metaethical expressivism is the combination of expressivism and noncognitivism. Expressivism is the view that the meaning of a sentence is explained by the mental state it is conventionally used to express. Noncognitivism is the view that the mental state expressed by a moral (...) sentence is a noncognitive, desire-like state. The central question of this dissertation is this: "How can metaethical expressivists explain that we are able to express moral beliefs?" The shortest answer this dissertation will give is the following: "by becoming `dispositional expressivists´." Dispositional expressivists hold that moral beliefs are dispositions to be in certain mental states that resemble desires. More precisely, a central thesis of this study is that to hold a moral belief is to be disposed to be in a desire-like, noncognitive state where this state has some specific structure. This dissertation arrives at and justifies this thesis by making a number of steps. It starts with preparing the ground by developing a general theory of what it is for a sentence to express a mental state. Roughly, this theory states that for a sentence to express a mental state is for that sentence to be conventionally used to perform a speech act which counts as sincere only if the user or speaker is in that mental state. This theory of expressing satisfies all the constraints that must be satisfied for it to be compatible with the purposes of metaethical expressivism. The next step is to provide a solution to the most notorious semantic problem confronting metaethical expressivism, the problem of explaining how sentences containing moral words can have compositional meaning and stand in logical relations also known as the Frege-Geach or embedding problem. Building on the research of Mark Schroeder, a new expressivist semantics, called `Attitude Semantics´, will be developed for a first-order language. The motivating idea underlying attitude semantics is that expressivists can avoid the problematic feature of so-called `biforcation´ if they treat the structures of the mental states expressed by moral sentences as well as that expressed by non-moral sentences as being isomorphic. In order to give those structures a philosophically interesting interpretation, it will be suggested that we treat sentences as expressing dispositional states. As a toy example we will treat the sentence `murder is wrong´ as expressing the disposition to be for blaming for murder. After dealing with the Frege-Geach problem, I turn to the metaethical expressivists' project of quasi-realism. Two major goals of this project are to explain how moral sentence can be truth-apt and how the states expressed by them can be regarded as moral beliefs. In order to establish those goals I critically examine a number of theories of truth-aptness by providing counterexamples to them. I then present my own theory which says that a sentence is truth-apt iff and because there is a conventional way to use it to express a belief. What is new about this theory is that it treats expressing a belief as being explanatorily more basic than being truth-apt. Belief, according to this theory, explains truth-aptness, not vice versa. This has the important consequence that it shows that what I call the `orthodox strategy´ of quasi-realists is mistaken and needs to be inverted: expressivists must `earn the right´ to moral belief before they can establish that moral sentences are truth-apt, and not the other way round. The second part of this dissertation is then concerned with the question of what it is to hold a moral belief and how they can figure in rational reasoning. In short: it is concerned with earning the right to moral belief. After discussing the major problems confronting a noncognitivist theory of belief, most importantly what I call the `tightrope problem´, I argue for a dispositional form of noncognitivism about moral belief that combines elements from Eric Schwitzgebel's dispositionalism about belief and Sebastian Köhler's `conceptual role expressivism´. This dispositional noncognitivism perfectly matches and justifies the dispositionalist interpretation of the expressivist semantics developed before. In order to fully earn the right to moral belief, however, metaethical expressivists must also deal with several epistemological objections concerning the rationality of moral belief. The first objection I deal with comes from Derek Baker and is directed against Mark Schroeder's specific development of expressivism. I deal with this objection because my own theory bears close resemblance to Schroeder's theory and so it might be worried that it confronts the same issues. Fortunately, we will see that Baker's objection rest on a number of mistaken assumptions and so neither applies to Schroeder's expressivism nor to the version which I defend in this study. The other epistemological objection to metaethical expressivism I deal with is Cian Dorr's famous wishful-thinking objection. In contrast to previous replies to this objection I attack it by directly questioning the central assumption on which it rests, namely that reasoners can never be justified in basing a factual belief on a desire-like state. I argue that this is false, and that under certain conditions reasoning from desire-like mental states to factual beliefs is justified. My argument shows that even if holding a moral belief is being in a desire-like noncognitive state, basing factual belief on a moral belief can give one an epistemic reason for the factual belief. Hence, even if moral thoughts are wishes, moral thinking is not wishful thinking. I take my arguments to earn the dispositional expressivist the right to moral belief. By the previously defended theory of truth-aptness in terms of belief, this allows the dispositional expressivists to conclude that moral sentences are also truth-apt. This finishes the quasi-realist project of this dissertation. In the last part of this dissertation I switch gears and present two unnoted problems for metaethical expressivists having to do with explaining moral motivation. Both arguments are surprisingly simple. The first one is this: if motivation requires a belief as well as a desire (Humeanism), and moral judgments motivate by themselves (internalism), then moral judgments cannot be desires only. If this argument is sound, it shows that the famous `motivation argument´ not only entails the falsity of cognitivism, but also the falsity of noncognitivism. The second argument is the following: moral judgments when combined with suitable external desires motivate to act (externalism), but if motivation requires belief as well as desire (Humeanism), then moral judgments cannot be desires only. I discuss several ways in which noncognitivists might want to deal with those problems, but leave it as an open question whether those solutions will ultimately be viable. Overall, this dissertation makes serious contributions to the research program of metaethical expressivism. It makes proposals for how to solve a number of notorious problems, such as the Frege-Geach problem in semantics and the wishful-thinking problem in epistemology. Besides this, however, it also raises two new problems for metaethical expressivism in the philosophy of action. For those reasons this dissertation will be of interest for anyone who wants to defend or criticize expressivism in metaethics, and more generally, anyone who wants to better understand our astonishing ability to express moral beliefs. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson (...) is speaking of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

This dissertation begins by addressing the question of when a causal model is apt for deciding questions of actual causation with respect to some target situation. I first provide relevant background about causal models, explain what makes them promising as a tool for analyzing actual causation, and motivate the need for a theory of aptness as part of such an analysis (Chapter 1). I then define what it is for a model on a given interpretation to be accurate of, that (...) is, say only true things about, some target situation. This involves a systematization of various representational principles mentioned and/or discussed throughout the literature into a method of interpretation, which I propose be taken as standard (Chapter 2). Next, I explain and address two reasons for which accuracy as I’ve defined it is insufficient for aptness. The first reason – already discussed in the literature – is the problem of structural isomorphs. In response, I propose the aptness condition of Explicit Partial Mediation (Chapter 3). The second reason – which has yet to be noticed – is the problem of the indeterminacy of accuracy. As I demonstrate, a model is accurate of a target situation only relative to a set of background possibilities – what I call a modal profile. It follows that a model represents a situation only relative to some modal profile or other. I go on to discuss the ramifications of this observation for a theory of actual causation in terms of models. I argue that the relativity be taken at face value and built into our metaphysical account of causation, resulting in a view that I call causal relativism (Chapter 4). I explore one advantage of this view in detail: that the resulting account can defend the principle of strong proportionality against several objections (Chapter 5). Finally, I apply the earlier discussion of aptness to attempts to provide a semantics of counterfactuals in terms of causal models – an interventionist semantics. I show how just as a similarity semantics relies on an opaque notion of similarity, an interventionist semantics relies on an analogous notion of aptness. The challenge of articulating aptness thus undermines the claim that an interventionist semantics avoids representational problems inherent in a similarity semantics (Chapter 6). I close with a recap and suggestions for future research (Chapter 7). -/- . (shrink)

In Nicomachean Ethics V.6 Aristotle contrasts political justice with household justice, paternal justice, and despotic justice. My paper expands upon Aristotle’s sometimes enigmatic remarks about political justice through an examination of his account of justice within the oikia or ‘household’. Understanding political justice requires explicating the concepts of freedom and equality, but for Aristotle, the children and wife within the household are free people even if not citizens, and there exists proportionate equality between a husband and wife. Additionally, Aristotle’s articulation (...) and defence of political justice arises out of his examination of despotic justice in the first book of the Politics. Not only are the polis and the oikia similar insofar as they are associations, but Nicomachean Ethics VIII.9–11 suggests they are even isomorphic with respect to justice and friendship. Thus, in this paper I explore the relationships between father and son, husband and wife, master and slave, and between siblings in order to see what they tell us about Aristotle’s understanding of freedom, equality, and justice. (shrink)

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous lattices and (...) “pointless” topology.

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Expressions are synonymous if they have the same semantic content. Complex expressions are synonymously isomorphic in Alonzo Church’s sense if one is obtainable from the other by a sequence of alphabetic changes of bound variables or replacements of component expressions by syntactically simple synonyms. Synonymous isomorphism provides a very strict criterion for synonymy of sentences. Several eminent philosophers of language hold that synonymous isomorphism is not strict enough. These philosophers hold that ‘Greeks prefer Greeks’ and ‘Greeks prefer Hellenes’ express (...) different propositions even if they are synonymously isomorphic. They hold that the very recurrence (multiple occurrence) of ‘Greeks’ contributes to the proposition expressed something that indicates the very recurrence in question. Kit Fine argues that this thesis, which he labels semantic relationism calls for a radically new conception of semantics. I have argued that the relevant phenomenon is wholly pragmatic, entirely non-semantic. Here I supplement the case with a new argument. No cognition without recognition—or almost none. With this observation, standard Millianism has sufficient resources to confront Frege’s puzzle and related problems without injecting pragmatic phenomena where they do not belong. (shrink)

In this paper I show that the problem of temporary intrinsics and a fundamental philosophical problem concerning the doctrine of the incarnation are isomorphic. To do so, I present the problem of temporary intrinsics, along with five responses to the problem. I then present the fundamental problem for Christology, which I call the problem of natural intrinsics. I present six responses to that problem, all but the last analogous to a response to the problem of temporary intrinsics. My goal (...) is not to argue that any individual response to either problem is correct. Instead, my goal is to present and defend an interesting and unnoticed similarity between two different problems, and to note how work on one problem can help with work on the other. -/- This is a penultimate manuscript, and not a final version, of a forthcoming article in Oxford Studies in Philosophy of Religion. As such, please do not cite this version; cite the official version, due out in OSPR VII in 2015. (shrink)

The common account of the analog vs digital distinction is based on features of physical systems, being related to the usage of continuous vs discrete supports respectively. It is proposed here to alternatively characterize the concepts of analog and digital as related to coding systems, of which a formal definition is given, by suggesting that the distinction refers to the strategy adopted to define the coding function: extensional in digital systems, isomorphic intensional in analog systems. This thesis is supported (...) by examples, in particular of analog systems exploiting discrete supports, and is discussed to explain why digital coding is currently so widespread in technological and social practice. (shrink)

At the close of Philosophy and the Mirror of Nature Richard Rorty lays out a contrast between what he calls 'systematic' and 'edifying' philosophical anthropologies. Whereas the systematic philosopher aims to speak for the ages, the edifying philosopher addresses herself to issues of her day, often by way of shattering conventional idols. Rorty sees these two approaches as mutually exclusive. The aim of this paper is to defend a conception of philosophy as both systematic and edifying in the relevant senses. (...) I first respond to Rorty's argument that Wilfrid Sellars' account of picturing, as an isomorphic relation between the world and states of the central nervous system, involves an illicit 'mind as mirror' metaphor. I then lay out some of the philosophical anthropology that motivates Sellars' account of picturing, and I connect this anthropology to philosophical and scientific work undertaken in Europe and America in the nineteenth and early twentieth centuries. On this basis I argue that we are left with a (systematic) project handed down to us by earlier generations, and that in taking up this project we are in the (edifying) business of creating new ways of thinking about ourselves as part of the natural world. (shrink)

In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic (...) multiverse to be the best framework for mathematical practice. According to UNIFY, an adequate set theory should be foundational, in the sense that it should allow one to represent all the currently accepted mathematical theories. As for MAXIMIZE, this states that any adequate set theory should be as powerful as possible, allowing one to prove as many results and isomorphisms as possible. In a recent paper, Maddy (2017) has argued that this two principle justify ZFC as the best framework for mathematical practice. I argue that, pace Maddy, these two principles justify a multiverse conception of set theory, more precisely, the generic multiverse with a core (GMH). (shrink)

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory. In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this paper clarifies von Neumann's contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the paper is that von Neumann's argument described in his letter to Ulam is the very first proof that (...) spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic. (shrink)

Homo deceptus is a book that brings together new ideas on language, consciousness and physics into a comprehensive theory that unifies science and philosophy in a different kind of Theory of Everything. The subject of how we are to make sense of the world is addressed in a structured and ordered manner, which starts with a recognition that scientific truths are constructed within a linguistic framework. The author argues that an epistemic foundation of natural language must be understood before laying (...) claim to any notion of reality. This foundation begins with Ludwig Wittgenstein’s Tractatus Logico-Philosophicus and the relationship of language to formal logic. Ultimately, we arrive at an answer to the question of why people believe the things they do. This is effectively a modification of Alfred Tarski’s semantic theory of truth. The second major issue addressed is the ‘dreaded’ Hard Problem of Consciousness as first stated by David Chalmers in 1995. The solution is found in the unification of consciousness, information theory and notions of physicalism. The physical world is shown to be an isomorphic representation of the phenomenological conscious experience. New concepts in understanding how language operates help to explain why this relationship has been so difficult to appreciate. The inclusion of concepts from information theory shows how a digital mechanics resolves heretofore conflicting theories in physics, cognitive science and linguistics. Scientific orthodoxy is supported, but viewed in a different light. Mainstream science is not challenged, but findings are interpreted in a manner that unifies consciousness without contradiction. Digital mechanics and formal systems of logic play central roles in combining language, consciousness and the physical world into a unified theory where all can be understood within a single consistent framework. (shrink)

This paper explores the defects in fuzzy (hyper) graphs (as complex (hyper) networks) and extends the fuzzy (hyper) graphs to fuzzy (quasi) superhypergraphs as a new concept.We have modeled the fuzzy superhypergraphs as complex superhypernetworks in order to make a relation between labeled objects in the form of details and generalities. Indeed, the structure of fuzzy (quasi) superhypergraphs collects groups of labeled objects and analyzes them in the form of the part to part of objects, the part of objects to (...) the whole group of objects, and the whole to the whole group of objects at the same time. We have investigated the properties of fuzzy (quasi) superhypergraphs based on any positive real number as valued fuzzy (quasi) superhypergraphs, considering the complement of valued fuzzy (quasi) superhypergraphs, the notation of isomorphism of valued fuzzy (quasi) superhypergraphs based on the permutations, and we have presented the isomorphic conditions of (self complemented) valued fuzzy (quasi) superhypergraphs. The concept of impact membership value of fuzzy (quasi) superhypergraphs is introduced in this study and it is applied in designing the real problem in the real world. Finally, the problem of business superhypernetworks is presented as an application of fuzzy valued quasi superhypergraphs in the real world. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

A rather promising value theory for environmental philosophers combines the well-known fitting attitude (FA) account of value with the rather less well-known account of value as richness. If the value of an entity is proportional to its degree of richness (which has been cashed out in terms of unified complexity and organic unity), then since natural entities, such as species or ecosystems, exhibit varying degrees of richness quite independently of what we happen to feel about them, they also possess differing (...) degrees of mind-independent and subject-independent value. In particular, their value is not dependent on the desires or preferences of humans. The fitting attitudes account of value, at least as it is standardly developed, demands isomorphic evaluative responses on the part of all valuers. In particular, it entails that all valuers should have isomorphic preferences. But this seems absurd. I consider three different strategies with which the fitting attitude theorist can deflect this challenge. The first makes use of an account of non-standard value relations in terms of permissible preference orderings. The second appeals to value appearances and the associated notions of value distance and value perspective. The third involves an account of the ultimate bearers of value as properties, rather than as propositions or states of affairs. These strategies are not all mutually incompatible. While it isn’t possible to combine the first and second strategies, it is possible to combine the first and third strategies, and also to combine the second and the third. (shrink)

In the practical philosophy of the Discours de la Méthode, before the theoretical metaphysics of Part Four and the Meditationes, Descartes gives us an inductive argument that his method, the procedure and cognitive psychology, is veracious at its inception. His evidence, akin to his Scholastic predecessors, is God, a maximally perfect being, established an ontological foundation for knowledge such that reason and nature are isomorphic. Further, the method, he tells us, is a functional definition of human reason; that is, (...) like other rationalists during this period, he holds the structure of reason maps onto the world. The evidence for this thesis is given in what I call the groundwork to Descartes’ philosophical system, essentially the first half of the Discours, where, through a series of examples in the preamble of Part Two, he, step-by-step, ascends from the perfection of artifacts through the imposition of reason to the perfection of a constituent’s use of her cognitive faculties, to God perfecting and ordering reality. Finally, he descends, establishing the structure of human reason, which undergirds and entails the procedure of the method. (shrink)

In Philosophical Investigations, Wittgenstein proposed the notion of meaning that accounts for the large variety of contexts in which we apply the term “meaning”. This paper agreement with the manner in which Wittgenstein enhance his conception of meaning emphasizing his methodology of observation and description of particular cases. By applying a descriptive approach, Wittgenstein demonstrated that meaning of the term does not reside in physical or mental objects as well as in its correlations. As a result of contrasting denotative theory (...) as well as the correspondence theory of meaning and recognizing the inadequacy of the accounts of meaning, he has proposed earlier in his Tractatus. But in later work, he has suggested that only one conception of meaning which could not be invalidated, at least for a large class of cases. This is none other than the notion of meaning which is regarded as public in nature. Consequently, the meaning of a term is not its denotation but its “use” in the language. Hence, by upholding the slogan i.e., “meaning is use” here I want to illustrate the supremacy of use over “naming” concerning to Wittgenstein’s Tractatus and Philosophical Investigations. (shrink)

Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary (...) quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π. (shrink)

Along with “epoché” or his “reductions”, Husserl’s “noema” and “noesis”, being neologisms invented by him, are main concepts in phenomenology able to represent its originality. Following the trace of a recent paper (Penchev 2021 July 23), its formal and philosophical approach is extended to both correlative notions, in the present article. They are able to reveal the genesis of the world from consciousness in a transcendental method relevant to Husserl, but furthermore described formally as a process of how subjective temporality (...) appears being isomorphic to objective temporality of the “world by itself” (an abstraction meaning it out of consciousness or transcendental consciousness): thus, it shares the same mathematical structure, which is embodied in the physical process of decoherence by the physical quantity of quantum information. The temporal world is able to appear naturally (rather as a ridiculous effect of the mythical “Big Bang”). The same process translated by formal and mathematical tools as interpreted in terms of “noema”, “noesis”, or transcendental consciousness is isomorphic to how “Self” (including in an individual and psychological sense) appears in virtue of transcendental consciousness. (shrink)

In the real word, in most cases, everything (an attribute, event, proposition, theory, idea, person, object, action, etc.) is evaluated in general by many sources (called experts), not only one. The more sources evaluate a subject, the better accurate result (after fusioning all evaluations). That’s why, in this paper, we straightforwardly extend the Refined Neutrosophic Set to the MultiNeutrosophic Set, and we show that the last two are isomorphic. A MultiNeutrosophic Set is a Neutrosophic Set whose all elements’ degrees (...) of truth/indeterminacy/falsehood are evaluated by many (Multi) sources. Afterwards, we introduce a total order on the set of n-plets of the form (p, r, s), we build the operators on the (p, r, s)-plets, and show several applications of the MultiNeutrosophic Sets. Several particular cases of the MultiNeutrosophic Sets are presented: such as MultiFuzzy Set, MultiIntuitionistic Fuzzy Set, MultiPicture Fuzzy Set, and other Multi(Fuzzy Extension) Set. (shrink)

The paper takes a novel approach to a classic problem—Hempel’s Raven Paradox. A standard approach to it supposes the solution to consist in bringing our inductive logic into “reflective equilibrium” with our intuitive judgements about which inductive inferences we should license. This approach leaves the intuitions as a kind of black box and takes it on faith that, whatever the structure of the intuitions inside that box might be, it is one for which we can construct an isomorphic formal (...) edifice, a system of inductive logic. By popping open the box we can see whether that faith is misplaced. I aim, therefore, to characterize our pre-theoretical, intuitive understanding of generalizations like “ravens are black” and argue that, intuitively, we take them to mean, for instance: “ravens are black by some indeterminate yet characteristic means.” I motivate and explicate this formulation and bring it to bear on Hempel’s Problem. (shrink)

We study the algorithmic complexity of isomorphic embeddings between computable structures.

The alien in science fiction has not often been seen as part of an imperial colonial discourse. By examining John W. Campbell’s founding golden age SF text, “Who Goes There?” (1938), this paper explores the ways in which the alien adheres to an invisible mythos of whiteness that has come to be seen through a colonizing logic as isomorphic with the human. Campbell’s alien-monster comes to disseminate and invade both self and world and as such serves as an interrogation (...) of what whites have done through colonization. It is thus part and parcel of imperial domination and discourse and appears as the very nightmare of whiteness in the form of its liminal and estranged shadow side. Part of what has made Campbell’s text so influential is that it offers a new type of alien invasion in the figure of “contagion,” which speaks “to the transition from colonial to postcolonial visions of modernity and its attendant catastrophes” (Rieder, 124), and which can be further examined as a race metaphor in American SF—indeed, as the white man’s fear of racial mixing that has a long and dehumanizing history. Through its threat of mixture, I read the alien as a creolizing figure that both troubles and undoes the white/black, human/nonhuman binary in science fiction, which I also read as being a creolizing, i.e., hybrid and plastic, genre. (shrink)

According to the received view of scientific theories, a scientific theory is an axiomatic-deductive linguistic structure which must include some set of guidelines (“correspondence rules”) for interpreting its theoretical terms with reference to the world of observable phenomena. According to the semantic view, a scientific theory need not be formulated as an axiomatic-deductive structure with correspondence rules, but need only specify models which are said to be “isomorphic” with actual phenomenal systems. In this paper, I consider both the received (...) and semantic views as they bear on the issue of how a theory relates to the world (Section 1). Then I offer a critique of some arguments frequently put forth in support of the semantic view (Section 2). Finally, I suggest a more convincing “meta-methodological” argument (based on the thought of Bernard Lonergan) in favor of the semantic view (Section 3). (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.