The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case (...) of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. (...) A tableau system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences. (shrink)

One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial (...) languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

One well known problem regarding quantifiers, in particular the 1st order quantifiers, is connected with their syntactic categories and denotations.The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for (...) categorial languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these (...) classes of trees in terms of the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analysing first order definable and undefinable paths in trees. (shrink)

The higher-order thought (HOT) theory of consciousness is a reductive representational theory of consciousness which says that what makes a mental state conscious is that there is a suitable HOT directed at that mental state. Although it seems that any neural realization of the theory must be somewhat widely distributed in the brain, it remains unclear just how widely distributed it needs to be. In section I, I provide some background and define some key terms. In section II, I (...) argue against the view that HOT theory should treat first-order (i.e. world-directed) conscious states as requiring prefrontal cortical activity though it is reasonable to suppose that conscious states are realized in the brain. In section III, I then explore some of the key background metaphysical issues involved in understanding the nature of consciousness, such as the debate between realism and idealism as well as the prospects for solving the so-called “hard problem” of consciousness. Some of the differences in question often mirror the traditional differences between Western and Eastern perspectives on the nature of consciousness. Overall, I argue that some form of realism and physicalism is more plausible than the opposing views. I also argue that materialists (and especially HOT theorists) can offer plausible replies to the hard problem. (shrink)

We propose a generalization of Sahlqvist formulas to polyadic modal languages by representing such languages in a combinatorial PDL style and thus, in particular, developing what we believe to be the right syntactic approach to Sahlqvist formulas at all. The class of polyadic Sahlqvist formulas PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due (...) in large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Most descriptions of higher-order vagueness in terms of traditional modal logic generate so-called higher-order vagueness paradoxes. The one that doesn't is problematic otherwise. Consequently, the present trend is toward more complex, non-standard theories. However, there is no need for this.In this paper I introduce a theory of higher-order vagueness that is paradox-free and can be expressed in the first-order extension of a normal modal system that is complete with respect to single-domain Kripke-frame semantics. This is (...) the system QS4M+BF+FIN. It corresponds to the class of transitive, reflexive and final frames. With borderlineness defined logically as usual, it then follows that something is borderline precisely when it is higher-order borderline, and that a predicate is vague precisely when it is higher-order vague.Like Williamson's, the theory proposed here has no clear borderline cases in Sorites sequences. I argue that objections that there must be clear borderline cases ensue from the confusion of two notions of borderlineness—one associated with genuine higher-order vagueness, the other employed to sort objects into categories—and that the higher-order vagueness paradoxes result from superimposing the second notion onto the first. Lastly, I address some further potential objections. (shrink)

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view \(O\) and wants to interpret a formal language \(L\) , a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for \(L\) in such a way that its ontological commitment will be in perfect accordance with \(O\) . Basically, that is what I call constructing formal semantics from (...) an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. From those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, showing thus that we can perfectly quantify over properties and relations while being ontologically committed only to individuals. I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and thus why endorsing an ontological view should have an impact on the kind of logic one should use. (shrink)

This paper deals with the logical form of quantified sentences. Its purpose is to elucidate one plausible sense in which quantified sentences can adequately be represented in the language of first-order logic. Section 1 introduces some basic notions drawn from general quantification theory. Section 2 outlines a crucial assumption, namely, that logical form is a matter of truth-conditions. Section 3 shows how the truth-conditions of quantified sentences can be represented in the language of first-order logic consistently (...) with some established undefinability results. Section 4 sketches an account of vague quantifier expressions along the lines suggested. Finally, section 5 addresses the vexed issue of logicality. (shrink)

In this paper, we define for the first time three neutrosophic actions and their properties. We then introduce the prevalence order on {T, I, F} with respect to a given neutrosophic operator “o”, which may be subjective - as defined by the neutrosophic experts; and the refinement of neutrosophic entities <A>, <neutA>, and <antiA> . Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, and we define the refinement neutrosophic (...) literal space. (shrink)

In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize (...) main ideas and results, and outline further research perspectives. (shrink)

This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d,α ,[ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,~). The relation ~ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,~) come along with naturally defined (...) maps h and s that define a Galois connection on the power set PW of W. These maps can be used to define two distinct boundary operators bd and BD on W. The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection. (shrink)

The concept of Generative Artificial Intelligence (GenAI) is ubiquitous in the public and semi-technical domain, yet rarely defined precisely. We clarify main concepts that are usually discussed in connection to GenAI and argue that one ought to distinguish between the technical and the public discourse. In order to show its complex development and associated conceptual ambiguities, we offer a historical-systematic reconstruction of GenAI and explicitly discuss two exemplary cases: the generative status of the Large Language Model BERT and the (...) differences between protein structure predictions from AlphaFold 2 and 3. Our analysis shows that there is no unique and unambiguous definition of GenAI based on a purely technical account of the term. Following this conclusion, we argue that the public discourse is not simply a less complex way of speaking, but instead transcends its technical basis. As a means to structure this newly emerging discussion landscape we introduce a non-exhaustive list of four central aspects of GenAI: (multi-)modality, interaction, flexibility, and productivity. These dimensions constitute a first step towards defining GenAI beyond its technical basis. (shrink)

Contemporary semantic description of logic is based on the ontology of all possible interpretations, an insufficiently clear metaphysical concept. In this article, logic is described as the internal organization of language. Logical concepts -- logical constants, logical truths, and logical consequence -- are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential (...) quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language. (shrink)

The conflation of two fundamentally distinct issues has generated serious confusion in the philosophical and biological literature concerning the units of selection. The question of how a unit of selection of defined, theoretically, is rarely distinguished from the question of how to determine the empirical accuracy of claims--either specific or general--concerning which unit(s) is undergoing selection processes. In this paper, I begin by refining a definition of the unit of selection, first presented in the philosophical literature by William Wimsatt, (...) which is grounded in the structure of natural selection models. I then explore the implications of this structural definition for empirical evaluation of claims about units of selection. I consider criticisms of this view presented by Elliott Sober--criticisms taken by some (for example, Mayo and Gilinsky 1987) to provide definitive damage to the structuralist account. I shall show that Sober has misinterpreted the structuralist views; he knocks down a straw man in order to motivate his own causal account. Furthermore, I shall argue, Sober's causal account is dependent on the structuralist account that he rejects. I conclude by indicating how the refined structural definition can clarify which sorts of empirical evidence could be brought to bear on a controversial case involving units of selection. (shrink)

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity are in play (...) (Williamson-style "luminosity" or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys axiom 4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone axiom 4 (by adopting what elsewhere we call columnar higher-order vagueness), and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject axiom 4. Third, we rebut a number of arguments that have been produced by opponents of axiom 4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)

Timothy Williamson has argued that in the debate on modal ontology, the familiar distinction between actualism and possibilism should be replaced by a distinction between positions he calls contingentism and necessitism. He has also argued in favor of necessitism, using results on quantified modal logic with plurally interpreted second-order quantifiers showing that necessitists can draw distinctions contingentists cannot draw. Some of these results are similar to well-known results on the relative expressivity of quantified modal logics with so-called inner and (...) outer quantifiers. The present paper deals with these issues in the context of quantified modal logics with generalized quantifiers. Its main aim is to establish two results for such a logic: Firstly, contingentists can draw the distinctions necessitists can draw if and only if the logic with inner quantifiers is at least as expressive as the logic with outer quantifiers, and necessitists can draw the distinctions contingentists can draw if and only if the logic with outer quantifiers is at least as expressive as the logic with inner quantifiers. Secondly, the former two items are the case if and only if all of the generalized quantifiers are first-order definable, and the latter two items are the case if and only if first-order logic with these generalized quantifiers relativizes. (shrink)

Dretske makes arguments in which he suggests three levels of the intentionality of knowledge: (1) a low level belonging to law-like causal relationships between physical properties, (2) a middle level defined in terms of the intensionality of sentences describing knowledge of these properties, and (3) a highest level of human cognition. Acknowledging the need to explain humans’ analytic knowledge, however, he proposes that we know a proposition P analytically when we know that P entails Q, even though P and Q (...) are not identical. -/- I explore examples of deduction involving properties of every-day life, such as being a bachelor, being a stay-at-home mom, and being a consultant. I argue that to make these common sense inferences, the average person has to be aware of both analytical entailments and propositional entailments. While the former simply is knowing that if Emily is a stay-at-home mom, then Emily is female, for example, the latter requires us to be conscious of the (often complex) logical structure given to our beliefs by the deductive connectives AND, NOT, and OR. From an empirical standpoint, there are two general possibilities for reductive accounts of these kinds of awareness. One would be a counterfactually defined set of causal criteria, such as Dretske’s. The other would involve statistical patterns of synaptic connections between neural phenomena, whose firings represent our concepts for AND, NOT, and OR. I argue that neither of these reductive explanations is viable; and that in order to explain the success of our inferences, we must appeal to two kinds of higher cognitive intentionality. First, we usually have some conscious awareness of analytical and propositional entailments when we engage in common sense-based deduction. Second, we often have a conscious intention to think deductively during these inferences, which drives our success. (shrink)

We introduce a new product bilattice con- struction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lat- tice factors. Finally, we show how (...) to employ our product construction to define first-order definable classes of bi- lattices corresponding to any first-order definable subclass of residuated lattices. (shrink)

Arriving at definitions in philosophy is as time-honored as it is controversial. Although learned reflection in the west about sport goes back at least to the time of ancient Greece, the sub-discipline of the philosophy of sport emerged in the world of Anglophone analytic philosophy in the 1970s. Shawn Klein’s edited volume, Defining Sport: Conceptions and Borderlines, is both the fruit of and a valuable contribution to such an emerging field (indeed, it is the first book-length study of its (...) topic within philosophy of sport). Although Huizinga had sought to define the phenomenon of play very broadly in his Homo Ludens (1938), investigation of the overlapping questions of what is sport? what is a game?, and what is play? were central to the sub-discipline at its inception. Foundational was the work of Bernard Suits, whose Grasshopper: Games, Life, and Utopia (1978) sought to refute Wittgenstein’s claim that the notion of game was indefinable. In order to refute Wittgenstein, Suits sought to establish the conditions or properties both necessary and sufficient of the concepts of game and sport such that all instances of those concepts share such properties. (shrink)

Will I die? As a hypothesis, in my natural scientific understanding, the psyche, is nothing more than, and exclusively just some states of my living brain – I will die as a result of his death. -/- In presented answer, psyche – itself own immediate reality itself, that is – undoubted. -/- This work was performed in reality “in the first person” (“subjective reality”, “phenomenal consciousness”). To realize, how, what it is the reality of the “in the first (...) person” let’s imagine ourselves as an experimenter, who does not replace reality with reasoning about it. – Now for us, as experimenters, how (what it is) psychic is an immediate and undoubted reality. And the awareness of “I am” is a direct reality, and not knowledge about it, which is always doubtful. That is, in psychic reality, nothing prevents us from answering undoubtedly and definitively the question, asked by this psychic reality itself about itself. The author acts from the position of “being in reality – not going beyond its limits, for example, into philosophy” (philosophy, including methodology, is the replacement of reality itself with always dubious reasoning about reality) – explores phenomenal consciousness (“psychic”) directly and as an immediate reality. That is, this work is the mastery of psychic reality through subjective-psychical experimentation – don’t philosophize, but experiment! -/- For example, Descartes, introspectively experimenting in his own psyche reality with his own psyche, informed us about the immutability of the “mind” and proposed to understand this immutability as our own immortality: “the mind does not represent any combination of accidents, it is a pure substance, and although all its accidents are subject to change, it understands some things, it desires others or feels others, etc. – nevertheless, in itself it does not change; and as for the human body, it changes, if only because they are subject to a change in the shape of some of its parts. It follows from this that the body perishes very easily, while the mind, by its very nature, is immortal” (Декарт Р., p. 13). Note that, Descartes did not realize that he was immortal, but logically deduced his own immortality from the immutability of the mind. However, Descartes’ unchanging “mind” itself is such that there is something to die. And in my natural scientific understanding, the immutability of the mind does not imply its immortality – it will die without changing in its subjective self-perception, while instantly fading away as a result of brain death. Psyche (psychical) – a designation of the fact that this is one’s own immediate reality (that is, undoubtedly) – how (what) I myself am immediately, undoubtedly, and that immediate reality of mine, how and through which I myself and everything in general for me there is, and which is exclusively the only one for me. It should be noted that while I am unconscious (brain injury or dreamless sleep), there is nothing at all. -/- But this does not mean that “I”, having completely gone into oblivion, I'll die. This does not exclude that, I will disappear completely into oblivion and does not exist, but (at the same time) “I” – am. So, let’s define immortality as the impossibility of dying because there is nothing to die (psychic, but is such that there is nothing to die – is immortal) and will make a conclusion important for this work: if I am, as psyche, exclusively only is such that there is something to die and nothing else, then, therefore, my death is not excluded. But am I like that? The idea of this work is that I, as a psyche can to manifest (reveal) and realize myself as such (I such), that there is nothing to die, but (at the same time) I – am. I will realize of myself as immortal – such that to die (to decease) nothing. In other words, if I will approach with a bias and, acting purposefully, will reveal the subjective reality “I (psyche) such is that to die (to decease) nothing, but I – am”, that is – I am immortal, that the hypothesis “I will die” will falsified and excluded. -/- To realize this in the work, it is proposed to use the subject-object model, shift the boundary of “I” to the pole of the subject and become objectless, manifesting (revealing) oneself as such that there is nothing to die. Therefore, the aim (goal) of the work – to realize: I – am, but such that there is to die (decease) nothing. To become objectless and wondering about the possibility to dying, as a result, or I will exclude the immortality of the manifested state, or I will manifest (will reveal) immortal undoubtedly and definitively. -/- At the same time, I, as a subject, am “a necessary pole of subject-object relations. … An object is what the activity of … a subject is directed at. … In fact, everything that exists can become an object. … it is important to keep in mind the fundamental fact that the object is always outside the subject, does not merge with it. This externality also takes place when the subject deals with the states of his own consciousness, his Self ... it is important to emphasize that the relation of the subject and the object is not the relation of two different worlds, but only two poles as part of some unity” (Лекторский В.А., p. 6-7). That is, everything that I find existing in myself, this (such) is as an object. “The subject always refers to all “objects”, but it can never become an object for itself” (Титлин Л.И., p. 23). “… you will not be able to see the seeing vision ...” (Титлин Л.И., p. 40), and in this model this implies the “limit” of the subject. As the limit of the subject, I am “objectless”. And, therefore, in its own pole the subject is “ideal” – in its (own) limit (as its own limit, its own limit), I do not exist for myself (non-existent). -/- Using the “fact of the mobility of the boundaries of the subject” (for example, Тхостов А.Ш., p. 16), I have the opportunity to “pull” the boundary of the “I” into myself and become my own limit. At the same time, it is necessary to realize and accept that “inside” the limit of the subject and behind – “beyond of the subject”, and in general the very “inside” and “behind” in this version – unreality. In such a subject limit, the required paradoxical subjective state is assumed – I exist and see, but at the same time I am such that I do not exist for myself, that is, I am, but (subjectively such that) there is nothing to die (“immortality”). -/- Accordingly, the following tasks are indicated in the experiment: 1. In the object-subject sphere of visual perception, shift (and shift) the boundary of “I” into oneself –» to run into one’s own limit and become one’s own limit (without thickness: without depth, without “from within” and without reverse side) –» 2. Having become your own limit, which in its own limit is, but for myself I do not exist (non-existent), realize: • Myself – I am and this is real; • Will I die? – or there is nothing to die, which of the options is to me, as a psyche (as itself the psychic “I”), is applicable: • or I am such that it is not impossible that I will die, and the realized state is not immortal; • or “I” am such that “I” – am (is reality), but such that myself “I” to die nothing, that is, my own immortality – I am immortal; • Is this real? -/- Instruction for reproducing the target result: sequence of actions and results in a subjective-psychic experiment (hereinafter referred to as SPE) -/- At a distance of 25-40 cm from the wall (so that the wall occupies the entire field of view or its overwhelming part), • Here and now, focus on your face: “I am” – reality (and not is the knowing about it). • In yourself step back and look out of yourself at this wall. • Without moving your head, cover the field of vision with your attention as wide as possible and at the same time perceive everything at once (all this) as a single whole. • Perceive a high and wide wall (option: look up at an angle of 135-150 degrees in front of you) in such a way that the space (volume) surveyed by you without turning your head is a closed asymmetrical “lens” • limited in a circle in the periphery of the field of view, • convex in you; You – the beholder, rearwards concave in self (“subject”) – which one of you are watching? • Without moving your head, perceive the field of view as a single whole –» • Into yourself “draw in” –» “backing away” –» step back from everything –» • lean back into yourself –» up to dead-end limit –» • become a dead end itself – a concave (reflector-like) limit-plane. • You – into yourself – backward concave – a perceiving visual screen (limit-dead end), plane –» • lean back and press –» • something in general somehow inside the dead-end plane and behind the dead-end plane, as well as “inside” and “behind”, “behind the dead-end plane” – unreality; • myself – a dead-end plane without thickness (depth), without “inside”, without a reverse side for oneself. • –» That is, as a psychic reality (what is it psychically) will I die? –» or • Something there is, and that is I will die (it is possible); or • I am such that there is nothing to die, but I – am. or • Subjective-psychic experiment is incomplete… • –» Realize that this is reality (realize the reality of it) and emphasize (answer). -/- As a result, “I” am, but there is nothing to die/not to die, that is, the hypothesis “I will die” is unrealistic. -/- Remarks -/- • Each completed SPE is completed with the target result: “I am such that there is nothing to die, but I – am”. And the that what I will die (not excluded) does not correspond to this, that is unreality and is excluded by this. • But I am not such that I will not die. The concepts “I will die/ I will not die” to me, as a psyche are not applicable. That is, I, as a psyche, is immortal, but I am not eternal (not eternal – endlessly lasting, such that there is something to last) – to last, to be eternal and to end there is nothing. This is “immortality”. • In the target result of this work, I – am, but such that there is nothing to die/not die (“immortality”). And such a “I” is not the Atman of the Hindu schools of Nyaya, Vaisheshika and Purva Mimamsa, which E.A. Torchinov. Since the “I” – psychic, and not a substance with the psyche as an attributes. • According to illusionism, what I am as a result of the experiment given in the work is an illusion. And the impression of immortality is a product of the brain. But in reality I am such that I will die (it is not excluded). However, it is necessary to realize that illusionism is a detached reasoning about the “I”, but not is the self itself. And to what I am (as psiche), reasoning and questions about the reality of the “I” (for example, question: am I really like this, what am I like as psyche?) and other expressing illusionism do not correspond. That is, subjectively (psychic, in the psychic reality of the “first person”, in first person perspective), the illusionism of the “I” is unreality. Subjectively (psychic, in the psychic reality of the “first person”, in first person perspective), this is undeniable and final. • Reasoning about what such a “I” is in this (mine) or not in this brain, as well as (for example, holding a real skull in my hands), questions like “how (or, but now where is it here) such “I” – how is it after brain death?” – To the fact that is such an “I” these questions do not correspond and is falsified by this. • After being unconscious, the impression remains that I'm is not and that there is nothing at all. That is, my “unconsciousness” is psychic indistinguishable from my non-existence. However, science has not shown, but and it is not excluded that before and after I exist physically (there is something to die), I still or I no longer exist, there is still or nothing to die as a soul, substance, or otherwise. But what I am in the target result of this work is what I – am, and not “died and already now or still unreality” and I am such that there is nothing to die, this is – reality. That is, I will die? – there is nothing (to die), but I – am (reality), even if I am not (don’t exist). Or I do not exist as something. • I am such that there is nothing to be and die, but I am. And subjectively (psychic, in the psychic reality of the “first person”, in first person perspective), this immortality is – reality. But this reality eludes my rational explanation. -/- Conclusion -/- Will I die? – In the target result the I am, but at the same time I am such that there is nothing to die. That is, I am immortal, but at the same time, I am not eternal (“I” not something eternal (endlessly continuing)) – I am such that to be eternal there is nothing. And, if we accept the subjective (psychic, “first person”, in first person perspective) reality without trying to understand it rationally, then I am exactly like that – I realize that this is my undoubted nature. That is, “I will die / I will not die” is an unreality. Subjectively (psychic, in the psychic reality of the “first person”, in first person perspective), it is undoubted and final. Despite the unscientific nature, this work is a way of mastering reality. The author would be grateful for the identified and shown unreality of the fact that the “I” – am, but such that there is nothing to die, and ideally, the awareness: “I will die (not excluded).” -/- Bibliography -/- Декарт Р., Размышления о первой философии //Декарт Р. Соч.: В 2 т. Т. 2. М., 1994. Лекторский В.А. Субъект в истории философии: проблемы и достижения // Методология и история психологии. 2010. Том 5. Выпуск 1. С. 5-18. Титлин Л.И. Проблема субъекта в индийской философии («Пудгалавинишчая» Васубандху) [Текст] / [Л. И. Титлин]; Федеральное государственное бюджетное учреждение науки Институт философии Российской академии наук. – Москва: ОнтоПринт: Сделано-Сказано, 2018. – 361, [1] с. Тхостов A. Ш. Культурно-историческая патопсихология: монография, – М.: Канон+ РООИ «Реабилитация», 2020. – 320 с. Торчинов Е.А. «Словарик индуизма» (Дата написания: 2003; дата файла: 15.06.2007). Торчинов Е.А. Пути философии Востока и Запада: познание запредельного — СПб.: «Азбука-классика», «Петербургское Востоковедение», 2005. — 480 с. ISBN 5-352-01371-5 (Азбука-классика) ISBN 5-85803-258-3 (Петербургское Востоковедение). -/- Скончаюсь? – я бессмертен (как осознать бессмертие «от первого лица») -/- Скончаюсь? В качестве гипотезы в моем естественно-научном понимании психическое (каково психически), это не более чем, и исключительно только всего лишь некоторые состояния моего живого мозга – скончаюсь в результате его смерти. -/- В представленном ответе психическое – сама собственная непосредственная реальность, то есть, то, каково психически, это – несомненно. -/- Данная работа выполнена в реальности «первого лица» («субъективная реальность», «феноменальное сознание»). Чтобы осознать то, как, каково это, представим себя экспериментатором, который не подменяет реальность рассуждениями о ней. – Теперь для нас, как экспериментаторов, то, как (каково) психически, это непосредственная и несомненная реальность. И осознание «я есть», это непосредственная реальность, а не знание об этом, которое всегда сомнительно. То есть, в психической реальности, на вопрос, заданный самой этой психической реальностью самой себе о себе самой, нам ничто не запрещает ответить несомненно и окончательно. Автор действует с позиции «быть в реальности – не выходить за ее пределы, например, в философию» (философия, включая методологию, это подмена самой реальности всегда сомнительными рассуждениями о реальности) – исследует феноменальное сознание («психическое») непосредственно и в качестве непосредственной реальности. То есть, данная работа, это овладение психической реальностью посредством субъективно-психического экспериментирования – не философствуй, а экспериментируй! -/- Например Декарт, интроспективно экспериментируя в собственной психической реальности с собственным психическим, сообщил нам о неизменности «ума» и предложил эту неизменность понять в качестве нашего собственного бессмертия: «ум не представляет какого-то соединения акциденций, являет собой чистую субстанцию и, хотя все его акциденции подвержены изменению — он то понимает какие-то вещи, то желает другие или чувствует третьи и т.д.,— тем не менее сам по себе он не изменяется; а что касается тела человека, то оно изменяется хотя бы уже потому, что подвержены изменению формы некоторых его частей. Из этого следует, что тело весьма легко погибает, ум же по самой природе своей бессмертен» /Декарт Р., стр. 13. Заметим, Декарт не осознавал себя бессмертным, а логически вывел собственное бессмертие из неизменности ума. Однако, неизменный «ум» Декарта таков, что скончаться есть чему. А в моем естественно-научном понимании из неизменности ума не следует его бессмертие – он, не изменяясь в своем субъективном само-восприятии, при этом мгновенно угаснув в результате смерти мозга, скончается. Психически (психическое) – обозначение того, что это сама собственная непосредственная реальность (то есть – несомненно) – то, как (каково) я сам непосредственно, несомненно, и та моя непосредственная реальность, как и посредством чего я сам и всё вообще для меня есть, и которое для меня исключительно единственное. Следует заметить, что пока бываю без сознания (травма мозга или сон без сновидений), ничего и никак нет вообще. -/- Но это не значит, что «я» ушедши в небытие окончательно – скончаюсь. Этим не исключено, что таковой ушедший в небытие «я» не существую, но (при этом) есть. Итак, определим бессмертие, как невозможность скончаться потому, что скончаться нечему (например, если «умерший» психически есть, но уже таков, что скончаться больше нечему, то теперь он бессмертен) и сделаем важный для данной работы вывод: если психически я исключительно только таков, что скончаться есть чему и ни каков иначе, то, следовательно, то, что скончаюсь, это не исключено. Но таков ли я? Идея данной работы в том, что мне удастся проявить (выявить) и осознать себя таковым (я таков), что скончаться нечему, но (при этом) я есть. Иначе выражаясь, если я подойду предвзято и действуя целеустремленно, выявлю субъективную реальность «я есть, но таков, что скончаться нечему» – то смерть мне, как психическому несвойственна, то есть, я бессмертен – гипотеза возможности субъективно скончаться («я скончаюсь») будет фальсифицирована и исключена. -/- Чтобы это реализовать в работе предлагается воспользоваться субъект-объектной моделью, сдвинуть границу «я» в полюс субъекта и стать безобъектным, проявившись (выявившись) собой таковым, что скончаться нечему. Следовательно, цель работы – осознать: я таков, что есть, но скончаться нечему. Стать безобъектным и задаваясь вопросом о возможности скончаться, или исключу бессмертие проявленного состояния, или проявлюсь (выявлюсь) бессмертным несомненно и окончательно. -/- При этом, я, как субъект, являюсь «необходимым полюсом субъектно-объектных отношений. … Объект – то, на что направлена активность … субъекта. … В действительности объектом может стать все, что существует. … важно иметь в виду тот принципиальный факт, что объект всегда внеположен субъекту, не сливается с ним. Эта внеположность имеет место и тогда, когда субъект имеет дело с состояниями собственного сознания, своим Я … важно подчеркнуть, что отношение субъекта и объекта – это не отношение двух разных миров, а лишь двух полюсов в составе некоторого единства» /Лекторский В.А., стр. 6-7. То есть, всё, что нахожу в себе существующим, это (таковое) есть в качестве объекта. «Субъект – всегда относится ко всем «объектам», но он никогда не может стать объектом для самого себя» /Титлин Л.И., стр. 23. Иначе это выражая – «ты не сможешь увидеть видящего видение…» /Титлин Л.И., стр. 40, и в данной модели таковое предполагает «предел» субъекта. Как предел субъекта я «безобъектен». То есть, «идеален» – своим (собственным) пределом (в качестве собственного предела) самим для себя не существую (несуществующий). -/- Используя «факт подвижности границ субъекта» /например, Тхостов А.Ш., стр. 16, я имею возможность границу «я» в себя-назад «втянуть» и стать собственным пределом. При этом необходимо осознать и принять, что внутри предела субъекта и сзади – «за пределом субъекта», и вообще само «внутри» и «сзади» в этой модели – нереальность. Таким субъектным пределом предполагается требуемое парадоксальное субъективное состояние – я есть и вижу, но при этом я таков, что для себя не существую, то есть, я есть, но (психически таков, что) умереть нечему («бессмертие»). -/- Соответственно, в эксперименте обозначаются следующие задачи: 1. В объект-субъектной сфере зрительного восприятия сдвинуть (и сдвинув) границу «я» в себя-назад –» упереться в собственный предел и стать самим собственным пределом (без толщены: без глубины, без «внутри» и без обратной стороны) –» 2. Став собственным пределом, которым есть, но для себя не существую (несуществующий), осознать: • Себя – я есть и это реально; • Скончаюсь? – есть ли чему скончаться или скончаться нечему, какой из вариантов мне (как психическому «я») свойственен: • или я таков, что то, что скончаюсь не исключено, и реализованное состояние не бессмертно; • или я есть, но таков, что скончаться нечему, то есть, собственное бессмертие – я бессмертен; • Реальность ли это. -/- Инструкция воспроизведения целевого результата: последовательность действий и результатов в субъективно-психическом эксперименте (далее СПЭ) -/- На расстоянии 25-40 см от стены (так, чтобы стена занимала всё поле зрения или его подавляющую часть), • Здесь и сейчас сосредоточьтесь на своём лице: «я есть» – реальность (а не знание об этом). • В себе отстранитесь назад и смотрите из себя на эту стену. • Не двигая головой, охватите поле зрения вниманием предельно широко и одновременно всё сразу (всё это) воспримите единым целым. • Высокую и широкую стену (вариант: смотрите вверх под углом 135-150 градусов перед собой) воспримите так, что обозреваемое вами без поворота головы пространство (объем), это – ограниченная по окружности в периферии поля зрения замкнутая несимметричная «линза» • плоская с противоположной вам стороны • и выпуклая в вас; Вы – вогнутый в себя-назад смотрящий («субъект») – каков? – кто из вас сейчас на всё это смотрит? • Не двигая головой, поле зрения воспримите единым целым –» • В себя-назад «втянитесь» –» и «пятьтесь» –» «за мысли» и за всё вообще –» из этого всего назад вытесняясь –» от всего вообще взад отстранитесь –» • собой в себя-назад подайтесь и упершись в собственный (себя) задний к объектному обращённый (развернутый) тупиковый предел –» • станьте самим пределом-тупиком – в себя-назад вогнутой (рефлектор-подобной) пределом-плоскостью смотрящей – воспринимающим зрительным экраном –» станьте плоскостью-тупиком –» • «проскользните» по себе – плоскости-тупику и охватите вниманием целиком –» • собой – самим плоскостью-тупиком назад надавить, нажать –» • осознать: что-то как-то внутри плоскости-тупика и сзади за плоскостью-тупиком, и там вообще «что-то как-то», а также само «внутри» и «сзади», «за плоскостью-тупиком» и вообще «там» – нереальность; • самого себя – плоскости-тупика без толщины (глубины), без «внутри», без обратной стороны для себя нет. • –» То есть – в качестве психической реальности (каково психически) я скончаюсь? –» или • скончаться есть чему – скончаюсь (не исключено). или • я таков, что скончаться нечему, но я есть. или • субъективно-психический эксперимент не завершён. • –» Осознайте, что это – реальность (осознать реальность этого) и подчеркните (ответ). -/- В результате я есть, но скончаться/не скончаться нечему, то есть гипотеза «скончаюсь» – нереальность. -/- Замечания -/- • Каждый завершённый СПЭ завершён целевым результатом: «я таков, что скончаться нечему, но я есть». И тому, каков я в результате СПЭ, то, что скончаться есть чему – скончаюсь (не исключено) не соответствует, то есть нереальность и этим исключено. • То есть, я и не таков, что не скончаюсь. «Я скончаюсь/не скончаюсь» ко мне, как к психике, неприменимо. Иначе это выражая, тому, каков я, смерть несвойственна. То есть, я, как психика (психическое), бессмертен, но я не вечен (не вечный – бесконечно длящийся, таковой, что есть чему длиться) – длиться, быть вечным и не скончаться нечему. Таково «бессмертие». • В целевом результате данной работы я есть, но таков, что скончаться/не скончаться нечему («бессмертие»). И таковой «я» не Атман индуистских школ ньяя, вайшешика и пурва-миманса, о котором сообщает Е.А. Торчинов. Так как «я» психичен, а не субстанция с психическим в качестве атрибута. • Согласно иллюзионизму, то, каков я в результате данного в работе эксперимента, это иллюзия. И впечатление бессмертия, это порождение мозга. А реально я таков, что скончаюсь (не исключено). Однако, необходимо осознать, что иллюзионизм, это отстраненные рассуждения о «я», но не сам я непосредственно. И тому, каков я психически (каково психически), рассуждения и вопросы о реальности «я» (например, таков ли я реально, каков психически?) и другие выражающие иллюзионизм «я» не соответствуют. То есть, каково психически, иллюзионизм «я» – нереальность. Психически, это несомненно и окончательно. • Рассуждения о том, что таковой «я» в этом (моем) или не в этом мозге, а также (например, держа в руках реальный череп), вопросы, типа «а как же (или, но теперь где же здесь) таковой «я» – как это после смерти мозга?» тому, каков таковой «я» эти вопросы не соответствуют. • После пребывания без сознания остается впечатление, что без сознания меня нет и ничего никак нет вообще. То есть, моё «бессознание» психически неотличимо от моего несуществования (небытия). Однако, наукой не показано и не исключено то, что до и после того, как я есть физически (скончаться есть чему), меня ещё или уже нет, скончаться ещё или уже нечему в качестве души, субстанции или как-то иначе. Но, то, каков я в целевом результате данной работы, таковой я есть, а не «скончался и уже теперь или ещё нереальность» и я таков, что скончаться нечему – реален (реалия). То есть, скончаюсь? – нечему (скончаться), но я есть, даже если меня нет (не существую). • Я таков, что быть и скончаться нечему, но я есть. И то, как (каково) психически, это бессмертие – реальность. Но эта реальность ускользает от моего рационального её объяснения. -/- Заключение -/- Скончаюсь? – в целевом результате я есть и вижу, но при этом я таков, что умереть (скончаться) нечему. То есть, я бессмертен, но при этом я не вечен (не нечто вечное (бесконечно длящееся)) – быть вечным нечему. И, если принять психическое реальностью не пытаясь понять это рационально, то я именно таков – осознаю, что такова моя несомненная для меня природа. То есть, «скончаюсь/не скончаюсь» – нереальность. Психически, это несомненно и окончательно. Несмотря на ненаучность, данная работа – способ овладения реальностью. Автор был бы признателен за выявленную и показанную нереальность целевого результата данной работы, а в идеале, за осознание: «скончаюсь (не исключено)». -/- Annex 2. The key essence of this work (only in Russian to convey the nuances without distortion) -/- Performed, expressed and transmitted initially by means of Russian, the native language of the author, and, in order to avoid distortions in the correspondence of the linguistic nuances of the reproduced subjective reality, it assumes an independent translation into the language of the experimenter by the experimenter himself in the process of reproducing the target result of this work. -/- «Я» скончаюсь? – нечему (скончаться), но «Я» есть. Will “I” die (decease)? – to die (decease) nothing, but “I” am. -/- Заметить: • Выполнено, выражено и передано изначально посредством русского – родного для автора языка и, чтобы избежать искажений соответствия языковых нюансов воспроизводимой психической реальности, предполагается самостоятельный перевод на язык экспериментатора самим экспериментатором в процессе воспроизведения целевого результата данной работы. • В «кавычки» взяты слова и выражения, требующие «субъективного» приведения в соответствие психическому содержанию. Подготовка: • Вам потребуется уединённое место (там, где вас не отвлекут). • На первоначальном этапе понадобится некоторое (возможно длительное) время для того, чтобы усвоить предложенный алгоритм действий в психической реальности в процессе реализации и воспроизвести целевой результат впервые. • Возможно, результату поспособствует отклонение от предложенного алгоритма «психических действий», импровизация в направлении целевого результата. • В процессе эксперимента количество попыток не ограничено и затраченное на это время не имеет значения. Необходим исключительно только именно целевой результат. Реализация: • В уединении встать перед плоской широкой и высокой (достаточно большой, чтобы её границы были на периферии вашего поля зрения) стеной, например, покрытой однородной светлой (но не белой) керамической, не глянцевой (не отражающей на вас свет или отвлекающих внимание изображений, не дающей бликов) плиткой с лёгким текстурным рисунком (позволяющим глазам «зацепиться» за стену и отчётливо воспринимать перед собой поверхность, а не «утонуть» в однородном (монотонном) пространстве), на расстоянии 25-40 см от неё (так, чтобы стена занимала всё поле зрения или его подавляющую часть) –» • Скончаюсь (1)? – каково психически (психическое), в моем естественно-научном понимании – это (я) не более чем и исключительно только некоторые состояния функционирующего (живого) мозга – скончаюсь, как и уже скончался каждый умерший мозг – того каждого, кто скончался, как реальности (реального, реалии) (2) уже теперь нет окончательно. Теоретически, если психически я исключительно только таков, что скончаться есть чему и ни каков иначе, то, следовательно, что (я) скончаюсь, это не исключено. • Глядя на стену, здесь и сейчас сосредоточиться на своём лице (и осознать): «я есть» – реальность (а не знание об этом) –» • В себе «отстраниться» назад – смотрю из себя на эту стену –» • Не двигая головой, охватить поле зрения вниманием предельно широко (поднять (вздернуть) брови, если это расширит поле зрения) и одновременно всё сразу (всё это) воспринять единым целым –» • Высокую и широкую стену (вариант: смотреть вверх под углом 135-150 градусов перед собой) воспринять так, что обозреваемое без поворота головы пространство (объем), это – ограниченная по окружности в периферии поля зрения замкнутая несимметричная «линза» • плоская (ограниченная стеной) с противоположной мне стороны • и «выпуклая в меня» –» • Я – «вогнутый» в «себя-назад» смотрящий, каков? – кто из меня сейчас на всё это смотрит? –» • Не двигая головой, поле зрения воспринять единым целым –» • в себя-назад «втянуться» –» и «пятиться» –» «за мысли» и за всё вообще –» из этого всего назад «вытесняясь» –» от всего вообще «взад отстраниться» –» • «собой» в себя-назад «податься» и «упершись» в «собственный (себя) задний к объектному обращённый (развернутый)» «предел тупиковый» («тупо-уперто») –» • «стать» самим «пределом-тупиком» – «в себя-назад вогнутой (рефлектор-подобной)» «пределом-плоскостью» смотрящей – воспринимающим «зрительным экраном» –» стать «плоскостью-тупиком» –» • «проскользнуть» по себе – плоскости-тупику и охватить вниманием целиком –» • собой — самим плоскостью-тупиком «назад надавить, нажать»: • осознать: «что-то как-то внутри плоскости-тупика» и «сзади – за плоскостью-тупиком», и «там вообще» «что-то как-то», а также само «внутри» и «сзади», «за плоскостью-тупиком» и вообще «там» – нереальность; • самого себя – плоскости-тупика без толщины (глубины), без «внутри», без обратной стороны для себя нет –» Собой – плоскостью-тупиком: • без глубины (толщины), без «внутри», без обратной стороны таков, что скончаться есть чему – скончаюсь (не исключено) –» • смотрю –» смотря –» «Непосредственно-психически» -/- Заметить: • Специфику используемых в данной работе слов (терминов), названий и выражений необходимо осознать в соответствии с (4.1.) (данный пункт смотрите ниже). • Необходимо от себя – (как) «смотрящего плоскости-тупика», такового, что скончаться есть чему, оттождествиться собой — «самим психическим» (каково психически) (3) –» • –» «отличить (4)» «Я» –» «Я-лик». • Отличив, и себя такового обозначая, но не имея в языке отличающего местоимения, именую «Я» (5). • «Соединённое-через-дефис-в-кавычках» реализовать и/или воспринять «одним-единым-целостным»; • Последовательность слов и знаков предполагает последовательность психических действий (алгоритм действий в собственном психическом); • Вопросительность слова, фразы или отдельной её, именно вопросительной части задаётся знаком «?» расположенным перед и после них. Требование к действию задаётся знаком «!» расположенным перед и после слова (фразы); • Субъективно-психический эксперимент завершён, если воспроизведен (4.1.), иначе субъективно-психический эксперимент не завершён. -/- Собой – плоскостью-тупиком … смотрю –» смотря –» -/- «Психически» (6) 1. ?кто? видит –» !отличить «Я»! –» 2. –» !«Я-видящий-лик»! –» 3. –» !«Я-лик»! –» 4. –» !«Я-ликом» всем «психически-назад-надавить(нажать)»: «Я» ?скончаюсь?! –» !«Я-ликом» всем «психически-назад-давить(жать)»: (4.1.) – нечему (7), но «Я» есть.! -/- Признательность: Согласно иллюзионизму, то, каков «Я» (как (каково) «скончаться нечему, но «Я» есть.»), реально, это не так (не таково). То есть, таковой «Я» нереален (нереальность), а реально – таков, что скончаюсь (не исключено). Однако, необходимо осознать, что это «отстраненные» от «Я» рассуждения о «Я», но не сам «Я» непосредственно. И тому, каков «Я» (как (каково) скончаться нечему, но «Я» есть.) таковое (рассуждения) не соответствует. То есть, эти рассуждения о «Я» – заблуждение. Это (как осознание) несомненно и окончательно. Не так? – за выявленную и показанную нереальность (того, что) «Скончаюсь? – нечему (скончаться), но «Я» есть.», за показанную реальность того, что скончаться не исключено, а в идеале, за осознание: «скончаюсь (не исключено)» я был бы признателен. -/- (Сноски) (1) Именно «скончаюсь, скончался», то есть – как реальности (реального, реалии) нет окончательно. (2) Реалия, реально, реален, реальность, реальное – само то, что (каково это) непосредственно, как само таковое – само то, каково познаваемое и как-либо моделируемое вне его познания и иного (вообще) моделирования. То есть, само само-осознание себя реальностью, например, «я-реалия», «я есть» или «скончаться – нечему, но «Я» есть.» – реальность. То есть – нам ничто не запрещает ответить на вопрос «скончаюсь?» несомненно и окончательно. (3) Психически (психическое) – то, как (каково) я сам непосредственно, несомненно, и та моя непосредственная реальность, как и посредством чего я сам и всё вообще для меня есть, и которое для меня исключительно единственное. Следует заметить, что пока бываю без сознания (травма мозга или сон без сновидений), ничего и никак нет вообще. Также само обозначение того, что это сама собственная непосредственная реальность (то есть – несомненно). И чтобы её осознать, представьте себя экспериментатором, который не подменяет реальность рассуждениями о ней. – Теперь для вас, как экспериментатора, то, как (каково) психически, это – несомненно (непосредственная несомненная реальность). Не философствуй (не рассуждай, не моделируй её), а экспериментируй! (4) «Я-Лик» – специфическое выражение, соответствующее психическому в данный момент СПЭ. (5) «Я» – целевой (конечный, «предельный») и «ключевой» результат данной работы. «Я» в данной работе не склоняется, выделено полужирным подчеркнутым шрифтом и заключено в кавычки. (6) «Каково психически» – то, как (каково) «Я» сам непосредственно, несомненно – само «первое лицо» (субъект психического), а также обозначение принятия позиции «первого лица» и сообщения «от первого лица». (7) Пока я бываю без сознания («бессознание», например, сплю без сновидений), моего сознания нет (оно сведено на нет, угасло) – ничего и никак нет вообще (это моё абсолютное небытие) – (каково) психически, мне скончаться нечем. А в целевом результате данной работы, психически (то есть – непосредственно), «Я» таков, что скончаться не нечем, а именно нечему, но при этом «Я» есть. (shrink)

ABSTRACT: This paper offers a generic revenge-proof solution to the Sorites paradox that is compatible with several philosophical approaches to vagueness, including epistemicism, supervaluationism, psychological contextualism and intuitionism. The solution is traditional in that it rejects the Sorites conditional and proposes a modally expressed weakened conditional instead. The modalities are defined by the first-order logic QS4M+FIN. (This logic is a modal companion to the intermediate logic QH+KF, which places the solution between intuitionistic and classical logic.) Borderlineness is introduced (...) modally as usual. The solution is innovative in that its modal system brings out the semi-determinability of vagueness. Whether something is borderline and whether a predicate is vague or precise is only semi-determinable: higher-order vagueness is columnar. Finally, the solution is based entirely on two assumptions. (1) It rejects the Sorites conditional. (2) It maintains that if one specifies borderlineness in terms of the ‒suitably interpreted‒ modal logic QS4M+FIN, then one can explain why the Sorites appears paradoxical. From (1)+(2) it results that one can tell neither where exactly in a Sorites series the borderline zone starts and ends nor what its extension is. Accordingly, the solution is also called agnostic. (shrink)

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second- (...) class='Hi'>order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms. (shrink)

This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account (...) provided which bear on three debated issues concerning quantification. (shrink)

We adjust the notion of typicality originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class ${\rm NT}$ of nontypical sets comes out as a natural strengthening of Russell's initial definition, which employs properties of small (minority) extensions, when the latter are restricted to the various levels $V_\zeta$ of $V$. This strengthening leads to defining ${\rm NT}$ (...) as the class of sets that belong to some countable ordinal definable set. It follows that ${\rm OD}\subseteq {\rm NT}$ and hence ${\rm HOD}\subseteq {\rm HNT}$. It is proved that the class ${\rm HNT}$ of hereditarily nontypical sets is an inner model of ${\rm ZF}$. Moreover the (relative) consistency of $V\neq {\rm NT}$ is established, by showing that in many forcing extensions $M[G]$ the generic set $G$ is a typical element of $M[G]$, a fact which is fully in accord with the intuitive meaning of typicality. In particular it is consistent that there exist continuum many typical reals. In addition it follows from a result of Kanovei and Lyubetsky that ${\rm HOD}\neq {\rm HNT}$ is also relatively consistent. In particular it is consistent that ${\cal P}(\omega)\cap {\rm OD}\subsetneq{\cal P}(\omega)\cap {\rm NT}$. However many questions remain open, among them the consistency of ${\rm HOD}\neq {\rm HNT}\neq V$, ${\rm HOD}={\rm HNT}\neq V$ and ${\rm HOD}\neq {\rm HNT}= V$. (shrink)

This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction of premises to a conclusion is valid when the corresponding conditional is true. The (...) validity predicate so defined significantly increases our expressive resources and provides a coherent formal treatment of paradoxical arguments. (shrink)

The goal of this short chapter, aimed at philosophers, is to provide an overview and brief explanation of some central concepts involved in predictive processing (PP). Even those who consider themselves experts on the topic may find it helpful to see how the central terms are used in this collection. To keep things simple, we will first informally define a set of features important to predictive processing, supplemented by some short explanations and an alphabetic glossary. -/- The features described (...) here are not shared in all PP accounts. Some may not be necessary for an individual model; others may be contested. Indeed, not even all authors of this collection will accept all of them. To make this transparent, we have encouraged contributors to indicate briefly which of the features are necessary to support the arguments they provide, and which (if any) are incompatible with their account. For the sake of clarity, we provide the complete list here, very roughly ordered by how central we take them to be for “Vanilla PP” (i.e., a formulation of predictive processing that will probably be accepted by most researchers working on this topic). More detailed explanations will be given below. Note that these features do not specify individually necessary and jointly sufficient conditions for the application of the concept of “predictive processing”. All we currently have is a semantic cluster, with perhaps some overlapping sets of jointly sufficient criteria. The framework is still developing, and it is difficult, maybe impossible, to provide theory-neutral explanations of all PP ideas without already introducing strong background assumptions. (shrink)

Gödel’s slingshot-argument proceeds from a referential theory of definite descriptions and from the principle of compositionality for reference. It outlines a metasemantic proof of Frege’s thesis that all true sentences refer to the same object—as well as all false ones. Whereas Frege drew from this the conclusion that sentences refer to truth-values, Gödel rejected a referential theory of definite descriptions. By formalising Gödel’s argument, it is possible to reconstruct all premises that are needed for the derivation of Frege’s thesis. For (...) this purpose, a reference-theoretical semantics for a language of first-order predicate logic with identity and referentially treated definite descriptions will be defined. Some of the premises of Gödel’s argument will be proven by such a reference-theoretical semantics, whereas others can only be postulated. For example, the principle that logically equivalent sentences refer to the same object cannot be proven but must be assumed in order to derive Frege’s thesis. However, different true (or false) sentences can refer to different states of affairs if the latter principle is rejected and the other two premises are maintained. This is shown using an identity criterion for states of affairs according to which two states of affairs are identical if and only if they involve the same objects and have the same necessary and sufficient condition for obtaining. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (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)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction (...) rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

This paper introduces two languages and associated logics designed to afford perspicuous representations of a range of natural language arguments involving adverbs and the like: first-order logic with basic adverbs (FOL-BA) and first-order logic with scoped adverbs (FOL-SA). The guiding logical idea is that an adverb can come between a term and the rest of the statement it is a part of, resulting in a logically stronger statement. I explain various interesting challenges that arise in the (...) attempt to implement the guiding idea, and provide solutions for some but not all of them. I conclude by outlining some directions for further research. (shrink)

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two (...) kinds of import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the (...) full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical. (shrink)

A central dispute in social ontology concerns the existence of group minds and actions. I argue that some authors in this dispute rely on rival views of existence without sufficiently acknowledging this divergence. I proceed in three steps in arguing for this claim. First, I define the phenomenon as an implicit higher-order disagreement by drawing on an analysis of verbal disputes. Second, I distinguish two theories of existence—the theory-commitments view and the truthmaker view—in both their eliminativist and their (...) constructivist variants. Third, I examine individual contributions to the dispute about the existence of group minds and actions to argue that these contributions have an implicit higher-order disagreement. This paper serves two purposes. First, it is a study to apply recent advances in meta-ontology. Second, it contributes to the debate on social ontology by illustrating how meta-ontology matters for social ontology. (shrink)

Since the noun phrase `artificial intelligence' (AI) was coined, it has been debated whether humans are able to create intelligence using technology. We shed new light on this question from the point of view of themodynamics and mathematics. First, we define what it is to be an agent (device) that could be the bearer of AI. Then we show that the mainstream definitions of `intelligence' proposed by Hutter and others and still accepted by the AI community are too weak (...) even to capture what is involved when we ascribe intelligence to an insect. We then summarise the highly useful definition of basic (arthropod) intelligence proposed by Rodney Brooks, and we identify the properties that an AI agent would need to possess in order to be the bearer of intelligence by this definition. Finally, we show that, from the perspective of the disciplines needed to create such an agent, namely mathematics and physics, these properties are realisable by neither implicit nor explicit mathematical design nor by setting up an environment in which an AI could evolve spontaneously. (shrink)

In Groenendijk & Stokhof [1989] a system of dynamic predicate logic (DPL) was developed, as a compositional alternative for classical discourse representation theory (DRT ). DPL shares with DRT the restriction of being a first-order system. In the present paper, we are mainly concerned with overcoming this limitation. We shall define a dynamic semantics for a typed language with λ-abstraction which is compatible with the semantics DPL specifies for the language of first-order predicate logic. We shall (...) propose to use this new logical system as the semantic component of a Montague-style grammar (referred to as dynamic Montague grammar, DMG), which will enable us to extend the compositionality of DPL to the subsentential level. Furthermore, we shall extend this analysis also in this sense that we shall add new, dynamic interpretations for logical constants which in DPL were treated in a static fashion. This will substantially increase the descriptive coverage of DMG. (shrink)

We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction (...) and truth to the compound formulas of PA over N---I_PA(N; SV ) and I_PA(N; SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both I_PA(N; SV ) and I_PA(N; SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under I_PA(N; SV ), then this assignment corresponds to the classical non-finitary putative standard interpretation I_PA(N; S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment I_PA(N; SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Goedel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which---under interpretation---are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas' Goedelian argument is validated if the assignment I_PA(N; SV ) can be treated as circumscribing the ambit of human reasoning about `true' arithmetical propositions, and the assignment I_PA(N; SC) as circumscribing the ambit of mechanistic reasoning about `true' arithmetical propositions. (shrink)

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw (...) suggestions from his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

This is a contribution to the idea that some proofs in first-order logic are synthetic. Syntheticity is understood here in its classical geometrical sense. Starting from Jaakko Hintikka’s original idea and Allen Hazen’s insights, this paper develops a method to define the ‘graphical form’ of formulae in monadic and dyadic fraction of first-order logic. Then a synthetic inferential step in Natural Deduction is defined. A proof is defined as synthetic if it includes at least one synthetic (...) inferential step. Finally, it will be shown that the proposed definition is not sensitive to different ways of driving the same conclusion from the same assumptions. (shrink)

In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several (...) extensions of \SQEMA\ where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension \SSQEMA\ we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon's monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of \SSQEMA\ which guarantee canonicity, too. (shrink)

The basic principles of Cipra's metaphysics (according to his book "Metamorphoses of Metaphysics") are analyzed with respect to Cipra's request for the revision of classical logical principles (of identity, excluded middle and contradiction). In Cipra's metaphysics, the principle of identity holds for being, necessity and past only, the principle of excluded middle does not hold for coming-to-be, possibility and present, and the principle of contradiction does not hold for the actuality, reality (freedom) and future. A propositional and first-order (...) temporal model and semantics are defined for such a concept of metaphysical time. The propositional ockhamistic branching time model is extended, in a first-order model, with a non-empty set of object shapes and an equivalence relation between object shapes in time, so that, for example, different, previously equivalent object shapes can in future contradict one another. (shrink)

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that (...) only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)

In this paper authors for the first time define a new notion called neutrosophic lattices. We define few properties related with them. Three types of neutrosophic lattices are defined and the special properties about these new class of lattices are discussed and developed. This paper is organised into three sections. First section introduces the concept of partially ordered neutrosophic set and neutrosophic lattices. Section two introduces different types of neutrosophic lattices and the final section studies neutrosophic Boolean algebras. (...) Conclusions and results are provided in section three. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...) ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)