The categoricity problem for a system of logic reveals an asymmetry between the model-theoretic and the proof-theoretic resources of that logic. In particular, it reveals prima facie that the proof-theoretic instruments are insufficient for matching the envisaged model-theory, when the latter is already available. Among the proposed solutions for solving this problem, some make use of new proof-theoretic instruments, some others introduce new model-theoretic constrains on the proof-systems, while others try to use instruments from both sides. On the proof-theoretical side, (...) two main approaches for solving the categoricity problem for propositional classical logic consist in the enforcement of the formal systems of this logic by introducing rules of inference of a new kind. A multiple-conclusions logic allows rules of inference with more than one conclusion, while a bilateralist system contains rules of inference formulated in terms of assertion and rejection. Both these approaches reveal interesting formal features of logical reasoning. This paper analyses some of the advantages and limitations of these two approaches as solutions for a full formalization of logic, i.e., for a categorical formalization. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s (...) bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system---ST---that conservatively extends (...) classical logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered. (shrink)

This paper provides a proof-theoretic account of imperative logical consequence by generalising Greg Restall’s multiple conclusion bilateralism for declarative logic. According to imperative bilateralism, a sequent Γ ⊢ Δ is valid iff jointly commanding all the imperatives Φ ∈ Γ and prohibiting all the imperatives Ψ ∈ Δ clashes. This account has three main virtues: (1) it provides a proof-theoretic account of imperatives; (2) it does not rely on the controversial notion of imperative inference; and (3) it is neutral (...) regarding cognitivism about imperatives. (shrink)

Consequence rleations over sets of "judgments" are defined by using "overdetermined" as well as "underdetermined" valuations. Some of these relations are shown to be categorical. And generalized soundness and completeness results are given for both multiple and single conclusion consequence relations.

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)

Gideon Rosen has recently sketched an argument which aims to establish that the notion of metaphysical modality is systematically ambiguous. His argument contains a crucial sub-argument which has been used to argue for Metaphysical Contingentism, the view that some claims of fundamental metaphysics are metaphysically contingent rather than necessary. In this paper, Rosen’s argument is explicated in detail and it is argued that the most straight-forward reconstruction fails to support its intended conclusion. Two possible ways to save the argument are (...) rebutted and it is furthermore argued that the crucial sub-argument only supports a rather particular variant of Metaphysical Contingentism. (shrink)

In the fifth version of our reply article [26] to Imamura's critique, we recall that Neutrosophic Non-Standard Logic was never used by the neutrosophic community in any application, that the quarter-century old (1995-1998) neutrosophic operators criticized by Imamura were never used as they were improved soon after, but omits to talk about their development, and that in real-world applications we need to convert/approximate the hyperreals, monads and bi-nads of Non-Standard Analysis to tiny intervals with the desired precision; otherwise (...) they would be inapplicable. We pointed out several errors and false statements by Imamura [21] regarding the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is incorrect, as is his definition of nonstandard unit interval, and we showed that there is no total order in Neutrosophic Computing and Machine Learning , Vol. 23, 2022 Florentin Smarandache, Definición mejorada de la lógica neutrosófica no estándar e introducción a los hiperreales neutrosóficos (Quinta versión) 2 the set of hyperreals (due to the recently introduced Neutrosophic Hyperreals which are indeterminate), so the Transfer Principle from R to R* is questionable. After his critique, several reply posts on non-standard theoretical neutrosophy followed in 2018-2022. As such, I extended the Nonstandard Analysis by adding the right-closed left monad, the left-closed right monad, the punctured binad (which we introduced in 1998), and the nonpunctured binad - all in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of the Nonstandard Unitary Interval and Nonstandard Neutrosophic Logic are presented, along with Nonstandard Neutrosophic Operators. (shrink)

Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate (...) a kind of converse to Carnap’s result. (shrink)

Bilateralism is the view that the speech act of denial is as primitive as that of assertion. Bilateralism has proved helpful in providing an intuitive interpretation of formalisms that, prima facie, look counterintuitive, namely, multiple-conclusion sequent calculi. Under this interpretation, a sequent of the form $\Gamma \vdash \Delta$ is regarded as the statement that it is incoherent, according to our conversational norms, to occupy the position of asserting all the sentences in $\Gamma$ and denying all the sentences in $\Delta$. (...) Some have argued, based on this interpretation, that the notion of invalidity is as conceptually primitive and important as the notion of validity: whereas the latter is couched in terms of what positions are incoherent and hence untenable, the former is couched in terms of what positions are coherent and hence tenable. My ultimate goal in this paper is to contest this view. Based on a novel technical account of the two notions—one that I find more accurate than the existing accounts in the literature—I shall argue that the notion of incoherence takes conceptual priority over the notion of coherence. (shrink)

A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of connectives and truth values, (...) therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (shrink)

This paper is based on the comments I gave to Alfred Mele regarding his original Zygote Argument during my presentation at a small workshop on manipulation arguments in Budapest back in 2012. After those comments, Mele changed the conclusion of his original Zygote Argument (OZA) from a positive, explanatory conclusion to a negative, non-explanatory conclusion--and, correspondingly, redefined 'incompatibilism' so that it would no longer refer in his work to the view that determinism precludes (undermines, eliminates, destroys, etc.) free will, but (...) refers narrowly to the perhaps spurious association claim that appears in the conclusion of the revised Zygote Argument (Mele 2013+). Yet, over the last decade, Mele has made no comment in print about the changes he made, the invalidity objection that led to them, nor to the fact that his revised Zygote Argument was originally presented by me as a (non-optimal) reaction to my invalidity objection to his original Zygote Argument. To my knowledge, Mele has only mentioned this 2015 paper once, at it was not in the context of understanding manipulation arguments; it was, rather, in the context of defending his revised, non-explanatory definition of 'incompatibilism' at the opening of Aspects of Agency (p. 6, n. 4). There, Mele singles out *just this 2015 paper* as a place where they'll find what Mele degrades as a type of nonstandard/nontraditional definition of 'incompatibilism' that he rejects. This is notable because Mele provides no empirical evidence that his preferred definition is standard, and most (including Lehrer, who coined the term) reject the negative, non-explanatory, association-based definition that Mele came to endorse in the aftermath of my invalidity objection to OZA. So, if Mele had a genuine concern about philosophers who use the term 'incompatibilism', then Mele could have--and should have--named almost any leading figure in free will for this purpose (since most--including Pereboom, Vihvelin, Haji, etc.) reject the non-explanatory definition that Mele-2013+ now endorses. Yet, for untold reasons, Mele isolated the working definition of 'incompatibilism' used in one paper by one unemployed junior researcher to use as his example--even though Neil Levy had used this same definition in previously published work (eg. Levy 2011). Moreover, if Mele were genuinely interested in helping his readers understand the ambiguity of terms like 'compatibilism' and 'incompatibilism' it is a mystery that he referenced a paper in which nothing of substance hung on the definitions given to these terms -- especially considering that he might have instead pointed them to my OTHER 2015 paper, "A Critique of Vihvelin's Three-Fold Classification" in which the MAIN AIM of the paper is to expose the difficulties facing anyone who wants to define the terms 'compatibilism' and 'incompatibilism' in a way that everyone will accept (including a critique of the negative, non-explanatory "incompossibilist" definition that Mele adopted after my objection to OZA). By all appearances, Mele's main aim in citing specifically THIS paper and none of my other work was to give readers the false impression that my working definition of 'incompatibilism' played some role in my charge that his argument is invalid (which it doesn't, given that the term doesn't appear in either the premises or conclusion of that argument). Mele adds to this unseemly rhetorical campaign in his book *Manipulation Agents* where he mentions an invalidity objection in the intro (p. 84) and then discusses that objection in the final lines of Chapter 5 of the book (p. 120-121). The invalidity objection discussed here is completely pedantic, has never been seriously raised in the primary literature (and was previously addressed in multiple places my multiple philosophers, including me), and has nothing whatever to do with my invalidity objection. In other words, Mele was content to discuss an invalidity objection that no one has ever had to an argument that is clearly valid while completely ignoring the ONLY substantive invalidity objection to his argument in print. This is unfortunate, because it has led to serious confusion --including the patently false description of my invalidity objection to OZA as an objection to the revised version of the Zygote Argument that Mele adopted in *response* my objection (e.g., Capes mistakenly claims "Mickelson (2015) contends that arguments like this [i.e. like the revised Zygote Argument] are invalid. But that’s because she defines “incompatibilism” differently than I’ve defined it here" in his "Manipulation Arguments and Direct Arguments" chapter in the 2023 Wiley-Blackwell Companion to Free Will; I did not say this, and the definition of 'incompatibilism' is irrelevant to my objection). In "The Zygote Argument Is Invalid--Now What?", I simply use the invalidity of the simple 2-premise Zygote Argument as a foil to make a series of more philosophically significant points (e.g. regarding the dialectical structure of the free-will debate, the role of best-explanation reasoning, and to develop the first-ever "master manipulation argument"). Mele has yet to address my reasons for calling attention to the non-trivial invalidity of his OZA or to any of my answers to the "Now What?" question in this paper, even though this question and my proposed answers to this question are the paper's main point. For those interested in a more recent discussion of how the logical and metaphilosophical observations made in this "Now What?" paper have helped to move the free-will debate forward, I suggest looking at the discussions of the "paradigm shift" in the free-will debate that helps to explain why the free-will debate has become mired in stalemates and empty verbal disputes that are found, for example, in my contributions to the Wiley-Blackwell Companion to Free Will (Intro and Chapter 4, 2023) and my paper "Free Will, Self-Creation, and the Paradox of Moral Luck" (2019). (shrink)

This book is an anthology with the following themes. Non-European Tradition: Bussanich interprets main themes of Hindu ethics, including its roots in ritual sacrifice, its relationship to religious duty, society, individual human well-being, and psychic liberation. To best assess the truth of Hindu ethics, he argues for dialogue with premodern Western thought. Pfister takes up the question of human nature as a case study in Chinese ethics. Is our nature inherently good (as Mengzi argued) or bad (Xunzi’s view)? Pfister ob- (...) serves their underlying agreement, that human beings are capable of becoming good, and makes precise the disagreement: whether we achieve goodness by cultivating autonomous feelings or by accepting external precepts. There are political consequences: whether government should aim to respect and em- power individual choices or to be a controlling authority. Early Greek Thinking: Collobert examines the bases of Homeric ethics in fame, prudence, and shame, and how these guide the deliberations of heroes. She observes how, by depending upon the poet’s words, the hero gains a quasi- immortality, although in truth there is no consolation for each person’s inevi- table death. Plato: Santas examines Socratic Method and ethics in Republic 1. There Socrates examines definitions of justice and tests them by comparison to the arts and sciences. Santas shows the similarities of Socrates’ method to John Rawls’ method of considered judgments in reflective equilibrium. McPherran interprets Plato’s religious dimension as like that of his teacher Socrates. McPherran shows how Plato appropriates, reshapes, and extends the religious conventions of his own time in the service of establishing the new enterprise of philosophy. Ac- cording to Taylor, Socrates believes that humans in general have the task of helping the gods by making their own souls as good as possible, and Socrates’ unique ability to cross-examine imposes on him the special task of helping others to become as good as possible. This conception of Socrates’ mission is Plato’s own, consisting in an extension of the traditional conception of piety as helping the gods. Brickhouse and Smith propose a new understanding of Socratic moral psychology—one that retains the standard view of Socrates as an intellectualist, but also recognizes roles in human agency for appetites and passions. They compare and contrast the Socratic view to the picture of moral psychology we get in other dialogues of Plato. Hardy also proposes a new, non-reductive understanding of Socratic eudaimonism—he argues that Socrates invokes a very rich and complex notion of the “Knowledge of the Good and Bad”, which is associated with the motivating forces of the virtues. Rudebusch defends Socrates’ argument that knowledge can never be impotent in the face of psychic passions. He considers the standard objections: that knowledge cannot weigh incom- mensurable human values, and that brute desire, all by itself, is capable of moving the soul to action. Aristotle: Anagnostopoulos interprets Aristotle on the nature and acquisition of virtue. Though virtue of character, aiming at human happiness, requires a complex awareness of multiple dimensions of one’s experience, it is not properly a cognitive capacity. Thus it requires habituation, not education, according to Aristotle, in order to align the unruly elements of the soul with reason’s knowledge of what promotes happiness. Shields explains Aristotle’s doctrine that goodness is meant in many ways as the doctrine that there are different analyses of goodness for different types of circumstance, just as for being. He finds Aristotle to argue for this conclusion, against Plato’s doctrine of the unity of the Good, by applying the tests for homonymy and as an immediate cons- equence of the doctrine of categories. Shields evaluates the issue as unresolved at present. Russell discusses Aristotle’s account of practical deliberation and its virtue, intelligence (phronesis). He relates the account to contemporary philo- sophical controversies surrounding Aristotle’s view that intelligence is neces- sary for moral virtue, including the objections that in some cases it is unnecessary or even impedes human goodness. Frede examines the advantages and disadvantages of Aristotle’s virtue ethics. She explains the general Greek con- ceptions of happiness and virtue, Aristotle’s conception of phronesis and compares the Aristotle’s ethics with modern accounts. Liske discusses the question of whether the Aristotelian account of virtue entails an ethical-psy- chological determinism. He argues that Aristotle’s understanding of hexis allows for free action and ethical responsibility : By making decisions for good actions we are able to stabilize our character (hexis). Hellenistic and Roman: Annas defends an account of stoic ethics, according to which the three parts of Stoicism—logic, physics, and ethics—are integrated as the parts of an egg, not as the parts of a building. Since by this analogy no one part is a foundation for the rest, pedagogical decisions may govern the choice of numerous, equally valid, presentations of Stoic ethics. Piering interprets the Cynic way of life as a distinctive philosophy. In their ethics, Cynics value neither pleasure nor tradition but personal liberty, which they achieve by self-suffi- ciency and display in speech that is frank to the point of insult. Plotinus and Neoplatonism: Gerson outlines the place of ordinary civic virtue as well as philosophically contemplative excellence in Neoplatonism. In doing so he attempts to show how one and the same good can be both action-guiding in human life and be the absolute simple One that grounds the explanation of everything in the universe. Delcomminette follows Plotinus’s path to the Good as the foundation of free will, first in the freedom of Intellect and then in the “more than freedom” of the One. Plotinus postulates these divinities as not outside but within each self, saving him from the contradiction of an external foundation for a truly free will. General Topics: Halbig discusses the thesis on the unity of virtues. He dis- tinguishes the thesis of the identity of virtues and the thesis of a reciprocity of virtues and argues that the various virtues form a unity (in terms of reciprocity) since virtues cannot bring about any bad action. Detel examines Plato’s and Aristotle’s conceptions of normativity : Plato and Aristotle (i) entertained hybrid theories of normativity by distinguishing functional, semantic and ethical normativity, (ii) located the ultimate source of normativity in standards of a good life, and thus (iii) took semantic normativity to be a derived form of normativity. Detel argues that hybrid theories of normativity are—from a mo- dern point of view—still promising. Ho ̈ffe defends the Ancient conception of an art of living against Modern objections. Whereas many Modern philosophers think that we have to replace Ancient eudaimonism by the idea of moral obligation (Pflicht), Ho ̈ffe argues that Eudaimonism and autonomy-based ethics can be reconciled and integrated into a comprehensive and promising theory of a good life, if we enrich the idea of autonomy by the central elements of Ancient eudaimonism. Some common themes: The topics in Chinese and Hindu ethics are perhaps more familiar to modern western sensibilities than Homeric and even Socratic. Anagnostopoulos, Brickhouse and Smith, Frede, Liske, Rudebusch, and Russell all consider in contrasting ways the role of moral character, apart from intellect, in ethics. Brickhouse / Smith, Hardy, and Rudebusch discuss the Socratic con- ception of moral knowledge. Brickhouse / Smith and Hardy retain the standard view of the so called Socratic Intellectualism. Shields and Gerson both consider the question whether there is a single genus of goodness, or if the term is a homonym. Bussanich, McPherran, Taylor, and Delcomminette all consider the relation between religion and ethics. Pfister, Piering, Delcomminette, and Liske all consider what sort of freedom is appropriate to human well-being. Halbig, Detel, and Ho ̈ffe propose interpretations of main themes of Ancient ethics. (shrink)

When logicians work with multiple-conclusion systems, they use a metalinguistic comma ‘,’ to aggregate premises and/or conclusions. In this note, I present an analogy between this comma and Prior’s infamous connective tonk. The analogy reveals that these expressions have much in common. I argue that, indeed, the comma can be seen as a structural incarnation of tonk. The upshot is that, whatever story one has to tell about tonk, there are good reasons to tell a similar story about the (...) comma in typical multiple-conclusion systems, and vice versa. (shrink)

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

I discuss Prawitz’s claim that a non-reliabilist answer to the question “What is a proof?” compels us to reject the standard Bolzano-Tarski account of validity, andto account for the meaning of a sentence in broadly verificationist terms. I sketch what I take to be a possible way of resisting Prawitz’s claim---one that concedes the anti-reliabilist assumption from which Prawitz’s argument proceeds.

This paper presents a new argument to defend the normativity of meaning, specifically the thesis that there are no meanings without norms. The argument starts from the observation inferentialists have emphasized that incompatibility relations between sentences are a necessary part of meaning as it is understood. We motivate this approach by showing that the standard normativist strategy in the literature, which is developed in terms of veridical reference that may swing free from the speaker’s understanding, violates the ought-implies-can principle, (...) but ours does not. In addition, our approach is superior because, unlike the representationalist approach, it can be extended from declarative sentences to non-representational uses of language. In this paper, however, we only formulate the argument for the base case that involves incompatibility relations between declarative sentences. The goal is not to derive norms from something that is not normative, but to explicate the distinctive type of normativity that is built into meaning as it is understood by language-users. The explication proceeds in two steps. (1) For any sentence s a speaker understands, there is an incompatible sentence s’ the speaker understands to be incompatible with s. (2) In virtue of understanding this incompatibility of meanings, she ought not to be committed to both s and s’. This prohibition is not derived from instrumental practical reason, nor is it based on representational correctness, but its source is the incompatibility of meanings. (shrink)

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation of those strategic subtleties working within science. By "Theory T thinking," we mean to describe the simplistic methodology in which mathematical science allegedly supplies ‘processes’ that parallel nature's own in a tidily isomorphic fashion, wherein "Theory T’s" feigned rigor and methodological dogmas advance inadequate discrimination that fails to distinguish between explanatory structures (...) that are architecturally distinct. One of Wilson's main goals is to reverse such premature exclusions and, thus, early on Wilson returns to John Locke's original physical concerns regarding material science and the congeries of descriptive concern insofar as capturing varied phenomena (i.e., cohesion, elasticity, fracture, and the transmission of coherent work) encountered amongst ordinary solids like wood and steel are concerned. Of course, Wilson methodologically updates such a purview by appealing to multiscalar techniques of modern computing, drawing from Robert Batterman's work on the greediness of scales and Jim Woodward's insights on causation. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things (...) return. It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)

Though the duties of care owed toward innocents in war and in civil life are at the bottom univocally determined by the same ethical principles, Bazargan-Forward argues that those very principles will yield in these two contexts different “in-practice” duties. Furthermore, the duty of care we owe toward our own innocents is less stringent than the duty of care we owe toward foreign innocents in war. This is because risks associated with civil life but not war (a) often increase the (...) expected welfare of the individuals upon whom the risk is imposed, (b) are often imposed with consent, and (c) are often imposed reciprocally. The conclusion—that we have a pro tanto reason for adopting a more stringent standard of risk imposition toward foreign innocents in war—has implications for not only what standards of risk we should adopt in war, but also how we should weigh domestic versus foreign civilian lives. (shrink)

I discuss methodology in epistemology. I argue that settling the facts, even the epistemic facts, fails to settle the questions of intellectual policy at the center of our epistemic lives. An upshot is that the standard methodology of analyzing concepts like knowledge, justification, rationality, and so on is misconceived. More generally, any epistemic method that seeks to issue in intellectual policy by settling the facts, whether by way of abductive theorizing or empirical investigation, no matter how reliable, is inapt. (...) The argument is a radicalization of Moore’s Open Question Argument. I conclude by considering the ramifications of this conclusion for the debate surrounding “Modal Security”, a proposed necessary condition on undermining defeat. (shrink)

We present a model of the distribution of labour in science. Such models tend to rely on the mechanism of the invisible hand . Our analysis starts from the necessity of standards in distributed processes and the possibility of multiple standards in science. Invisible hand models turn out to have only limited scope because they are restricted to describing the atypical single-standard case. Our model is a generalisation of these models to J standards; single-standard models such as (...) Kitcher are a limiting case. We introduce and formalise this model, demonstrate its dynamics and conclude that the conclusions commonly derived from invisible hand models about the distribution of labour in science are not robust against changes in the number of standards. (shrink)

Formal ontologies are nowadays widely considered a standard tool for knowledge representation and reasoning in the Semantic Web. In this context, they are expected to play an important role in helping automated processes to access information. Namely: they are expected to provide a formal structure able to explicate the relationships between different concepts/terms, thus allowing intelligent agents to interpret, correctly, the semantics of the web resources improving the performances of the search technologies. Here we take into account a problem (...) regarding Knowledge Representation in general, and ontology based representations in particular; namely: the fact that knowledge modeling seems to be constrained between conflicting requirements, such as compositionality, on the one hand and the need to represent prototypical information on the other. In particular, most common sense concepts seem not to be captured by the stringent semantics expressed by such formalisms as, for example, Description Logics (which are the formalisms on which the ontology languages have been built). The aim of this work is to analyse this problem, suggesting a possible solution suitable for formal ontologies and semantic web representations. The questions guiding this research, in fact, have been: is it possible to provide a formal representational framework which, for the same concept, combines both the classical modelling view (accounting for compositional information) and defeasible, prototypical knowledge ? Is it possible to propose a modelling architecture able to provide different type of reasoning (e.g. classical deductive reasoning for the compositional component and a non monotonic reasoning for the prototypical one)? We suggest a possible answer to these questions proposing a modelling framework able to represent, within the semantic web languages, a multilevel representation of conceptual information, integrating both classical and non classical (typicality based) information. Within this framework we hypothesise, at least in principle, the coexistence of multiple reasoning processes involving the different levels of representation. (shrink)

In a possible world framework, an agent can be said to know a proposition just in case the proposition is true at all worlds that are epistemically possible for the agent. Roughly, a world is epistemically possible for an agent just in case the world is not ruled out by anything the agent knows. If a proposition is true at some epistemically possible world for an agent, the proposition is epistemically possible for the agent. If a proposition is true at (...) all epistemically possible worlds for an agent, the proposition is epistemically necessary for the agent, and as such, the agent knows the proposition. -/- This framework presupposes an underlying space of worlds that we can call epistemic space. Traditionally, worlds in epistemic space are identified with possible worlds, where possible worlds are the kinds of entities that at least verify all logical truths. If so, given that epistemic space consists solely of possible worlds, it follows that any world that may remain epistemically possible for an agent verifies all logical truths. As a result, all logical truths are epistemically necessary for any agent, and the corresponding framework only allows us to model logically omniscient agents. This is a well-known consequence of the standard possible world framework, and it is generally taken to imply that the framework cannot be used to model non-ideal agents that fall short of logical omniscience. -/- A familiar attempt to model non-ideal agents within a broadly world involving framework centers around the use of impossible worlds where the truths of logic can be false. As we shall see, if we admit impossible worlds where “anything goes” in epistemic space, it is easy to avoid logical omniscience. If any logical falsehood is true at some impossible world, then any logical falsehood may remain epistemically possible for some agent. As a result, we can use an impossible world involving framework to model extremely non-ideal agents that do not know any logical truths. -/- A much harder, and considerably less investigated challenge is to ensure that the resulting epistemic space can also be used to model moderately ideal agents that are not logically omniscient but nevertheless logically competent. Intuitively, while such agents may fail to rule out impossible worlds that verify complex logical falsehoods, they are nevertheless able to rule out impossible worlds that verify obvious logical falsehoods. To model such agents, we need a construction of a non-trivial epistemic space that partly consists of impossible worlds where not "anything goes". This involves imposing substantive constraints on impossible worlds to eliminate from epistemic space, say, trivially impossible worlds that verify obvious logical falsehoods. -/- The central aim of this dissertation is to investigate the nature of such non-trivially impossible worlds and the corresponding epistemic spaces. To flag my conclusions, I argue that successful constructions of epistemic spaces that can safely navigate between the Charybdis of logical omniscience and the Scylla of of “anything goes” are hard, if not impossible to find. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals (...) are Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

In considering relations between science and theology, the discussion of the Big Bang model plays a significant role. Amongst the sources of this model there are not only scientific achievements of recent decades taken as objective knowledge as seen in modern methodology, but also many non-scientific factors. The latter is connected with the quite obvious fact that the authors, as well as the recipients of the Model, are people who are guided in their activity - including obtaining their rational knowledge (...) - by non-rational motives. Those motives appear on the one hand in the very process of creation of the Model. Different scientific theories as well as unverified hypotheses are being joined in one "picture" called The Standard Model. It seems that it is being done on the grounds of various factors that lie outside the field of science. Among them there are the different convictions of the persons constructing this view of the world. However, those convictions, commonly shared by the authors and recipients of the Model, are not based on the rational criterion of scientific knowledge. On the other hand, the Big Bang model may be interpreted in opposite ways by its recipients. The influences of religious and other beliefs are so essential, that they may lead to extremely different conclusions though based on the same ground. It is demonstrated that, due to the epistemological status of cosmology, such a situation is inevitable, and no final verdict with regard to the idea of the cause of the world can be reached. (shrink)

Description In this paper, we propose an advanced mathematical framework centered around the Energy Number Field (E), which fundamentally avoids the conventional concept of zero by introducing a neutral ele- ment, νE. Through this approach, we redefine core mathematical constructs, including limits, continuity, differentiation, integration, and series summation, ensuring they operate seamlessly within a zero-less paradigm. We address and redefine matrix operations, topology, metric spaces, and complex analysis, aligning them with the principles of E. Additionally, we explore non-mappable properties of (...) energy numbers in the context of symmetry groups, non-standard fields, fractional calculus, and non-Euclidean geometries. By employing νE as the central element, our framework resolves conventional zero-related sin- gularities and computational anomalies, offering a novel perspective that bridges advanced mathematical theories with practical applications in physics, computational algorithms, and topological transforma- tions. This work paves the way for future research to experimentally validate these formulations and explore potential applications in advanced physics, quantum mechanics, and beyond. (shrink)

In book 10 of the Republic we find a new argument for the division of the soul. The argument’s structure is similar to the arguments in book 4 but, unlike those arguments, it centres on a purely cognitive conflict: believing and disbelieving the same thing, at the same time. The argument presents two interpretive difficulties. First, it assumes that a conflict between a belief and an appearance—e.g. disbelieving that a stick partially immersed in water is, as it appears, bent—entails a (...) conflict between beliefs. Prima facie, there is only one belief, the belief that contradicts the appearance. Second, it is unclear what parts of the soul Plato intends to divide between: some argue that it is, as in book 4, a partition between a rational and a non-rational part; others argue that it is a new partition between a higher and a lower subdivision of the rational part. This paper offers solutions to both difficulties through an analysis of what Plato means by φαινόμενα, ‘appearances’, and δόξαι, ‘beliefs’. It is argued, first, that the relevant appearances are entirely sensory but nonetheless sufficiently belief-like to (a) warrant being called δόξαι and (b) oppose, by themselves, our beliefs; there is no need for a third mental state, a belief that assents to the appearance. A second claim concerns a central line in the argument, 602e4–6, that has served as the primary evidence that the partition is within the rational part of the soul. Those who wish to avoid this conclusion generally resort to alternative, and less natural, translations of 602e4–6. It is argued that this is unnecessary: once we have correctly understood sensory appearances, we see that the standard translation of 602e4–6 in fact entails a division between a rational and a non-rational part of the soul. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the (...) categoricity problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the (...) categoricity problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (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)

L'évaluation des écosystèmes est l'un des pivots essentiels pour l'élaboration de moyens adap- tés permettant de lutter contre la diminution massive de la biodiversité. Pour la première fois, elle a fait l'objet d'une analyse à l'échelle mondiale dans le cadre de l'Evaluation des écosys- tèmes en début de millénaire (EM). Le rassemblement de plus d’un millier de chercheurs et de plusieurs organismes internationaux durant quatre années ont permis de dessiner la carte nécessaire à toute action efficace. L'article expose les éléments (...) principaux de l'EM : l'évaluation des écosystèmes en tant que tels, mais surtout des services écosystémiques, dans toutes leurs dimensions, en ce que leur évolution affecte le bien-être humain. Il analyse ensuite les quatre points principaux de l'apport de l'EM, des avantages de l'utilisation croissante des services éco- logiques à sa non viabilité. Des scénarios, modèles et outils sont proposés pour inverser la courbe négative d'appauvrissement de la biodiversité et des services écosystémiques dans un premier bilan des retombées de l'EM. (shrink)

*These notes were folded into the published paper "Probability and nonclassical logic*. Revising semantics and logic has consequences for the theory of mind. Standard formal treatments of rational belief and desire make classical assumptions. If we are to challenge the presuppositions, we indicate what is kind of theory is going to take their place. Consider probability theory interpreted as an account of ideal partial belief. But if some propositions are neither true nor false, or are half true, or whatever—then (...) it’s far from clear that our degrees of belief in it and its negation should sum to 1, as classical probability theory requires (?, cf.). There are extant proposals in the literature for generalizing (categorical) probability theory to a non-classical setting, and we will use these below. But subjective probabilities themselves stand in functional relations to other mental states, and we need to trace the knock-on consequences of revisionism for this interrelationship (arguably, degrees of belief only count as kinds of belief in virtue of standing in these functional relationships). (shrink)

This paper explores two non-standard supermajority rules in the context of judgment aggregation over multiple logically connected issues. These rules set the supermajority threshold in a local, context sensitive way—partly as a function of the input profile of opinions. To motivate the interest of these rules, I prove two results. First, I characterize each rule in terms of a condition I call ‘Block Preservation’. Block preservation says that if a majority of group members accept a judgment set, then (...) so should the group. Second, I show that one of these rules is, in a precise sense, a judgment aggregation analogue of a rule for connecting qualitative and quantitative belief that has been recently defended by Hannes Leitgeb. The structural analogy is due to the fact that Leitgeb sets thresholds for qualitative beliefs in a local, context sensitive way—partly as a function of the given credence function. (shrink)

We report the results of an exploratory study that examines the judgments of climate scientists, climate policy experts, astrophysicists, and non-experts (N = 3367) about the factors that contribute to the creation and persistence of disagreement within climate science and astrophysics and about how one should respond to expert disagreement. We found that, as compared to non-experts, climate experts believe that within climate science (i) there is less disagreement about climate change, (ii) methodological factors play less of a role in (...) generating disagreements, (iii) fewer personal or institutional biases influence climate research, and (iv) there is more agreement about which methods should be used to examine relevant phenomena we also observed that the uniquely American political context predicted experts’ judgments about some of these factors. We also found that, in regard to disagreements concerning cosmic ray physics, and commensurate with the greater inherent uncertainty and data lacunae in their field, astrophysicists working on cosmic rays were generally more willing to acknowledge expert disagreement, more open to the idea that a set of data can have multiple valid interpretations, and generally less quick to dismiss someone articulating a non-standard view as non-expert, than climate scientists were in regard to climate science. (shrink)

Logics based on weak Kleene algebra (WKA) and related structures have been recently proposed as a tool for reasoning about flaws in computer programs. The key element of this proposal is the presence, in WKA and related structures, of a non-classical truth-value that is “contaminating” in the sense that whenever the value is assigned to a formula ϕ, any complex formula in which ϕ appears is assigned that value as well. Under such interpretations, the contaminating states represent occurrences of a (...) flaw. However, since different programs and machines can interact with (or be nested into) one another, we need to account for different kind of errors, and this calls for an evaluation of systems with multiple contaminating values. In this paper, we make steps toward these evaluation systems by considering two logics, HYB1 and HYB2, whose semantic interpretations account for two contaminating values beside classical values 0 and 1. In particular, we provide two main formal contributions. First, we give a characterization of their relations of (multiple-conclusion) logical consequence—that is, necessary and sufficient conditions for a set Δ of formulas to logically follow from a set Γ of formulas in HYB1 or HYB2 . Second, we provide sound and complete sequent calculi for the two logics. (shrink)

This article argues that the standard model of political representation mischaracterises the structure of representation. After surveying the classical types of representation and their application to non-humans, the basic nature of representation is shown to have been unduly centred on interests, responsiveness and unidirectional protocols. It proposes a different structure by drawing inspiration from recent scholarship and developments in political philosophy, as well as the representation of non-human actors. It proposes an ontological grounding of representation in ‘irreducible multiplicity’, and (...) a structural analysis based on the concepts of claim and relation. This abstract form of representation can take into account both human and non-human cases, and works to ground different typologies. The relational structure of representation creates interests and preferences, subjects and actors, power dynamics and seemingly immutable identities. (shrink)

Which do we conceptualize like Human in opposition to non Human ? The institution of “large shares” or “The Great Divide”, in terms of categories between the Human one and the non Human one, is far from to be always established in various times and Human spaces, such as we generally think it. This apparently natural institution, even expresses, appears after examination much less obviates that we thought it traditionally. For this reason, it constitutes an object of intellectual investigations of (...) choice for many traditional knowledge such anthropology, the history, philosophy, theology, but also for the whole of contemporary sciences. Because, this category institution questions the identity even the EC what we indicate like human, and consequently, like non human. Indeed, about what and which precisely speak us when it is question of Human or the Human ones, about humanity in generic term? What do we seek to describe and qualify under the registers of Human and non Human? Which are the non Human, of the binarism category ones presupposed which institute the registers of conceptual dualism Human/humanity/inhumanity? What do we seek to think in the idea of one “beyond the human one”, which the latter institutes a differentialism between the beings or entities or contrary, seeks with the indifférencier as in certain news of anthropology who challenges any anthropocentrée & humanistic design? Most traditional definitions tend to affirm that this we claim to conceptualize & to subsume under the concept of “Human” constitutes something like a certain exception in the order of the alive one. What is it exactly of this alleged “human exception”? What does it recover precisely? In addition, to answer the interrogation: “That are the non Human ones and which is the human ones? ”, does not have anything manifesto apart from this we can crudely observe and who allows us to or not retain, more or less, certain features of appearance and the behavior. We could also answer in a provocative and relativistic way: “With each one its Human and its non Human, its humanity and its nonhumanity”, according to the configurations which we institute to return account of the whole of the interactions that we let us have or not with our multiple, material & immaterial, visible & invisible environments! There thus do not exist standards which would tend in manner more or less final and universal to govern the uses category of Human and of the Human one. For these reasons, the challenge of our short communication will be to expose presupposed and reasons of such an interest category, conceptual and descriptive fundamentally Human for the non Human one which gives an account of the multiple forms of the institution of human the versus the nonhuman one, while founding, in addition, a whole range of beings or of entities going from the one with the other, thus questioning the imaginary and real bases of our multidimensional assignments in category terms, of identity, anybody, cognitive faculties, various heritages, sensitivity, etc. (shrink)

In the following essay I will describe the cultural and disciplinary areas in which Open Theism has been developing and deal with the main authors, who has defended this new doctrine, and their main works. In the second section I will analyse their main theses about divine attributes, some theological questions, several objections to this new non-standard theism and their rebuttals. In the conclusion I will highlight the problems still open and evaluate the overall Open Theism’s theoretical work. At (...) the end, also the text "Omniscience, Freedom, and Mystery", a part of the article TRANSLATED into english. The issue of omniscience is one of the most debated in contemporary Analytical Philosophy of Religion. However, what is often lacking in this discussion is a deep understanding of the dilemma of omniscience and human freedom within a complete epistemological (what can we really say about the divine and the world), metaphysical and theological framework. For example, it is often forgotten to frame some issues within a clear definition of the notion of mystery. I defined what we can mean by “mystery” in this forthcoming article: "Trinity and Mystery. Three Models: Aquinas, Leibniz, and Hegel" In the same article (and also in the first article mentioned above) can be found a reflection on the analogical use of terms, which involve the terms (the notions) of “freedom” and “omniscience”. This use, therefore, could make possible to develop the argument I propose. (shrink)

In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of—primarily state-supported—mathematics: (a) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that first-order Set Theory can be treated as the lingua franca of mathematics, since its theorems—even if unfalsifiable—can be treated as ‘knowledge’ because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (b) the slowly emerging, but (...) powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as ‘knowledge’ which are, further, categorically communicable as Factually Grounded Beliefs—hence as comprehensible pre-formal ‘mathematical truths’—by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing under-estimation of the value to society of ‘semantic arithmetical truth’, and an increasing over-estimation of the value to society of ‘syntactic set-theoretical provability’; a chasm which must be narrowed and bridged explicitly. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical ‘provability’ and mathematical ‘truth’ need to be interdependent and complementary, ‘evidence-based’, assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical ‘proofs’ may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical ‘truths’ must be viewed as seeking to ensure that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the post-graduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent—and hitherto unsuspected—unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a ‘mathematical truth’; (ii) what we can evidence as a ‘mathematical truth’; and (iii) what we ought not to believe is a ‘mathematical truth’; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be ‘arithmetical truth’, not ‘set-theoretical proof’. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula φ⁠, any complex formula in which φ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple (...) contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

The paper discusses from a metaphysical standpoint the nature of the dependence relation underpinning the talk of mutual action between material and spatiotemporal structures in general relativity. It is shown that the standard analyses of dependence in terms of causation or grounding are ill-suited for the general relativistic context. Instead, a non-standard analytical framework in terms of structural equation modeling is exploited, which leads to the conclusion that the kind of dependence encoded in the Einstein field equations is (...) a novel one. (shrink)

Inferentialism is the view that representational content is explained by lingual or mental states interacting according to inferential rules. Mendelovici and Bourget have argued against inferentialism that rules of inference do not sufficiently constrain content. This paper argues that their argument can be further strengthened such that its conclusion yields that content and inferential roles are strictly independent. It will then be argued that this conclusion is untenable and that the argument, rather than undermining inferentialism, corrodes the model theoretic foundations (...) of standard philosophical semantics. The paper concludes with some hints towards an alternative semantic paradigm. (shrink)

One of the advantages of my account in the essay “Instantiation as Partial Identity” was capturing the contingency of instantiation—something David Armstrong gave up in his experiment with a similar view. What made the contingency possible for me was my own non-standard account of identity, complete with the apparatus of counts and aspects. The need remains to lift some obscurity from the account in order to display its virtues to greater advantage. To that end, I propose to respond to (...) those who have grappled with it in print. There are various criticisms by commentators: that it is rendered absurd by the transitivity of identity, that it makes instantiation necessary instead of contingent, that it is unclear what counts are, that aspects are simply tropes, that my view does not capture multiple location, that I make an unclear reference to a theory of composition as identity, that the account suffers from problems with polyadicity, and that it is not a realist account of universals after all. I give responses to these objections. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

What is the connection between modelling thought and modelling the brain? In a model (as understood here), we strip away from the modelled system some non-essential features and retain some essential ones. What are the essential features of thought that are to be re- tained in the model, and conversely, what are its inessential features, that may be stripped away in the model? According to a prevalent view in contemporary science and philoso- phy, thought is a computation, and therefore its (...) essential features are its computational features. A necessary part of the computational view of thought is the idea that the same computation can be realised by, or implemented in, physically heterogeneous systems, an idea known as “Multiple Realizability” of the computational features or properties by the physical ones. I will describe why the very idea of Multiple Realizability, especially in the case of mental computation, entails mind-body dualism, and explore some implications of this conclusion concerning the question of which are the essential features of thought to be retained in modeling it. (shrink)

The text is a continuation of the article of the same name published in the previous issue of Philosophical Alternatives. The philosophical interpretations of the Kochen- Specker theorem (1967) are considered. Einstein's principle regarding the,consubstantiality of inertia and gravity" (1918) allows of a parallel between descriptions of a physical micro-entity in relation to the macro-apparatus on the one hand, and of physical macro-entities in relation to the astronomical mega-entities on the other. The Bohmian interpretation ( 1952) of quantum mechanics proposes (...) that all quantum systems be interpreted as dissipative ones and that the theorem be thus derstood. The conclusion is that the continual representation, by force or (gravitational) field between parts interacting by means of it, of a system is equivalent to their mutual entanglement if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. General relativity can be interpreted as a superluminal generalization of special relativity. The postulate exists of an alleged obligatory difference between a model and reality in science and philosophy. It can also be deduced by interpreting a corollary of the heorem. On the other hand, quantum mechanics, on the basis of this theorem and of V on Neumann's (1932), introduces the option that a model be entirely identified as the modeled reality and, therefore, that absolutely reality be recognized: this is a non-standard hypothesis in the epistemology of science. Thus, the true reality begins to be understood mathematically, i.e. in a Pythagorean manner, for its identification with its mathematical model. A few linked problems are highlighted: the role of the axiom of choice forcorrectly interpreting the theorem; whether the theorem can be considered an axiom; whether the theorem can be considered equivalent to the negation of the axiom. (shrink)

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem of (...) Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

Andreas & Günther have recently proposed a difference-making definition of actual causation. In this paper I show that there exist conclusive counterexamples to their definition, by which I mean examples that are unacceptable to everyone, including AG. Concretely, I show that their definition allows c to cause e even when c is not a causal ancestor of e. I then proceed to identify their non-standard definition of causal models as the source of the problem, and argue that there is (...) no viable strategy open to AG to fixing it. I conclude that their definition of causation is damaged beyond repair. (shrink)

Abstract. The article is devoted to the possibilities of the formation and development of engineering thinking. The paper considers the features of engineering thinking, compares various concepts that characterize engineering activities. The authors compare the concepts of technical, economic, research thinking, identifying the principles of engineering thinking. The need for a humanitarian component in engineering thinking is noted. Consistency and multidimensionality are considered by the authors as the most important concepts for the formation of engineering thinking. In conclusion, the authors (...) point out that engineering thinking universally orients the employee when solving different types of tasks and contributes to greater variability in making standard and non-standard decisions in the production process. -/- . (shrink)