We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form$L = \langle \omega, \wedge, \vee,0,1,{ \le _L}\rangle$whereωdenotes the set of natural numbers and the following four conditions hold: (1)$\wedge, \vee$are binary computable operations; (2)${ \le _L}$is a computably enumerable preordering relation, with$0{ \le _L}x{ \le _L}1$for everyx; (3) the equivalence relation${ \equiv _L}$originated by${ \le _L}$is a congruence onLsuch that the corresponding quotient structure is a nontrivial bounded lattice; (4) the${ \equiv _L}$-equivalence classes of 0 and 1 (...) form an effectively inseparable pair of sets). Solving a problem in (Montagna & Sorbi, 1985) we show (Theorem 4.2), that ifLis an e.i. prelattice then${ \le _L}$is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relationRthere exists a computable functionfreducingRto${ \le _L}$, i.e.,$xRy$if and only if$f\left( x \right){ \le _L}f\left( y \right)$, for all$x,y$. In fact (Corollary 5.3)${ \le _L}$is locally universal, i.e., for every pair$a{ < _L}b$and every c.e. preordering relationRone can find a reducing functionffromRto${ \le _L}$such that the range offis contained in the interval$\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also (Theorem 5.7)${ \le _L}$is uniformly dense, i.e., there exists a computable functionfsuch that for every$a,b$if$a{ < _L}b$then$a{ < _L}f\left( {a,b} \right){ < _L}b$, and if$a{ \equiv _L}a\prime$and$b{ \equiv _L}b\prime$then$f\left( {a,b} \right){ \equiv _L}f\left( {a\prime,b\prime } \right)$. Some consequences and applications of these results are discussed: in particular (Corollary 7.2) for$n \ge 1$the c.e. preordering relation on${{\rm{\Sigma }}_n}$sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s systemRorQis locally universal and uniformly dense; and (Corollary 7.3) the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

« Nous sommes embarqués », certainement ; le tout est de savoir comment. On peut bien invoquer une condition commune, qui est celle des habitants d’une planète exposée aux risques de transformations soudaines. Mais cela ne nous autorise pas à dire que cette condition est celle de l’inséparation. Ce qu’il y a d’inséparé entre les êtres est toujours localisé. Il correspond à ce que Gilbert Simondon appelle le « transindividuel ». L’inséparé, qui existe localement entre quelques êtres, est l’enjeu d’un (...) travail dialectique – un travail d’inséparation. Un travail qui se reconnaît à ceci qu’il permet d’abriter l’expérience d’un temps commun. L’existence même du temps commun est l’enjeu essentiel de la politique aujourd’hui, et la condition d’une action à la mesure de la situation. Car ce qu’impose avant tout l’ennemi aujourd’hui, c’est bien une certaine forme du temps, qu’il s’agit de briser. (shrink)

Berkeley and Hume object to Locke's account of abstraction. Abstraction is separating in the mind what cannot be separated in reality. Their objection is that if a is inseparable in reality from b, then the idea of a is inseparable from the idea of b. The former inseparability is the reason for the latter. In most interpretations, however, commentators leave the former unexplained in explaining the latter. This article assumes that Berkeley and Hume present a unified front against Locke. (...) Hume supplements Berkeley's argument just where there are gaps. In particular, Hume makes explicit something Berkeley leaves implicit: The argument against Locke depends on the principle that things are inseparable if and only if they are identical. Abstraction is thinking of one of an inseparable pair while not thinking of the other. But doing so entails thinking of something while not thinking of it. This is the fundamental objection. (shrink)

In [7] and [8], it is established that given any abstract countable structure S and a relation R on S, then as long as S has a recursive copy satisfying extra decidability conditions, R will be ∑0α on every recursive copy of S iff R is definable in S by a special type of infinitary formula, a ∑rα() formula. We generalize the typ e of constructions of these papers to produce conditions under which, given two disjoint relations R1 and R2 (...) on S, there is a recursive copy of S in which R1 and R2 are 0α inseparable. We then apply these theorems to specific everyday structures such as linear orderings, boolean algebras andvector spaces. (shrink)

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, (...) if X intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset. (shrink)

In this guest column, the author argues against Wittgenstein's aphorism that the human body is the best picture of the human soul. Such a picture, she shows, would have to include all that a person is, in some sense, inseparable from. Drawing from Plato's analogy between city and soul, and from Socrates's description of a luxurious city, this piece argues that what we are inseparable from are mostly things and persons that our body can go on without. To the extent (...) that such things and persons, and the relationship in which we stand to them, are distinct from our bodies, what we are inseparable from is also a sign that we are not self-sufficient creatures. Thus, the piece concludes, inseparability is evidence of the species' gregariousness, and the body is only one among many tokens of a person's soul. (shrink)

In this paper I present the Ingardenian distinction between inseparability and dependence. My considerations are both historical and systematic. The historical part of the paper accomplishes two goals. First, I show that in the Brentanian tradition the problem of existential conditioning was entangled into parts—whole theories. The best examples of such an approach are Kazimierz Twardowski’s theory of the object and Edmund Husserl’s theory of parts and wholes. Second, I exhibit the context within which Ingarden distinguished inseparability and (...) dependence. Moreover, Ingarden’s motivations are presented: the problem of understanding the Husserlian concept of “immanent transcendence,” the issue of the existence of purely intentional objects, and finally the problem of the relationship between individual objects and ideas. The systematic part deals with the ambiguity of Ingarden’s definition of inseparability. I seek to improve this definition by reference to the distinction between absolute and summative wholes. I also present some divisions of inseparability and dependence and investigate whether these types of existential conditioning are reflexive, symmetric, or transitive. (shrink)

An elementary theory T in a language L is (strongly) finitely inseparable if the set of logically valid sentences of L and the set of T-finitely refutable sentences are recursively inseparable. In §1 we establish a sufficient condition for the elementary theory of a class of BA's with operators to be finitely inseparable. This is done using the methods developed independently by M. Rabin and D. Scott (see [6]) on the one hand and by Ershov on the other (see [2]).

This paper will first introduce first-order mereotopological axioms and axiomatized theories which can be found in some recent literature and it will also give a survey of decidability, undecidability as well as other relevant notions. Then the main result to be given in this paper will be the finite inseparability of any mereotopological theory up to atomic general mereotopology (AGEMT) or strong atomic general mereotopology (SAGEMT). Besides, a more comprehensive summary will also be given via making observations about other (...) properties stronger than undecidability. (shrink)

We deal with ontological problems concerning basic systems of explicit mathematics, as formalized in Jäger's language of types and names. We prove a generalized inseparability lemma, which implies a form of Rice's theorem for types and a refutation of the strong power type axiom POW + . Next, we show that POW + can already be refuted on the basis of a weak uniform comprehension without complementation, and we present suitable optimal refinements of the remaining results within the weaker (...) theory. (shrink)

The essay examines the impact of thought insertion on typical conceptions of self-consciousness. Stephens and Graham have recently argued that thought insertion is compatible with the inseparability thesis, which maintains that with regard to self-consciousness subjectivity is a proper part of introspection--introspection and subjectivity are inseparable. They argue that thought insertion is an error of agency and not an error of subjectivity. The essay contends that even if they are correct in their interpretation that thought insertion is an error (...) of agency rather than subjectivity, which is unlikely, they are incorrect to maintain that it is not also an error of subjectivity. Evidence is put forth to indicate that thought insertion is, at best, a mistake of agency and subjectivity. It is concluded that thought insertion is incompatible with the inseparability thesis, and a new inseparability thesis is thereby postulated. (shrink)

The present article argues that the doctrine of the inseparable external operations of the Trinity is consistent with the doctrine of the incarnation of the Son alone. To demonstrate this, it will be shown, first, that the assumption of human nature can be ascribed to the Son alone when taken as a state, as opposed to an action. Secondly, I will defend John Owen’s claim that the Son is not the “immediate” agent of Christ’s actions. Finally, an appeal is made (...) to Trinitarian “missions” to show the coherence of OAE with our having communion with distinct divine persons. (shrink)

Résumé La publication, en 2020, de la nouvelle de Beauvoir, désormais intitulée Les Inséparables, constitue une bonne surprise pour ses lecteur.trice.s. Jugée, en 1954, sans intérêt par son auteure (dont il faut interroger les motivations), elle se révèle passionnante pour comprendre la genèse des Mémoires d’ une jeune fille rangée, autobiographie qu’ elle annonce (sans l’ égaler). Elle intéresse également par elle-même, car elle apporte un éclairage supplémentaire sur l’ histoire tragique de Zaza, et sur l’ amitié qui l’ unit (...) à la jeune Simone. (shrink)

One of the most important theses of Existential Thomism is that contingent things are composed of essence and existence. The thesis is immediately supplemented by a proviso that these components are not parts in the regular sense of the word. Essence and existence are not extended pieces of the thing which can be detached from it. They are inseparable aspects of the thing wherein inseparability is understood as a sort of dependence. In my paper I analyze the thesis of (...) the inseparability of existence. For me this is untenable. I argue that this inseparability blocks up the essential function ascribed by Thomists to existence: the function of making the thing real. Thus to save this function we are forced to export existence to outside the thing. It contradicts our deep belief that existence is the most intimate aspect of the thing. Therefore the Thomistic analysis of existence seems to be invalid. (shrink)

At Aristotle,MetaphysicsE.1, 1026a14, Schwegler’s conjectural emendation of the manuscript reading ἀχώριστα to χωριστά has been widely adopted. The objects of physical science are therefore here ‘separate’, or ‘independently existent’. By contrast, the manuscripts make them ‘not separate’, construed by earlier commentators as dependent on matter. In this paper, I offer a new defense of the manuscript reading. I review past defenses based on the internal consistency of the chapter, explore where they have left supporters of the emendation unpersuaded, and attempt (...) to strengthen their appeal. I challenge Schwegler’s central case, developed by Ross and others, that the construction μὲν ἀλλ’ οὐκ demands an implausible ‘logical antithesis’ between inseparability and mobility. This is arguably the fundamental obstacle to the manuscript reading, and counterexamples have not to date convinced emenders. I offer a new, systematic review of Aristotle’s use of the phrase, including relevant cases from theRhetoric, to show how its usual meaning in Aristotle supports the transmitted text; I also reply to possible objections. Finally, I explore the implications of this defense for the classification of the sciences in E.1. (shrink)

We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of (...) interpretability relies on an adaptation of techniques leading to Nies’ theorem. (shrink)

Emmanuel Levinas is variously used to provide a conceptualization of ethics from which to deduce an ethical politics, an account of the movement from ethics to politics or an exhortation to continually interrupt politics in the name of ethics. What all these approaches share is a reading of Levinas where ethics and politics are separated and ethics is prioritized. My argument in this article is that if the concept of the Third is given due weight in Levinas's work then this (...) separation and prioritization is untenable. The reading advanced foregrounds the Third and in doing so demonstrates that the unproblematized ‘ethics’ often drawn from Levinas is more complex than might initially appear. I argue that if the Third is taken seriously then Levinas's work leads to a requirement to think in terms of the ethico-political, so complicating any attempt to deduce politics from ethics which draws on this. (shrink)

We exhibit a construction which produces for every Turing machine T with two halting states μ 0 and μ -1 , an algebra B(T) (finite and of finite type) with the property that the variety generated by B(T) is residually large if T halts in state μ -1 , while if T halts in state μ 0 then this variety is residually bounded by a finite cardinal.

Résumé La parution posthume des Inséparables invite à distinguer les deux modalités d’ écriture par lesquelles Beauvoir a fait revivre la figure majeure pour elle de Zaza (1907-1929), son amie d’ enfance et de jeunesse : la fiction puis l’ autobiographie ; et cela à la lumière des appréciations de l’ autrice et du pacte de lecture propre à chaque modalité d’ écriture. Ces approches successives sur plusieurs décennies témoignent de la fascination constante de Beauvoir pour Zaza et incitent le (...) lecteur à prolonger l’ exploration beauvoirienne par les écrits mêmes de Zaza. (shrink)

We investigate the Peres–Horodecki positive partial transpose criterion in the context of conserved quantities and derive a condition of inseparability for a composite bipartite system depending only on the dimensions of its subsystems, which leads to a bi-linear entanglement witness for the two qubit system. A separability inequality using generalized Schrodinger–Robertson uncertainty relation taking suitable operators, has been derived, which proves to be stronger than the bi-linear entanglement witness operator. In the case of mixed density matrices, it identically distinguishes (...) the separable and non separable Werner states. (shrink)

Several noted political theorists have argued that a state can be legitimate even if it does not generate in its citizens an obligation to obey the law. I argue that this claim is false. All plausible analyses of political legitimacy either build in the concept of political obligation, or else incorporate claims that require some account of political obligation. In either case, political legitimacy is possible only when a state successfully generates in its citizens an obligation to obey the law.

Nous sommes désormais entrés dans l’ontologie de l’inséparation. Un mouvement de fond nous a fait passer d’un univers humaniste composé d’entités séparables à un réel inséparé où tous les phénomènes devenus globalisés sont liés a priori (et non a posteriori), en intra- relation de co-évolution et de co-dépendance. Notre monde élabore une nouvelle condition d’existence, d’où la figure de l’Autre a disparu. Est-ce bien? Est-ce mal? Là n’est pas la question. Cela est.

In this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q₊. The theory, subset of Presburger theory of addition S₊, is the additive fragment of Robinson system Q. We prove that every set that separates Q₊` from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models. The (...) result implies also that any theory of addition that is consistent with Q₊—in particular any theory contained in S₊—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories. Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S₊. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable. (shrink)

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

In a recent paper, Montagna proved the undecidability of the first-order theory of diagonalisable algebras. This result is here refined — the set of finitely refutable sentences is shown effectively inseparable from the set of theorems. The proof is quite simple.

In this paper we explore the hypothesis that constrained uses of imagination are crucial to economic modeling. We propose a theoretical framework to develop this thesis through a number of specific hypotheses that we test and refine through six new, representative case studies. Our ultimate goal is to develop a philosophical account that is practice oriented and informed by empirical evidence. To do this, we deploy an abductive reasoning strategy. We start from a robust set of hypotheses and leave space (...) for the generation of further hypotheses and theoretical claims based on the qualitative analysis of new empirical data. (shrink)

Sources for Stoicism present conflicting accounts of the Stoic principles. Some suggest that the principles are inseparable from each other. Others suggest that they are separable. To resolve this apparent interpretive dilemma, I distinguish between the functions of the principles and the bodies that realize those functions. Although the principles cannot separate when realizing their roles, the Stoic theory of blending entails that the bodies that realize those roles are physically separable. I present a strategy for further work on the (...) principles in light of this interpretation. (shrink)

This book explores the nature of human freedom, or what Crosby calls genuine freedom. He argues at length for the crucial importance of genuine freedom for responsible and meaningful human life and takes extended issue, on practical as well as theoretical grounds, with those who argue for the compatibility of freedom with causal determinism.

Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results (...) for the apodictic syllogistic in the Prior Analytics but it also generates rather than assumes Aristotle's distinctions among 'necessary', 'essential' and 'accidental'. By developing a system around tests that are in Aristotle and basic to ancient Greek philosophy, the system is linked to a history of practices, providing a platform for future work on the origins of logic. (shrink)

Disputes about how to understand intersectional relations often pivot around the tension between separateness and inseparability, where some scholars emphasize the need to separate between different intersectional categories while others claim they are inseparable. In this article the author takes issue with the either/or thinking that underpins an unnecessary and unproductive polarization in the debate over the in/separability of intersectional categories. Drawing on Roy Bhaskar’s dialectical critical realist philosophy, the author argues that we can think of intersectional categories as (...) well as different ontological levels as both distinct and unified and elaborates on the issue of how significance of the dialectical notion of unity-in-difference for intersectional studies. As part of the argument the author addresses the issue of what it actually means for something to be distinct or separate as opposed to inseparable or unified with something else, demonstrating that lack of clarity about this is at the heart of polarized arguments about separateness versus inseparability in intersectionality theory. (shrink)

The most efficient way of obtaining information about the state of a quantum system is not always a direct measurement. It is sometimes preferable to extend the original Hilbert space of states into a larger space, and then to perform a quantum measurement in the enlarged space. Such an extension is always possible, by virtue of Neumark's theorem. The physical interpretation usually given to that theorem is the introduction of an auxiliary quantum system, prepared in a standard state, and the (...) execution of a quantum measurement on both systems together. However, this widespread interpretation is unacceptable, because the statistical properties of the supposedly standard auxiliary system are inseparably entangled with those of the original, unknown system. A different method of preparing the auxiliary system is proposed, and shown to be physically acceptable. (shrink)