This chapter contains sections titled: Introduction The Stoic Logical System Conclusion Bibliography.

It has been noted before in the history of logic that some of Frege's logical and semantic views were anticipated in Stoicism. In particular, there seems to be a parallel between Frege's Gedanke (thought) and Stoic lekton; and the distinction between complete and incomplete lekta has an equivalent in Frege's logic. However, nobody has so far claimed that Frege was actually influenced by Stoic logic; and there has until now been no indication of such a causal connection. (...) In this essay, we attempt, for the first time, to provide detailed evidence for the existence of this connection. In the course of our argumentation, further analogies between the positions of Frege and the Stoics will be revealed. The classical philologist Rudolf Hirzel will be brought into play as the one who links Frege with Stoicism. The renowned expert on Stoic philosophy was Frege's tenant and lived in the same house as the logician for many years. In der Geschichte der Logik ist häufig bemerkt worden, dass einige der logischen und semantischen Auffassungen Freges in der Stoa antizipiert worden sind. Genannt wurden insbesondere die Parallelen zwischen dem Fregeschen Gedanken und dem stoischen Lekton sowie die Unterscheidung zwischen vollständigen und unvollständigen Lekta, die bei Frege ihre Entsprechung hat. Ein Wirkungszusammenhang ist allerdings nicht behauptet worden. Dazu gab es bislang auch keinen Anlass. Der vorliegende Beitrag versucht erstmalig, einen detaillierten Indizienbeweis für das Bestehen eines solchen Zusammenhangs vorzulegen. Dabei werden weitere charakteristische Übereinstimmungen zwischen Frege und der Stoa aufgewiesen. Als Mittelsmann wird der Altphilologe Rudolf Hirzel vorgestellt. Er wohnte lange Jahre als Mieter zusammen mit Frege im selben Haus und war ein anerkannter Experte der stoischen Philosophie. (shrink)

In this wide-ranging study, Ada Bronowski reconstructs the Stoic account of lekta. Usually translated into English as "sayables," lekta—Bronowski transliterates all key Greek terms—number among the Stoics' most contentious proposals in antiquity and remain the subject of interpretative disagreement today. One clear Stoic commitment is that lekta are what is signified by human speech: my utterance "Dion walks" in normal circumstances signifies the lekton 'Dion walks.' Are lekta to be understood, then, simply as meanings? And (...) is the existence of lekta dependent on token utterances? No, claims Bronowski. 'Dion walks' is part of reality whether or not one utters "Dion walks." Lekta are "there to be said, but need not... (shrink)

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version (...) of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models. (shrink)

In 1973, Dag Prawitz conjectured that the calculus of intuitionistic logic is complete with respect to his notion of validity of arguments. On the background of the recent disproof of this conjecture by Piecha, de Campos Sanz and Schroeder-Heister, we discuss possible strategies of saving Prawitz's intentions. We argue that Prawitz's original semantics, which is based on the principal frame of all atomic systems, should be replaced with a general semantics, which also takes into account restricted frames of (...) atomic systems. We discard the option of not considering extensions of atomic systems, but acknowledge the need to incorporate definitional atomic bases in the semantic framework. It turns out that ideas and results by Westerståhl on the Carnap categoricity of intuitionistic logic can be applied to Prawitz semantics. This implies that Prawitz semantics has a status of its own as a genuine, though incomplete, semantics of intuitionstic logic. An interesting side result is the fact that every formula satisfiable in general semantics is satisfiable in an axioms-only frame (a frame whose atomic systems do not contain proper rules). We draw a parallel between this seemingly paradoxical result and Skolem's paradox in first-order model theory. (shrink)

Dirac's approach to gauge symmetries is discussed. We follow closely the steps that led him from his conjecture concerning the generators of gauge transformations {\it at a given time} ---to be contrasted with the common view of gauge transformations as maps from solutions of the equations of motion into other solutions--- to his decision to artificially modify the dynamics, substituting the extended Hamiltonian for the total Hamiltonian. We show in detail that Dirac's analysis was incomplete and, in completing (...) it, we prove that the fulfilment of Dirac's conjecture ---in the ``non-pathological" cases--- does not imply any need to modify the dynamics. We give a couple of simple but significant examples. (shrink)

This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of (...) the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature. (shrink)

Kurt Gödel’s Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Gödel was able to find a way for any given P (UTM), (read as, “P of UTM” for “Program of Universal Truth Machine”), actually to write down a complicated polynomial that has a solution iff (=if and only if), G is true, where G stands for a Gödel-sentence. So, if G’s truth is a necessary condition for the truth of a given polynomial, then P (UTM) has to answer first that (...) G is true in order to secure the truth of the said polynomial. But, interestingly, P (UTM) could never answer that G was true. This necessarily implies that there is at least one truth a P (UTM), however large it may be, cannot speak out. Daya Krishna and Karl Potter’s controversy regarding the construal of India’s Philosophies dates back to the time of Potter’s publication of “Presuppositions of India’s Philosophies” (1963, Englewood Cliffs Prentice-Hall Inc.) In attacking many of India’s philosophies, Daya Krishna appears to have unwittingly touched a crucial point: how can there be the knowledge of a ‘non-cognitive’ mokṣa? [‘mokṣa’ is the final state of existence of an individual away from Social Contract]—See this author’s Indian Social Contract and its Dissolution (2008) mokṣa does not permit the knowledge of one’s own self in the ordinary way with threefold distinction, i.e., subject–knowledge-object or knower–knowledge–known. But what is important is to demonstrate whether such ‘knowledge’ of non-cognitive mokṣa state can be logically shown, in a language, to be possible to attain, and that there is no contradiction involved in such demonstration, because, no one can possibly express the ‘experience-itself’ in language. Hence, if such ‘knowledge’ can be shown to be logically not impossible in language, then, not only Daya Krishna’s arguments against ‘non-cognitive mokṣa’ get refuted but also it would show the possibility of achieving ‘completeness’ in its truest sense, as opposed to Gödel’s ‘Incompleteness’. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of mokṣa is construed as the state of complete knowledge in Advaita. This possibility of ‘completeness’ is set in this paper in the backdrop of Śrī Śaṅkarācārya’s Advaitic (Non-dualistic) claim involved in the mahāvākyas (extra-ordinary propositions). (Mahāvākyas that Śaṅkara refers to are basically taken from different Upaniṣads. For example, “Aham Brahmāsmi” is from Bṛhadāraṇyaka Upanisad, and “Tattvamasi” is from Chāndogya Upaniṣad. Śrī Śaṅkarācārya has written extensively. His main works include his Commentary on Brahma-Sūtras, on major Upaniṣads, and on ŚrīmadBhagavadGītā, called Bhāṣyas of them, respectively. Almost all these works are available in English translation published by Advaita Ashrama, 5 Dehi Entally Road, Calcutta, 700014.) On the other hand, the ‘Incompleteness’ of Gödel is due to the intervening G-sentence, which has an adverse self-referential element. Gödel’s incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer (Kurt Gödel: on formally undecidable propositions of principia mathematica and related systems. Oliver & Boyd, Edinburgh, 1962). The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says,—“Gödel’s novel metamathematical method is that of attaching numbers to the signs, to the series of signs (formulae) and to the series of series of signs (“proof-schemata”) which occur in his formal system…Gödel invented what might be called co-ordinate metamathematics…”) Meltzer (1962 p. 7). In Antone (2006) it is said “The problem is that he (Gödel) tries to replace an abstract version of the number (which can exist) with the concept of a real number version of that abstract notion. We can state the abstraction of what the number needs to be, [the arithmoquining of a number cannot be a proof-pair and an arithmoquine] but that is a concept that cannot be turned into a specific number, because by definition no such number can exist.”.), especially so where first-hand personal experience is called for. Therefore, what ultimately rules is the semanticity as in a first-hand experience. Similar points are voiced, albeit implicitly, in Antone (Who understands Gödel’s incompleteness theorem, 2006). (“…it is so important to understand that Gödel’s theorem only is true with respect to formal systems—which is the exact opposite of the analogous UTM (Antone (2006) webpage 2. And galatomic says in the same discussion chain that “saying” that it ((is)) only true for formal systems is more significant… We only know the world through “formal” categories of understanding… If the world as it is in itself has no incompleteness problem, which I am sure is true, it does not mean much, because that is not the world of time and space that we experience. So it is more significant that formal systems are incomplete than the inexperiencable ‘World in Itself’ has no such problem.—galatomic”) Antone (2006) webpage 2. Nevertheless galatomic certainly, but unwittingly succeeds in highlighting the possibility of experiencing the ‘completeness’ Second, even if any formal system including the system of Advaita of Śaṅkara is to be subsumed or interpreted under Gödel’s theorem, or Tarski’s semantic unprovability theses, the ultimate appeal would lie to the point of human involvement in realizing completeness since any formal system is ‘Incomplete’ always by its very nature as ‘objectual’, and fails to comprehend the ‘subject’ within its fold. (shrink)

A good division of biology is important for clarity of thought, presentation of the problems and nowadays also for science policy. Various classifications are in use, depending upon different parameters. A division can be based on objects and on aspects. A modern classification on objects is based on the levels of organisation. Methodological principles play a role in the background.

In their recent book Every Thing Must Go, Ladyman and Ross claim: (i) Physics is analytically complete since it is the only science that cannot be left incomplete. (ii) There might not be an ontologically fundamental level. (iii) We should not admit anything into our ontology unless it has explanatory and predictive utility. In this discussion note I aim to show that the ontological commitment in implies that the completeness of no science can be achieved where no (...) fundamental level exists. Therefore, if claim requires a science to actually be complete in order to be considered as physics,, and if Ladyman and Ross's “tentative metaphysical hypothesis ... that there is no fundamental level” is true,, then there simply is no physics. Ladyman and Ross can, however, avoid this unwanted result if they merely require physics to ever strive for completeness rather than to already be complete. (shrink)

Russell’s use of incomplete symbols constituted progress in philosophy. They allowed Russell to make true negative existential claims, like ‘the present King of France does not exist’, and to analyse away logical constructs like tables. Russell’s view rested on the availability of complete symbols, logically proper names, which single out objects which we know by acquaintance, which we are committed to, and to whose existence discourse about apparent complexes can be reduced. Susan Stebbing enthusiastically embraced incomplete symbols (...) for use in negative existentials and in her innovative theory of metaphysical analysis. Yet she also raised trenchant objections to Russell’s assumption that complete symbols are purely referring expressions, equivalent to mere demonstration or pointing. Stebbing thought it could not be assumed that analysis terminates in objects of acquaintance. In this chapter I lay out Stebbing’s arguments and argue that they bear a striking similarity to those made by Stebbing’s former college’s Mistress, Constance Jones, whose influence on Russell is now largely forgotten but worth recovering. Jones defended a sense-reference (signification-denotation) distinction before Frege. Russell’s objections to that distinction in ‘On Denoting’ appear to borrow from Jones’s presentation of it. In Russell and Jones’s 1911 exchange at the Aristotelian Society, Jones’s arguments that names have signification and are more than mere noises or shapes, and that unless both signification and denotation are taken into account, all identity statements reduce to the trivial ‘a=a’ form, foreshadow Stebbing’s later arguments against complete symbols. (shrink)

This book covers completeness of first-order logic, some model theory, Gödel's incompleteness theorems and related results, and a smattering of computability theory. The text is self-contained and provides full proofs of the main facts. Ideally, the reader of this work has already taken at least one introductory logic course; however, everything needed to understand the syntax and semantics of first-order logic is presented herein. Students from philosophy, linguistics, computer science, physics, and other related subjects will find this work useful and (...) enlightening. (shrink)

What is the best way to make sustained societal progress over time? Non-ideal theory done on its own faces the problem of second best, but ideal theory seems unable to cope with disagreement about how to make progress. If ideal theory gives up its claims to completeness, then we can use the method of incompletely theorized agreements to make progress over time.

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; (...) 4. Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

John Rawls’s use of the “fully cooperating assumption” has been criticized for hindering attempts to address the needs of disabled individuals, or non-cooperators. In response, philosophers sympathetic to Rawls’s project have extended his theory. I assess one such extension by Cynthia Stark, that proposes dropping Rawls’s assumption in the constitutional stage (of his four-stage sequence), and address the needs of non-cooperators via the social minimum. I defend Stark’s proposal against criticisms by Sophia Wong, Christie Hartley, and Elizabeth Edenberg and Marilyn (...) Friedman. Nevertheless, I argue that Stark’s proposal is crucially incomplete. Her formulation of the social minimum lacks accompanying criteria with which the adequacy of the provisions for non-cooperators may be assessed. Despite initial appearances, Stark’s proposal does not fully address the needs of non-cooperators. I conclude by considering two payoffs of identifying this lack of criteria. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...) establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

The Stoics’ guiding principle in ontology is the Eleatic principle. Their existents are bodies that have the power to act and be acted upon. They account both for the constitution of material objects and the causal interactions among them in terms of such dynamic bodies. Blending is the physical mechanism that explains both constitution and causation; and is facilitated by the fact that for the Stoics all bodies exist as unlimited divided. In this paper I offer a novel analysis of (...) this Stoic stance, and of the role it plays in their metaphysics. I argue that when blended, the substances involved become sharing subjects of qualities and structural features. (shrink)

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at (...) least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC. (shrink)

This article considers the virtue of tolerance as it is found in Epictetus and MarcusAurelius. It defines the virtue of tolerance and links it to the Stoic idea of proper control of the passions in pursuit of both self-sufficiency and justice. It argues that Stoic tolerance is neither complete in difference nor a species of relativism. Finally, it discusses connections between the moral virtue of Stoic tolerance and the idea of political toleration found in modern liberalism.

This note does examine a recent volume by Ada Bronowski on the Stoic doctrine of the lekta. The first part summarizes its content, while the second discusses some critical and controversial points, in particular the Platonic tenor of the interpretation of the Stoic lekta offered by the Author. This interpretation gives an extra-linguistic and realistic function to items that the Stoics cautiously left under-determined from the ontological point of view.

Christopher Gill ABSTRACT: How much theory is enough for a complete account of ancient Stoic ethics and for modern life-guidance? Stoic ethics was presented either purely in its own terms or combined with the idea of human or universal nature. Although the combination of ethical theory with human and universal nature provides the ….

Tristan J. Rogers ABSTRACT: What might a Stoic approach to politics look like? David Goodhart aptly describes the political divide pervading Western societies in terms of the ‘somewheres,’ who are communitarian, rooted in particular places, and resistant to social and political change, versus the ‘anywheres,’ who are cosmopolitan, mobile, and enthusiastic embracers of change. Stoicism ….

What can Stoicism offer to contemporary debates about forgiveness? Given their outright rejection of a reactive attitudes framework for responding to wrongdoing and their bold suggestions of how to revise our moral practices, the Stoics provide a valuable lens through which to re-evaluate various central claims in the debates about forgiveness. In this chapter, I highlight four common assumptions that the Stoics would consider problematic: firstly, that forgiveness is opposed to justice; secondly, that anger and resentment are necessary for registering (...) wrongdoing; thirdly, that anger and resentment are generally reliable at tracking the severity of wrongdoing; fourthly, that reconciliation with wrongdoers is an option rather than an imperative of virtue. Insofar as the Stoics provide defensible and often compelling alternatives to these positions, Stoicism offers a number of philosophical resources to re-conceptualize the way we think about forgiveness and suggests ways in which forgiveness might be better integrated into a virtue ethical framework. (shrink)

Brouwer's papers after 1945 are characterized by a technique known as the method of the creating subject. It has been supposed that the method was radically new in his work, since Brouwer seems to introduce an idealized mathematician into his mathematical practice. A newly opened source, the unpublished text of a lecture of Brouwer from 1934, fully supports the conclusions of our analysis that: - There is no idealized mathematician involved in the method;- The method was not new at (...) all;- Brouwer uses an incomplete sequence, also known as choice sequence, in this method, which is special. The method does not take its place in the standard works on choice sequences. (shrink)

If r is a reducibility between sets of numbers, a natural question to ask about the structure ? r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that (...) every tt-incomplete computably enumerable truth-table degree a is branching in ? tt . The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof. (shrink)

The Stoic understanding of virtue is often taken to be a non-starter. Many of the Stoic claims about virtue – that a virtue requires moral perfection and that all who are not fully virtuous are vicious – are thought to be completely out of step with our commonsense notion of virtue, making the Stoic account more of an historical oddity than a seriously defended view. Despite many voices to the contrary, I will argue that there is a way of making (...) sense of these Stoic claims. Recent work in linguistics has shown that there is a distinction between relative and absolute gradable adjectives, with the absolute variety only applying to perfect exemplars. I will argue that taking virtue terms to be absolute gradable adjectives – and thus that they apply only to those who are fully virtuous – is one way to make sense of the Stoic view. I will also show how interpreting virtue theoretic adjectives as absolute gradable adjectives makes it possible to defend Stoicism against its most common objections, demonstrating how the Stoic account of virtue might once again be a player in the contemporary landscape of virtue theorizing. (shrink)

Maya Deren was a Russian-born American filmmaker, theorist, poet, and photographer working at the forefront of the American avant-garde in the 1940s and 1950s. Influenced by Jean Cocteau and Marcel Duchamp, she is best known for her seminal film Meshes of the Afternoon, a dream-like experiment with time and symbol, looped narrative and provocative imagery, setting the stage for the twentieth-century's groundbreaking aesthetic movements and films. Maya Deren assesses both the filmmaker's completed work and her numerous unfinished projects, arguing Deren's (...) overarching aesthetic is founded on principles of incompletion, contingency, and openness. Combining the contrasting approaches of documentary, experimental, and creative film, Deren created a wholly original experience for film audiences that disrupted the subjectivity of cinema, its standards of continuity, and its dubious facility with promoting categories of realism. This critical retrospective reflects on the development of Deren's career and the productive tensions she initiated that continue to energize film. (shrink)

Fregean logic has difficulty with certain arguments in which there is cross-reference between premises and conclusion. In this paper I describe a method of handling arguments of the troublesome kind: It involves replacing standard quantifiers with explicit existential statements, and turns standard logic into a free one. A validation procedure is provided for the logic.

We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.

Kripke bundle and C-set semantics are known as semantics which generalize standard Kripke semantics. In [4] and in [1, 2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics . Modal predicate logic Q-S4.1 is not Kripke bundle complete of the present paper). This is shown by using difference of Kripke bundle semantics and C-set semantics. In this (...) paper, by using the same idea we show that incompleteness results in Kripke bundle semantics which are extended versions of [2]. (shrink)

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however (...) complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π. (shrink)

This paper provides an algebraic reconstruction of Einstein’s argument for the incompleteness of quantum mechanics, in order to clarify the assumptions that underlie an understanding of Einstein completeness as categoricity.

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions have been considered and axiomatic systems which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we (...) use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one. (shrink)

BB′IW logic (or T→ is known to be D-complete. This paper shows that there are infinitely many weaker D-complete logics and it also examines how certain D-incomplete logics can be made complete by altering their axioms using simple substitutions.

In this work I have tried to justify the principle theses of Meinong's Theory of Objects and to criticize some ontological theses that do not seem to derive from the Theory of Objects. This work has been divided into three parts according to the fundamental classification of objects given by Meinong: object of representation, object of thought, and objects of emotion. ;In the first part I have analysed the doctrine of intentionality in relation with intentional act and content, and (...) their sorts of presentation in discussion with Brentano. ;In following I have given an interpretation of "pure object" and its status of "Assersein" by which pure object is beyond being and non-being. This is developed in close connection to the doctrine of impossible objects and the Theory of Objects as the most universal science inasmuch as it harbors both real and unreal worlds . I have built an extensive critique of Russell's theory of Descriptions and Denoting, and other opposing views. The rest of this part is devoted to both incomplete objects and objects of higher order . The theory of incomplete object is connected to our way of knowing things, and it expresses the incompleteness of our intentional attitudes. The theory of objects of higher order tries to explain the problem of space-time continuum. ;The second part began with the demonstration of "existence" and nature of Objectives . The Objective is grasped by either a judgment or an assumption, whether its true/false value is included in it or not. I have developed an extensive analysis of modalities--factuality, necessity and possibility--, and quasi-modalities--probability and truth--in order to give a complete Theory of Knowledge integrated in an Ontology. ;The last part was an analysis the objects of emotional lived experiences: the "dignitative" and the "desiderative." The dignitative harbors ethical and esthetical values, while the desiderative contains moral and esthetical laws, and in general, the "duties." The main analysis of this part treats the question of justification of the ethical objects and their grounding in intellectual objects. (shrink)

It is shown that one common formulation of Stalnaker's semantics for conditionals is incomplete: it has no sound and (strongly) complete proof system. At first, this seems to conflict with well-known completeness results for this semantics (e.g., Stalnaker and Thomason 1967; Stalnaker 1970 and Lewis 1973, ch. 6). As it turns out, it does not: these completeness results rely on another closely-related formulation of the semantics that is provably complete. Specifically, the difference comes down to how the (...) Limit Assumption is stated. I close with some remarks about what this means for the logic of conditionals. (shrink)

Husserl’s notion of definiteness, i.e., completeness is crucial to understanding Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating definiteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads Husserl to (...) a ‘computational’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently renounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly original; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results. (shrink)

The tremendous influence Stoicism has exerted on ethical thought from early Christianity through Immanuel Kant and into the twentieth century is rarely understood and even more rarely appreciated. Throughout history, Stoic ethical doctrines have both provoked harsh criticisms and inspired enthusiastic defenders. The Stoics defined the goal in life as living in agreement with nature. Humans, unlike all other animals, are constituted by nature to develop reason as adults, which transforms their understanding of themselves and their own true good. (...) The Stoics held that virtue is the only real good and so is both necessary and, contrary to Aristotle, sufficient for happiness; it in no way depends on luck. The virtuous life is free of all passions, which are intrinsically disturbing and harmful to the soul, but includes appropriate emotive responses conditioned by rational understanding and the fulfillment of all one's personal, social, professional, and civic responsibilities. The Stoics believed that the person who has achieved perfect consistency in the operation of his rational faculties, the "wise man," is extremely rare, yet serves as a prescriptive ideal for all. The Stoics believed that progress toward this noble goal is both possible and vitally urgent. (shrink)

In the early seventies, several logicians developed a semantics for propositional systems of relevance logic. The essential ingredients of this semantics were a privileged point o, an ‘accessibility’ relation R and a special operator * for evaluating negation. Under the truth- conditions of the semantics, each formula A(Pl,…,Pn) could be seen as expressing a first order condition A+(pl,…,pn, o, R,*) on sets p1,…,pn and o, R, *, while each formula-scheme could be regarded as expressing the second-order condition ∀p1,…,∀pn A+(p1,…,pn, o, (...) R, *). It could then be shown that many standard systems of propositional relevance logic were complete in the sense that their theorems were just those formulas true in all models whose components o, R and * conformed to the second-order conditions expressed by the axioms of the system. (shrink)

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any (...) class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind. (shrink)

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...) mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

Peirce did not achieve a final systematization of his work. Beyond the difficulties in explaining so many philosophical tools that he introduced—suffice it to mention semiotic, abductive logic, a heuristic based on continuity, scholastic realism—, there is a theoretical reason for this incompletion. All those new philosophical tools indicated a conception of synthesis very different from the one he received from Kant. Peirce did not realize the profound direction of his enquiry so that he did not directly question neither Kant’s (...) legacy on this issue nor the idea of necessity that presides over it. Starting from Peirce’s conception of continuity and change, this paper will give a new definition of synthetic and analytic judgments and reasoning, completing the picture with a third “vague” judgment and reasoning. In this new definition a synthetic judgment is a judgment that recognizes identity through changes. An analytical judgment is a judgment that loses identity through changes. A vague judgment is a judgment that it is blind to identity through changes. How do we perform synthetic reasoning? Following Peirce’s semiotic study of elements of Gamma Graphs as the sheet of assertion and the line of identity, the paper will first individuate the semiotic characteristics necessary for the recognition of identity. These characteristics lead us to discover “complete gesture” as the tool that we use in our every-day reasoning in order to acquire new knowledge synthetically. “Complete gestures” are actions through which we carry and recognize significant meanings. This new paradigm should provide an improved account of common-sense knowledge as well as of particular creative and hypothetic stages of conception in both scientific and humanistic thought. (shrink)

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table (...) class='Hi'>complete degree by identifying an acceptable set of domain-random strings within each degree. (shrink)

In [5], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete ‐sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.

John Lachs, one of American philosophy's most distinguished interpreters, turns to William James, Josiah Royce, Charles S. Peirce, John Dewey, and George Santayana to elaborate stoic pragmatism, or a way to live life within reasonable limits. Stoic pragmatism makes sense of our moral obligations in a world driven by perfectionist human ambition and unreachable standards of achievement. Lachs proposes a corrective to pragmatist amelioration and stoic acquiescence by being satisfied with what is good enough. This personal, yet modest, philosophy offers (...) penetrating insights into the American way of life and our human character. (shrink)

Russell's notion of an incomplete symbol has become a standard against which philosophers compare their views on the relationship between language and the world. But Russell's exact characterization of incomplete symbols and the role they play in his philosophy are still disputed. In this paper, I trace the development of the notion of an incomplete symbol in Russell's philosophy. I suggest – against Kaplan, Evans, and others – that Russell's many characterizations of the notion of an (...) class='Hi'>incomplete symbol are compatible. To this end, I examine and reject arguments for the purported incompatibility between declaring an expression to be incomplete and incorporating that symbol into a compositional semantic theory. I then examine how Russell puts the notion of an incomplete symbol to use in metaphysics. (shrink)

Einstein made several attempts to argue for the incompleteness of quantum mechanics, not all of them using a separation principle. One unpublished example, the box parable, has received increased attention in the recent literature. Though the example is tailor-made for applying a separation principle and Einstein indeed applies one, he begins his discussion without it. An analysis of this first part of the parable naturally leads to an argument for incompleteness not involving a separation principle. I discuss the argument and (...) its systematic import. Though it should be kept in mind that the argument is not the one Einstein intends, I show how it suggests itself and leads to a conflict between QM’s completeness and a physical principle more fundamental than the separation principle, i.e. a principle saying that QM should deliver probabilities for physical systems possessing properties at definite times. (shrink)

The purpose of this book is to trace the main developments in Greek philosophy during the period which runs from the death of Alexander the Great in 323 B.c. to the end of the Roman Republic. These three centuries, known to us as the Hellenistic Age, witnessed a vast expansion of Greek civilization eastwards, following Alexander's conquests; and later, Greek civilization penetrated deeply into the western Mediterranean world assisted by the political conquerors of Greece, the Romans. But philosophy throughout (...) this time remained a predominantly Greek activity. The most influential thinkers in the Hellenistic world were Stoics, Epicureans and Sceptics. This book gives a concise critical analysis of their ideas and their methods of thought. The last book in English to cover this ground was written sixty years ago. In the interval the subject has moved on, quite rapidly since the last war, but most of the best work is highly specialized. There is a clear need for a general appraisal of Hellenistic philosophy which can provide those who are not specialists with an up-to-date account of the subject. Hellenistic philosophy is often regarded as a dull product of second-rate thinkers who are unable to stand comparison with Plato and Aristotle. This book will help to remove such misconceptions and arouse wider interest in a field which is fascinating both historically and conceptually. (shrink)

After Plato's Forms, and Aristotle's substances, the Stoics posited the fundamental reality of lekta - the meanings of sentences, distinct from the sentences themselves. This volume analyses the resulting unique, complex, and consistent cosmic view in which lekta are the keystones of the structure of reality: they are all there is to say.

Alleged counter-examples deployed in Park [Erkenntnis 60: 229–240] against the account of selective hypothetico-deductive confirmation offered in Gemes [Erkenntnis 49: 1–20] are shown to be ineffective. Furthermore, the reservations expressed in Gemes [ibid] and [Philosophy of Science 62: 477–487] about hypothetico-deductivism are retracted and replaced with the conclusion that H-D is a viable account of confirmation that captures much of the practice of working scientists. However, because it cannot capture cases of inference to the best explanation and cases of the (...) observational confirmation of statistical hypotheses, it is concluded that H-D cannot supply a complete theory of confirmation. (shrink)