Susanne Bobzien

Since Mates’ seminal Stoic Logic there has been uncertainty and debate about how to treat the term anapodeiktos when used of Stoic syllogisms. This paper argues that the customary translation of anapodeiktos by ‘indemonstrable’ is accurate, and it explains why this is so. At the heart of the explanation is an argument that, contrary to what is commonly assumed, indemonstrability is rooted in the generic account of the Stoic epistemic notion of demonstration. Some minor insights into Stoic logic ensue.

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness.

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

This paper shows that, for the Hertz–Gentzen Systems of 1933, extended by a classical rule T1 and using certain axioms, all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. OPEN ACCESS.

ABSTRACT: This chapter offers a revenge-free solution to the liar paradox (at the centre of which is the notion of Gestalt shift) and presents a formal representation of truth in, or for, a natural language like English, which proposes to show both why -- and how -- truth is coherent and how it appears to be incoherent, while preserving classical logic and most principles that some philosophers have taken to be central to the concept of truth and our use of that notion. The chapter argues that, by using a truth operator rather than truth predicate, it is possible to provide a coherent, model-theoretic representation of truth with various desirable features. After investigating what features of liar sentences are responsible for their paradoxicality, the chapter identifies the logic as the normal modal logic KT4M (= S4M). Drawing on the structure of KT4M (=S4M), the author proposes that, pace deflationism, truth has content, that the content of truth is bivalence, and that the notions of both truth and bivalence are semideterminable.

Central passages: Alexander of Aphrodisias On Fate 181–3, 189, 192, 205, Nemesius On Human Nature 105.6–106.13; Although in the first three centuries AD there were several Stoic theories of fate in circulation, only one systematic exposition of a later Stoic compatibilist theory has survived. This is found in Alexander of Aphrodisias’ On Fate, with parallels in Nemesius. These are the only sources that provide a Stoic account of that which depends on us and its relation to Stoic fate theory. This chapter is devoted to this theory and its reception by opponents in later antiquity. It is shown that this theory goes back to a particular second century AD Stoic view, which, although ‘orthodox’, differs noticeably from Chrysippus’. The reason for the modifications can be found in the confrontation with the newly developed Middle Platonist and Peripatetic theories of fate and freedom, and the shift of emphasis to the predetermination of ‘mental events’ as opposed to actions only, which forced the discussion of Stoic type determinism and that which depends on us to a new level. It is in the context of the confrontation of these theories with the Stoic ones that problems similar to modern problems of free will and determinism arise, and both the Stoics and their opponents introduce new arguments designed to cope with the new state of the debate.

Central passages: Plutarch, Aud. Poet. 33d, Epictetus, Enchiridion 53.1, Seneca Letters 107.10; Hippolytus, Refutation of all Heresies 1.21.2 Eleutheria was an indispensable philosophical concept in early Stoic ethics, and became central in the moral philosophy of the Roman Stoa. The Stoic notion of eleutheria has been a source of modern misinterpretation and mis‐assessment of the early Stoic theory of fate and of Chrysippus’ compatibilism. In particular, it has been illegitimately confounded with Chrysippus’ notion of that which depends on us. There is no evidence that the Stoic concept of eleutheria and the question of that which depends on us were connected in any way before the Roman Stoa. This chapter contains first an analysis of Epictetus’ concept of that which depends on us and how it relates to the early Stoic concept; then an outline of the Stoic notion of eleutheria, its role in early Stoic philosophy and in Epictetus, and its connection with the concept of that which depends on us, and the specific philosophical contexts in which the two concepts belong. Finally, the chapter provides a discussion of a couple of passages, including the famous dog and cart simile, which are shown to be often wrongly associated with Chrysippus’ compatibilism, to belong rather in the wider context of Epictetan ethics, and to have had their origin in Cleanthes’ philosophy.

Central passages: Gellius Attic Nights 7.2; Cicero On Fate 8 11, 39–45; Plutarch On Stoic Self‐Contradictions 1055f –1056d. There are only three sources that attest undoubtedly that Chrysippus, in some way, dealt with the problem of causal determinism and moral responsibility. They report the so‐called cylinder analogy and a Chrysippan distinction of causes, and present the core of Chrysippus’ compatibilism. The discussion of these passages in this chapter shows that they fit in smoothly with Chrysippus’ other arguments, adding to the ‘preserved’ concepts of contingency and intentional action finally that of moral responsibility—all fully in accord with his theory of fate and causal determinism. The chapter describes the specific problem of moral responsibility and determinism that Chrysippus and his opponents encountered, and shows how it differs from problems that are today subsumed under labels such as ‘the free‐will problem’. In particular, it emerges that the conception of moral responsibility of Chrysippus and his contemporaries is grounded not on any concept of freedom to do otherwise, but on a kind of ‘agent autonomy’. This conception of moral responsibility leads to different problems for its justification within a deterministic system than a theory of freedom to do otherwise. Based on their theories of causation, of mind, of decision making, and of character determination, the Stoics provide a sophisticated solution to these difficulties, showing among other things how it is possible that the rational and moral aspects of human agents can manifest themselves in human actions. Chrysippus’ claim that human actions are, though fated, not necessary, becomes comprehensible if one takes on his distinction between the modal concept of something's being necessary, on the one hand, and the physical power ‘Necessity’, on the other. In Chrysippus’ arguments for the compatibility of fate and moral responsibility, we encounter no word for freedom. The freedom that is identified as relevant to the arguments is freedom from external hindrances and from force. This freedom was referred to generally by negative phrases such as ‘not externally hindered’ or ‘not necessitated’, but not by a single term or noun phrase; and it was regarded as a necessary condition for something's depending on us and happening because of us. By ‘that which depends on us’ and ‘that which happens because of us’ Chrysippus seems to have understood simply the things of which we, qua rational beings, are the possible or actual cause. Such causal origination is brought about by the faculty of assent. The freedom from force and external hindrances is guaranteed by the utilization of this very mechanism through which we become the cause of our actions. For it is part of the nature of this human mental capacity of assent that it is neither forced nor fully externally determined.

The Idle Argument is the classical argument for fatalism and the futility of action: ‘If it is fated that you will recover, you will, regardless of whether you consult a doctor. If it is fated that you won’t recover, you won’t, regardless... Either it is fated that you will recover or that you won’t. Therefore it is pointless to consult a doctor.’ In the first part of this chapter, the sources that preserve this argument are analysed in detail, and the various ways of understanding the argument, and the ambiguities in its expressions, are exposed. The main distinction turns out to be that between a reading of the argument as manifesting logical determinism, and a reading of it as manifesting causal determinism. Depending on which version one considers, in which way one understands the ambiguous phrases, and whether one takes them throughout in the same way, the argument may turn out as valid or invalid, its premisses as true or false, and it may or may not justify idleness. The second part of the chapter is dedicated to Chrysippus’ reply: He takes the Idle Argument to concern causal determinism, and to be a fallacy, and his refutation of it is designed to make clear the difference between common and garden fatalism and Stoic causal determinism, which is consistent and does not lead to idleness, since human action and inaction are just as fated as everything else. It is shown that Chrysippus’ refutation does not require the existence of an empirically accessible, universal relation of necessitation, nor a causal theory with universal laws of nature.

Central passages: Cicero On Fate 11–17; Alexander of Aphrodisias On Fate 208.15–21. Critics of the Stoics tried to show up an inconsistency between Chrysippus’ acceptance of divination, on the one hand, and his conception of contingency, on the other. The argument claims that, since theorems of divination connect, in a conditional, a proposition about the past with one about the future, and since the necessity of the former is transferred to the latter, future occurrences—as far as covered by divination—are all necessary. In the first part of this chapter, the argument is analysed in detail, and several logical difficulties contained in it are unravelled. The second part is concerned with Chrysippus’ counter‐argument. By determining that divinatory theorems are correctly expressed as negated conjunctions rather than conditionals, Chrysippus successfully blocks his opponents’ argument: what is genuinely predicted by diviners is not necessitated through the prediction, because the relation between sign and predicted future event does not ‘transmit’ necessity. Universal regularity between sign and predicted event does not suffice to make a future event necessary. There is no ‘causal chain’ connecting them. Theorems of other sciences, for example, medical theorems that express an empirical connection between things/events, make true Chrysippean conditionals. However, human freedom as conceived by the Stoics is not thereby threatened.

Central passages: Boethius On Interpretation II 234–5; Diogenes Laertius 7.75; Plutarch, On Stoic Self‐contradictions 1055d–f; Alexander of Aphrodisias On Fate 176.14–23. In the Stoic debate over fate and determinism, the modalities played a role in a number of different contexts. Chrysippus rejected Diodorus Cronus’ modal theory because of its built‐in necessitarian consequences. He developed his own set of modal notions, which, in themselves, do not lead to necessitarianism and which secure a necessary condition for free action—namely, the existence of occurrences that are both possible and non‐necessary. Chrysippus’ modalities are not epistemic. Some critics of the Stoics developed arguments to show that there is a conflict between Chrysippus’ modal notions and the Stoic theory of fate. Later Stoics replied to such objections by giving an epistemic interpretation of Chrysippus’ modal concepts. Critics of the Stoics objected that fate, qua Necessity, renders all events necessary. However, this objection is not justified within Chrysippus’ philosophy.

Central passages: Cicero On Fate 20–21, 26–28, 37, On Divination I 125–6. The first argument: Chrysippus assumes as true a weak, non‐modal form of logical determinism. From this assumption he argues for the truth of his theory of causal determinism, by showing that the latter is a necessary condition of the former. This argument differs considerably from the argument for the truth of future propositions to their necessity, which Epicurus is said to have attacked. Cicero confuses the two. The second argument: Chrysippus, as most Stoics, believed in divination as a science. He construes an argument from the existence of this science to his type of causal determinism by showing that causal determinism is a necessary condition for the existence of the science of divination.

Central passages: Plutarch On Stoic Self‐Contradictions 1045b–c; 104950, 1056; Stobaeus Ecl. I 79.1–12. The physical and ontological foundations of the Stoic theory of determinism are investigated: the active principle, causation, motion, and qualitative states and how they relate to the Stoic concept of propositions. Stoic teleological determinism grows out of the basic assumptions of Stoic cosmology and is thus firmly anchored in early Stoic physics. Stoic physics stands out in antiquity not so much because it is a deterministic system, but because it contains a worked‐out theory of universal, mechanical, causal determinism. In the Stoic theory of universal fate, the teleological and mechanical aspects of Stoic determinism are combined.

Methodological information and historical background: The book focuses on problems of determinism and freedom as they were conceived in antiquity, this being a necessary condition for a full understanding of the Stoic theory. The Stoic theory of determinism and freedom is central to their philosophy as a whole. The history of works on fate, determinism, and freedom from Plato to the end of antiquity is sketched. The difficulty of the fragmentary state of our sources and the lack of first‐hand sources, the inadequacy of von Arnim's collection of Stoic fragments on fate, and the limits of the present book are discussed.

This is a comprehensive introduction to the two great logical systems of antiquity, Aristotelian logic and Stoic logic. Although advanced undergraduate and graduate students are the intended primary readership, classical and philosophy scholars with non-specialist interests in the subject will find the fresh approach useful and stimulating. A conscious effort has been made to create a fruitful dialogue between the Aristotelian and Stoic systems both in terms of theoretical content and in terms of methodological issues and approaches. In all sections emphasis is placed on broader questions concerning the distinctive nature of ancient logic, its relation with modern logic and the study of the history of logic, and the methodological difficulties posed by the reconstruction, analysis and assessment of ancient views, both in themselves and in the light of modern logic.

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and modal theorems, and to make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy.

ABSTRACT: One-page entry on freedom in the philosophical (as opposed to political) sense in antiquity, noting (among other things) that a notion of freedom of choice that requires that the person not be causally predetermined in his/her actions is developed only in the 1st-3rd cents. CE in Alexander of Aphrodisias, building on elements of Aristotelian ethics and logic, Stoic psychology and perhaps Christian and Middle Platonic influences. Both German version (1998) and English translation (2011).

ABSTRACT: 1. This paper argues that Epicurus had a notion of moral responsibility based on the agent’s causal responsibility, as opposed to the agent’s ability to act or choose otherwise; that Epicurus considered it a necessary condition for praising or blaming an agent for an action, that it was the agent and not something else that brought the action about. Thus, the central question of moral responsibility was whether the agent was the, or a, cause of the action, or whether the agent was forced to act by something else. Actions could be attributed to agents because it is in their actions that the agents, qua moral beings, manifest themselves. 2. As a result, the question of moral development becomes all important. The paper collects and discusses the evidence for Epicurus views on moral development, i.e. (i) on how humans become moral beings and (ii) on how humans can become morally better. It becomes clear that Epicurus envisaged a complex web of hereditary and environmental factors to shape the moral aspect of a human being. 3. In line with Epicurus’ theory of moral responsibility and moral development, Epicurus ethics does not have the function of developing or justifying a moral system that allows for the effective allocation of praise and blame. Rather, for him the function of ethics – and in fact of the whole of philosophy – is to give everyone a chance to morally improve.

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘Aristotelian logic’, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as ‘hypothetical syllogisms’. However, Aristotle did not discuss such arguments, nor did he call any arguments ‘hypothetical syllogisms’. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them ‘hypothetical syllogisms’; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle’s logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called ‘hypothetical syllogisms’? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle’s dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle’s logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories.

ABSTRACT: Stewart Shapiro recently argued that there is no higher-order vagueness. More specifically, his thesis is: (ST) ‘So-called second-order vagueness in ‘F’ is nothing but first-order vagueness in the phrase ‘competent speaker of English’ or ‘competent user of “F”’. Shapiro bases (ST) on a description of the phenomenon of higher-order vagueness and two accounts of ‘borderline case’ and provides several arguments in its support. We present the phenomenon (as Shapiro describes it) and the accounts; then discuss Shapiro’s arguments, arguing that none is compelling. Lastly, we introduce the account of vagueness Shapiro would have obtained had he retained compositionality and show that it entails true higher-order vagueness.

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

ABSTRACT: Alexander of Aphrodisias’ commentaries on Aristotle’s Organon are valuable sources for both Stoic and early Peripatetic logic, and have often been used as such – in particular for early Peripatetic hypothetical syllogistic and Stoic propositional logic. By contrast, this paper explores the role Alexander himself played in the development and transmission of those theories. There are three areas in particular where he seems to have made a difference: First, he drew a connection between certain passages from Aristotle’s Topics and Prior Analytics and the Stoic indemonstrable arguments, and, based on this connection, appropriated at least four kinds of Stoic indemonstrables as Aristotelian. Second, he developed and made use of a specifically Peripatetic terminology in which to describe and discuss those arguments – which facilitated the integration of the indemonstrables into Peripatetic logic. Third, he made some progress towards a solution to the problem of what place and interpretation the Stoic third indemonstrables should be given in a Peripatetic and Platonist setting. Overall, the picture emerges that Alexander persistently (if not always consistently) presented passages from Aristotle’s logical œuvre in a light that makes it appear as if Aristotle was in the possession of a Peripatetic correlate to the Stoic theory of indemonstrables.

NOTE: The English translation is listed separately. ABSTRACT: A general interpretation and close textual analysis of Kant’s theory of the categories of freedom (or categories of practical reason) in his Critique of Practical Reason. My main concerns in the paper are the following: (1) I show that Kant’s categories of freedom have primarily three functions: as conditions of the possibility for actions (i) to be free, (ii) to be comprehensible as free and (iii) to be morally evaluated. (2) I show that for Kant actions, although qua theoretical objects they are always already constituted by means of the theoretical categories, qua practical objects (objects of reason in its practical use, i.e. objects qua possibly good or bad) they are constituted by means of the categories of freedom; and that it is only in this way that actions, qua phenomena, can be a consequence of freedom, and can be understood and evaluated as such. (3) Since Kant's presentation of his theory of the Categories of Freedom is extremely brief, Kant's parallel theory of the theoretical categories in his Critique of Pure Reason is used as a guide for the interpretation of the practical categories and their systematic relevance.

ABSTRACT: In contemporary discussions of freedom in Stoic philosophy we often encounter the following assumptions: (i) the Stoics discussed the problem of free will and determinis; (ii) since in Stoic philosophy freedom of the will is in the end just an illusion, the Stoics took the freedom of the sage as a substitute for it and as the only true freedom; (iii) in the c. 500 years of live Stoic philosophical debate, the Stoics were largely concerned with the same philosophical problems of freedom. In this paper I argue that (i) can be upheld only in a very restricted way; (ii) is altogether untenable; and regarding (iii), that, although there may have occurred little change in the Stoic philosophical position on freedom over the centuries, we can detect more than one transformation of the philosophical problems that were at the forefront of the discussion. Moreover, that all the conceptions and problems of freedom were linked to Stoic ethics, and that the differences between them become transparent when one considers their various roles in this context.