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.

ABSTRACT: In this paper I argue that the ‘discovery’ of the problem of causal determinism and freedom of decision in Greek philosophy is the result of a combination and mix-up of Aristotelian and Stoic thought in later antiquity; more precisely, a (mis-)interpretation of Aristotle’s philosophy of deliberate choice and action in the light of Stoic theory of determinism and moral responsibility. The (con-)fusion originates with the beginnings of Aristotle scholarship, at the latest in the early 2nd century AD. It undergoes several developments, absorbing Epictetan, Middle-Platonist, and Peripatetic ideas; and it leads eventually to a concept of freedom of decision and an exposition of the ‘free-will problem’ in Alexander of Aphrodisias’ On Fate and in the Mantissa ascribed to him.

ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of deduction can be found in S. Bobzien, Stoic Syllogistic, OSAP 1996.

Much of chapters 2 to 6 of this book is in agreement with publications from the last twenty years (including those of the reviewer); so for example Frede’s points that neither Aristotle nor the Stoics had a notion of free-will; that in Epictetus (for the first time) the notions of freedom and will were combined; that an indeterminist notion of free-will occurs first in Alexander. The achievement of these chapters lies in the way Frede carefully joins them together and uses them as a basis for some substantive criticism and rewriting of the history of free-will regarding late antique Pagan and Christian authors, in particular Plotinus, Origen and Augustine.

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

ABSTRACT: Galen’s Institutio Logica is the only introduction to logic in Greek that has survived from antiquity. In it we find a theory that bears some resemblance to propositional logic. The theory is commonly understood as being essentially Stoic. However, this understanding of the text leaves us with a large number of inconsistencies and oddities. In this paper I offer an comprehensive alternative interpretation of the theory. I suggest that it is Peripatetic at base, and has drawn on Stoic elements, but adapted them to an overall decidedly non-Stoic conception of logic and language, a conception indebted to Aristotelian logic in many respects. This interpretation makes it possible to reduce the seeming inconsistencies dramatically. The Peripatetic theory on which Galen draws was possibly developed in the first century BCE. Importantly, it differs from Stoic logic in that it shuns the latter’s syntactic approach, and considers certain linguistic assumptions and language conventions as part of the logical theory itself. My reconstruction of the theory results in a logic of propositions which differs wildly both from Stoic logic and from the ‘classical’ propositional logic of the 20th century. Interestingly, though, the theory in Galen shows that the ancients grappled with a number of logico-linguistic problems that over the last two decades have again become a matter of debate among contemporary logicians and linguists.

The ‘Dialectical school’ denotes a group of early Hellenistic philosophers that were loosely connected by philosophizing in the — Socratic — tradition of Eubulides of Megara and by their interest in logical paradoxes, propositional logic and dialectical expertise.. Its two best known members, Diodorus Cronus and Philo the Logician, made groundbreaking contributions to the development of theories of conditionals and modal logic. Philo introduced a version of material implication; Diodorus devised a forerunner of strict implication. Each developed a system of modal notions that satisfies the basic logical requirements laid down by modern standard modal theories. In antiquity, Diodorus Cronus was famous for his so called Master Argument, which aims to prove that only the actual is possible.

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity.

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.

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.

The first monograph on Stoic modal logic. Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; the relation between the Stoic modal notions and those of Diodorus Cronus and Philo of Megara; the role of ‘external hindrances’ for the modalities; the temporal dependency of the modalities; propositions that change their modalities; the principle that something possible can follow from something impossible; the interpretations of the Stoic modal system by B. Mates, M. Kneale, M. Frede, J. Vuillemin and M. Mignucci are evaluated. For a shorter, updated, English version of Part 1 of the book see S. Bobzien, ‘Stoic Logic’, in K. Algra et al. (eds), The Cambridge History of Hellenistic Philosophy, Cambridge 1999, 92-157. For a shorter, updated, English version of Part 2 of the book see S. Bobzien 'Chrysippus' Modal Logic and its Relation to Philo and Diodorus', in K. Doering / Th. Ebert (eds) Dialektiker und Stoiker (Stuttgart 1993) 63-84.