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: 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. (Translation by Alex Worsnip, reviewed and approved by the author.)

ABSTRACT: Recently a bold and admirable interpretation of Chrysippus’ position on the Sorites has been presented, suggesting that Chrysippus offered a solution to the Sorites by (i) taking an epistemicist position1 which (ii) made allowances for higher-order vagueness. In this paper I argue (i) that Chrysippus did not take an epistemicist position, but − if any − a non-epistemic one which denies truth-values to some cases in a Sorites-series, and (ii) that it is uncertain whether and how he made allowances for higher-order vagueness, but if he did, this was not grounded on an epistemicist position.

ABSTRACT: Although from the 2nd century BC to the 3rd AD the problems of determinism were discussed almost exclusively under the heading of fate, early Stoic determinism, as introduced by Zeno and elaborated by Chrysippus, was developed largely in Stoic writings on physics, independently of any specific "theory of fate ". Stoic determinism was firmly grounded in Stoic cosmology, and the Stoic notions of causes, as corporeal and responsible for both sustenance and change, and of effects as incorporeal and as predicates, are indispensable for a full understanding of the theory. Stoic determinism was originally not presented as causal determinism, but with a strong teleological element, in the context of a theory of natural motions, which makes use of a distinction between a global and an inner-worldly perspective on events. However, Chrysippus also employed his conception of causality in order to explicate his determinism, and can be shown to have maintained a universal causal determinism in the modern sense of the erm. The teleological and mechanical elements of early Stoic determinism were brought together in Chrysippus' conception of fate, which places elements of rationality in every cause.

ABSTRACT: This paper serves two purposes: (i) it can be used by students as an introduction to chapters 1-5 of book iii of the NE; (ii) it suggests an answer to the unresolved question what overall objective this section of the NE has. The paper focuses primarily on Aristotle’s theory of what makes us responsible for our actions and character. After some preliminary observations about praise, blame and responsibility (Section 2), it sets out in detail how all the key notions of NE iii 1-5 are interrelated (Sections 3-9). The setting-out of these interconnections makes it then possible to provide a comprehensive interpretation of the purpose of the passage. Its primary purpose is to explain how agents are responsible for their actions not just insofar as they are actions of this kind or that, but also insofar as they are noble or base: agents are responsible for their actions qua noble or base, because, typically via choice, their character dispositions are a causal factor of those actions (Section 10). The paper illustrates the different ways in which agents can be causes of their actions by means of Aristotle’s four basic types of agents (Section 11). A secondary purpose of NE iii 1-5 is to explain how agents can be held responsible for consequences of their actions (Section 12), in particular for their character dispositions insofar as these are noble or base, i.e. virtues or vices (Section 13). These two goals are not the only ones Aristotle pursues in the passage. But they are the ones Aristotle himself indicates in its first sentence and summarizes in its last paragraph; and the ones that give the passage a systematic unity. The paper also briefly consider the issues of freedom-to-do-otherwise, free choice and free-will in the contexts in which they occur (i.e. in the final paragraphs of Sections 6, 7, 12, 13).

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. OPEN ACCESS LINK.

ABSTRACT: In antiquity we encounter a distinction of two types of hypothetical syllogisms. One type are the ‘mixed hypothetical syllogisms’. The other type is the one to which the present paper is devoted. These arguments went by the name of ‘wholly hypothetical syllogisms’. They were thought to make up a self-contained system of valid arguments. Their paradigm case consists of two conditionals as premisses, and a third as conclusion. Their presentation, either schematically or by example, varies in different authors. For instance, we find ‘If (it is) A, (it is) B; if (it is) B, (it is) C; therefore, if (it is) A, (it is) C’. The main contentious point about these arguments is what the ancients thought their logical form was. Are A, B, C schematic letters for terms or propositions? Is ‘is’, where it occurs, predicative, existential, or veridical? That is, should ‘A esti’ be translated as ‘it is an A’, ‘A exists’, ‘As exist’ or ‘It is true/the case that A’? If A, B, C are term letters, and ‘is’ is predicative, are the conditionals quantified propositions or do they contain designators? If one cannot answer these questions, one can hardly claim to know what sort of arguments the wholly hypothetical syllogisms were. In fact, all the above-mentioned possibilities have been taken to describe them correctly. In this paper I argue that it would be mistaken to assume that in antiquity there was one prevalent understanding of the logical form of these arguments - even if the ancients thought they were all talking about the same kind of argument. Rather, there was a complex development in their understanding, starting from a term-logical conception and leading to a propositional-logical one. I trace this development from Aristotle to Philoponus and set out the deductive system on which the logic of the wholly hypothetical syllogisms was grounded.

ABSTRACT: This paper is distinctly odd. It demonstrates what happens when an analytical philosopher and historian of philosophy tries their hand at the topic of reception. For a novice to this genre, it seemed advisable to start small. Rather than researching the reception of an author, book, chapter, section or paragraph, the focus of the paper is on one sentence: Aristotle’s Nicomachean Ethics 3.5, 1113b7-8. This sentence has markedly shaped scholarly and general opinion alike with regard to Aristotle’s theory of free will. In addition, it has taken on a curious life of its own. Part one of the paper examines the text itself. Part two explores its reception from antiquity to the present day, including present-day popular culture, later ancient, Byzantine, Arabic, Latin Medieval, Renaissance, Victorian and contemporary scholarship. There are some surprises on the way. (The paper also serves as an introduction to the reception of the Nicomachean Ethics from its beginnings to the present.)

The paper presents a new theory of higher-order vagueness. This theory is an improvement on current theories of vagueness in that it (i) describes the kind of borderline cases relevant to the Sorites paradox, (ii) retains the ‘robustness’ of vague predicates, (iii) introduces a notion of higher-order vagueness that is compositional, but (iv) avoids the paradoxes of higher-order vagueness. The theory’s central building-blocks: Borderlinehood is defined as radical unclarity. Unclarity is defined by means of competent, rational, informed speakers (‘CRISPs’) whose competence, etc., is indexed to the scope of the unclarity operator. The unclarity is radical since it eliminates clear cases of unclarity and, that is, clear borderline cases. This radical unclarity leads to a (bivalence-compatible, non-intuitionist) absolute agnosticism about the semantic status of all borderline cases. The corresponding modal system would be a non-normal variation on S4M.

ABSTRACT: This is a short companion piece to my ‘Found in Translation – Aristotle’s Nicomachean Ethics III.5 1113b7-8 and its Reception’ in which I examine in close textual analysis the philosophical question whether these two lines from the Nicomachean Ethics provide any evidence that Aristotle discussed free choice – as is not infrequently assumed. The result is that they do not, and that the claim that they do tends to be based on a mistranslation of the Greek. (There is some inevitable overlap with Part 1 of the 'Found in Translation' paper.)

ABSTRACT: I argue that there is no evidence that Epicurus dealt with the kind of free-will problem he is traditionally associated with; i.e. that he discussed free choice or moral responsibility grounded on free choice, or that the "swerve" was involved in decision processes. Rather, for Epicurus, actions are fully determined by the agent's mental disposition at the outset of the action. Moral responsibility presupposes not free choice but that the person is unforced and causally responsible for the action. This requires the agent's ability to influence causally, on the basis of their beliefs, the development of their behavioral dispositions. The "swerve" was intended to explain the non-necessity of agency without undermining Epicurus' atomistic explanation of the order in the universe, viz. by making the mental dispositions of adults non-necessary.

ABSTRACT: This paper discusses the Stoic treatment of fallacies that are based on lexical ambiguities. It provides a detailed analysis of the relevant passages, lays bare textual and interpretative difficulties, explores what the Stoic view on the matter implies for their theory of language, and compares their view with Aristotle’s. In the paper I aim to show that, for the Stoics, fallacies of ambiguity are complexes of propositions and sentences and thus straddle the realms of meaning (which is the domain of logic) and of linguistic expressions (which is the domain of linguistics), but also involve a pragmatic element; that the Stoics believe that the premises of the fallacies, when uttered, have only one meaning and are true, and thus should be conceded; that hence there is no need for a mental process of disambiguation in the listeners; that Aristotle, by contrast, appears to assume that the premises always have all their meanings, and accordingly recommends that the listeners explicitly disambiguate them, which presupposes a process of mental disambiguation. I proffer two readings of the Stoic advice that we ‘be silent’ when confronted with a fallacy of ambiguity in dialectical discourse, and explicate how each leads to an overall consistent interpretation of the textual evidence. Finally, I demonstrate that the method advocated by the Stoics works in all cases of fallacies of lexical ambiguity.

This paper is an expanded written version of my reply to Rosanna Keefe’s paper ‘Modelling higher-order vagueness: columns, borderlines and boundaries’ (Keefe 2015), which in turn is a reply to my paper ‘Columnar higher-order vagueness, or Vagueness is higher-order vagueness’ (Bobzien 2015). Both papers were presented at the Joint Session of the the Aristotelian Society and the Mind Association in July, 2015. At the Joint Session meeting, there was insufficient time to present all of my points in response to Keefe’s paper. In addition, the audio of the session, which is available online, becomes inaudible at the beginning of my reply to Keefe’s comments due to a technical defect. The following is a full version of my remarks.