Stoic Logic Research Articles

Frege, Hirzel, and Stoic logic

This paper is a discussion of Gabriel, Hülser and Schlotter’s pioneering 2009 article on a possible causal relation between Stoic logic and Frege. It provides detailed argument for why Rudolf Hirzel should not be taken as the qualified middleman in philosophical discussion with whom Frege learned what he ‘borrowed’ without acknowledgement from Stoic logic. Additionally, this paper offers some very modest findings about some aspects of Frege's and Hirzel's lives and work habits, which may help us understand a little better Frege's connection to Hirzel and to Stoic logic as well as Frege’s failure to acknowledge the Stoics. This paper is a purely historical offshoot of my essay ‘Frege plagiarized the Stoics’. Zero direct insights into either Frege’s or Stoic philosophy are offered.

Frege, Sigwart, and Stoic Logic

This very brief paper provides plausible answers to the two residual questions that Jamie Tappenden states, but leaves unanswered, in his 2024 paper ‘Following Bobzien: Some notes on Frege’s development and engagement with his environment’, namely, why Frege read Sigwart’s Logik and what caused Frege to read Prantl.

Why Frege Did Not Plagiarize the Stoics: More on the Relationship Between Fregean and Stoic Logic

In this paper we give a detailed comparison of the key elements of Frege’s formal language of thought and apparently similar views in Stoic formal logic. That is, we compare their views on the following topics: connectives, negation, simple sentences, propositional content, predicates and their incompleteness, and quantifications. We show that in most of these cases the similarities between Frege’s views and the Stoic views are only superficial. Frege’s views are far more systematic, better developed and can in no case directly or fully be traced back to Stoic ideas. Furthermore, we show that Prantl, pace Bobzien 2021, is in most of these cases not a reliable source with respect to Stoic logic and that it is very unlikely that Frege’s views on logic were influenced by Prantl’s interpretation of Stoic formal logic.

A Chrysippean Modality

Abstract In this paper, I attempt to explain one of the most controversial views attributed to the Stoic Chrysippus: that the impossible can follow from the possible. My solution finds in Chrysippus a distinction later made by the medieval logician John Buridan: that between being possible (there being a state of affairs that may occur) and being possibly-true (there being some proposition whose truth-conditions are that state of affairs). Buridan and Chrysippus have radically opposing views on the nature of propositions. What their conceptions share is the conclusion that at least some propositions must be contingent beings. They argue for this while maintaining a rigorous commitment to the view that propositions are strictly bivalent. In 2. I explain the Chrysippean passage in terms of a distinction Buridan makes explicitly. In 3. I show how the distinction follows implicitly from the Stoic theory of quantification. In 4. I compare the modality with other aspects of Stoic logic. In 5. I discuss how the distinction behaves in the future tense.

Mathematical Logic in the History of Logic: Łukasiewicz’s Contribution and Its Reception

Łukasiewicz introduced a new methodological approach to the history of logic. It consists of the use of modern formal logic in the research of the history of logic. Although he was not the first to use formal logic in his historical research, Łukasiewicz was the first who used it consistently and formulated it as a requirement for a historian of logic. The aim of this paper is to present Łukasiewicz's contribution and the history of its formulation. In addition, the paper will present its reception. There are two main themes in Łukasiewicz’s work on the history of logic. He discussed Stoic logic and Aristotle’s syllogistic. The second part of the paper focuses on current reception of Łukasiewicz’s contribution and on the question of whether there is a difference between knowledge about his contribution to Stoic logic and to Aristotle’s syllogistic, given that it has been hypothesised that Łukasiewicz’s contribution is currently more known among Aristotelian scholars than among those who focus on Stoic logic. Based on the papers that have appeared in the last five years at Web of Science and Scopus this hypothesis has not been confirmed.

Are Ancient Logics Explosive?

The twentieth-century logical mainstream, derived from works by Łukasiewicz and Scholz, pictures the history of logic for the most part as the prehistory of Boolean–Fregean mathematical logic. Particularly, with respect to classical propositional calculus, the Stoic logic has been pictured as an early stage of it and Aristotle's or the Peripatetics' logic as a theory that assumes it. Although it was not emphasised, it follows that the ancient logics contain the principle of explosion. In the endmost quarter of the twentieth century, a competitive view began to spread to the effect that the ancient systems of logic were paraconsistent or relevantistic. In the twenty-first century, the latter view prevails and has every chance of becoming a new orthodoxy. It is claimed that although in Łukasiewicz's argument for the classicality of ancient logics, there are gaps, it may be demonstrated that the ancient logics contain the principle of explosion.

The Principle of Explosion in the Stoic Logic

I argue that the Stoic logic is explosive. The claim applies to the Stoics' syllogistic in the strictest sense, because there is a provable syllogism which qualifies as a principle of explosion. It applies also to the general consequence operation, in the sense that every sentence is derivable from any pair containing both a sentence and the negation of the sentence. Finally, it applies to the connective of implication (conditional), in the sense that any conditional is derivable, providing its antecedent is a conjunction of a sentence and the negation of the sentence. All three claims allow weakening, i.e., additions of extra premises to an inference or extra conjuncts to the antecedent of an implication, respectively. Consequently, no concept of relevance, let alone paraconsistency or connexivity is applicable to the Stoic logic; in particular, the Stoics' connective of implication is either material (Boolean) or formal (strict).

The Stoic Distinction between Syllogisms and Subsyllogisms

Abstract This paper aims to explain the distinction between syllogisms and subsyllogisms in Stoic logic. Subsyllogisms replace at least one premise in a syllogism with a premise that is, according to Galen and Alexander, equipollent to the respective syllogistic premise. This equipollence is not synonymy of meaning between two linguistic expressions, but obtains between two propositions when they are true or false by the same standard. Subsyllogistic premises are simple propositions that are equipollent to the non-simple premises of the respective syllogisms. For subsyllogistic premises that replace syllogistic premises consist of predicates combining with cases rather than of connectives or negations governing propositions. Because of this difference in their premises, subsyllogisms have a different logical form than syllogisms and are not formally valid.

Sign-inferences in Greek and Buddhist Logic

The Yogācāra school of logic developed a theory of sign-inferences that has many features of the Stoic and Epicurean logical teachings with small inclusions of Aristotelian ideas. In the Nyāyabindu of Dharmakīrti, we can find the following schemes of formal reasoning: modus Barbara (Figure I) and modus Camenes (Figure IV) of the Aristotelian syllogistic, and all the inference rules of the Stoic logic: modus ponens, modus tollens, modus ponendo tollens, modus tollendo ponens I, modus tollendo ponens II. The three premises of Yogācāra inference (if p, then q; there is an example that p and q occur together; p; then q) corresponds to the Epicurean doctrine of three premises with a necessary exemplification. Thus, the paper is a comparative study of Greek and Buddhist theory of sign-inferences, showing a deep similarity between them.

The Stoic Theory of Sign and the Semantic Modulation of Models

AbstractThe theory of mental models is a current cognitive approach trying to describe the way people make inferences. According to this theory, people reason from possibilities or models linked to sentences. Sometimes, such possibilities or models are transformed by the action of a semantic modulation. The point this paper is intended to make is that Stoic logic also has the machinery to explain semantic processes such as that of modulation. This is shown by means of the criterion Chrysippus of Soli provided for conditional sentences and the theory of sign related to that criterion.

Stoic Logic from the Theory of Mental Models

An essential point about Stoic philosophy is why certain arguments and rules are basic in their logic. That is the case of the indemonstrables and the themata. It has been proposed that assuming the theory of mental models, one can think that the five indemonstrables and two of the themata are easy to understand for the human mind. This can explain why those arguments and rules are essential components in Stoic logic. In addition, it is relevant because, given that the theory of mental models tries to capture the real way people reason, it can show that Stoic logic is closer to the manner individuals naturally make inferences than modern propositional calculus. The present paper is intended to move forward in this direction. It has two aims: one of them is to give an account from the theory of mental models of all of the themata. The other one is to argue that a simple schema that is correct in modern propositional calculus, and which, however, is not deemed as a true syllogism in Stoic logic, is difficult for people according to the theory of mental models. Those are further pieces of evidence that Stoic logic describes the way human beings think to a greater extent than modern logic.

The Ultimate Meaning of Counter-Actualisation: On the Ethics of the Univocity of Being in Deleuze'sLogic of Sense

As is well known, Deleuze says in Difference and Repetition that ‘the task of contemporary philosophy has been defined: to reverse Platonism’. This task is then continued in Logic of Sense, through its discussion of Stoic logic. Deleuze says there that ‘the Stoics are the first to reverse Platonism’. And, at the same time, in the big Spinoza book, we see Deleuze present Spinoza's ‘anti-Cartesian reaction’. This anti-Cartesian reaction is equivalent to the reversal of Platonism. We can say then that the task of the reversal of Platonism unifies the three books Deleuze published at the end of the Sixties. The thesis of this essay will consist in arguing that ethics we find in Logic of Sense – which we can call ‘the ethics of the univocity of being’ – completes, in a constructive way, Deleuze's early project of the reversal of Platonism. The ethics centres on the idea of counter-actualisation. We shall argue that there are four senses of counter-actualisation. Here is a summary of the four senses of counter-actualisation: (1) beatitude against resentment (against presentist modes of measurement); (2) caring for the sense against neglecting the sense (denouncing any one determinate answer); (3) the embodiment of plural hopings and rememberings against the embodiment of one hope and one memory (against the character); and (4) ‘just enough’ against ‘too much’ (against suicide). Yet, as we shall see, there is a fifth meaning of counter-actualisation, the ultimate meaning. The ultimate sense of counter-actualisation is universal freedom. This is where my essay will end.

Antiochus’ and Cicero’s different theories of memory in the Lucullus

Abstract The essay deals with an important epistemological debate in the Lucullus: Can there be remembrance of false beliefs, as Cicero argues against his interlocutor, who defends Antiochus’ position? It is shown that Antiochus, like Aristoteles, considers ‘remember’ to be a double achievement verb: Remembrance occurs only if a correct past perception is faithfully transmitted to the present. Cicero, on the other hand, insists that faithful transmission can also occur with false beliefs. The distinction seems to be analogous to that between valid and sound inference in contemporary Stoic logic. Finally, I discuss differences between ancient and modern forms of skepticism, address the issue of why misremembering plays no role in the Lucullus, and find an answer in Cicero’s doctrine of the orator.

Stoic Logic Allows Understanding a Priori Falsity in the Conditional

An issue to explain in cognitive science is nowadays the case of certain conditionals that people seem to deem as a priori false. Those conditionals appear to be false by virtue of semantics: the meanings of their antecedents and their consequents seem not to admit any link between them. This is a problem because, from the point of view of classical logic, they are not always false; there can be situations in which they are true (as classical logic provides, whenever their antecedents are false, those conditionals in entirety are true). There are contemporary frameworks explaining this phenomenon (e.g., the theory of mental models). However, this paper tries to make the point that the solution might be already in ancient philosophy: in particular, in Chrysippus’ logic. Thus, the paper describes in details (1) why those conditionals are controversial in classical logic and (2) the account that can be given for them from Chrysippus’ philosophy. That account is based mainly on the Stoic idea that the negation of the second clause of a conditional should not be compatible with its first clause.

Sociological Stoicism

This article initiates a conversation between sociological theory and the contemporary revival of Stoicism. Identifying four problems common to historical and contemporary incarnations of Stoicism and tracing them to their shared individualism, I contend that only a sociological Stoicism is viable. I then sketch this sociological Stoicism by redefining key Stoic terms in the collective register; outlining a Stoic logic of case selection; assessing the fit of redefined Stoic concepts and logic of case selection with Marxian, Weberian, and Bourdieuvian frameworks; and developing a Stoic research agenda. These exercises culminate in the proposal to significantly alter sociology’s methods and epistemology.

The Metaphysics of Stoic Corporealism

Abstract The Stoics are famously committed to the thesis that only bodies are, and for this reason they are rightly called “corporealists.” They are also famously compared to Plato’s earthborn Giants in the Sophist, and rightly so given their steadfast commitment to body as being. But the Stoics also notoriously turn the tables on Plato and coopt his “dunamis proposal” that being is whatever can act or be acted upon, to underwrite their commitment to body rather than shrink from it as the Giants do. The substance of Stoic corporealism, however, has not been fully appreciated. This paper argues that Stoic corporealism goes beyond the dunamis proposal, which is simply an ontological criterion for being, to the metaphysics of body. This involves, first, an account of body as metaphysically simple and hence fundamental; second, an account of body as malleable and continuous, hence fit for blending (krasis di’ holou) and composition. In addition, the metaphysics of body involves a distinction between this composition relation seen in the cosmology, and the constitution relation by which the four-fold schema called the Stoic Categories proceeds, e.g. the relation between a statue and its clay, or a fist and its underlying hand. It has not been appreciated that the cosmology and the Categories are distinct — and complementary — explanatory enterprises, the one accounting for generation and unity, the other taking those individuals once generated, and giving a mereological analysis of their identity and persistence conditions, kinds, and qualities. The result is an elegant division of Plato’s labor from the Battle of Gods and Giants. On the one hand, the Stoics rehabilitate the crude cosmology of the Presocratics to deliver generation and unity in completely corporeal terms, and that work is found in their Physics. On the other hand, they reform the Giants and “dare to corporealize,” delivering all manner of predication (from identity to the virtues), and that work is found in Stoic Logic. Recognizing the distinctness of these explanatory enterprises helps dissolve scholarly puzzles, and harmonizes the Stoics with themselves.

Demonstration and the Indemonstrability of the Stoic Indemonstrables

Abstract 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 (apodeixis). Some minor insights into Stoic logic ensue.

The Propedeutic of the Theory of Judgment in Ancient Philosophy from the Sophists to Plato’s Theaetetus

In the ancient epistemology, precisely stated definition of judgment (axioma) appears only in the 3rd century B.C. It was formulated by Chrysippus of Soli, the founder of the Stoic logic. However, on the other hand, the analysis of the extant utterances in which the knowledge had been objectified since the first Greek thinkers, allows us to state that the evolution of the theory of judgment was a long process. In this development, Greek epistemology had to deal with a number of problems connected with the object of the judgment –– knowledge, with the form of its objectification –– predication, and also with the predicates of the true and false judgment –– categories of “truth” (aletheia) and “falsehood” (pseudos). The first definition of the false judgement (logos pseudes) and the true judgment (logos alethes) can be found only in the late dialogue of Plato, Sophist, which delivers precisely established terminology of the theorem. Yet, such a definition could be formulated only when Greek epistemology re-defined the scope of the meaning of the key terms-concepts, aletheia and pseudos. The term-concept aletheia was identified with the term-concept being, functioning in the ontological-axiological sphere. On the other hand, pseudos did not mean false in the sense of negating the truth, but something, which is different than truth, is its imperfect copy. Thus, the pre-Platonic philosophy has not yet formulated the terminology in which predication of something inconsistent with the actual state of being, with the truth, could be verbalized. Often to express such a form of predication, a phrase “to utter things, which are not” was used. The other problem was connected with –– characteristic ofthe Greek language –– dual function of the verb to be/einai, which included both existential and truthful function. Accordingly, every utterance, in which the predicate was the verb einai or its derivates, was ex definitione a true predication –– “it spoke beings (things, which are).” In such a situation, there was noneed in epistemology to precisely define judgment as such, and to state the conditions which the true judgment hadto meet. The problem is definitely solved by Plato in his dialogue Theaetetus, in which the philosopher defines the object of the judgment, which is knowledge (however, its object is not stated yet) and introduces the project of verification of the utterances/opinion, thanks to which an opinion ––doxa can reach the status of judgment ––logos. An opinion needs to be verified with the dialectical procedures.

Stoic Sequent Logic and Proof Theory

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.

On Boccaccio's Debt to Stoicism

On Boccaccio's Debt to Stoicism Michael Papio (bio) An examination of remnants of Stoic thought that could be teased from Boccaccio's works may at first blush seem easy. After all, Stoicism was one of the principal doctae religiones of the ancient world and adherents to the philosophy (or those presumed to be) include Vergil, Cicero, Seneca and Servius. However, this task presents three problems that are difficult to mitigate. First, most of what we know about the Stoa comes from fragments (and medieval thinkers knew even less); second, their unique ideas were unattributable in many cases because they had been influenced by Aristotle, were inherited from Platonism or were accessible in a modified form in works of Neoplatonists1; and third, everything Boccaccio knew of the Stoics had been acquired in Latin, not Greek.2 He mentions the "setta" four times: once equating it quite directly with "filosofia morale" in the case of Seneca's studies with the (semi-)Stoic Sotion (Esp. 4.lit.336); once (by way of the Third Vatican Mythographer to whom Boccaccio refers as Alberic) to explain that the Stoics had an aversion to carnal pleasure (Genealogie 3.22.18); a third time in an inexact reference to the doctrine of pneuma (Gen. 9.8.1);3 and, lastly, suggesting incorrectly—in my reading—that the Stoics were precursors to Plato.4 In other words, although Boccaccio [End Page S-152] did not know enough about Stoicism to be able to disentangle its tenets from those of other ancient philosophies, he had unquestionably given most of them serious consideration as the personal reflections of some of his favorite authors. Excellent work has been done on the Stoics' medieval reception, and much of it examines a host of characters whom Boccaccio knew well,5 but our author is unfortunately never mentioned. In the few pages that follow, it is my intention to point out, albeit briefly and with no claim to comprehensiveness, a small number of Stoic themes that in my opinion were significant to Boccaccio, whether he knew where they came from or not. The Stoics themselves established three fundamental areas of inquiry: logica, ethica and physica (φύσις), and we would do well to divide our quick study accordingly. Stoic logic seems to have left no unambiguous traces in Boccaccio's works, but their moral philosophy affected him deeply, especially through the writings of "Seneca moralis."6 It seems fair to say that Boccaccio understood Seneca's ethical values and respected them, especially because he viewed them as being largely compatible with Christian constancy, which—although Boccaccio probably did not know it—was not a coincidence.7 Jerome and Augustine had studied Seneca, as Boccaccio knew, and spoke more highly of him than others. Boccaccio quotes freely from Seneca (the tragedian) in the Elegia di Madonna Fiammetta and from Seneca (the moralist) even during the period in which he was writing the Decameron.8 It is not difficult to imagine Natan (Dec. 10.3) as ἐλεύθερος in his generosity, King Carlo (10.6) as an eventual example of ἀπάθεια, recalling Augustine's De civitate Dei,9 and, of course, Griselda herself as a paragon of ἀταραξία (10.10). These cases of admirable behavior are hardly out of place in the giornata dedicated to those who, like the Stoics' ideal sage, have acted "magnificamente," but it is uncertain whether Boccaccio would have accurately considered them instances of Stoicism or something more along the lines of a sort of "secular" hagiography. At any rate, it is clear in historical retrospect that Stoic ideas about ethics (in addition to several other important subjects) dovetail nicely with a series [End Page S-153] of notions derived from the New Testament.10 It is also interesting to note that Barlaam of Seminara (d. 1348), with whom both Petrarch and Leontius Pilatus studied, had underscored constancy in his treatise known by its Latin title Ethica secundum Stoicos as a defining characteristic of Stoicism, using beatitudo as a synonym of εὐδαιμονία: "we know that the blessed man is always constant, yet all perturbations are enemies of constancy; thus, the blessed man avoids them and, in so doing, loses no virtue."11 Virtue, he explains, "is an...