The Stoics

The founder of the Stoa, Zeno of Citium (335–263 BCE), studied with Diodorus. His successor Cleanthes (331–232) tried to solve the Master Argument by denying that every past truth is necessary and wrote books—now lost—on paradoxes, dialectics, argument modes and predicates. Both philosophers considered knowledge of logic as a virtue and held it in high esteem, but they seem not to have been creative logicians. By contrast, Cleanthes' successor Chrysippus of Soli (c. 280–207) is without doubt the second great logician in the history of logic. It was said of him that if the gods used any logic, it would be that of Chrysippus (D. L. 7.180), and his reputation as a brilliant logician is amply attested. Chrysippus wrote over 300 books on logic, on virtually every topic logic today concerns itself with, including speech act theory, sentence analysis, singular and plural expressions, types of predicates, indexicals, existential propositions, sentential connectives, negations, disjunctions, conditionals, logical consequence, valid argument forms, theory of deduction, propositional logic, modal logic, tense logic, epistemic logic, logic of suppositions, logic of imperatives, ambiguity and logical paradoxes, in particular the Liar and the Sorites (D. L. 7.189–199). Of all these, only two badly damaged papyri have survived, luckily supplemented by a considerable number of fragments and testimonies in later texts, in particular in Diogenes Laertius (D. L.) book 7, sections 55–83, and Sextus Empiricus Outlines of Pyrrhonism (S. E. PH) book 2 and Against the Mathematicians (S. E. M) book 8. Chrysippus' successors, including Diogenes of Babylon (c. 240–152) and Antipater of Tarsus (2nd cent. BCE), appear to have systematized and simplified some of his ideas, but their original contributions to logic seem small. Many testimonies of Stoic logic do not name any particular Stoic. Hence the following paragraphs simply talk about ‘the Stoics’ in general; but we can be confident that a large part of what has survived goes back to Chrysippus.

5.1 Logical Achievements Besides Propositional Logic

The subject matter of Stoic logic is the so-called sayables (lekta): they are the underlying meanings in everything we say and think, but—like Frege's 'senses'—also subsist independently of us. They are distinguished from spoken and written linguistic expressions: what we utter are those expressions, but what we say are the sayables (D. L. 7.57). There are complete and deficient sayables. Deficient sayables, if said, make the hearer feel prompted to ask for a completion; e.g. when someone says ‘writes’ we enquire ‘who?’. Complete sayables, if said, do not make the hearer ask for a completion (D. L.7.63). They include assertibles (the Stoic equivalent of propositions), imperativals, interrogatives, inquiries, exclamatives, hypotheses or suppositions, stipulations, oaths, curses and more. The accounts of the different complete sayables all had the general form ‘a so-and-so sayable is one in saying which we perform an act of such-and-such’. For instance: ‘an imperatival sayable is one in saying which we issue a command’, ‘an interrogative sayable is one in saying which we ask a question’, ‘a declaratory sayable (i.e. an assertible) is one in saying which we make an assertion’. Thus, according to the Stoics, each time we say a complete sayable, we perform three different acts: we utter a linguistic expression; we say the sayable; and we perform a speech-act. Chrysippus was aware of the use-mention distinction (D. L. 7.187). He seems to have held that every denoting expression is ambiguous in that it denotes both its denotation and itself (Galen, On ling. soph. 4; Aulus Gellius 11.12.1). Thus the expression ‘a wagon’ would denote both a wagon and the expression ‘a wagon’.[2]
Assertibles (axiômata) differ from all other complete sayables in their having a truth-value: at any one time they are either true or false. Truth is temporal and assertibles may change their truth-value. The Stoic principle of bivalence is hence temporalized, too. Truth is introduced by example: the assertible ‘it is day’ is true when it is day, and at all other times false (D. L. 7.65). This suggests some kind of deflationist view of truth, as does the fact that the Stoics identify true assertibles with facts, but define false assertibles simply as the contradictories of true ones (S. E.M 8.85).
Assertibles are simple or non-simple. A simple predicative assertible like ‘Dion is walking’ is generated from the predicate ‘is walking’, which is a deficient assertible since it elicits the question ‘who?’, together with a nominative case (Dion's individual quality or the correlated sayable), which the assertible presents as falling under the predicate (D. L. 7.63 and 70). There is thus no interchangeability of predicate and subject terms as in Aristotle; rather, predicates—but not the things that fall under them—are defined as deficient, and thus resemble propositional functions. It seems that whereas some Stoics took the—Fregean—approach that singular terms had correlated sayables, others anticipated the notion of direct reference. Concerning indexicals, the Stoics took a simple definite assertible like ‘this one is walking’ to be true when the person pointed at by the speaker is walking (S. E. M 100). When the thing pointed at ceases to be, so does the assertible, though the sentence used to express it remains (Alex. Aphr. An. Pr. 177–8). A simple indefinite assertible like ‘someone is walking’ is said to be true when a corresponding definite assertible is true (S. E. M 98). Aristotelian universal affirmatives (‘Every A is B’) were to be rephrased as conditionals: ‘If something is A, it is B’ (S. E. M 9.8–11). Negations of simple assertibles are themselves simple assertibles. The Stoic negation of ‘Dion is walking’ is ‘(It is) not (the case that) Dion is walking’, and not ‘Dion is not walking’. The latter is analyzed in a Russellian manner as ‘Both Dion exists and not: Dion is walking’ (Alex. Aphr. An. Pr. 402). There are present tense, past tense and future tense assertibles. The—temporalized—principle of bivalence holds for them all. The past tense assertible ‘Dion walked’ is true when there is at least one past time at which ‘Dion is walking’ was true.

5.2 Syntax and Semantics of Complex Propositions

Thus the Stoics concerned themselves with several issues we would place under the heading of predicate logic; but their main achievement was the development of a propositional logic, i.e. of a system of deduction in which the smallest substantial unanalyzed expressions are propositions, or rather, assertibles.
The Stoics defined negations as assertibles that consist of a negative particle and an assertible controlled by this particle (S. E. M8.103). Similarly, non-simple assertibles were defined as assertibles that either consist of more than one assertible or of one assertible taken more than once (D. L. 7.68–9) and that are controlled by a connective particle. Both definitions can be understood as being recursive and allow for assertibles of indeterminate complexity. Three types of non-simple assertibles feature in Stoic syllogistic. Conjunctions are non-simple assertibles put together by the conjunctive connective ‘both … and …’. They have two conjuncts.[3]Disjunctions are non-simple assertibles put together by the disjunctive connective ‘either … or … or …’. They have two or more disjuncts, all on a par. Conditionals are non-simple assertibles formed with the connective ‘if …, …’; they consist of antecedent and consequent (D. L. 7.71–2). What type of assertible an assertible is, is determined by the connective or logical particle that controls it, i.e. that has the largest scope. ‘Both not p and q’ is a conjunction, ‘Not both p and q’ a negation. Stoic language regimentation asks that sentences expressing assertibles always start with the logical particle or expression characteristic for the assertible. Thus, the Stoics invented an implicit bracketing device similar to that used in Łukasiewicz' Polish notation.
Stoic negations and conjunctions are truth-functional. Stoic (or at least Chrysippean) conditionals are true when the contradictory of the consequent is incompatible with its antecedent (D. L. 7.73). Two assertibles are contradictories of each other if one is the negation of the other (D. L. 7.73); that is, when one exceeds the other by a—pre-fixed—negation particle (S. E. M 8.89). The truth-functional Philonian conditional was expressed as a negation of a conjunction: that is, not as ‘if p, q’ but as ‘not both p and not q’. Stoic disjunction is exclusive and non-truth-functional. It is true when necessarily precisely one of its disjuncts is true. Later Stoics introduced a non-truth-functional inclusive disjunction (Aulus Gellius, N. A. 16.8.13–14).
Like Philo and Diodorus, Chrysippus distinguished four modalities and considered them modal values of propositions rather than modal operators; they satisfy the same standard requirements of modal logic. Chrysippus' definitions are (D. L. 7.75): An assertible is possible when it is both capable of being true and not hindered by external things from being true. An assertible is impossible when it is [either] not capable of being true [or is capable of being true, but hindered by external things from being true]. An assertible is necessary when, being true, it either is not capable of being false or is capable of being false, but hindered by external things from being false. An assertible is non-necessary when it is both capable of being false and not hindered by external things [from being false]. Chrysippus' modal notions differ from Diodorus' in that they allow for future contingents and from Philo's in that they go beyond mere conceptual possibility.

The Stoics

5.1 Logical Achievements Besides Propositional Logic

5.2 Syntax and Semantics of Complex Propositions

The Stoics :: تعاليق

The Stoics