Stoicism

11. Inferential logic

Propositions are the bearers of truth and falsity, and simple propositions are the atomic units of Stoic logic. One symptom of the latter is that in syllogistic the negation sign ‘Not…’ is properly prefixed to the entire proposition, rather than (as in Aristotle’s logic) to its predicate. On the other hand, propositions are classified partly according to the subject terms which they contain: ‘This individual is walking’ (ideally accompanied by pointing) is a ‘definite’ proposition; ‘Someone is walking’ is ‘indefinite’; and propositions expressed with a noun in the subject position, for example, ‘A human being is walking’, ‘Dion is walking’, are ‘intermediate’. This triple distinction, like most aspects of Stoic logic, is adopted ultimately for the sake of validating the arguments in which these simple propositions feature. Indefinite propositions are verified by the corresponding intermediate or definite ones, but not vice versa. Hence in a syllogism whose major premise begins ‘If x is walking…’ and whose minor premise has the form ‘But y is walking…’, validity will normally be preserved if ‘x is walking’ is indefinite, but neither if it is definite while ‘y is walking’ is indefinite or intermediate, nor if it is intermediate while ‘y is walking’ is indefinite.

Complex propositions are those compounded of two or more simple propositions with the help of connectives like ‘and’ or ‘if’. Here again, they are selected for their role in syllogistic theory. Thus they comprise: conjunctive propositions, for example, ‘It is day and it is light’, which are important mainly in negated conjunctions, for example, ‘Not: it is day and it is dark’; disjunctive propositions, for example, ‘Either it is day or it is night’; and conditionals, for example, ‘If it is day, it is light’.

Simple and complex propositions are the basic components of syllogisms. These are held either to have, or to be reducible to by four rules of analysis known as the themata (see Logic, ancient §6), one of the following five forms: (1) ‘If the first, the second; but the first; therefore the second’; (2) ‘If the first, the second; but not the second; therefore not the first’; (3) ‘Not: the first and the second; but the first; therefore not the second; (4) ‘Either the first or the second; but the first; therefore not the second’; (5) ‘Either the first or the second; but not the second; therefore the first’. These are considered irreducibly primitive argument forms, hence called the five ‘indemonstrables’.

A great deal of work went into the diagnosis of argument validity (see Logic, ancient §5). Both formal rigour and inferential cogency were demanded. A valid argument, according to Stoic logic, is one in which the conclusion follows from the conjunction of the premises in just the way in which in a true conditional the consequent follows from the antecedent. Hence the notion of ‘following’ became a focal point of debate. Some Stoics adopted a truth-functional analysis of a true conditional: the second proposition follows from the first just if ‘Not (the first and not the second)’ is true (see Philo the Dialectician for the background to this). But Chrysippus adopted a much stronger criterion, called synartēsis or ‘cohesion’: ‘If the first, the second’ is true just if the negation of the second is incompatible with the first. This restricts ‘following’ to a strongly conceptual kind of implication.

The restriction in turn led Chrysippus to the view that many ordinary-language uses of ‘if’ do not represent real conditionals, and should properly be expressed by the negated conjunction: not ‘If the first, the second’, but ‘Not both: the first and not the second’. He applied this, for example, to the rules of empirical sciences such as divination (see §20); and analogous doubts led at least some Stoics to deny the validity of inductive inference. It was also applied, importantly, to the individual steps of the Little-by-little argument or Sorites (see Vagueness §2). The Academic sceptics plagued the Stoics with this puzzle, aimed at challenging important philosophical distinctions by asking to be told the exact cut-off point. The archetypal Sorites is ‘If two grains are not a heap, three aren’t; if three aren’t, four aren’t;…’ – and so on to the conclusion that if two grains are not a heap then 10,000 are not. Chrysippus, who wrote extensively on this problem, advised that at some point in the procedure there will be premises of this form which it is proper neither to affirm nor to deny. But he also seems to have authorized the Stoic practice of insisting that the individual steps be formulated not as conditionals, as above, but as negated conjunctions: for example, ‘Not both: seven grains are not a heap and eight grains are a heap’. Part of the point is clearly that, whatever the grounds for asserting such a premise may be, there is no actual incompatibility between ‘Seven grains are not a heap’ and ‘Eight grains are a heap’, so that assent to the premise must remain optional.

A wide range of other logical puzzles exercised Chrysippus and other Stoics. Some turned on ambiguities, and they developed a sophisticated classification of ambiguity types. The most persistent thorn in their flesh, however, was the liar paradox: ‘I am lying’ is, if false, true, and, if true, false (see Semantic paradoxes and theories of truth §1). This, along with the Sorites, was wielded by the Academics as a potentially lethal weapon against Stoic logic. It appeared to undermine the most basic tenet – that every proposition is either true or false. Chrysippus wrote many books in refutation of the liar paradox, but if he had his own favoured solution it is impossible now to recover it from our sources.

Inferential logic is ubiquitously employed in Stoic texts on physics and ethics. These regularly purport to demonstrate their tenets, and a demonstration is itself defined as an argument in which evident premises serve to reveal a non-evident conclusion.