Arguments

Arguments are—normally—compounds of assertibles. They are defined as a system of at least two premises and a conclusion (D. L. 7.45). Syntactically, every premise but the first is introduced by ‘now’ or ‘but’, and the conclusion by ‘therefore’. An argument is valid if the (Chrysippean) conditional formed with the conjunction of its premises as antecedent and its conclusion as consequent is correct (S. E. PH 2.137; D. L. 7.77). An argument is ‘sound’ (literally: ‘true’), when in addition to being valid it has true premises. The Stoics defined so-called argument modes as a sort of schema of an argument (D. L. 7.76). The mode of an argument differs from the argument itself by having ordinal numbers taking the place of assertibles. The mode of the argument

اقتباس :: If it is day, it is light.
But it is not the case that it is light.
Therefore it is not the case that it is day.

is

اقتباس :: If the 1st, the 2nd.
But not: the 2nd.
Therefore not: the 1st.

The modes functioned first as abbreviations of arguments that brought out their logically relevant form; and second, it seems, as representatives of the form of a class of arguments.

5.4 Stoic Syllogistic

In terms of contemporary logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural-deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata. An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of thethemata (D. L. 7.78). Thus syllogisms are certain kinds of formally valid arguments. The Stoics explicitly acknowledged that there are valid arguments that are not syllogisms; but assumed that these could be somehow transformed into syllogisms.
All basic indemonstrables consist of a non-simple assertible as leading premiss and a simple assertible as co-assumption, and have another simple assertible as conclusion. They were defined by five standardized meta-linguistic descriptions of the forms of the arguments (S. E. M 8.224–5; D. L. 7.80–1):

A first indemonstrable is an argument that concludes from a conditional and its antecedent the consequent
A second indemonstrable is an argument that concludes from a conditional and the contradictory of the consequent the contradictory of the antecedent .
A third indemonstrable is an argument that concludes from the negation of a conjunction and one of the conjuncts the contradictory of the other conjunct.
A fourth indemonstrable is an argument that concludes from a disjunction and one of the disjuncts the contradictory of the other disjunct.
A fifth indemonstrable is an argument that concludes from a disjunction and the contradictory of one of its disjuncts the other disjunct.

Whether an argument is an indemonstrable can be tested by comparing it with these meta-linguistic descriptions. For instance,

اقتباس :: If it is day, it is not the case that it is night.
But it is night.
Therefore it is not the case that it is day.

comes out as a second indemonstrable, and

اقتباس :: If five is a number, then either five is odd or five is even.
But five is a number.
Therefore either five is odd or five is even.

as a first indemonstrable. For testing, a suitable mode of an argument can also be used as a stand-in. A mode is syllogistic, if a corresponding argument with the same form is a syllogism (because of that form). However in Stoic logic there are no five modes that can be used as inference schemata that represent the five types of indemonstrables. For example, the following are two of the many modes of fourth indemonstrables:

اقتباس :: Either the 1st or the 2nd.
But the 2nd.
Therefore not the 1st.
Either the 1st or not the 2nd.
But the 1st.
Therefore the 2nd.

Although both are covered by the meta-linguistic description, neither could be singled out as themode of the fourth indemonstrables: If we disregard complex arguments, there are thirty-two modes corresponding to the five meta-linguistic descriptions; the latter thus prove noticeably more economical. The almost universal assumption among historians of logic that the Stoics represented their five (types of) indemonstrables by five modes is false and not supported by textual evidence.[4]
Of the four themata, only the first and third are extant. They, too, were meta-linguistically formulated. The first thema, in its basic form, was:

When from two [assertibles] a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows (Apuleius Int. 209.9–14).

This is an inference rule of the kind today called antilogism. The third thema, in one formulation, was:

When from two [assertibles] a third follows, and from the one that follows [i.e. the third] together with another, external assumption, another follows, then this other follows from the first two and the externally co-assumed one (Simplicius Cael. 237.2–4).

This is an inference rule of the kind today called cut-rule. It is used to reduce chain-syllogisms. The second and fourth themata were also cut-rules, and reconstructions of them can be provided, since we know what arguments they together with the third thema were thought to reduce, and we have some of the arguments said to be reducible by the second thema. A possible reconstruction of the second thema is:

When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two.

A possible reconstruction of the fourth thema is:

When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s). (Cf. Bobzien 1996.)

A Stoic reduction shows the formal validity of an argument by applying to it the themata in one or more steps in such a way that all resultant arguments are indemonstrables. This can be done either with the arguments or their modes (S. E. M 8.230–8). For instance, the argument mode

اقتباس :: If the 1st and the 2nd, the 3rd.
But not the 3rd.
Moreover, the 1st.
Therefore not: the 2nd.

can be reduced by the third thema to (the modes of) a second and a third indemonstrable as follows:
When from two assertibles (‘If the 1st and the 2nd, the 3rd’ and ‘But not the 3rd’) a third follows (‘Not: both the 1st and the 2nd’—this follows by a second indemonstrable) and from the third and an external one (‘The 1st’) another follows (‘Not: the 2nd’—this follows by a third indemonstrable), then this other (‘Not: the 2nd’) also follows from the two assertibles and the external one.
The second thema reduced, among others, arguments with the following modes (Alex. Aphr. An. Pr. 164.27–31):

اقتباس :: Either the 1st or not the 1st.
But the 1st.
Therefore the 1st.
If the 1st, if the 1st, the 2nd.
But the 1st.
Therefore the 2nd.

The Peripatetics chided the Stoics for allowing such useless arguments. In agreement with contemporary logic, the Stoics insisted that, if the arguments can be reduced, they are valid.
The four themata can be used repeatedly and in any combination in a reduction. Thus propositional arguments of indeterminate length and complexity can be reduced. Stoic syllogistic has been formalized, and it has been shown that the Stoic deductive system shows strong similarities with relevance logical systems like those by Storrs McCall. Like Aristotle, the Stoics aimed at proving non-evident formally valid arguments by reducing them by means of accepted inference rules to evidently valid arguments. Thus, although their logic is a propositional logic, they did not intend to provide a system that allows for the deduction of all propositional-logical truths, but rather a system of valid propositional-logical arguments with at least two premises and a conclusion. Nonetheless, we have evidence that the Stoics expressly recognized many simple logical truths. For example, they accepted the following logical principles: the principle of double negation, stating that a double negation (‘not: not: p’) is equivalent to the assertible that is doubly negated (i.e. p) (D. L. 7.69); the principle that any conditional formed by using the same assertible as antecedent and as consequent (‘if p, p’) is true (S. E. M 8.281, 466); the principle that any two-place disjunctions formed by using contradictory disjuncts (‘either p or not: p’) is true (S. E. M 8.282, 467); and the principle of contraposition, that if ‘if p, q’ then ‘if not: q, not:p’ (D. L. 7.194, Philodemus Sign., PHerc. 1065, XI.26–XII.14).

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Arguments

5.4 Stoic Syllogistic

Arguments :: تعاليق

Arguments

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