1 Introduction

In Stoic logic, not all valid arguments are also syllogisms. Syllogisms form a special class of valid arguments that are distinct from other valid arguments on account of certain formal features. Within the class of non-syllogistically valid arguments, there are ὑποσυλλογιστικοὶ λόγοι – subsyllogistic arguments. Whereas syllogisms are composed of specific kinds of propositions with specific formulations, subsyllogistic arguments deviate in their expression from those formulations. For instance, a Stoic syllogism would be formulated as ‘If A, then B; but A; therefore, B’ while an argument that is formulated as ‘B follows A; but A; therefore, B’ is subsyllogistic.

Due to the lack of a Stoic source that explains this distinction, its interpretation must rely on the criticism of the distinction by Alexander of Aphrodisias. Alexander finds it nonsensical to distinguish between syllogisms and subsyllogisms on the basis of their formulations. Similarly, Jonathan Barnes has argued that there can be no subsyllogisms (Barnes 2007, 321). Most recently, however, Benjamin Morison has proposed two ways of understanding the distinction between syllogisms and subsyllogisms, the first of which I wish to defend here. His first proposal is that syllogisms and subsyllogisms have different logical forms such that only some logical forms render an argument syllogistic and only syllogisms are valid in virtue of their form (Morison 2011, 183). Morison, however, rejects this proposal because the distinction between syllogisms and subsyllogisms is made among arguments that are formulated in a natural language, Greek. The difference between syllogistic and subsyllogistic formulations is a difference among ordinary Greek expressions. One would normally think that those expressions are different ways of stating the same logical form rather than one expression corresponding to a syllogistic form and the other to another form (Morison 2011, 184). As a consequence, he endorses a second view according to which syllogisms and subsyllogisms have the same logical form. The difference between syllogisms and subsyllogisms would consist in the fact that subsyllogisms use non-canonical expressions. The use of non-canonical expressions is supposed to be significant because (i) arguments expressed in this way do not admit of proof-theoretical analysis as straightforwardly as syllogistic expressions, and (ii) it is less clear that the respective conclusion follows from its premises (Morison 2011, 184). On this view, the distinction concerns a preference for some linguistic expressions over others and would not amount to a logically significant difference between types of arguments.

Morison’s first proposal, which he himself rejects, has been proposed in earlier treatments. Most notably, Susanne Bobzien maintained that the differently formulated premises in a syllogism and the respective subsyllogistic argument are distinct kinds of proposition in so far as the relevant subsyllogistic premises are metalinguistic propositions (Bobzien 1999, 152). This paper aims to defend this analysis.^[1]

Barnes and Morison have each raised a central difficulty in maintaining that subsyllogistic arguments differ from syllogisms on account of their logical form.^[2] In this paper I resolve these difficulties in order to support Bobzien’s analysis of subsyllogisms. I thereby arrive at a more complete picture of what exactly differentiates a subsyllogism from a corresponding syllogism.

The difficulty articulated by Barnes begins with the assumption that a subsyllogistic premise is expressed through a linguistic formulation that means the same as the formulation of a corresponding syllogistic premise. From that assumption it follows that the subsyllogistic premise is either identical to the syllogistic premise, or, if it is distinct from its syllogistic counterpart, then it is not subsyllogistic at all. I will address this difficulty by showing that the relation between syllogistic and the corresponding subsyllogistic premises is not synonymy. The relevant requirement for subsyllogisms is instead ‘equipollence,’ which requires that a subsyllogistic premise and its syllogistic counterpart be evaluated by the same standard. This conception of the subsyllogism can be saved from Barnes’ charge of incoherence.

Morison, by contrast, indicates a difficulty in motivating the view that subsyllogisms have a distinct logical form from that of the respective syllogism. However, I argue that the relevant premises of a subsyllogistic argument, although they are superficially similar to non-simple propositions, are in fact simple propositions. This is because they consist of cases and predicates rather than propositions and propositional operators. For that reason, they do not have the same logical form as the premises of a syllogism. As a result, Morison’s preferred view, the second option above, is ruled out.

Judging by the extant examples, the respective subsyllogistic simple premises even provide the semantic accounts of their syllogistic counterparts. For this reason, I agree with Bobzien’s interpretation of the relevant subsyllogistic premises as metalinguistic propositions; and I build on that view by underlining the distinctness of those metalinguistic premises in virtue of their being simple propositions. The resulting view is that a subsyllogism differs from a corresponding syllogism by having (at least) one premise that is itself simple but is evaluated by the same standard as the corresponding non-simple premise in a syllogism.

Section 2 introduces the accounts of the non-simple propositions that can be found in syllogisms. The main result here is that those non-simple propositions and negated propositions contain propositional operators. Section 3 introduces the notion of subsyllogistic arguments with the help of Alexander of Aphrodisias. This leads to an objection, voiced by Alexander and Barnes, against the very notion of a subsyllogism. The objection is based on an interpretation of the requirement of equipollent premises. Section 4 addresses the crucial assumption in Alexander’s and Barnes’ critique and offers a different reading of the requirement of equipollence. Section 5 prepares my explanation of the difference between syllogisms and subsyllogisms by arguing that the Stoics analyze propositions about propositions as simple propositions consisting of a case that corresponds to a proposition and a predicate about that proposition. Section 6 applies this analysis of propositions about propositions to the extant examples of subsyllogisms: they have simple premises where their syllogistic counterparts have non-simple ones because the relevant premises do not use propositional operators. Finally, section 7 addresses the further question of why subsyllogisms are not another kind of syllogism: subsyllogisms have distinct logical forms in virtue of having those simple premises in place of non-simple ones.

2 Non-simple Propositions

For the Stoics, linguistic expressions signify λεκτά (sayables), which provide the meaning of those expressions. The status and properties of λεκτά are not essential to the aim of this paper. These λεκτά are divided into complete and incomplete ones, where complete λεκτά correspond to complete sentences of various types and incomplete λεκτά correspond to components of sentences and are components of complete λεκτά; at least predicates belong in this class (DL 7.63).

Within the class of complete λεκτά, there are ἀξιώματα that I shall call ‘propositions’ because that term marks out the function ἀξιώματα play in Stoic logic: they are the truth-bearers (DL 7.65).^[3] These ἀξιώματα are further subdivided into simple and non-simple ones: non-simple ones are composed of several ἀξιώματα or the same ἀξίωμα taken twice, while simple ones are composed of one ἀξίωμα that is not taken twice (DL 7.68).

Finally, arguments in general consist of ἀξιώματα (Bobzien 1999, 121) and never of linguistic expressions, since syllogisms (a kind of argument) are listed as things signified (DL 7.63).

Syllogisms are either so-called indemonstrables or reducible to those indemonstrables according to certain rules that are called θέματα (cf. DL 7.78).^[4] In short, this means there are basic kinds of syllogisms into which more complex syllogisms can be decomposed.^[5] Hence, I only need to consider the propositions out of which the indemonstrables are composed. According to DL 7.80–81, which lists descriptions of these five indemonstrables, the indemonstrables employ negated propositions and three types of non-simple proposition: conditional, disjunction, and conjunction.

According to SE M 8.108, non-simple propositions “are composed of a proposition taken twice or of different propositions and in which a connective or connectives govern.”^[6] The crucial point is that a non-simple proposition is governed by a connective (σύνδεσμος). The requirement of a connective that joins propositions is also present in the accounts of conditionals, disjunctions, and conjunctions at DL 7.71–73. A conditional is “that which is composed through the conditional-forming connective ‘if’”^[7] (DL 7.71, cf. SE M8.109–110). At DL 7.72, the conjunction is specified through the connectives ‘both … and ---’ (καὶ … καὶ ---). The disjunction is specified through ἤτοι, which means ‘either’ in ‘either … or ---’, and presumably ‘ἤτοι … ἤ ---’ is meant, as indicated by the example at DL 7.72.

This talk of connectives involves a difficulty in so far as the only explicit characterization of connectives (DL 7.57–58) treats them as words that connect sentences. Surely, however, the connectives that govern non-simple propositions cannot be words. For words, i.e., sounds or ink spots, are corporeal and λεκτά incorporeal (SE M 8.12). It would be absurd if something incorporeal should consist of something corporeal. It is not explicit in the texts how to deal with this, but it is plausible that the connectives which govern non-simple propositions are particles on the level of λεκτά that join propositions.

Despite the lack of information about the precise relation between linguistic connectives and λεκτά, if the Stoics accept these connectives joining propositions, it is certainly plausible that these should have linguistic analogues in sentences (namely, the words ‘and’ or ‘if’) that express non-simple propositions. On this view, different linguistic connectives stand for different connectives on the level of λεκτά. For example, the word ‘and’ will not mean the same as ‘if.’ In order to express a certain non-simple proposition, one must use a word that stands for the correct connective between the relevant propositions. The connective for conditionals, for instance, is specified as having a certain function: it is a conditional-forming connective. This leaves open the possibility that another Greek word, such as εἴπερ at SE M 8.109, or a word that means ‘if’ in another language, could be used to express a conditional-forming connective.^[8] Hence, it is not enough to point to the use of specific words to decide whether an utterance expresses a conditional, since the expression of a conditional, on the view developed here, depends on words that stand for connectives with a certain function and which seem to be able to be expressed by several different words. To decide whether an utterance expresses a conditional, one needs to determine the function of the connectives expressed in the utterance.

In fact, the connectives in conditionals, disjunctions, and conjunctions are operators that take two or more propositions and form a non-simple proposition. The connectives are distinct in that they produce different kinds of propositions with different kinds of truth-condition.

Aside from conditionals and disjunctions, the third indemonstrable makes use of the negations of conjunctions. At DL 7.69–70, there are three types of negation, one of which is the negation of an entire proposition (rather than a component of it). This type is relevant to the third indemonstrable and the negations of conjunctions and it is called ἀποφατικὸν ἀξίωμα.^[9] The description of these negations reads in Dorandi’s text: “<…> and of a proposition, such as ‘not: it is day’” (DL 7.69).^[10] The other types of negative proposition are specified as being composed of (συνεστὸς ἐκ) a certain type of negative particle (μόριον) and the relevant element of the λεκτόν that is negated (DL 7.70). It is, hence, safe to assume that the ἀποφατικὸν ἀξίωμα is composed of a negative particle and a proposition.^[11] These negative particles can be assumed to take a proposition and to form another proposition. In the case of negations, there is no explicit mention of specific types of words. But here it seems to be a plausible assumption that negative particles on the level of λεκτά are expressed by specific words that combine with whole sentences to produce a negated sentence, such as οὐχί in Diogenes’ example.

Summarizing these results, the major premises of the five indemonstrables are of certain types that are explained through operators which take one (negation) or more (conditional, disjunction) propositions and produce another proposition. These operators appear to be associated with certain kinds of linguistic expressions.

3 Alexander and Barnes against Subsyllogistic Arguments

Diogenes Laertius reports at DL 7.78 a distinction between syllogistically valid arguments and ‘specifically’ valid ones (i.e., a species of valid arguments that has the same name as the genus). The only explanation Diogenes provides is that the latter class of arguments does conclude but not in a syllogistic manner. Then he cites an example:

Example 1: ‘It is day and it is night’ is false; but it is day; therefore, not: it is night.

ψεῦδός ἐστι τὸ ἡμέρα ἐστὶ καὶ νύξ ἐστιν· ἡμέρα δέ ἐστιν· οὐκ ἄρα νύξ ἐστιν.

At first glance, this argument might look like a third indemonstrable, but the major premise for a third indemonstrable would be: ‘not (it is day and it is night).’ The difference between a syllogism and Example 1 is visible in the linguistic formulation of a premise.

According to Alexander and Galen, the label for this type of argument is ὑποσυλλογιστικός λόγος, subsyllogistic argument.^[12] Galen’s account of such arguments is that “the so-called subsyllogistic arguments [are] said in equipollent formulations” (Gal. Inst. log. xix 6).^[13] According to Galen, the distinctive feature of such arguments is some sort of equipollence in formulations in which the arguments are said. He does not elaborate on what is equipollent to what and wherein the equipollence consists.

Alexander makes the following remark in the context of two different formulations for Aristotle’s mood Baroco:

Such [an argument] is subsyllogistic, as it is called by the more recent ones, which takes that which is equipollent to a syllogistic premise and infers the same through this premise. For ‘not belonging to all’ has been changed to ‘not belonging to some’ to which it is equipollent. (Alex. In An. Pr. 84, 12–15)^[14]

In this passage, Alexander describes what the “more recent ones” (i.e., Stoic logicians) call a subsyllogistic argument. According to him, it has the following features: (i) it has a premise (presumably at least one) that is in some sense equipollent (ἰσοδυναμοῦν) with, but distinct from, a syllogistic premise; and (ii) a subsyllogistic argument entails the same conclusion as the corresponding syllogism. So, if a given syllogism is valid, a corresponding subsyllogism is valid as well.

On the basis of (i) and (ii), Alexander proceeds to construe an objection against the distinction between syllogisms and subsyllogisms. He accepts that equipollent formulations of premises can be used to infer the same conclusion, but he does not want to say that the difference in formulations has any impact on the kind of argument that is used (Morison 2011, 185). Alexander’s critique is perhaps clearest in another context:

[1] So that whenever the same things are signified by different formulations primarily and taken similarly, the syllogism will be the same. The formulation that says ‘animal is what is substance’ signifies the same as the formulation that says ‘animal is in the genus substance.’ And in this way a change from a statement to a statement has come about: therefore, also their change into each other will not at all make the syllogisms different. […] Aristotle, then, deals in this way with changes according to formulations. [2] But the more recent ones, since they attend to the formulations and no longer to the things that are signified, do not say that the same happens in the changes into equipollent formulations of the terms. For even though ‘If A, then B’ signifies the same as ‘B follows A,’ they say that an argument is syllogistic if such a formulation is taken ‘If A, then B; but A; therefore, B’, and that ‘B follows A; but A; therefore, B’ is no longer syllogistic, but valid [in the specific sense]. (Alex. In An. Pr. 373, 19–35)^[15]

This text stems from Alexander’s commentary on An. Pr. I.39 49b3 – “δεῖ δὲ καὶ μεταλαμβάνειν ἅ τὸ αὐτὸ δύναται.” Aristotle recommends switching between formulations that ‘have the same power’ and Alexander explains [1] that this is reasonable, as it does not change the argument of which the formulations are a part.

Then he accuses [2] the “more recent ones” (i.e., the Stoics) of paying attention only to the “formulations” (λέξεις), and not also to that which is signified by them, when they distinguish between syllogisms and subsyllogisms. At first glance, there is some merit to this accusation: the accounts of the types of non-simple proposition specify connectives and there are connectives which are parts of λόγος (i.e., speech), and every λόγος is also a λέξις (i.e., an utterance or formulation; DL 7.57). So it is not entirely unmotivated for Alexander to claim that they pay attention to words. But, as indicated in section 2, it seems plausible for the Stoics also to consider connectives on the level of what is signified and it will turn out that they do not disregard that which is signified in their account of subsyllogisms.^[16]

Alexander’s objection consists in claiming that there is a tension between asserting that the two example formulations signify the same and distinguishing, as the Stoics do, between syllogisms and subsyllogisms on the basis of the differences between those two formulations.

First, ‘If A, then B’ is said to signify the same as ‘B follows A’ (ταὐτὸν γὰρ σημαίνοντος τοῦ ‘εἰ τὸ Α, τὸ Β’ τῷ ἀκολουθεῖν τῷ Α τὸ Β). There is a difference only in the words, but these words mean the same. Alexander calls the phenomenon at issue ‘change into equipollent formulations’ (ἐν ταῖς εἰς τὰς ἰσοδυναμούσας λέξεις μεταλήψεσι). Equipollent (ἰσοδυναμοῦσαι) λέξεις are ones that signify the same, in Alexander’s usage.^[17] For Alexander introduces his objection by saying that they only pay attention to the λέξεις in these changes and then explains that they make the distinction between syllogisms and subsyllogisms even though the relevant λέξεις signify the same (cf. Barnes 2007, 317).^[18] According to Alexander, for the Stoics the type of argument

Example 2: B follows A; but A; therefore, B

ἀκολουθεῖ τῷ Α τὸ Β, τὸ δὲ Α, τὸ ἄρα Β

is not syllogistic, even though ‘B follows A’ is equipollent to the syllogistic premise ‘If A, then B.’

Alexander is not explicit on what exactly the problem is with paying attention to words rather than the things they signify. He might think that it is inconsistent for the Stoics to distinguish between syllogisms and subsyllogisms because they cannot both think that arguments consist of λεκτά (i.e., meanings of expressions), and that a mere difference in words makes a difference in types of argument.

At any rate, Barnes (2007, 320–321) has a very clear objection on this point, which is a variant of Alexander’s objection. In Examples 1 and 2 and the corresponding syllogisms, the ink spots that express these arguments either signify the same arguments or different arguments. If the ink spots for the examples and the corresponding syllogisms signify the same arguments, then there can be no distinction between the two types of argument. But if the ink spots signify two distinct arguments, then that is because the ink spots for the different syllogistic and subsyllogistic premises express two distinct premises. But then the examples are not subsyllogistic anymore because, Barnes maintains, the expressions of subsyllogistic major premises are synonymous with the expressions of syllogistic major premises. Hence, Barnes concludes that there can be no subsyllogistic arguments.

The crucial assumption in Barnes’ argument (as in Alexander’s) is that subsyllogistic and syllogistic formulations have the same meaning (i.e., signify the same proposition). If the equipollence required for subsyllogisms is the synonymy of linguistic expressions, then it is indeed difficult to see why the Stoics would make the distinction between syllogisms and subsyllogisms. In fact, that distinction would seem to be arbitrary.^[19]

4 Equipollence

The central question for evaluating these criticisms of the Stoics is how the Stoics understand the equipollence between syllogistic and subsyllogistic formulations. I contend that Alexander’s and Barnes’ assumption that for the Stoics equipollent formulations express the same proposition is false. Subsyllogistic formulations express different propositions than their syllogistic counterparts and the equipollence consists in a feature these propositions share rather than in the shared meaning of the respective formulations. For the indemonstrables are described as having specific types of major premises and these types of propositions are defined through connectives and negations that appear to have specific linguistic expressions (see section 2). Two sentences, even when they are equipollent, ought to express two different propositions if they contain expressions which each stand for different kinds of elements of a λεκτόν. The claim here is not merely that ‘If …, then ---’ is a different linguistic expression than ‘--- follows …,’ but that it signifies a λεκτόν with a different function. For the accounts of the types of propositions under consideration require connectives with specific functions. If the relevant premises of subsyllogisms can be shown to contain elements with different functions, then they must be different kinds of proposition. As I argue in sections 5 and 6, the elements of λεκτά that differ in the relevant cases are, on the one hand, connectives and negative particles and, on the other hand, predicates.

In order to maintain the distinction between syllogisms and subsyllogisms, the Stoics should claim that Example 1 does not contain the negation of a conjunction and Example 2 does not contain a conditional. This requires that the relevant premises in these examples are of distinct types from the major premises in the corresponding syllogisms. Now I aim to show that the Stoics do make a distinction between different but equipollent propositions.

Alexander and Galen use the verb ἰσοδυναμεῖν to describe the phenomenon at issue. For Alexander it amounts to claiming synonymy between two expressions. There is some indication that the Stoics use that expression, too:

In as many ways as it is possible to change the equipollent [things] into each other, it is possible to change the forms of the epicheiremes and enthymemes in the domain of arguments. (Epict. Diss. I.8.1)^[20]

Epictetus here speaks of τὰ ἰσοδυναμοῦντα in the context of arguments – practically the same context as in Alexander. It is safe to assume that he is talking about the changes between certain formulations and corresponding changes in the relevant arguments.

Yet as a later author, Epictetus might have picked up this talk of equipollence from later developments or the Peripatetics. However, after the quoted text he gives a Stoic syllogism as an example, so he is probably not relying on Peripatetic discussions. And when Epictetus mentions logical issues, his remarks can usually be connected to Chrysippus or, at any rate, older developments in Stoic logic, as indicated by Jonathan Barnes (1997, 77–125). Thus, Epictetus seems to have adopted this use of the verb ἰσοδυναμεῖν from early Stoics. Accordingly, it is plausible that when Alexander uses this verb in a similar context, namely in his report on the Stoic concept of the subsyllogism, he adopts a term the Stoics used themselves.

Now the expression ἰσοδυναμεῖν when used in the context of arguments cannot for the Stoics have the same meaning as it has for Alexander: the δύναμις in question cannot be what a linguistic expression signifies. This becomes clear from the following text:

For the writers of handbooks say that a definition differs only in composition from the universal, whereas it is the same according to the power. And reasonably so: for someone who says ‘Man is a rational mortal animal’ says the same with respect to the power and something different with respect to the sound as the one who says ‘If something is a human being, it^[21] is a rational mortal animal.’ (SE M 11.8, trans. modified from Ruge 2022, 112)^[22]

The context of the passage mentions Chrysippus (SE M 11.11), so this should be Stoic doctrine.^[23] A universal proposition is (here, at least) a conditional that consists of an indefinite proposition as the antecedent and a proposition which is expressed with an anaphoric pronoun as the consequent,^[24] such as (the proposition signified by) ‘If something is a human being, it is a rational mortal animal.’ An alternative designation for such propositions is ‘indefinite conditional’ (infinitum conexum, Cic. Fat. 15). Definitions are here apparently taken to be predicative statements, such as (the proposition signified by) ‘Man is a rational mortal animal.’ The Stoic claim is that these definitions which are formulated in a predicative manner are equipollent to, and can be formulated as, such universal conditionals. The two items mentioned differ, according to the source, only in composition (or syntax) but have the same ‘power’ (δύναμις).

This has been taken to mean that the two items are linguistic expressions which are distinct, but that they have the same meaning or δύναμις.^[25] I contend that this reading is impossible.^[26]

First, the two items are introduced with εἰπῶν (or the dative thereof), the aorist active participle of λέγειν (‘to say’ or ‘to speak’). It is unlikely that in Stoic terminology a form of λέγειν would be used to introduce an utterance:

And also saying differs from uttering: for sounds, on the one hand, are uttered; things, on the other hand, which are certainly also sayables, are said. (DL 7.57, trans. modified from Ruge 2022, 113)^[27]

The Stoics draw a sharp distinction in terminology between uttering a sound and saying a λεκτόν. Hence, a form of λέγειν should signal that a λεκτόν is meant rather than its linguistic formulation. Thus, one would not expect the two items that are different in composition to be formulations but the propositions themselves.

Second, these examples provide the clearest case of what I mentioned above: according to SE M 8.108, a connective governs a non-simple proposition. The proposition ‘Man is a rational mortal animal’ does not contain a connective; ‘If something is a human being, it is a rational mortal animal’ does contain a connective. Hence, the one proposition is simple and the other non-simple.

However, at this point it has been argued that the indefinite conditional ‘If something is a human being, it is a rational mortal animal’ is not a conditional at all, nor a non-simple proposition (Bobzien and Shogry 2020, 16–18). Bobzien and Shogry contend that this proposition is governed by the indefinite particle and that the ‘if’ is part of the predicate. Due to the importance of this passage for my purposes and the potency of this challenge, I now digress in order to consider Bobzien and Shogry’s two arguments against regarding indefinite conditionals as non-simple propositions.

Their second argument (2020, 17) concerns the passage M 11.8: according to the Stoics, the expressions that are mentioned there have the same meaning. The proposition ‘Man is a rational mortal animal’ must be simple. Since the other proposition must be the same because the expressions are synonymous, the indefinite conditional must be simple. This argument is the inversion of my own, so that one is confronted with two opposing readings of this text.

In order to support Bobzien and Shogry’s version, one might adduce the fact that the passage M 11.8 contrasts δύναμις with φωνή:^[28] the example items are distinct with respect to the φωνή (i.e., the sound or expression), and the same with respect to the δύναμις. An admittedly natural reading of this contrast is that the expressions are different, but they have the same signification (i.e., one and the same proposition).

But that need not be the intention in this contrast, since the difference in the sound that is mentioned need not be the only difference between the two items.^[29] The difference in the sound could well correspond to a difference between two propositions when one of them is non-simple and the other simple if the suggestion from section 2 is correct that connectives on the level of λεκτά always correspond to connectives in the linguistic expressions. The focus might be on the sound because that is a perceptible and immediately recognizable difference.

Therefore, the wording of M 11.8 is at least not an indisputable reason for Bobzien and Shogry’s reading, since alternative readings are possible. Nor is the wording definitive proof against Bobzien and Shogry, since the contrast between δύναμις and φωνή does admit of the reading they prefer. Further considerations are needed.

Their first argument (2020, 16–17) introduces the putative requirement, called ‘anaphora removal,’ that when the antecedent and consequent in a conditional have the same subject, they must contain the same term. However, the non-anaphoric conditional ‘If someone is walking, someone is moving’ is different from ‘If someone is walking, he is moving.’ Since the examples for indefinite conditionals cannot repeat the antecedent’s subject term in the consequent, they are not genuine conditionals. They justify this requirement through examples such as ‘If Plato lives, then Plato breathes.’

This is a problematic interpretation because indefinite conditionals are prima facie conditionals and could be the counterexample such that anaphora removal is not required for all conditionals. To be sure, Bobzien and Shogry claim that ‘indefinite’ could be an alienans adjective (2020, 16), which means that indefinite conditionals need not be conditionals merely because they are called conditionals. But granted that ‘indefinite’ could be intended in this way, it could also be intended as demarcating a species of conditionals. Thus, they need to show that ‘indefinite’ is in fact an alienans adjective. However, the observation that all the conditionals which are not called ‘indefinite’ display anaphora removal is consistent with the view that indefinite conditionals are conditionals. This is precisely because ‘indefinite’ could demarcate a species of conditionals in which the consequent needs to be anaphoric. Thus, Bobzien and Shogry’s observations do not seem to be clinching.

The requirement of anaphora removal is supposed to make it apparent that the consequent in a conditional can be detached through an application of a first indemonstrable. And the inapplicability of anaphora removal for indefinite conditionals is supposedly associated with the impossibility of detaching the pseudo-consequent of such propositions (Bobzien and Shogry 2020, 17). I suppose they mean, first, that in their example ‘If someone is walking, he is moving,’ the pseudo-consequent ‘He is moving’ could not be asserted on its own because the anaphora cannot be removed. Second, every conditional must be such that it can function in a modus ponens argument. Hence, indefinite conditionals cannot be genuine conditionals. I shall grant the second assumption, namely that every conditional can feature in a first indemonstrable, since it seems to follow from the description of the first indemonstrable (cf. DL 7.80; SE PH 2.157).^[30] I will instead question the first assumption that these putative pseudo-consequents cannot be detached because the anaphora cannot be removed.

It is not obvious that every detached occurrence of anaphora (i.e., in an independent simple proposition) must be able to be removed. Maybe it is only necessary that anaphoric pronouns take their reference from the linguistic context. Given that condition, one could detach ‘He is moving’ from ‘If someone is walking, he is moving.’ In fact, there is reason to think the Stoics considered this latter condition to be necessary and sufficient for anaphoric pronouns. At DL 7.70, there are these two examples of indefinite propositions in the transmitted text:

• Someone is walking (τις περιπατεῖ)

• He is moving (ἐκεῖνος κινεῖται).

Thus, the Stoics appear to regard ‘He is moving’ with an anaphoric ‘he’ as an indefinite proposition. Crivelli (1994, 189–190) accordingly introduced a distinction between non-anaphoric and anaphoric simple indefinite propositions to make sense of these examples.^[31] An anaphoric proposition here is a proposition that employs anaphora in its subject, which means at least that it is expressed with an anaphoric pronoun in subject position. The argument below indicates that in such propositions the subjects on the level of the λεκτά are themselves anaphoric as well.

Now ‘He is moving’ might be indefinite already because of the anaphora itself and not because the anaphoric pronoun can be substituted with an indefinite pronoun.^[32] In that case, anaphoric propositions would not be stand-ins for another type of proposition; they would seem to be a type of proposition in their own right. They would have to be stand-ins for other propositions if the applicability of anaphora removal were necessary for detached anaphoric propositions. For example, for Bobzien and Shogry’s argument to work, ‘He is moving’ would only be a detachable, if suboptimal, consequent if it was really the proposition ‘Plato is moving’ in the conditional ‘If Plato is walking, Plato is moving.’

It is reasonable to classify anaphoric simple propositions as indefinite because they should count as indefinite by the criteron of SE M 8.97: in that text, ‘Someone is walking’ counts as indefinite “since it does not determine one of the particular walkers” (ἐπεὶ οὐκ ἀφώρικέ τινα τῶν ἐπὶ μέρους περιπατούντων). Such propositions accordingly do not have a referent (Bobzien and Shogry 2020, 15). But this criterion applies to ‘He is moving’ as well because the expression of the proposition does not contain a noun that would determine the reference, nor is it accompanied by an act of indication (δεῖξις) that would determine the reference.^[33] For the other two types of affirmative simple propositions are intermediate or categorical propositions (e.g., ‘Socrates is walking’; cf. DL 7.70; SE M 8.97), and definite or categoreutical propositions (e.g., ‘This one is walking’; cf. DL 7.70; SE M 8.96). Definite propositions are the ones accompanied by a δεῖξις. The difference between Crivelli’s anaphoric and non-anaphoric indefinite propositions then concerns the question of whether the reference is determined by the linguistic context or left open entirely.^[34]

Hence, it seems that the subject of an anaphoric simple proposition does not itself determine a reference to a specific object and that the linguistic context determines the reference. But then there is no reason to deny that the consequent of ‘If someone is walking, he is moving’ can be detached even though the anaphora cannot be removed. Hence, it is still possible for indefinite conditionals to be genuine conditionals.

Thus, Bobzien and Shogry’s arguments do not seem convincing. To return to M 11.8, I prefer the view I introduced. That passage does not compare two expressions of the same proposition, but two distinct propositions. So the δύναμις at issue is not the meaning of an expression, but rather a feature of the propositions themselves. In this Stoic usage, propositions have the same δύναμις and yet different linguistic expressions (τῇ μὲν δυνάμει τὸ αὐτὸ λέγει, τῇ δὲ φωνῇ διάφορον).^[35] And these propositions must be of distinct types because the one contains a connective and the other does not. The fact that two distinct propositions share the same δύναμις could be paraphrased as the statement that they are equivalent, which means at least that the one is true if and only if the other one is true.^[36]

Now the passage is not explicit on what this equivalence consists in.^[37] However, material^[38] and strict^[39] equivalence can be relatively safely excluded. Material equivalence obtains between two propositions as soon as both of them happen to have the same truth-value. Strict equivalence obtains between any two necessarily false or true propositions.^[40] If either of these notions of equivalence were at issue, there would be nothing about ‘Man is a rational mortal animal’ and ‘If something is a human being, then it is a rational mortal animal’ that would render these two propositions in particular equivalent. But the point seems to be specifically about propositions of these forms and not just any propositions that happen to be true or false at the same time. Since the equivalence should rely on specific features of these particular kinds of proposition, it can be plausibly viewed as an intensional equivalence.^[41]

The most likely candidate for the δύναμις shared by the two propositions would be their truth-conditions. Such a δύναμις would be a suitably specific feature of the two propositions. This is supported by the way these two types of propositions are discussed at M 11.9: both the universal conditional and the corresponding definitory proposition are said to encompass all the species that fall under the definiendum. Moreover, they are said to be false when there can be a human being that is not a rational mortal animal; this presumably means that they are true when there could not be such a counterinstance.^[42] The two examples in this text are clearly meant to be evaluated by the same standard (which is sufficient to consider them equivalent). Therefore, the two propositions in question are equivalent for the Stoics in the sense that their truth is evaluated by the same standard, namely their shared δύναμις. The equivalence here thus amounts to a special case of an intensional relation that holds because the propositions in question are evaluated by the same standard.^[43]

If one applies this interpretation of how the Stoics use the term δύναμις to Alexander’s discussion, the relation between syllogistic and subsyllogistic major premises is not synonymy in expression but equivalence of the propositions themselves. On this view, Alexander used an authentic account of subsyllogistic arguments but interpreted the crucial term ἰσοδυναμεῖν differently.

This notion of equivalence can be applied to Examples 1 and 2 and their syllogistic counterparts. Supposing for a moment that the expression ‘B follows A’ (from Example 2) indeed signifies a different kind of proposition than ‘If A, then B,’ it is easy to see that they are equivalent in virtue of being evaluated by the same standard. For the description of conditionals also contains the following claim about the conditional-forming connective:

And this connective announces that the second follows the first, such as ‘If it is day, it is light.’ (DL 7.71)^[44]

The connective is associated with a kind of truth-condition. This kind of truth-condition is whatever ‘following’ means. Hellenistic logicians were in agreement that a conditional is true when the consequent follows the antecedent, and they had some controversies on when this following obtains (cf. SE PH 2.110–113; M 8.110–117), but that is not the issue here. A conditional ‘If P, then Q’ is true if and only if Q follows P, which is the case only when ‘Q follows P’ is true. Hence, the relevant premise in Example 2 is equivalent to a conditional in virtue of the account of conditionals.

Similarly, ‘‘It is day and it is night’ is false’ (from Example 1) employs the term ‘false’ that also appears in the account of opposition with respect to truth and falsity for propositions and their contradictory opposites at DL 7.73. And the negation of a proposition is explicitly identified as one in a pair of contradictory opposites there. ‘Not-P’ is true if and only if it is false that P, which is the case only when ‘P is false’ is true.

Therefore, the subsyllogistic premises under consideration constitute the semantic accounts of their syllogistic counterparts and are for that reason equivalent to them.^[45]

However, even if equipollence is equivalence and not synonymy, the distinct but equivalent premises in syllogisms and the respective subsyllogisms might be distinct tokens of the same type of proposition. For instance, Example 2 could contain a conditional, but a different conditional than the one in the corresponding syllogism. But then Example 2 would still fulfil the description of the first indemonstrable (cf. DL 7.80; SE PH 2.157). The next task is to show that these subsyllogistic and syllogistic formulations do indeed express different kinds of proposition, and that the distinction between syllogisms and subsyllogisms hinges on such differences in kinds of proposition.

5 Cases of Propositions

I contend that the expressions ‘… is false’ and ‘--- follows …’ stand for predicates rather than operators (connectives, negative particles) that act on propositions.^[46] According to DL 7.64, a predicate (κατηγόρημα) is “an incomplete λεκτόν that can be coordinated with a nominative case for the generation of a proposition.”^[47] This coordination of a predicate with a nominative case surely results in a type of simple proposition. In fact, two types of simple proposition are explicitly described as consisting of a predicate and a nominative case at DL 7.70: the categorical (or intermediate) propositions, such as ‘Dion is walking,’ and the categoreutical (or definite) proposition, such as ‘This one is walking.’ The ἀποφατικόν (i.e., the negation of a proposition, or negated proposition), is listed among the simple propositions (DL 7.69), but it does not result from the combination of a predicate and a nominative case, but rather the combination of a proposition with a negative particle.

Predicates differ from connectives and negative particles in that predicates are combined with cases to form a proposition, whereas connectives and negative particles act on propositions to form propositions.

The decisive question concerning the subsyllogistic formulations of Examples 1 and 2 is whether they are deviant expressions for propositional operators or instead meant to be a different kind of expression altogether, namely expressions of predicates. I argue that the formulations of the subsyllogistic major premises express the application of a predicate to a case of a proposition.

To defend this claim, it is necessary to consider cases more closely. DL 7.64 (see above) indicates the function of nominative cases: they fill, so to speak, the empty slot of a predicate to form a complete λεκτόν. Thus, they are components of complete λεκτά and especially of propositions. The Stoics also recognize other cases, namely genitive, dative and accusative (DL 7.75). There is evidence that the Stoics also recognized incomplete λεκτά that take several cases to form a proposition, such as ‘loves’ which needs a subject and an object (Amm. In Int. 44, 32–45, 3).^[48] This most likely applies to the λεκτόν expressed by ‘--- follows …’ as well.

Since cases are components of incorporeal λεκτά, one would expect them to be incorporeal and λεκτά themselves.^[49] Clement of Alexandria claims that “the case is agreed to be incorporeal” (ἡ πτῶσις δὲ ἀσώματος εἶναι ὁμολογεῖται, LS 33O). In addition, Gourinat (2000, 128) has pointed out strong evidence against the corporeality of cases based on a clearly Stoic text: in the list of complete λεκτά at DL 7.67, there is an item called προσαγορευτικόν (greeting) and its example is expressed exclusively by nouns. Nouns should stand for cases so that the example is a complete λεκτόν composed only of cases. If cases were corporeal, this complete λεκτόν would consist only of bodies. But it is supposed to be incorporeal and cannot consist only of bodies (even if one granted that an incorporeal proposition could have some corporeal components). Thus, cases should be incorporeal.

Cases are components of propositions that stand for some object, but there is a question as to what sorts of object are eligible for falling under a case. According to SE M 8.11–13, the Stoics link three elements together in a semantic relation: a sound, the meaning of the sound, and the object to which the sound refers.^[50] Here the third item, which is called the τυγχάνον, is most relevant, for the τυγχάνον is that which falls under a case.^[51]

But Sextus says at M 8.12 that the τυγχάνον is corporeal. If this means that everything falling under a case is corporeal, then there can be no cases of propositions and my proposed analysis of Examples 1 and 2 does not work. However, prima facie there should also be cases for λεκτά, since it is possible to speak about what someone has said as much as about physical objects. Sextus’ statement may only apply to the examples that he has in mind. The ontological status of the τυγχάνον is not central to his concerns anyway, as he focuses on the notion of the λεκτόν in his discussion of the Stoics.

Furthermore, the Stoics most likely do make statements about propositions and, thus, should accommodate such statements in their semantical theories. At M 8.255, Sextus reports a requirement on signs, namely that the sign be present and the sign of something present. A sign is a proposition that is the true antecedent in a true conditional and reveals the consequent (SE PH 2.104). In his discussion of that requirement, he says the following:

The ones who say such things evidently do not know that past and future things are different, but that the sign and signified are even in these cases something present of something present. For in the first example ‘If this one has a scar, this one has had a wound’ the wound has already occurred and is past, but that this one has had a wound, since it is a proposition, is present, as it is being said of something that has happened. And in ‘If this one has a wound in the chest, this one is going to die’ the death is imminent, but that this one is going to die is present as a proposition, as it is being said about something that will happen. For that reason it is now also true. (SE M 8.255)^[52]

The point of the passage is that it is a previously agreed upon requirement of signs that they be simultaneous with that which they signify. Yet, the Stoics use examples where sign and signified occur at different times. The response is that physical events which are related in a sign-conditional may occur at different times in these examples, but the propositions are true simultaneously.

More pertinently, the passage contains two ways of referring to propositions. Both employ the neuter singular definite article as a quotational device. The first way is to reformulate an assertion into an articulated infinitive as indirect speech:

• τὸ δὲ ἕλκος ἐσχηκέναι τοῦτον (‘that this one has had a wound’)

• τὸ δὲ ἀποθανεῖσθαι τοῦτον (‘that this one is going to die’)

That this is a device specifically for referring to propositions rather than sentences is clear from the fact that the passage calls these items ἀξιώματα and that these are the things signified by signs. That which is signified by a sign is the consequent in a kind of conditional and, hence, a proposition.

But the passage contains another way of using the definite article as a quotational device, namely in direct speech:

• ἔν τε τῷ “εἰ οὐλὴν ἔχει οὗτος, ἕλκος ἔσχηκεν οὗτος” (in ‘If this one has a scar, this one has had a wound’)

• ἔν τε τῷ “εἰ καρδίαν τέτρωται οὗτος, ἀποθανεῖται οὗτος” (in ‘If this one has a wound in the chest, this one is going to die’)

Here again, the quotational device is used for the proposition and not the sentence. The consequents in these examples are called ἀξιώματα, and so the items which contain them should be ἀξιώματα as well.^[53]

The Stoics consciously formulate expressions that refer to propositions and fill the functions of nouns in a sentence, both in the nominative and the dative in these examples. When one says a proposition about a proposition – for instance, that it is true or that it is a proposition (as in the text from M 8.255) – that proposition seems to consist of a predicate and a case. This analysis of the quotational devices is preferable to an analysis in terms of operators that govern propositions (e.g., for ‘… is true’) because the examples in direct speech resist the latter type of analysis; they occur in the dative and are difficult to construe as being governed by operators. And it is plausible that the use of quotational devices is analyzed uniformly (i.e., as standing for a case) for all syntactical roles that such a device can occupy in a sentence. Therefore, a quotational device in a linguistic expression should stand for a case that corresponds to a proposition which, in turn, is the τυγχάνον.

6 Subsyllogistic Arguments Redeemed

Now I will argue that the subsyllogistic formulations in the examples stand for simple, predicative propositions and not for non-simple propositions. The major premises in the examples are formulated as follows:

Example 1: ‘It is day and it is night’ is false (ψεῦδός ἐστι τὸ ἡμέρα ἐστὶ καὶ νύξ ἐστιν)

Example 2: B follows A (ἀκολουθεῖ τῷ Α τὸ Β)

These formulations take two sentences that express propositions and form one sentence that expresses a proposition. However, the expressions ‘--- follows …’ and ‘… is false’ do not stand for operators that act on propositions.^[54]

This is clear in Example 1, as it uses a quotational device for expressing cases; ψεῦδός ἐστι τὸ ἡμέρα ἐστὶ καὶ νύξ ἐστιν is parallel to the examples from the previous section. It uses a quotation in direct speech for a non-simple proposition. Thus, the phrase ψεῦδός ἐστι expresses a predicate that applies to the case of that proposition. It cannot express a propositional operator if it is combined with an expression for a case rather than with an expression for a proposition.

In the case of ‘--- follows …’ the empty spaces can be filled with mere nouns and the resulting expression is still a complete sentence. This is because the verb ἀκολουθεῖν has several uses in Greek where the subject and indirect object are not propositional constructions. Consider, for instance, this testimony of Chrysippus and Cleanthes in the context of the ethical maxim to live in accordance with nature:

Chrysippus understands ‘nature,’ following which one must live, as both the common and the individual human nature. But Cleanthes takes the common nature, which one must follow, and no longer the individual nature. (DL 7.89)^[55]

Hence, it is accidental to the expression ‘--- follows …’ that it takes two sentences. That means either the Stoics think that ‘--- follows …’ has different functions in different contexts: it can signify a predicate when it takes two nouns or a connective when it takes two sentences. Alternatively, the Stoics might think that the expression always has the same function, namely as signifying a predicate with two cases – and these cases can be of propositions in the relevant examples. The use of the definite article in ἀκολουθεῖ τῷ Α τὸ Β, which might be taken to indicate that this is an example of the quotational device for expressing cases of propositions, is not decisive; this use of the definite article is already motivated by the use of the letters A and B as placeholders for propositions. Hence, Example 2 is by itself not decisive for the analysis of the difference between syllogistic and subsyllogistic formulations.

But the comparison with Example 1 heavily suggests that here, too, a predicate is used in connection with cases of propositions. Example 1 uses an expression that does not stand for a propositional operator, but it contains an equipollent proposition. It is plausible that the correct analysis of the major premise in Example 2, as well, should be the one according to which the expression ‘--- follows …’ stands for a predicate rather than a propositional operator.

Hence, it can be reasonably claimed that this is what renders the subsyllogistic formulations of Examples 1 and 2 distinct from syllogistic ones: they contain expressions that stand for predicates which take cases. As such, the subsyllogistic formulations in these examples do not express non-simple propositions but simple ones. Thus, Examples 1 and 2 are not just syllogisms with different expressions. They are distinct arguments.

Even though there are only two examples without an explicit Stoic account of what makes them subsyllogistic, it is plausible to generalize the analysis of these examples to the entire class of Stoic subsyllogisms.^[56] After all, Examples 1 and 2 are introduced by Diogenes and Alexander as arguments that are valid but not syllogisms (see section 3). Now Diogenes (DL 7.78) uses Example 1 to illustrate what it means for an argument to be non-syllogistically valid. But this example (as well as Example 2) turns out not to be a syllogism (instead of a syllogism with a non-standard expression). Thus, Diogenes gives the strong impression that being a non-syllogistically valid argument entails not being a syllogism (instead of permitting syllogisms with non-standard expressions to count as non-syllogistically valid). Now it is clear that subsyllogistic arguments in general are meant to be non-syllogistically valid (Gal. Inst. log. xix.6). Therefore, it is quite plausible that all subsyllogisms are distinct arguments from syllogisms. Judging by Examples 1 and 2, the difference between subsyllogisms and corresponding syllogisms always concerns the use of a simple premise in a subsyllogism instead of a non-simple syllogistic major premise.^[57]

7 The Difference between Syllogisms and Subsyllogisms

Granted that subsyllogisms have equivalent but distinct (in kind) premises, why can they not simply be a different kind of syllogism?

The difference between syllogistic and subsyllogistic premises affects the logical form of those premises and the arguments they constitute.^[58] The Stoics seem to have theorized about logical forms under the label τρόπος or ‘mood’ (cf. DL 7.76, SE M 8.227).^[59] There does not seem to be any evidence for the claim that the Stoics recognized moods of subsyllogistic arguments, since moods occur only in the contexts of syllogisms (cf. DL 7.76, SE M 8.227) and certain invalid arguments (cf. SE M.433). But it suffices for my purposes that if one applied the procedure for formulating syllogistic moods to subsyllogisms, the result would differ from a syllogistic mood in an important respect, as becomes clear below. That is, subsyllogisms do not have syllogistic moods. Even though subsyllogisms may not have moods at all (for the Stoics), one can certainly describe their formal features, and I speak of their logical form to discuss their formal differences to syllogisms.

Now, to explain the notion of a mood, in the argument

If it is day, it is light.

But it is day.

Therefore, it is light.

one could uniformly replace every occurrence of ‘It is day’ and ‘It is light’ with two other propositions like this:

If it is night, it is dark.

But it is night.

Therefore, it is dark.

The resulting argument is dissimilar from the first one in that it uses different simple propositions, but it also has something in common. What the two arguments have in common can be called the ‘form’ of the argument. This stands in contrast to the contents ‘It is day’ and ‘It is light,’ which have been changed in these examples. To mark out the common feature of these arguments, one can also do away with particular propositions entirely and use placeholders for ‘It is day’ and ‘It is light.’ The Stoics choose numerals, ‘the first’ and ‘the second’ in order to formulate moods in this way. The expression of a mood consists only of non-logical constants that are in need of interpretation and logical constants (cf. Barnes 2007, 307–314). This permits the evaluation of an argument irrespective of the content and truth-values of the propositions out of which it is composed.

But how does one arrive at ‘the’ mood in which an argument is formulated? Surely, there are different ways of formalizing the same argument.^[60] First, the examples of moods at SE M 8.235, 432–431 and DL 7.80–81 suggest that the Stoics tend to assign one constant to each proposition and one proposition to each constant. Second, the most specific mood results from assigning constants to all simple affirmative propositions (Bobzien 1999, 131). But the context in which an argument is discussed may warrant the assignment of constants to non-simple propositions as well.^[61] That is, which mood is used depends on which features of an argument one wants to discuss (Frede 1974, 146).

Most importantly, the non-logical constants in the expressions of moods are always placeholders for propositions (rather than subpropositional λεκτά). This can be seen at DL 7.76 and SE M 8.227–238, 432–433. That means the structure of simple propositions is not reflected in the moods of arguments. This is reasonable to the extent that the inferences of Stoic syllogistic depend only on relations between propositions.^[62] If all the non-logical constants stand for propositions, then all the logical constants must be propositional operators (i.e., connectives and negative particles).

Since the Stoics only treat propositional operators as logical constants, they do not consider subpropositional structures when formalizing arguments. But the major premises in Examples 1 and 2 are simple propositions, without propositional operators. Hence, those arguments do not have syllogistic moods and their validity is not visible in their logical form, as illustrated below. Consider this third indemonstrable:

Not (it is day and it is night).

But it is day.

Therefore, not: it is night.

This yields the following, using letters instead of numerals:

Not (P and Q).

But P.

Therefore, not-Q.

Now consider Example 1:

‘It is day and it is night’ is false.

But it is day.

Therefore, not: it is night.

The phrase ‘… is false’ does not express a negation but a predicate. And this predicate does not combine with a proposition as a proposition but with a case of a proposition. The conjunction ‘It is day and it is night’ is, hence, not present as a conjunction, but rather fills the role of the subject. Hence, ‘‘It is day and it is night’ is false’ is a simple proposition and can only be represented by one non-logical constant. Thus, Example 1 yields the following:^[63]

P.

But Q.

Therefore, R.

The validity of Example 1 is not visible in its logical form as one could substitute an argument with true premises and a false conclusion in this form.^[64]

The same goes for Example 2: ‘B follows A’ is a schematic representation that replaces cases (of propositions) with interpretable constants. As such, it would be a suitable form for making the validity of a subsyllogistic argument visible. But this form does not capture what the Stoics consider a logical constant and the logical relations they consider fundamental, namely the ones used in the indemonstrables. Hence, the validity of Example 2 (as well as Example 1) is not visible in its form. At least, this result holds based on the Stoic criteria for determining the logical form of an argument, which also means that a subsyllogism could be a syllogism in a different system, but not in Stoic logic.^[65]

8 Conclusion

A subsyllogistic argument is one in which at least one premise is a simple proposition that serves as a counterpart to a syllogistic non-simple premise in so far as it constitutes the semantic account of the syllogistic premise. The difference in wording between a syllogistic and subsyllogistic formulation indicates a difference in the propositions that are signified by the formulations, namely that one of them is non-simple and the other simple. This difference in the kind of premises between syllogisms and subsyllogisms amounts to a difference in the logical form of syllogisms and subsyllogisms. If one formulates the logical form of a subsyllogism by the rules of Stoic syllogistic, the result is not adequate to explain the validity of the argument. Syllogisms, on the other hand, are valid in virtue of their form. Hence, the Stoics do precisely what Morison thinks they would not do: they distinguish between these arguments because they are expressed through different natural language formulations that correspond to different kinds of proposition.