Aristotelian Modal Syllogistic

Origin. Aristotle (4th c. BCE). Prior Analytics A.8–22, with the modal conversion rules in A.3. The modal extension of the assertoric syllogistic: premises and conclusions qualified by necessity and contingency. The first system of modal logic. Disputed from the start — Theophrastus and Eudemus, Aristotle's own school, rejected his mixed apodeictic moods.

Models. Categorical propositions under modal qualification. Three premise qualities: apodeictic (necessary), assertoric (unqualified), problematic (contingent). Pure moods: both premises of one quality. Mixed moods: qualities combined — the contested part of the system. Aristotle's three figures (the fourth is post-Aristotelian; see Aristotelian Syllogistics.md). Modality attaches inside the categorical sentence, to the predication of terms — whether that is a de re or a de dicto claim is the question every reconstruction must settle.

Formalism.

Modal values: N: apodeictic — necessarily. X: assertoric — no modal qualification. Q: problematic — two-sided contingency, neither necessary nor impossible. M: one-sided possibility — not impossible. Aristotle's ἐνδεχόμενον covers both Q and M. The defined sense (An. pr. A.13, 32a18) is Q: "that which is not necessary and the supposed existence of which implies nothing impossible."

Mood notation: three letters, major premise – minor premise – conclusion. Barbara NXN: apodeictic major, assertoric minor, apodeictic conclusion.

Apodeictic syllogistic: the valid assertoric moods, with N added to both premises and conclusion. NNN Barbara, Celarent, Darii, Ferio.

Conversion of apodeictic propositions: exactly as assertoric. NE converts simply: NEba ↔ NEab. NI converts simply: NIba ↔ NIab. NA converts per accidens: NAba → NIab.

Conversion of problematic propositions: Complementary conversion (Ross's term): QA ↔ QE, QI ↔ QO — two-sided contingency is indifferent to negating the predicate. QE does not convert simply: Aristotle denies QEba → QEab. For one-sided possibility: MA → MIab, MI → MIab, ME ↔ MEab; MO does not convert.

The two Barbaras (An. pr. A.9, 30a15–25): Barbara NXN: Necessarily every b is an a. Every c is a b. ⊢ Necessarily every c is an a. — asserted. Barbara XNN: Every b is an a. Necessarily every c is a b. ⊬ — rejected; only the assertoric conclusion follows. Aristotle's ground for NXN: "since every b is necessarily an a or not an a, and c is one of the b's, it is evident that c too will be necessarily one of them." Theophrastus and Eudemus, against: the conclusion always has the modal character of the weaker premise — the in peiorem rule (frs. 106–107 FHS&G), with M weaker than X and X weaker than N. Under their rule NXX and XNX are the valid mixed Barbaras and the asymmetry disappears.

The minor-term ambiguity (A.13): "a may be predicated of everything of which b is predicated" reads either "every c that is a b" or "every c that may be a b" — Aristotle notes both readings.

Symbols.

Metatheory. Aristotle's own metatheory is the same as for the assertoric system: imperfect moods are reduced to the perfect moods of the first figure by conversion and reductio per impossibile, and invalidities are established by term counterexamples — the method survives the modal extension, but the conversion rules it rests on no longer agree across the qualities, since QE fails to convert. What the moderns established is mostly negative. Jan Łukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic (2nd enlarged edition, Clarendon Press 1957), whose added chapters treat the modal system inside his own modal propositional logic, delivered the standing verdict: "in contrast to the assertoric syllogistic which is perfectly clear and nearly free of errors, Aristotle's modal syllogistic is almost incomprehensible because of its many faults and inconsistencies", and of the contingency fragment, "Aristotle's theory of contingent syllogisms is full of grave mistakes." Albrecht Becker, Die Aristotelische Theorie der Möglichkeitsschlüsse (Berlin: Junker und Dünnhaupt, 1933), opened the diagnostic line: Łukasiewicz credits him with pointing out the ambiguity of ἐνδεχόμενον — "the merit of having pointed out this ambiguity is due to A. Becker." The reading of the modal sentences themselves is the other diagnostic axis, and Ulrich Nortmann's survey of the approaches runs through de dicto- and de re-necessity ("The Logic of Necessity in Aristotle", History and Philosophy of Logic 23, 2002). A very wide range of reconstructions has been proposed — Storrs McCall's Aristotle's Modal Syllogisms (North-Holland 1963) among the earliest — the majority reproducing only part of what Aristotle asserts. Marko Malink, Aristotle's Modal Syllogistic (Harvard University Press 2013), built a semantics whose primitives are predication relations between terms (aX- and aN-predication) and is the reconstruction that reproduces everything Aristotle says — though the model, as its reviewers note, is of high complexity and its sentence meanings are not compositional. The system's continuations are their own entries: Avicenna's temporal reading of the modalities, whose dhātī/waṣfī disambiguation of the subject term removes the difficulty of the mixed apodeictic-assertoric moods (see Avicenna Temporal.md), and Buridan's modal logic, which separates de dicto from de re questions, reads the divided modal proposition with its subject ampliated to possible beings, and drops the essentialist assumptions of the thirteenth-century tradition.

Applies to. History of modal logic. Arabic and Latin modal syllogistics. The de re/de dicto distinction. Aristotle scholarship, where the reconstruction programme is an active research area.

Limitations. No modern reconstruction is canonical: the apparatus that validates Barbara NXN while invalidating Barbara XNN, the apodeictic conversions, and the contingency moods together has to be built for the purpose, and Susanne Bobzien's assessment in the Stanford Encyclopedia states the standing possibility plainly — as Aristotle's theory is conceivably internally inconsistent, the formal models that have been suggested may all be unsuccessful. The problematic syllogistic is the worst case: complementary conversion makes the Q-forms interdeducible while QE refuses simple conversion, and this is where Łukasiewicz located the grave mistakes. The reasons for the two-Barbaras asymmetry have been found, in Robin Smith's words, obscure or unpersuasive or both, almost from Aristotle's own time. Modalities do not iterate, and there is no modal logic of propositions under the term logic — Łukasiewicz: "a modal logic of terms presupposes a modal logic of propositions; this was not clearly seen by Aristotle." And the system inherits every expressive limit of the assertoric syllogistic it extends: monadic terms only, no relations, no propositional connectives, no nested quantification (see Aristotelian Syllogistics.md).

© 2026 Lingenic LLC