Post-Avicennan Arabic Logic

Origin. The twelfth to fifteenth centuries, working from Avicenna rather than from Aristotle. Fakhr al-Dīn al-Rāzī (d. 1210) reopened the Avicennan modal syllogistic critically; Afḍal al-Dīn al-Khūnajī (d. 1248) rebuilt it in the Kashf al-asrār; Najm al-Dīn al-Kātibī al-Qazwīnī (d. 1276) compressed the result into al-Risāla al-Shamsiyya, which became the standard madrasa handbook and drew the commentary of Quṭb al-Dīn al-Rāzī al-Taḥtānī (d. 1365) and the glosses of Sa'd al-Dīn al-Taftāzānī (d. 1390) and al-Sayyid al-Sharīf al-Jurjānī (d. 1413). The commentary-and-gloss form carried the system into the Ottoman, Persian, and Indian traditions and kept it in teaching use into the nineteenth century.

Models. A proposition is fixed by three independent choices rather than by a single modal operator: a modality (necessary, actual, possible), a temporal quantification over the times at which the predicate is affirmed (always, sometimes, not always), and a condition delimiting the times in question (while the subject exists, or while it answers the description under which it is named). Crossing these yields the list of thirteen categorical forms canonized in the Shamsiyya, extended to fifteen in some later treatments. Each form has its own conversion behaviour and its own record of which syllogistic moods it sustains, so the system is a table of licensed combinations rather than a calculus with general rules.

Formalism.

Thirteen (later fifteen) propositions: Combinations of:

modality: necessary, possible, actual
temporality: always, sometimes, not-always
conditions: while-subject-exists, while-description-holds

Sample types: wujūdiyya lā-dā'ima: actual, not perpetual "A is B actually but not always"

ḍarūriyya mā dāma dhātuhu: necessary while essence "A is necessarily B while A exists"

ḍarūriyya mā dāma mawṣūfan: necessary while described "Writer necessarily moves while writing"

Al-Kātibī's Shamsiyya: Systematic treatment. Standard pedagogical text. Thirteen categorical propositions.

Two-sorted reading of a form: "Every J is necessarily B while J" ≈ ∀x (Jx → ∀t (Jxt → Bxt)) "Every J is B not always" ≈ ∀x (Jx → ∃t ¬Bxt) Modality and temporality both become quantification over times.

Conversion rules: Complex: type changes in conversion. E-conversion restricted by modality. Temporal conditions preserved differently. Converting a form need not return a form of the same type.

Syllogistic combinations: Not all moods valid for all types. Some Aristotelian moods fail. New moods emerge.

Conditions on validity: Actualist vs essentialist readings. While-existing vs while-described. Mixing types carefully.

Later refinements: Al-Taftāzānī: Commentary tradition. Al-Jurjānī: Further distinctions. Continues to Ottoman period.

Beyond the categorical: Conditionals (sharṭiyya) treated as propositions in their own right. luzūmī (implicative) vs ittifāqī (coincidental) connection. Khūnajī and successors develop a syllogistic for them.

Symbols.

Symbol	Unicode	Name	Meaning
ḍarūrī	—	ضروري, necessary	Holds at every time
wujūdī	—	وجودي, actual	Holds at some time, without necessity
dā'im	—	دائم, perpetual	Holds always, asserted without modality
lā-dā'im	—	لا دائم, not-perpetual	Fails at some time
mumkin	—	ممكن, possible	Neither necessary nor impossible
mawṣūf	—	موصوف, described	The subject taken under a description
mā dāma	—	ما دام, so long as	Prefix of the time-restricting condition
dhāt	—	ذات, essence	Condition "so long as the essence remains"
sharṭiyya	—	شرطية, conditional	Hypothetical proposition, opposed to categorical
luzūmī	—	لزومي, implicative	Connection holding by necessity
ittifāqī	—	اتفاقي, coincidental	Connection holding as a matter of fact
∀, ∃	U+2200, U+2203	Quantifiers	Used in the two-sorted rendering
ḍ, ṭ, ṣ	U+1E0D, U+1E6D, U+1E63	Dot-below letters	Transliteration of the Arabic emphatic consonants
ā, ī, ū	U+0101, U+012B, U+016B	Macron vowels	Transliteration of the Arabic long vowels

Metatheory. The tradition establishes no metatheorems of its own — there is no completeness proof, no independence argument, and no demonstration that the thirteen forms exhaust the possibilities — so what follows is what modern reconstruction has shown. Because both the modality and the temporal qualification of a categorical are quantifications over times, each form translates into a two-sorted first-order formula over individuals and times, and the syllogistic sits inside a decidable fragment: whether a given mood is valid is settled by a finite check. On that reading the disputes recorded in the commentaries — whether a mood holds, whether a conversion is legitimate — become determinate questions with answers, and Street, Thom, Hodges, and El-Rouayheb have reconstructed particular disputes this way. The treatment of conditionals is the other point of genuine advance: hypothetical propositions are handled as propositions rather than reduced to categoricals, and a syllogistic is developed for them.

Applies to. Arabic logic history. Modal logic. Syllogistic extensions. Islamic intellectual history.

Limitations. The system grows by enumeration rather than by rule: thirteen forms, then fifteen, each with its own conversions and its own table of admissible moods, and nothing in the framework generates them or bounds their number, so a new distinction can only be met with a new entry. Necessity remains omnitemporal truth, which leaves accidental universal regularities indistinguishable from necessary ones and unrealized possibilities indistinguishable from impossibilities. Time is quantified but not ordered — always, sometimes, not always, with no successor, no metric, and no relation between two times — so no temporal sequence can be stated. Propositions remain two-term categoricals: relations of higher arity, nested quantification, and iterated modalities are all outside the language, which is why the relational inferences the tradition itself recognized as valid could not be brought under the syllogistic. Existential import is assumed throughout, so the oppositions presuppose non-empty subject terms.