Medieval Consequentiae (Theory of Consequence)

Origin. 14th century. Buridan, Ockham, Burley. Inference theory. Valid consequence. Foundation of medieval proof theory.

Models. Consequentia: conditional or inference. Material vs formal. Rules of consequence. Propositional logic core. Inference analysis.

Formalism.

Consequentia structure: antecedent → consequent. "Si p, q" or "p, ergo q." Conditional or inference. Both analyzed.

Formal vs material: Formal: holds in virtue of form. Replacing categorematic terms preserves validity. Material: holds for this matter only. "A man runs, therefore an animal runs."

Formal consequence division: Simple: antecedent can't be true without consequent. As-of-now (ut nunc): true given current facts. Absolute vs temporal.

Rules (Buridan): From impossible, anything follows. Necessary follows from anything. From true, nothing false follows. To false, nothing true leads.

Propositional rules: Ex contradictoriis quodlibet. Modus ponens. Modus tollens. Hypothetical syllogism.

Distribution rules: What's distributed in consequent must be in antecedent. Descent under universal. Scope tracking.

Syncategorematic terms: omnis, nullus, aliquis (quantifiers). si, ergo (connectives). non, et, vel (operators). Logical constants.

Symbols.

Term	Origin	Gloss	Role
consequentia	Latin	Consequence	The inference or conditional taken as a whole
antecedens	Latin	Antecedent	What the consequence proceeds from
consequens	Latin	Consequent	What it proceeds to
ergo	Latin	Therefore	Marks the inferential, as against the conditional, form
si	Latin	If	Marks the conditional form
formalis	Latin	Formal	Holding under substitution for the categorematic terms
materialis	Latin	Material	Holding for this matter only
ut nunc	Latin	As of now	Holding given how things presently are
syncategorema	Latin	Syncategorematic term	The fixed vocabulary held constant under substitution
ex impossibili quodlibet	Latin	From the impossible, anything	Explosion, accepted by most and rejected by Abelard

Metatheory. The theory contains a criterion of formal validity and a live dispute about it. Buridan's Tractatus de Consequentiis (c. 1335) counts a consequence as formal when it holds "in all terms" — under every uniform substitution for the categorematic vocabulary, with the syncategoremata held fixed — which is the substitutional criterion later associated with Bolzano and Tarski. The English tradition of Ockham and Burley defined formality otherwise, by the presence of an intrinsic or extrinsic middle, so the two schools sort inferences differently rather than describing one notion in two idioms. Buridan also rejected the definition of validity as the impossibility of a true antecedent with a false consequent: propositions are tokens that exist only when formed, and one such as "no proposition is negative" is false whenever it exists, though things may be as it signifies. Material consequences he reduced to formal ones by adding a necessary premise, and ut nunc consequences by adding a true one.

Applies to. Medieval logic. Inference. History of logic. Propositional theory.

Limitations. There is no formal semantics and no completeness result. Rules are stated and defended by counterexample, and nothing shows any list of them exhaustive; the consequences are principles governing inferences rather than an axiomatized system whose derivable theorems could be surveyed. Because propositions are token sentences existing only when formed, the substitutional criterion is awkward to apply — an instance must actually be formed before it can be assessed, which is what forced Buridan's revision rather than a preference of style. Quantification is handled through supposition and descent instead of variable binding, so nested quantifiers are treated case by case and multiple generality stays out of reach. Even the propositional core was contested: Abelard's containment account of entailment rejects ex impossibili quodlibet, which the fourteenth-century authors take as a rule.