APo. 1.1 Prerequisites for scientific knowledge: prior knowledge required for learning
All teaching and learning presupposes antecedent cognition. Two modes:
(i) knowledge that x is
(ii) knowledge what x is
APo. 1.2 Unconditional scientific knowledge: definition of demonstrative knowledge
We have scientific knowledge of x when we know (i) the cause why x is, (ii) that this cause is the cause of x, and (iii) that this cannot be otherwise. Scientific knowledge is had in virtue of a demonstration—a deduction from things that are true, primary, immediate, better known than, prior to, and explanatory of the conclusion.
six conditions on premises of a demonstration
Premises of a demonstration must be: (i) true; (ii) primary (a starting-point); (iii) immediate (for which no premise is prior); (iv) better known than the conclusion; (v) prior to by nature to the conclusion; and (vi) causes of the conclusion.
PA 1.3 Against Circular Demonstration and Infinite Regress
If all knowledge required demonstration, demonstrations would regress infinitely or be circular — both absurd. Circular demonstration (antistrophē) proves nothing non-trivially. Aristotle concludes: there must be immediate (amesa) propositions, known non-demonstratively, that ground the chain. These are first principles (archai).
PA 1.4 Necessary Predication: Per Se and Universal
Demonstrative knowledge concerns what is necessarily the case. Two key modes of per se (kath’ hauto) predication: (i) the predicate is in the definition of the subject (e.g., line belongs to triangle per se); (ii) the subject is in the definition of the predicate (e.g., odd belongs to number per se). Per accidens predication cannot ground demonstration.
PA 1.4 (def.) Per Se (Kath’ Hauto) — Four Senses
Per se1: predicate is in the subject’s definition (e.g., ‘having angles’ of triangle). Per se2: subject is in the predicate’s definition (e.g., ‘odd’ of number). Per se3: existence is not predicated of another (substance). Per se4: that which belongs to something by reason of itself, not incidentally. Only per se1 and per se2 are formally relevant to demonstration.
PA 1.4 (def.) Katholou — Universal Predication
A predicate belongs katholou (universally) to a subject when it belongs (i) to every instance (kata pantos), (ii) per se, and (iii) qua itself (hē auto). The third condition excludes accidental universality: ‘having 2R’ belongs to every isosceles, but not qua isosceles — only qua triangle. Demonstration requires the widest subject of which the predicate holds kath’ hauto.
PA 1.5 The Error of Demonstrating Too Narrowly
A demonstration is defective if the predicate is shown for a narrower genus than the widest one for which it holds per se. E.g., proving ‘alternando’ separately for numbers, lines, times, solids, when it holds for all magnitudes qua magnitudes. The proper demonstration unifies under the most general relevant subject.
PA 1.6 Demonstrative Premises Must Be Necessary
Since the conclusion of a demonstration must be necessary (the fact cannot be otherwise), the premises must also be necessary. A syllogism from contingent premises may yield a contingent truth, but not episteme. Per se predications are necessary; per accidens predications are not, and therefore cannot serve as demonstrative premises.
PA 1.7 No Cross-Genus Demonstration
Demonstration cannot transfer across genera: one cannot prove arithmetic theorems by geometry or vice versa. The principles, subject genus, and attributes must all belong to the same genus. Exception: subalternation — optics is under geometry, harmonics under arithmetic; a science may borrow principles from its superior science (architektonikē).
PA 1.7 (def.) Subalternate Sciences
A subalternate (subordinate) science takes its explanatory principles from a superior science. E.g., optics borrows geometrical principles; harmonics borrows arithmetical principles. The superior science provides the ‘why’ (to dioti); the subalternate establishes ‘the that’ (to hoti). Knowledge of the fact vs. knowledge of the reasoned fact (1.13).
PA 1.8 Demonstration Concerns What Is Always True
Demonstration is of what holds always (aei), not of what holds sometimes or for the most part. Perishable particulars cannot be demonstrable objects; demonstration is of kinds and their necessary attributes. This does not mean only eternal entities are knowable, but that demonstration concerns the necessary features of natural kinds.
PA 1.9 Demonstrations Must Proceed from Appropriate Principles
Demonstrations must derive from the proper principles of the subject genus, not merely any true first principles. Using Bryson’s method (squaring the circle via a shared genus property) is sophistical — even if valid, it does not demonstrate the thing qua what it is, and so fails to produce scientific knowledge.
PA 1.10 Types of First Principles
Three types of archai: (1) Axioms (axiōmata/koinai archai): common to all sciences; must be known for any learning (e.g., LNC, LEM). (2) Theses: proper to a science; of two kinds — (a) Hypotheses (hypotheseis): assert existence of the subject genus; (b) Definitions (horismoi): assert what terms mean without asserting existence.
PA 1.10 (def.) Axiom / Hypothesis / Definition
Axiom: a principle that must be grasped by anyone who learns anything (LNC, LEM). Hypothesis: a first principle that asserts existence of the subject matter (e.g., ‘unit exists’ in arithmetic). Definition: stipulates the meaning of a term without asserting existence (e.g., ‘unit is indivisible quantity’). Only hypotheses make existential claims within a science.
PA 1.11 Role of Common Axioms in Demonstration
Common axioms (e.g., ‘if equals are subtracted from equals, the remainders are equal’) are used in all sciences but applied in restricted form to each genus. They do not need to be stated as premises in every demonstration; they function as enabling conditions that the demonstrator implicitly relies on.
PA 1.12 Scientific Questions and the Postulate
A postulate (aitēma) differs from an axiom in being assumed without demonstration but contrary to or without the learner’s opinion; it functions as a working assumption within a science. Questions in a science are of two kinds: about what belongs to the subject, and about which of two contradictories holds — i.e., scientific inquiry has a yes/no structure.
PA 1.13 Knowledge of the Fact vs. Knowledge of the Reason Why
To hoti (the fact) vs. to dioti (the reasoned fact / the why). A demonstration is of to dioti only when it proceeds through the proximate cause. Demonstrations of hoti exist: e.g., non-twinkling of planets proves their nearness — but this is the effect explaining the cause, not the cause explaining the effect. True scientific demonstration inverts this to give the dioti.
PA 1.13 (key) Sign-Reasoning vs. Causal Demonstration
Sign syllogisms (sēmeion): conclude from effect to cause (planets are near because they don’t twinkle). These yield to hoti but not to dioti. Causal demonstrations: conclude from cause to effect (planets don’t twinkle because they are near). Only the latter constitutes episteme in the strict sense. Epistemological asymmetry: causes are prior in nature; effects are often prior in perception.
PA 1.14 First Figure as the Figure of Scientific Demonstration
The first figure (Barbara, Celarent) is most suited to demonstration because (i) it alone produces universal affirmative conclusions, (ii) it is self-validating (can use itself to prove the value of universal demonstration), and (iii) the middle term is most transparently explanatory in this figure.
PA 1.15 Immediate Negative Propositions
Some negative propositions are immediate (have no middle term): e.g., ‘pleasure is not a good’ may be immediate if no middle connects them. Aristotle shows how negative demonstrations in Celarent and Camestres can be immediate in the relevant sense. Immediacy applies to negative as well as affirmative propositions.
PA 1.16–17 Ignorance Through Mediated Falsehood
Error (agnoia) arises either from the absence of knowledge or from a false deduction. False syllogistic conclusions result from false premises; since in demonstration the premises must be true and immediate, any false premise introduces a non-demonstrative syllogism. Analysis of error in affirmative and negative demonstrations across the three figures.
PA 1.18 Induction and the Necessity of Perception
If any perception is lacking, the corresponding universal cannot be known by induction, and therefore the relevant demonstrative knowledge is unavailable. Demonstration ultimately grounds out in perceptual particulars via induction (epagōgē), though demonstration itself is not of particulars. Perception is necessary but not sufficient for episteme.
PA 1.19–22 Finitude of Demonstrative Chains
Against infinite regress upward (ascending predications) and downward (descending predications): between any subject and predicate in a demonstration there are finitely many middle terms. The argument: if predications are infinite, demonstration is impossible (since we could never traverse the chain). Therefore the chain of predicates in any genus is finite in both directions.
