The Timpson Lyceum · History of Philosophy (2024)

Aristotle's Organon: Posterior Analytics & Science – Lecture 25 (History of Philosophy)

Overview

Where the Prior Analytics studied the form of valid inference, the Posterior Analytics turns to its matter — the content of the premises — and asks what it takes for a valid argument to yield genuine scientific knowledge (epistēmē). A syllogism can be flawlessly valid yet worthless if its premises are false ("all men are immortal; Socrates is a man; therefore Socrates is immortal" has perfect form and a false conclusion). Aristotle's term for a syllogism that is both valid and built from true premises is demonstration (apodeixis), and demonstration is the engine of every science. This work is, in effect, the first philosophy of science: it lays down what a proof must look like, what its starting points must be, and how those starting points are themselves known.

The central problem is the regress of justification. To know a conclusion scientifically is to know its cause and to see that it cannot be otherwise; that means deriving it from premises already known. But if every premise in turn required demonstration, justification would either run backward forever or curl into a circle — and neither yields knowledge. Aristotle's resolution is that demonstration must terminate in indemonstrable first principles: truths that are prior to, better known than, and explanatory of everything proved from them. These are not themselves demonstrated but grasped by nous, an intellectual insight abstracted from experience — the bedrock on which the whole edifice of science rests.

Key Concepts

• Demonstration (apodeixis) — A syllogism whose form is valid and whose premises are true, so that the conclusion is not merely validly drawn but actually established. Demonstration alone produces epistēmē, scientific knowledge.
• Validity vs. soundness — A valid argument with false premises is valid but not sound; "good form, bad matter." Science demands both good form and true matter.
• The three kinds of argument — By the quality of their premises: demonstrative (true, certain premises → science), dialectical (premises that are merely reputable or widely believed → the Topics), and sophistical (arguments that only appear valid or sound → Sophistical Refutations, which also catalogues the standard logical fallacies).
• To know is to know the cause — Having epistēmē of something means grasping (i) that it is so, (ii) why it is so — its cause (aitia) — and (iii) that it cannot be otherwise. Memorizing "$a^2+b^2=c^2$" is not knowing the Pythagorean theorem; knowing it is being able to demonstrate it from prior truths.
• Conditions on the premises of a demonstration — They must be true, primary and immediate (indemonstrable), better known than the conclusion, prior to it, and explanatory/causal of it. A conclusion can be no more certain than the premises it rests on, so the foundations must be the best-known things of all.
• Necessary, universal, eternal — The premises, conclusion, and every intermediate step of a genuine demonstration are necessary (could not be otherwise), universal (about kinds, not this individual triangle or that coin flip), and eternal (time- and place-independent). There is no science of chance events or particulars — only of what holds always and in every case.
• Induction (epagōgē) — First principles and observable facts are reached not by proof but by induction from repeated particular observations to a generalization (watching unsupported bodies fall, then framing "unsupported bodies fall"). Induction supplies the raw premises that demonstration then reasons from.
• The regress of justification — If every truth had to be demonstrated, proof would either regress infinitely or become circular ("a is true because a is true"). Since neither yields knowledge, some truths must be indemonstrable.
• First principles — The indemonstrable starting points, of two kinds: axioms / common principles shared by all sciences (the law of non-contradiction, "equals added to equals are equal," "a part of a finite whole is less than the whole"), and postulates / proper principles special to one science (in geometry, that two points determine a line segment). To these Aristotle adds definitions and the primitive terms they rest on.
• Nous (intellectual insight) — First principles are apprehended by nous, a grasp that is truer than and prior to demonstrated knowledge, because it is what demonstration presupposes. One does not have epistēmē of first principles; one has insight into them.
• Essential (per se) predication — Demonstration may use only essential attributes, which belong to a subject in its own right: either because the predicate is in the definition of the subject ("line" belongs to "triangle") or because the subject is a kind of the predicate ("triangle" belongs to "figure"). Accidental attributes (that all people happen to be standing right now) are barred.
• No cross-genus demonstration — A demonstration cannot pass from one genus to another; premises about mammals cannot conclude about reptiles. Hence there is no single universal super-science proving the principles of all the others — each science has its own proper starting points (as Euclidean and hyperbolic geometry rest on incompatible postulates).
• Proving a universal via an arbitrary particular — To prove a property of a whole kind, prove it of an arbitrary instance using only its essential attributes, then generalize (modern universal instantiation followed by universal generalization).
• Not innate — abstracted from experience — Against Plato's recollection, the mind is not born stocked with first principles; it has the capacity to recognize them once exposed to reality, as the eye has the capacity for color but must meet light. Basic concepts are abstracted from repeated encounters with particulars.

Worked Example

A demonstrative syllogism about the triangle, and why it must bottom out in first principles. Take the theorem that the interior angles of any triangle sum to two right angles. First cast the knowledge as a demonstration in Barbara, using only per se premises; then run the regress argument that forces the existence of indemonstrable starting points.

The demonstrative syllogism:

1. Major premise (necessary, universal, and explanatory — a property flowing from the essence of a rectilinear figure whose angles rest on parallels): every three-sided rectilinear plane figure has interior angles summing to two right angles.
2. Minor premise (per se, from the definition of the subject): every triangle is a three-sided rectilinear plane figure.
3. Conclusion (by Barbara): every triangle has interior angles summing to two right angles.

The property itself is genuinely derivable. For a triangle with interior angles $\alpha,\beta,\gamma$, draw through the apex a line parallel to the base; the two alternate interior angles equal $\alpha$ and $\beta$, and together with $\gamma$ they fill a straight angle:

$\alpha + \beta + \gamma = \pi = 180^{\circ} = 2 \times 90^{\circ}.$

1. But step 1 is not self-standing: it leans on the alternate-angle theorem and on the parallel postulate. Ask why those hold, and each demands its own prior premises.

Now the regress, formalized. Let "$D(p)$" mean "$p$ has been demonstrated," and note that a demonstration of $p$ requires premises $q$ prior to and better known than $p$.

1. Suppose, for contradiction, that every known truth must be demonstrated: $\forall p\,\big(\text{known}(p) \to D(p)\big)$.
2. Each demonstration $D(p)$ requires a prior demonstrated premise $q$ with $q \prec p$ (strictly better known, strictly prior).
3. Applying step 6 repeatedly generates an infinite descending chain $\cdots \prec q_3 \prec q_2 \prec q_1 \prec p$. Since $\prec$ is a strict order tracking increasing priority/knownness, no infinite descent can complete — the justification is never finished, so nothing is actually known. (Infinite regress: contradiction.)
4. The only way to avoid the infinite chain while still terminating is to reuse an earlier premise, giving a cycle $p \prec \cdots \prec p$. But $\prec$ is irreflexive ($p$ cannot be strictly better known than itself), so a cycle is impossible; a "proof" of $p$ ultimately from $p$ establishes nothing. (Circularity: contradiction.)
5. Both horns being impossible, the assumption in step 5 is false: not every known truth is demonstrated. Therefore there exist truths that are known but indemonstrable — the first principles.
6. These terminating principles cannot be objects of epistēmē (that would require demonstration); they are grasped instead by nous, and precisely because demonstration presupposes them they are better known than anything proved from them — including the theorem about the triangle. $\blacksquare$

Summary

• The Posterior Analytics studies the matter of argument: a valid syllogism becomes a demonstration only when its premises are also true, and demonstration is what yields scientific knowledge.
• To know something scientifically is to know its cause and that it cannot be otherwise; a demonstration's premises must be true, primary, indemonstrable, prior to, better known than, and explanatory of its conclusion, and must be necessary, universal, and eternal.
• Induction from observed particulars supplies the first premises; demonstration then reasons deductively from them, and the results must always answer to observable fact.
• The regress of justification shows that not everything can be proved — infinite regress and circularity are both fatal — so knowledge must rest on indemonstrable first principles: common axioms, proper postulates, and definitions.
• These first principles are apprehended not by demonstration but by nous, an insight abstracted from experience (contra Platonic recollection); each science keeps its own genus and principles, so there is no single super-science proving all the rest.