The Timpson Lyceum · History of Philosophy (2024)

Aristotle's Organon: Prior Analytics & Logic – Lecture 24 (History of Philosophy)

Overview

The Prior Analytics is the beating heart of Aristotle's Organon — the first systematic theory of deductive inference in the Western tradition, and the logic that stood essentially unchallenged from the fourth century BC until the rise of modern quantificational logic around AD 1900. Building on the Categories (simple concepts) and On Interpretation (propositions that can be true or false), the Prior Analytics asks a purely structural question: what is the form of valid reasoning? When someone grants "all men are mortal" and "Socrates is a man," why is the mind compelled to grant "Socrates is mortal" — and why do some rearrangements of the very same premises break that compulsion? Aristotle answers by isolating the syllogism and cataloguing exhaustively which arrangements are valid.

The machinery is built in stages. First the four categorical propositions (A, E, I, O) reduce every scientific premise to one of four canonical shapes relating a subject and predicate. These are organized by the square of opposition into contradictory, contrary, and subcontrary relations. Then the syllogism combines two categoricals sharing a middle term to yield a third, and the placement of that middle term generates the figures. Counting the ways to fill the figures with the four proposition-types produces the space of all possible moods — of which only a small handful are valid. This is logic pursued as a finite combinatorial search, and it is the direct ancestor of the modern predicate calculus that later replaced it.

Key Concepts

• The Organon — Aristotle's collected logical works (Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, Sophistical Refutations): the "instrument" (organon) for all reasoning, not a science of any particular subject but the tool used to build every science.
• Noun and verb (subject and predicate) — Aristotle's "noun" is what modern grammar calls the subject and his "verb" the predicate; the noun signifies a subject by convention without reference to time, while the verb carries a notion of time (tense). Neither alone is true or false.
• Sentence vs. proposition — A sentence is marks on paper (a "significant portion of speech"); a proposition is the fact-stating content — the thing that is true or false. Different sentences ("John is 18," in English, German, French) can encode one proposition. Only declarative sentences express propositions.
• Quantity of the subject — A subject is universal ("all/some men"), particular ("Socrates"), or indefinite ("men are mortal," with no quantifier). Science admits only definite propositions; indefinite ones are rejected because their interpretation is ambiguous.
• The four categorical propositions — Aristotle's canonical premise-forms, later labelled by the vowels of AffIrmo/nEgO: A universal affirmative (all S are P), E universal negative (no S is P), I particular affirmative (some S is P), O particular negative (some S is not P). In his own notation these are written P a S, P e S, P i S, P o S — read as "P belongs to all/no/some/not-some S."
• Modern translation (and the A/I trap) — The universals hide a conditional while the particulars hide a conjunction: A is $\forall x\,(Sx \to Px)$, E is $\forall x\,(Sx \to \lnot Px)$, but I is $\exists x\,(Sx \land Px)$ and O is $\exists x\,(Sx \land \lnot Px)$. A standing error is to negate a universal by keeping the arrow instead of switching to a conjunction.
• Square of opposition — Contradictories (A/O, E/I) cannot share a truth value: exactly one is true. Contraries (A/E) cannot both be true but can both be false. Subcontraries (I/O) cannot both be false but can both be true. The contradictory of "all swans are white" is "some swans are not white," not "no swans are white."
• Existential import — Aristotle assumes every term names something real, so he licenses the inference from a universal to the corresponding particular (subalternation, A $\Rightarrow$ I). Modern logic forbids this: from $\forall x\,(Sx \to Px)$ one cannot infer $\exists x\,(Sx \land Px)$ if S is empty (all dragons fly, all dragons are reptiles, yet there need be no flying reptile).
• The syllogism — A deduction of one categorical conclusion from two categorical premises sharing a common middle term. It contains exactly three terms: the middle term (in both premises, absent from the conclusion), the major term (predicate of the conclusion), and the minor term (subject of the conclusion).
• The four figures — Determined by where the middle term M sits. Figure 1: $M\!-\!P,\ S\!-\!M$. Figure 2: $P\!-\!M,\ S\!-\!M$. Figure 3: $M\!-\!P,\ M\!-\!S$. Figure 4: $P\!-\!M,\ M\!-\!S$. Aristotle worked three explicitly but was aware of the fourth.
• Moods and mnemonics — A mood is a choice of proposition-type (A/E/I/O) for each of the three lines within a figure. Medieval logicians gave the valid moods memory-names encoding their vowels: Barbara (AAA-1), Celarent (EAE-1), Darii (AII-1), Ferio (EIO-1), Cesare, Camestres, and so on.
• Modal logic — Aristotle also began the study of the possible and the necessary operators, an extension used later for such things as the Gödel ontological proof; it is a less precise system than the assertoric syllogistic and not the logic on which mathematics is built.
• The problem of future contingents — Holding that every proposition is determinately true or false, Aristotle worried that "there will be a sea battle tomorrow" would force the future to be predetermined; he set such future-tensed statements aside as a genuine puzzle he could not resolve.
• Validity vs. the truth of premises — What logic certifies is the implication: that if the premises hold then the conclusion follows. That an argument is valid says nothing about whether its premises are in fact true — the distinction the Posterior Analytics takes up next.

Worked Example

The mood Barbara, then a census of all possible moods. Barbara (AAA in the first figure) is the paradigm of valid deduction. Write the premises and conclusion in canonical form, translate to predicate logic, and derive the conclusion.

Premises and goal:

$\text{All } M \text{ are } P; \quad \text{All } S \text{ are } M; \quad \therefore\ \text{All } S \text{ are } P.$

1. Premise (major): $\forall x\,(Mx \to Px)$.
2. Premise (minor): $\forall x\,(Sx \to Mx)$.
3. Let $a$ be an arbitrary individual and suppose $Sa$.
4. Instantiating step 2 at $a$: $Sa \to Ma$; with $Sa$ this gives $Ma$ (modus ponens).
5. Instantiating step 1 at $a$: $Ma \to Pa$; with $Ma$ this gives $Pa$ (modus ponens).
6. Discharging the supposition: $Sa \to Pa$.
7. Since $a$ was arbitrary, generalize: $\forall x\,(Sx \to Px)$ — i.e. all $S$ are $P$. $\blacksquare$

Now count the space Aristotle had to search. Each of the three lines (major premise, minor premise, conclusion) can independently take one of the four proposition-types A, E, I, O, and the middle term can sit in any of the four figures. Hence the number of candidate moods is

$N = \underbrace{4}_{\text{figures}} \times \underbrace{4}_{\text{major}} \times \underbrace{4}_{\text{minor}} \times \underbrace{4}_{\text{conclusion}} = 4 \times 4^{3} = 4^{4} = 256.$

1. Of these $256$ combinations, only 24 are valid.
2. Of those $24$, five are subalternate (weakened) forms — a valid universal conclusion deliberately downgraded to the corresponding particular, legitimate only under Aristotle's existential-import assumption. Removing them leaves the 19 "principal" valid moods:

$24 - 5 = 19.$

1. The payoff: valid reasoning is not a matter of taste but a decidable, finite fact. From $256$ mechanically enumerable forms, fewer than one in ten preserve truth — and Barbara (AAA-1) is the perfect, self-evident exemplar to which the medieval tradition reduced the rest by conversion and reductio. $\blacksquare$

Summary

• The Prior Analytics founds formal deductive logic, studying the form of valid inference independently of any subject matter — a system that dominated for roughly two thousand years.
• Every scientific premise is regimented into one of four categorical propositions — A, E, I, O — whose logical relations are mapped by the square of opposition (contradictory, contrary, subcontrary).
• In modern notation the universals (A, E) are conditionals and the particulars (I, O) are conjunctions; conflating the two, or assuming existential import, are the classic pitfalls.
• A syllogism links two premises through a shared middle term; the middle term's placement yields the four figures, and filling the figures gives $4 \times 4^{3} = 256$ possible moods, of which only 24 (or 19 without subalternation) are valid.
• Barbara (AAA-1) is the canonical valid mood; Aristotle also opened modal logic and flagged the unresolved puzzle of future contingents, while insisting that logic certifies the implication, not the truth of the premises themselves.