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Ars Syllogistica

Exercises in Aristotelian–Scholastic Logic

How to Proceed

What to do: choose a difficulty, then select an exercise. Each correct answer earns points and each error costs them; a set is complete at 100 points. All moods and figures of the scholastic account are in play — including the fourth figure and the weakened (subaltern) moods.

Where schematic terms are possible, a small switch on the exercise page chooses letter or English terms — capital letters stand for general terms, lower-case letters and proper names for singular. Letters are the easier dress: finish a set in letters, and it will invite you to attempt the same in English.

You may well begin with the two Study sections, which teach without keeping score. The Diagram Workshop lets you mark Venn diagrams freely — shade regions, place ×s — while the program reads back what your marks assert. The Square of Opposition sets two terms in the four traditional relations and displays any immediate inference — converse, obverse, contrapositive, contradictory — drawn from whichever corner you choose.

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Every answer is illustrated with a Venn diagram of the correct analysis — shading declares a region empty; an × declares an occupant; ⊗ marks the existential import assumed in the traditional reading.

Study

The First Act of the Mind · Simple Apprehension

The mind’s first act, before all judging or reasoning: simply to grasp what a thing is. Here we learn to divide a whole into its kinds and to define what a thing essentially is — the two arts by which a concept is made clear and distinct, and the road to everything that follows.

The Second Act of the Mind · Judgment

The mind’s second act, joining or separating concepts to affirm or deny — the proposition. We picture propositions on Venn diagrams, turn one into another by immediate inference across the square of opposition, and qualify them by the modes of necessity and possibility.

The Third Act of the Mind · Reasoning

Ratiocinatio · The Work of Reason

The mind’s third act: from things already known, to reason out things unknown — the syllogism. An argument is valid when its conclusion must be true if its premises are, whatever they are about; it is invalid when the premises could be true and the conclusion still false. That single test is applied here to arguments of every kind — categorical, conjunctive and disjunctive, conditional, modal, and dressed in ordinary speech — the conclusion always following the weaker part.

Ad Demonstrationem · Approaching Science

Where valid form meets true matter, reasoning ripens into science — sure knowledge of why a thing must be so. Weighing an argument in both form and matter at once, this exercise does not yet raise complete demonstrations, but it prepares the mind for those later and higher proofs.

Rhetorica · The Art of Persuasion

The third act’s reasoning carried into the art of persuasion. The enthymeme is the orator’s syllogism — a premise left unsaid and silently supplied by the hearer. Learn to draw the hidden assumption into the open and look it in the face.

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The Diagram Workshop

Study

What the circles mean

A general term is a universal: one word standing over a host of individuals. Picture it looking down on them — the term above, and beneath it every single thing of which it can be truly said:

Every circle in these diagrams is such a universal seen from above, its individuals gathered inside. Two moves say everything: an × reaches down and picks one individual out at random — “some dog” — without saying which; shading declares that a region holds no individuals at all. When two or three circles overlap, their regions sort the individuals beneath — and the whole logic of propositions and syllogisms can be read off the picture. Use the buttons below to see any form drawn for you, or mark the regions yourself and let the readout say what your diagram asserts.

How to read a Venn diagram

Shading declares a region empty; an × declares that something lives there. Each categorical proposition makes exactly one mark:

“All S are P” — shade the part of S outside P (no S escapes P).
“No S are P” — shade the overlap (nothing is both).
“Some S are P” — place an × in the overlap.
“Some S are not P” — place an × in S outside P.

With three circles, a universal proposition empties a lens of two cells — one inside the third circle and one outside it. A single × placed in one cell asserts several “some”-propositions at once, as the readout below will show. In syllogism mode the readout also states what your marks force about the two outer terms — the conclusion, if one follows.

The hypothetical, the conjunctive, and the disjunctive. In the last three modes the circles no longer sort whole classes: they chart the possible standings of a single subject. Shading now means a region closed to the subject; the × fixes where it stands. “If Felix is a cat, then Felix is a mammal” closes off cat-but-not-mammal — shade that crescent. “Felix is not both a dog and a cat” closes the overlap. Shading the region outside both circles says “one or the other” — the broad disjunctive; shade the overlap as well, and it becomes strict: exactly one. The same picture bears all these voicings — conditional, conjunctive, and disjunctive differ in dress, not in force — and each mode reads your marks in its own idiom.

The moods then appear before the eyes. Shade the major; place the × where a minor premise puts the subject, and the readout draws the conclusion — ponendo ponens, tollendo tollens, ponendo tollens. Shade three regions, and the subject is driven into the fourth. But if your marks leave two regions open, nothing follows: the fallacies of affirming the consequent, denying the antecedent, and tollendo ponens are exactly the cases where the subject still has two homes.

Click a region: once to shade it (declared empty) · twice for an × (something exists) · a third time to clear

The Square of Opposition

Study

The square of opposition, in itself

Every simple categorical proposition has a quantity — universal or particular — and a quality — affirmative or negative. Cross these two and exactly four forms result, named by the vowels of affirmo (“I affirm”) and nego (“I deny”):

A — universal affirmative, All S are P. E — universal negative, No S are P. I — particular affirmative, Some S are P. O — particular negative, Some S are not P.

The square sets these four at its corners — A and E along the top, I and O beneath — so that the logical bonds between them may be seen at a glance. Four such bonds hold:

Contradictories (the diagonals, A–O and E–I) can never share a truth-value: of each pair, exactly one is true and one false. Contraries (A–E) cannot both be true, though both may be false. Subcontraries (I–O) cannot both be false, though both may be true. Subalterns (A above I, E above O): truth descends from the universal to its particular, and falsity climbs from the particular to its universal.

Why it is used. Given the truth or falsity of any one corner, the square lets you read off at once what must follow for the other three. It is the oldest engine of valid inference from a single premise — and the ground of the immediate inferences below, which you can try out on any pair of terms in the square at the foot of this page.

The immediate inferences, briefly

Conversion exchanges subject and predicate. E and I convert simply, and the result is equivalent — truth passes both ways: “No fish are birds” ⇄ “No birds are fish”; “Some singers are poets” ⇄ “Some poets are singers”.

A converts only per accidens — the quantity drops: “All foxes are animals” yields “Some animals are foxes”. Why the drop? The A proposition places the foxes wholly inside the animals, but it says nothing about the whole of the animals — “All animals are foxes” would claim far more than was given (the illicit conversion). Yet since there are foxes (every term is non-empty on the traditional account), that part of the animals which contains them must be foxes: hence “some animals are foxes”. Truth is preserved downward only — you cannot climb back from “Some animals are foxes” to “All foxes are animals”.

O has no converse at all: “Some animals are not foxes” is true, while “Some foxes are not animals” is false.

Obversion changes the quality and replaces the predicate with its complement. It works on all four forms and is fully truth-preserving — the obverse is equivalent to the original: “All foxes are animals” ⇄ “No foxes are non-animals”; “Some poets are singers” ⇄ “Some poets are not non-singers”.

Contraposition is the conversion of the obverse: the terms change places and each gives way to its complement. For A and O the result is equivalent: “All foxes are animals” ⇄ “All non-animals are non-foxes” (whatever fails to be an animal certainly fails to be a fox). E contraposes only per accidens, to an O; I has no contrapositive.

The contradictory is not a truth-preserving inference but a truth-reversing pairing: A with O, E with I, each always taking the opposite truth-value of the other.

In sum: simple conversion (E, I), obversion (all), and contraposition (A, O) are equivalences — truth passes both ways. Conversion of A and contraposition of E hold in one direction only, the universal weakening to a particular by existential import. And O cannot be converted, nor I contraposed, at all.

Select a corner of the square, then choose an immediate inference.

What Is Logic · An Orientation

Study

Before the Exercise · The Modes of Propositions

Introduction

A plain proposition says that something is. A modal proposition says how it is: of necessity, or impossibly, or possibly. Three doctrines govern them.

The four corners. Every modal claim reduces to one of four: necessary (cannot be false), impossible (cannot be true — the same as “necessarily not”), possible (can be true), and possible not (can be false). These four stand on a square just like A, E, I, O: necessary and impossible are contraries, never both true; possible and possible-not are subcontraries, never both false; each universal-strength corner is contradicted by the weak corner diagonally opposite; and truth flows downward — what is necessary is thereby possible, but never the reverse.

The equipollences. Negations move modal claims around the square by fixed laws: a NOT placed before the mode flips it to the opposite corner — “not possible” is “impossible,” while “not necessary” yields only the weak “possible not”; a NOT placed after the mode negates the content alone — “possible that not.” So “it is not possible that he is not running” folds up, step by step, into “it is necessary that he is running.” The commonest slip is hearing “not necessary” as “impossible”: denying the strong mode grants only the weak denial. The scholastics drilled these with the mnemonic vowels of Amabimus, Edentuli, Iliace, Purpurea.

The two senses — composite and divided. Take “the standing man can sit.” Read one way — the composite sense — the mode governs the whole statement bundled together: “it is possible that (he is standing and sitting at once)” — false, since the parts cannot be true together. Read the other way — the divided sense — the mode attaches to the man himself: “the man, who happens to be standing, has the power to sit” — true. Same words, two claims. It works for necessity too: “whatever runs is necessarily running” is true composed (while it runs, it cannot not be running) yet false divided (nothing runs by necessity — it could stop). Whenever a modal sentence puzzles you, ask first: is the mode governing the whole package, or the thing itself?

Before the Exercise · The Kinds of Division

Introduction

To divide is to part a whole into its members. But wholes are of different sorts, and so are divisions — know the four kinds, each shown here with two plain examples and one corrupt case.

Essential — a genus into its species. The whole is a common nature; the members are its kinds, cut by opposed differences.

e.g. a musical instrument into strings, winds, and percussion; a vertebrate into fish, amphibians, reptiles, birds, and mammals.

Corrupted: animals into the footed and the white — two different bases, and the members overlap.

Integral — a whole into its parts. The members compose the whole, and none of them bears its name.

e.g. a book into its chapters; a clock into its face, its hands, and its gears.

Corrupted: a tree into root, trunk, and the soil — the soil is no part of the tree; it falls outside the whole.

Division by Powers (Potestative) — a whole into its powers. One thing, parted by what it can do.

e.g. a computer into its power to calculate, to store, and to display; a person into the powers of body and of mind.

Corrupted: the soul of the beast into sense and the paw — a paw is a bodily part, not a power of the soul.

Accidental — a subject by its accidents. The cut touches nothing of the essence.

e.g. apples into the red and the green; cloaks into the new and the old.

Corrupted: stones into the precious and the heavy — a heavy gem falls under both members at once.

And the rules, whatever the kind. The members together must cover the whole, leaving nothing out; they must be mutually exclusive; no member may lie outside the whole; and one basis of division must govern throughout. Each question ahead sets one rule, or one kind, before your eyes.

Before the Exercise · The Kinds of Definition

Introduction

A definition answers the question what is it? — but there is more than one way of answering. The tradition distinguishes four kinds; know them by sight before you judge them.

Nominal — what the name signifies. It unfolds the word, not the thing, and must come first: we cannot ask what a thing is until we know what its name intends. “Astronomy” means the law of the stars.

Essential — genus and specific difference. The perfect definition, unfolding what the thing is: the nearest genus narrowed by the difference that converts with the defined alone. A circle is a plane figure bounded by one line, every point of which lies equally distant from the centre.

Descriptive — by proprium or telltale accident. It fixes the thing by a mark that flows from the essence, or merely accompanies it, without saying what the thing is. Man is a risible animal.

Causal — through a cause. It declares the thing by its maker, its matter, its form, or its end — the last being proper to every work of art. A cloak is a covering made to keep off the cold.

And the rules. Whatever its kind, a good definition is convertible with the defined — neither too broad nor too narrow; clearer than it — no metaphor; not circular; positive where a positive can be had; and brief, with nothing idle. Each question ahead will set one rule, or one kind, before your eyes.

Before the Exercise · Translating Ordinary Speech

Introduction

Ordinary language wears its logic loosely. Before judging the arguments, note where translation into standard form most often goes wrong.

Missing quantity. “Dogs are mammals” means all dogs; “dogs were barking in the alley” means some. When no sign appears, judge from the sense — and “a” swings both ways: “a whale is a mammal” is universal, “a man came to the door” particular.

“Not all” is not “none.” “Not all sailors are pirates” denies the universal only: it says Some sailors are not pirates. And the English “All S are not P” is treacherous — it usually intends No S are P, but sometimes only Some S are not P. Read twice.

“Only” reverses the terms. “Only citizens are voters” says All voters are citizens — not the converse. So too “none but” and “no one except.” What follows the “only” becomes the predicate.

“Few” against “a few.” “A few sailors are poets” affirms: Some sailors are poets. But “few sailors are poets” chiefly denies: Some sailors are not poets.

Existential turns. “There are no honest thieves” is No thieves are honest; “there are dogs that bite” is Some dogs are things that bite.

Find the conclusion, wherever it hides. “Since,” “because,” and “for” mark premises; “therefore,” “so,” “hence,” and “it follows that” mark the conclusion — which often comes first: “No oaks are shrubs, for every oak is a tree, and no tree is a shrub.”

Odd verbs want the copula. “All dogs bark” becomes All dogs are things that bark — supply “things that” and restore is or are.

Singulars stand whole. “Socrates is a man” takes its subject in its entirety: treat it as a universal with existential import — one thing, wholly included or wholly shut out.

Mark the connectives. “If… then” is the conditional (beware affirming the consequent); “not both” is the conjunctive, which concludes only by positing; “either… or” must confess itself — strict (“but not both”) licenses both moods, broad (“or perhaps both”) only tollendo ponens.

Mind the modes. “Necessarily” and “contingently” cling to the conclusion at their peril: the conclusion may never be stronger in mode than the weaker premise.

Before the Exercise · Immediate Inference

Introduction

An immediate inference draws a new proposition from a single one — no middle term, no second premise; the conclusion is read straight off the first. Each turns on the four categorical forms: A (All S are P), E (No S are P), I (Some S are P), and O (Some S are not P). You may experiment freely with every inference below in the Square of Opposition study section before you begin.

Conversion — exchange subject and predicate. E and I convert simply, truth passing both ways: “No fish are birds” ⇄ “No birds are fish.” A converts only per accidens, the quantity weakening: “All oaks are trees” yields only “Some trees are oaks.” O cannot be converted at all.

Obversion — change the quality, negate the predicate. It holds for all four forms and always preserves truth: “All oaks are trees” ⇄ “No oaks are non-trees”; “Some men are just” ⇄ “Some men are not unjust.”

Contraposition — swap the terms and negate each. It is the conversion of the obverse; for A and O it preserves truth: “All oaks are trees” ⇄ “All non-trees are non-oaks.” E contraposes only to an O, and I has no contrapositive.

The relations of the square. From one proposition’s truth the square yields the rest: contradictories (A–O, E–I) always take opposite values; contraries (A–E) are never both true; subcontraries (I–O) never both false; and by subalternation truth descends from a universal to its particular. “All swans are white,” if true, makes “Some swan is not white” false.

Before the Exercise · The Conjunctive and Disjunctive

Introduction

Beside the categorical syllogism stand arguments built not on terms but on whole propositions joined by a connective. Two are drilled here — the conjunctive and the disjunctive. Each has one lawful path to a conclusion; step off it and the argument fails.

The conjunctive — “not both.” It denies that two things hold together: “One cannot be both in Rome and in Athens at once.” From it you may conclude only by positing one member to remove the other — the mood ponendo tollens (“by positing, it takes away”): “He is in Rome; therefore he is not in Athens.” You may not reason the reverse: denying one member proves nothing of the other, for perhaps he is in neither.

The disjunctive — “either … or.” It offers alternatives: “The number is either odd or even.” Its sure mood is tollendo ponens (“by removing, it posits”): deny one member and the other stands — “It is not odd; therefore it is even.”

Strict against broad. Whether you may also reason ponendo tollens — positing one member to remove the other — depends on the “or.” A strict disjunction (“but not both”) excludes its members, so positing one does remove the other. A broad disjunction (“or perhaps both”) does not: from “He is either clever or lucky” you cannot infer that, being clever, he is not also lucky. Mark the connective before you conclude.

Before the Exercise · The Hypothetical Syllogism

Introduction

A hypothetical (conditional) proposition asserts not a fact but a connexion: “If it has rained, the ground is wet.” The clause after if is the antecedent; the clause it supports is the consequent. Two moods reason soundly from such a premise, and two tempting fallacies counterfeit them.

Ponendo ponens — affirm the antecedent. Posit the if-clause and the then-clause follows: “It has rained; therefore the ground is wet.” Always valid.

Tollendo tollens — deny the consequent. Remove the then-clause and the if-clause falls with it: “The ground is not wet; therefore it has not rained.” Always valid.

The two fallacies. Affirming the consequent — “The ground is wet; therefore it has rained” — fails, for a burst pipe would wet it too. Denying the antecedent — “It has not rained; therefore the ground is not wet” — fails for the same reason. A conditional runs one way only: from antecedent to consequent, never back.

Before the Exercise · The Modal Syllogism

Introduction

A modal syllogism is one whose premises are qualified by a mode — the manner in which the predicate belongs to the subject. Three modes are in play: the necessary (it cannot be otherwise), the assertoric or plain (it simply is so), and the contingent (it is so, yet might not be). The question is not only whether a conclusion follows, but in what mode.

The governing rule. Peiorem sequitur semper conclusio partem — the conclusion always follows the weaker premise. As a negative premise forces a negative conclusion, and a particular premise a particular one, so the weaker mode rules: join a necessary premise to a merely contingent one, and the conclusion can be no stronger than contingent.

The order of strength. Necessity is strongest, the plain assertion stands between, contingency is weakest. A conclusion may sink to the level of the weaker premise, but never rise above it.

An example. “Every man is necessarily mortal; every scholar is (in fact) a man; therefore every scholar is mortal” — yet the conclusion is drawn only as a plain assertion, not as a necessity, since the minor premise was merely assertoric. Judge each argument for the mode its weakest premise allows.

Before the Exercise · The Matter and the Form

Introduction

A demonstration is a syllogism that yields science — sure knowledge — by drawing its conclusion from premises that are not merely true but first, necessary, and better known than the conclusion, stating the very cause of the fact. To demonstrate is to show not only that a thing is so, but why it must be so.

Form and matter. Every argument may be weighed twice. Its form is its structure: an argument is valid when the conclusion must be true if the premises are, and invalid when it need not be. Its matter is the truth of the premises themselves. An argument both valid in form and true in matter is called sound; lacking either, unsound.

What this exercise trains. Here you judge both at once — is the form valid, and are the premises in fact true by the classical definitions? This is not yet full demonstration, for we do not ask whether the premises are first and causal; but it is its nearest gate. To require both a sound form and true matter is exactly the discipline that scientific knowledge demands, and so this exercise prepares the mind for those later and higher demonstrations.

Why it matters. The valid-but-unsound argument — flawless in form, false in a premise — is the commonest counterfeit of knowledge. To catch it is to glimpse what parts mere consistency from truth, and opinion from science.

Before the Exercise · The Enthymeme

Introduction

An enthymeme is a syllogism with a premise — or, at times, the conclusion — left unspoken: a “syllogism in the mind,” whose missing part the hearer is trusted to supply. “Socrates is mortal, for he is a man” says nothing of the tacit major, “all men are mortal,” yet the whole argument turns upon it.

Why the premise is left unsaid. Sometimes for brevity; sometimes because it is too obvious to state; and sometimes because it is weak, and would not survive being spoken aloud. The art of examining an enthymeme is to drag the hidden premise into the light and ask whether it is true.

How common it is. The enthymeme is the ordinary form of reasoning in real life — far more frequent than the full syllogism. Everyday speech runs on it (“You’ll like her — she’s a teacher”); the law lives by it (“He fled the scene, so he is guilty”); politics and advertising trade on it; and even philosophy, for all its rigor, argues enthymematically more often than not. Aristotle called it the very body of persuasion.

Your task. Supply the unspoken premise that would make the argument valid — or, where none can, say that none will serve. To see what is being taken for granted is to hold an argument to account.

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