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Ars Syllogistica

Exercises in Aristotelian–Scholastic Logic
How to Proceed

Where to begin. You may start with the Preliminary first step — The Art of Grammar — if you want a foundation in the parts of speech before logic. Or you may begin with the introduction to logic, What Is Logic · An Orientation. Or you may start with the two Study sections below, which teach without keeping score. Or you may go straight into the exercises: choose a difficulty, then select an exercise.

What to do in an exercise: each correct answer earns points and each error takes away points; a set is complete at 100 points. All moods and figures of the scholastic account are possible here, 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 form. When we finish a set in letters, the course invites us to attempt the same set in English.

Among the unscored options: The Diagram Workshop lets you mark Venn diagrams freely (shading regions and placing ×s) and shows 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.

Preliminary · Before Logic
Grammar is the first art of the trivium, and the one the other two are built on. Logic works upon statements, and a statement is made of words; so before reasoning can be judged, the parts of a sentence must be known. Begin here if you have had little grammar, or return to it for the medieval question of what grammar is for.
Difficulty
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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 first act of the mind, which comes before all judging or reasoning, is simply to grasp what a thing is. Here we learn the five predicables (the ways a universal is said of a subject) and the ten categories into which all that can be said of a thing falls; then we learn to divide a whole into its kinds and to define what a thing essentially is. These are the arts by which a concept is made clear and distinct, and everything that follows depends on them.
The Second Act of the Mind · Judgment
The second act of the mind joins or separates concepts in order to affirm or deny, and what it forms is 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 third act of the mind reasons from things already known to things unknown, and its form is 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 stated in ordinary speech), and in each of them the conclusion follows the weaker part.
Ad Demonstrationem · Approaching Science
When valid form is joined to true matter, reasoning becomes science, that is, sure knowledge of why a thing must be so. This exercise weighs an argument in form and in matter at once. It does not yet present complete demonstrations, but it prepares the mind for those later and higher proofs.
Dialectica · Reasoning from What Is Granted
Where we cannot demonstrate, we still need not guess, because dialectic reasons from what is probable — what seems so to everyone, or to most, or to the wise — and reaches an opinion held for good reasons. Here we learn the topics an argument may be drawn from, what standing a premise has before we offer it, and the four ways the schools answered an objection.
Rhetorica · The Art of Persuasion
Here the reasoning of the third act is carried into the art of persuasion. The enthymeme is the orator’s syllogism, in which a premise is left unsaid and silently supplied by the hearer. Here we learn to state the hidden assumption and examine it.
Sophistica · The Detection of Fallacy
Reasoning goes wrong, and it goes wrong in ways that can be named. A fallacy is an argument that looks sound and is not, and St Thomas insists that each has two causes: what makes it look good, and what makes it fail. Six come from the words and seven from outside them. We study them not to use them but to recognize them when others use them.
For instructors — build a printable quiz with answer key ⟶
For instructors — turn an exported progress file into a grade ⟶

These sets of exercises were designed by Timothy Kearns, PhD, created and maintained with the assistance of AI, and are extensively revised for clarity and precision.

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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. We may picture it as standing over them, with the term above and beneath it every single thing of which it can be truly said:

A general term above its individuals dogs the universal Fido Lassie Rex Spot … × — “some dog”: one picked out at random the singulars shading the region declares that no individuals are there

Every circle in these diagrams is such a universal seen from above, its individuals gathered inside. There are only two marks. An × picks out one individual 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 exists there. Each categorical proposition makes exactly one mark:

“All S are P”: shade the part of S outside P (no S lies outside 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.

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). When these two are combined, 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 relations between them may be seen at a glance. There are four such relations:

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 ascends 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 means of valid inference from a single premise, and the basis of the immediate inferences below, which you can try 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, so 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, that is, the quantity is reduced: “All foxes are animals” yields “Some animals are foxes”. Why is the quantity reduced? The A proposition places the foxes wholly inside the animals, but it says nothing about the whole of the animals, so “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 in one direction only: we cannot infer “All foxes are animals” back from “Some animals are foxes”.

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 applies to all four forms and is fully truth-preserving, since 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 gives, from an A or an O, an equivalent in which both terms have changed places and been replaced by their complements; but it is really three familiar steps in succession, not a single step: obvert, then convert, then obvert again. Consider “All foxes are animals.” (1) Obvert, changing the quality and negating the predicate: “No foxes are non-animals.” (2) Convert, since an E converts simply: “No non-animals are foxes.” (3) Obvert again: “All non-animals are non-foxes.” That is the contrapositive, equivalent to the original (whatever fails to be an animal certainly fails to be a fox). Converting first is a tempting shortcut, but it gives “All animals are foxes”, and an A does not convert. 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, so 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: “Some animals are not dogs” is true, yet its converse “Some dogs are not animals” is false — the O gives no purchase on the predicate beyond the part it names.

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. Four doctrines govern them: the four corners of the modal square, equipollence (when two expressions have the same force), the laws that produce equipollences, and the composite versus the divided sense.
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 passes downward, since what is necessary is thereby possible, but never the reverse.
Equipollence: the same force in different words. Two expressions are equipollent when they mean the same thing and must share the same truth-value; they are equal in force (aequipollentia), though the wording differs. In the modals, a claim is equipollent to another when both reduce to the same corner of the square: “it is not possible that he is not running” is equipollent to “it is necessary that he is running,” because both say necessary. If they reduce to different corners, they are not equipollent.
The laws that produce equipollences. Negations move modal claims around the square by fixed rules: a NOT placed before the mode changes 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”). We fold in the negations step by step until each expression shows its corner, and then we ask whether the two corners match. The commonest slip is reading “not necessary” as “impossible”, but 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. The difference is one of scope. Take “the standing man can sit.” In the composite sense the mode governs the whole proposition composed of its parts, taken together as one whole: “it is possible that (he is standing and sitting at once)”. This is false, since the parts cannot be true together. In the divided sense the mode’s scope is divided off and narrowed to the man himself: “the man, who happens to be standing, has the power to sit” — true. The words are the same, but the claims are two. It holds for necessity too: “whatever runs is necessarily running” is true composed (the whole “it runs and is running” cannot fail together) yet false divided (nothing runs by necessity, since it could stop). Whenever a modal sentence is puzzling, the first question is whether the mode governs the whole composed statement, or only the subject taken by itself.
Before the Exercise · The Five Predicables
Introduction
There are five ways a general term can be said of a subject, the five predicables of Porphyry. Three of them state part of what the thing is: genus, species, and difference. The other two do not: the property and the accident. We learn to recognize each before we try to name it.
Genus — the wider kind. It is said of many things of different kinds, naming the broader class they all share; it answers “what is it?” with the wider nature.
e.g. animal, said alike of man and of horse;   figure, said of the triangle and the square.
Species — the narrower kind under the genus. It is the genus narrowed by a difference, and it is said of the individuals that share that nature.
e.g. man, said of Socrates and of Plato;   man is a species of animal.
Difference (differentia) — the mark that sets a kind apart. It divides the genus and makes the species; it answers “what sort of thing, in its very nature?” (quale quid).
e.g. rational, which divides animal and makes man;   corporeal, which divides substance into body and spirit.
Property (proprium) — follows from what the thing is, without being part of it. It belongs to every member of the species, to that species alone, and at all times, so the two are always found together.
e.g. the power to laugh (risibility), in man;   a triangle’s having its angles equal to two right angles.
Accident: it may be present or absent, and the thing is still what it is. The subject may have it or lack it and still be the very same thing. It is either separable (sitting) or inseparable (the blackness of a raven), but it is never part of what the thing is.
e.g. Socrates is pale;   this man is musical.
Two questions sort them. Ask of any predicate: is it part of what the subject is, and is it said of that subject alone? Part of it, and said of more: the genus (figure, of the triangle). Part of it, and said of it alone: the difference, that is, the last difference (three-sided). Not part of it, yet said of it alone, of all, and always: the property (angles equal to two right angles). Not part of it, and not said of it alone: the accident (drawn in chalk). The first question concerns comprehension, the second extension.
The swap test. A property swaps with its species (Aristotle, Topics I.5). Every man is able to laugh, and whatever is able to laugh is a man. A genus does not swap: every man is an animal, but not every animal is a man. Neither does an accident, even one that never leaves its subject: every raven is black, but much that is black is not a raven (Porphyry).
Able to, and doing. A property is often a power. A man is always able to laugh, though he is not always laughing (Porphyry). So able to laugh is the property and is laughing an accident; able to learn grammar is the property and knows grammar an accident. One word changes, and so does the predicable.
And a caution on the relative terms. “Genus” and “species” shift along the tree of Porphyry: animal is a genus to man, a species to living body. Only substance is always a genus and never a species, and only the lowest kind (such as man) is always a species and never a genus; an individual like Socrates is neither, since he is only ever the thing being talked about. Each of the questions that follow takes up one predicable, or one point of the doctrine.
Supposition — what the term stands for here. Signification is what a word was imposed to mean, and it does not change from sentence to sentence. Supposition is what the term stands for in this proposition, and it does. In a man is running, man stands for some man. In Man is a species, man stands for the nature as known, not for any man. Grammar can say the sentence is well formed. Which supposition the noun is under is a question for logic, and this art is its home.
Before the Exercise · The Ten Categories
Introduction
Anything that can be said about a thing falls under one of ten highest kinds, the categories of Aristotle. One is substance, the thing itself; the other nine are accidents, which exist only in a substance. St Thomas shows how the nine follow from the different ways they relate to substance (Commentary on the Metaphysics V, lect. 9).
Substance — what exists in its own right. It is not said of anything else, nor does it exist in anything else: this man, this horse (primary substance); and the kinds they belong to, such as man and animal (secondary substance). Everything else depends on substance.
What BELONGS TO the substance itself. Quantity follows from its matter (how much? two feet long, a number); Quality follows from its form (of what sort? white, hot, just); and Relation is a bearing toward something else (double, half, a master).
What the substance DOES or UNDERGOES. Action is what it does to something else (cutting, burning), and Passion is what is done to it (being cut, being burnt).
What measures it from outside. When is its position in time (yesterday, now); Where is its position in place (in the school, in the marketplace); and Posture is how its parts are arranged in that place (he sits, he lies).
What it merely HAS ON it. Habit (having) is something attached to the substance without measuring it: he is shod, he is armed.
A last distinction. The ten categories sort what things are, as we first know them (first intentions). The five predicables are relations those things have only because they are known (second intentions), though they are grounded in what the things are. A first intention is a reality as known, not a thing apart from knowledge. A second intention is a relation things have as known, not a way of saying. Each of the questions that follow takes up one category, or one point of the doctrine.
Before the Exercise · The Kinds of Division
Introduction
To divide is to break a whole into its members. But wholes are of different sorts, and so are divisions. There are four kinds, and each is given here with two plain examples and one faulty case.
Essential — a general kind into the kinds beneath it. The whole is a shared nature; the members are its kinds, marked off from one another by opposed differences.
e.g. a musical instrument into strings, winds, and percussion;   a vertebrate into fish, amphibians, reptiles, birds, and mammals.
Faulty: animals into the footed and the white, which uses two different bases at once, so that the members overlap.
Integral — a whole into the parts that make it up. The members together compose the whole, and no one of them is called by the name of the whole.
e.g. a book into its chapters;   a clock into its face, its hands, and its gears.
Faulty: a tree into root, trunk, and the soil; the soil is not a part of the tree at all, so it does not belong to the whole being divided.
By its powers (potestative) — one thing according to what it can do. A single thing is divided not into pieces but into its capacities.
e.g. a bird into its power to fly, to sing, and to build a nest;   a person into the powers of body and of mind.
Faulty: the soul of an animal into sense and the paw, since a paw is a bodily part, not a power of the soul.
Accidental — a subject by its passing features. The division uses features the subject can gain or lose without becoming a different thing.
e.g. apples into the red and the green;   cloaks into the new and the old.
Faulty: stones into the precious and the heavy, since a heavy gem falls under both members at once.
And the rules, for every kind. The members together must cover the whole, leaving nothing out; they must not overlap, so nothing falls under two of them at once; every member must actually belong to the whole being divided; and the whole division must be made on a single basis. Each of the questions that follow takes up one rule, or one kind.
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, and we learn to recognize each before we judge it.
Nominal — what the name means. It explains the word, not the thing, and must come first: we cannot ask what a thing is until we know what its name refers to. “Astronomy” means the law of the stars.
Essential — nearest kind plus the difference. It is the complete definition, and it says what the thing is: the nearest wider kind (genus), narrowed by the difference that belongs to this thing and nothing else. A circle is a plane figure bounded by one line, every point of which lies equally distant from the center.
Descriptive — by a property or telltale feature. It picks the thing out by a mark that follows from its nature, or merely goes along with it, without saying what the thing is. Man is an animal that can laugh.
Causal — through one of its causes. It identifies the thing by what made it, what it is made of, its form, or what it is for; the last applies to everything made by human craft. A cloak is a covering made to keep off the cold.
And the rules. Whatever its kind, a good definition fits exactly what it defines, neither too broad nor too narrow; it is clearer than the thing defined, so it uses no metaphor; it does not contain the term being defined; it says what a thing is rather than what it is not, wherever it can; and it is brief, with no wasted words. Each of the questions that follow takes up one rule, or one kind.
Before the Exercise · Translating Ordinary Speech
Introduction
Ordinary language often states its logical form loosely. Before we judge the arguments, we should 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 of quantity appears, we judge from the sense, and “a” can be read either way: “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 ambiguous: it usually means No S are P, but sometimes only Some S are not P, so it should be 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.
Finding the conclusion. “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.”
Verbs other than the copula must be restated with it. “All dogs bark” becomes All dogs are things that bark; we supply “things that” and restore is or are.
Singular subjects are taken whole. “Socrates is a man” takes its subject in its entirety: we treat it as a universal with existential import, since it concerns one thing, which is either wholly included or wholly excluded.
Mark the connectives. “If… then” is the conditional (where the common error is affirming the consequent); “not both” is the conjunctive, which concludes only by positing; “either… or” must show which kind it is: the strict disjunction (“but not both”) licenses both moods, and the broad disjunction (“or perhaps both”) licenses only tollendo ponens.
The modes. “Necessarily” and “contingently” must be attached to the conclusion with care, because 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, with no middle term and no second premise, so the conclusion is read directly from 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: “Some animals are not dogs” is true, yet “Some dogs are not animals” is false.
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 — obvert, then convert, then obvert. For A and O it preserves truth: “All oaks are trees” → (obvert) “No oaks are non-trees” → (convert) “No non-trees are oaks” → (obvert) “All non-trees are non-oaks.” We cannot simply convert first, because an A does not convert. 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 practiced here, the conjunctive and the disjunctive. Each has one lawful way of reaching a conclusion, and any other way 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 we may conclude only by positing one member in order to remove the other, which is the mood ponendo tollens (“by positing, it takes away”): “He is in Rome; therefore he is not in Athens.” We may not reason the reverse, since 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”): if one member is denied, the other stands (“It is not odd; therefore it is even.”).
Strict against broad. Whether we 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” we cannot infer that, being clever, he is not also lucky. So we must note the connective before we conclude.
Before the Exercise · The Hypothetical Syllogism
Introduction
A hypothetical (conditional) proposition asserts not a fact but a connection: “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 validly from such a premise, and two tempting fallacies imitate them.
Ponendo ponens — affirm the antecedent. If the if-clause is posited, the then-clause follows: “It has rained; therefore the ground is wet.” This mood is always valid.
Tollendo tollens — deny the consequent. If the then-clause is removed, the if-clause is removed with it: “The ground is not wet; therefore it has not rained.” This mood is 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 holds in one direction only, from antecedent to consequent, and never from consequent to antecedent.
Before the Exercise · The Modal Syllogism
Introduction
A modal syllogism is one whose premises are qualified by a mode, that is, the manner in which the predicate belongs to the subject. Three modes are considered here: 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 requires a negative conclusion, and a particular premise a particular one, so the weaker mode governs the conclusion: if a necessary premise is joined to a merely contingent one, 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. So we judge each argument by the mode its weakest premise allows.
Before the Exercise · The Chain of Syllogisms
Introduction
Reasoning rarely stops at a single syllogism. More often one conclusion becomes a premise of the next, and so a chain is formed, which the old logicians called a polysyllogism, or, when the middle conclusions are left unspoken, a sorites. Here the links are shown in full, in schematic letters, so that you may judge the reasoning link by link.
How a chain is built. Each syllogism draws a conclusion; that very conclusion serves as a premise of the syllogism that follows, joined to a fresh premise, until the last conclusion is reached. All B are C; all A are B; ∴ all A are C. All C are D; ∴ all A are D.
When the chain holds. The whole chain is valid only if every link is a valid syllogism. A single invalid link makes the whole chain invalid, even if the last line happens to look plausible. So we follow each step and ask whether this conclusion truly follows from the two propositions above it.
What to do. Read the chain from top to bottom and judge it valid or invalid. There are no diagrams here, only the form of the reasoning. The chains grow longer at the higher levels: two or three links at first, and four or five at the Master level.
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; an argument that lacks either is unsound.
What this exercise trains. Here we judge both at once: whether the form is valid, and whether the premises are 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 the nearest step toward it. 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, which is valid in form but false in a premise, is the most common imitation of knowledge. To recognize it is to begin to see what separates 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 be accepted if it were stated. The art of examining an enthymeme is to state the hidden premise and ask whether it is true.
How common it is. The enthymeme is the ordinary form of reasoning in daily life, and it is far more frequent than the full syllogism. Everyday speech relies on it (“You’ll like her — she’s a teacher”); so does the law (“He fled the scene, so he is guilty”); politics and advertising depend 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 no premise can, say that none will serve. To see what is being taken for granted is to hold an argument to account.
Before the Exercise · Dialectic
Introduction
Most questions worth arguing about cannot be proved. Dialectic is the art of reasoning well about them nonetheless, not from what is demonstrated, but from what may reasonably be granted.
What is “probable” here. It is not a guess about the odds. A premise is probable when it seems so to everyone, or to most people, or to the wise, and among the wise, to all of them or to the best known. It is what we may fairly ask an opponent to grant.
e.g. a promise ought generally to be kept (granted by all);   a law nobody can obey is no true law — granted by the learned.
What dialectic reaches. It reaches a reasoned opinion, well supported and still open to challenge. Demonstration reaches science, which cannot be otherwise; dialectic reaches the better view. Knowing which of the two we are doing keeps us honest about how much we have shown.
The topics — where an argument comes from. A topic (Latin locus, a place) is a standing relation from which we can draw an argument: from the definition, from the wider kind, from the narrower kind, from a property, from opposites, from the more and the less, from likeness, from whole and parts, or from authority. Once the topic is named, we can usually see at once what would answer the argument.
e.g. every bird has feathers, and a robin is a bird (from the wider kind);   if courage is praised, cowardice is blamed — from opposites.
Authority is a topic, and the weakest one. That those who know a subject judge so is a real reason, and the tradition counts it among the topics. But it moves us by who says a thing rather than by what makes it so; hence it is the weakest topic, though it is still a real reason.
Answering an objection — the four moves. This is the discipline of the disputed question, and it is worth learning by heart. Distinguish the term (the distinguo) when a word is true in one sense and false in another. Deny the premise when it is simply false. Grant it all and deny the consequence when every premise stands and the conclusion still does not follow. Or grant the whole objection and state the claim again more narrowly, because sometimes the objection is right.
The topics are for FINDING, not for labeling. This is the point of the whole book, and it is easy to miss. Logic divides in two: the art of judging an argument we have been given, and the art of finding one we need. The Analytics teach the first. The Topics teaches the second. So the real use of a topic is not to name what somebody else did, but to answer the question where do I go to look? Given something we must show, and something already granted, a topic shows which place will make the one a reason for the other.
Every topic carries a maxim. The maxim (Latin maxima propositio) is the governing rule an argument from that place depends on. From the wider kind: whatever belongs to the whole family belongs to each kind within it. From opposites: what holds of one opposite, the contrary holds of the other. When the maxim is true the argument holds. Note well: a fallacy is the same art turned about — an argument that depends on a maxim which is false and merely looks true. Dialectic and the detection of fallacy are one doctrine learned from two ends.
Two people, two parts. A disputation takes place between the one who puts the case and the one who answers. In each of the questions that follow, you take one of those two parts.
Before the Exercise · The Fallacies
Introduction
A fallacy is an argument that looks sound and is not. St Thomas’s De fallaciis treats them better than any list does, because it explains where each one had to come from.
Every fallacy has two causes, not one. The cause of the appearance is what makes the argument look good, that is, what moves us to accept it. The cause of the failure is what actually breaks it. We are deceived by the two together, since something appears to be so and is not.
e.g. I know the man who is coming; Coriscus is the man coming; so I know Coriscus. It looks sound because the feature and the man really do go together here; it fails because a man and a feature he happens to have are not the same thing.
Two families, and only two. The appearance can arise in just two ways. From the words: one sound is taken for one thing, as the several things called “dog” look like one thing because “dog” is one word. From outside the words: two things that agree somehow are taken to be simply one. Six fallacies fall on the first side, seven on the second.
The six in the words. They follow from a division rather than from a list. Ambiguity is threefold. Actual ambiguity is one sound, unchanged, meaning several things: in a word, equivocation; in a phrase, amphiboly. Potential ambiguity is meaning several things according to how it is said: in a word, accent; in a phrase, composition and division. Apparent ambiguity is a word that truly means one thing and seems to mean another: figure of speech. So there are six fallacies in the words (two, three, and one), and the number follows from the division.
The seven outside the words. Accident (taking a passing feature for the thing); in a certain respect, taken flatly; missing the point; begging the question; not the cause as cause; the consequent (reading a one-way connection backwards); and many questions asked as one. These are the dangerous ones, because the language is faultless and only the thinking is at fault.
The hidden rule. Every argument depends on a rule, usually unspoken. Socrates is a man, therefore an animal depends on a true one: whatever the narrower kind is said of, the wider kind is said of too. Socrates is an animal, therefore a man depends on the same rule reversed, which is false. Both premises may be true and the argument still worthless, because the rule it depends on is false.
The same faults, under two sets of names. Modern books keep most of these and rename some. Accident is now called applying a general rule to a special case; in a certain respect taken flatly is hasty generalization, or ignoring the qualification; missing the point is the irrelevant conclusion, with the straw man and the red herring as species of it; not the cause as cause is false cause, or post hoc ergo propter hoc; many questions as one is the loaded question. And the consequent is what is now called affirming the consequent when it is stated with an “if,” and the undistributed middle when it is not; modern logic counts those one fault, as the schools did. Each answer here gives you both names, so that you can read either kind of book afterwards.
The other half of the same doctrine. In Dialectic we learn that every topic (every place an argument may be drawn from) carries a maxim, the rule the argument depends on. When that rule is true, the argument holds. A fallacy is the same art turned about: an argument that depends on a rule that is false and looks true. So the two exercises teach one doctrine from opposite sides, and the question below about the hidden rule connects them.
Why we study them. We study them not in order to use them. St Thomas notes that when we reason badly by ourselves, it always happens beside our intention, since nobody sets out to deceive himself. The honest man who has not noticed that his word shifted is commoner than the sophist who shifts it on purpose. So we should apply this study to our own arguments first.
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