Where to begin. The course begins with the Introduction: What Logic Is, and a study of each of the three acts of the mind stands at the head of that act’s exercises. Each study has a Review beside it, a short set of questions drawn from its cards. After the last section, Logic as a Science carries each act further, with a study and an exercise for each topic. The Study section below (the Diagram Workshop and the Square of Opposition) teaches without keeping score. It is also possible to go straight into the exercises by choosing a difficulty and then 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 the moods of the later textbook account are possible here, including the weakened moods and the fourth figure, which St. Thomas and John of St. Thomas do not count separately but treat as the indirect moods of the first.
Where schematic terms are possible, a small switch on the exercise page chooses letter or English terms. Capital letters stand for general terms, and lower-case letters and proper names for singular terms. 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 the contradictory, the converse, the obverse, or the contrapositive drawn from whichever corner you choose.
part one
part two
Perfecting the art, once the elements are in hand
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.
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:
Every circle in these diagrams is such a universal seen from above, its individuals gathered inside. There are two basic marks. An × says that at least one individual is there (“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.
Shading declares a region empty; an × declares that something exists there. Each categorical proposition makes exactly one mark in the diagram (what an affirmative universal also says about existence is shown by a ⊗ when it is needed):
“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, that is, the conclusion, if one follows.
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 contradictory, the converse, the obverse, and the contrapositive below, which you can try on any pair of terms in the square at the foot of this page.
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, because the O gives no purchase on the predicate beyond the part it names.