The logic section of the TOLC (the Italian university admission tests) is the one almost everyone underestimates (“it’s just reasoning, you can’t study for it”), and it is where students lose points that a little method would win back in twenty seconds. It is not a section for the naturally talented. The questions come in a handful of types, they repeat, and after about fifty of them you recognise the type before you have finished reading.
In this post you will find the six types that come up most, one worked example for each, and the typical mistake to avoid. The weight of the section changes from test to test: ten questions in twenty minutes in the TOLC-I (the engineering admission test) and the TOLC-SU (humanities), thirteen in thirty minutes in the TOLC-E (economics), only five in the healthcare professions test. The structure and timing of your own test are in the dedicated guides; the method here is the same for all of them.
The method, before the types
Three habits are worth more than any trick.
- Translate into symbols or a drawing before you answer. A circle diagram for “all/some/none”, a line of letters for orderings, a small table for who tells the truth. Answering “by eye” is the quickest way to fall for the distractor.
- The answer is what follows necessarily, not what is plausible. In the logic of the test a conclusion either follows for certain or it does not. “Probably true” counts as false.
- Two minutes is the ceiling. If after two minutes you have no scheme, mark the question and move on: with the −0.25 penalty, a question left blank costs less than one answered wrongly in a hurry. The rules on blank answers are in the post on how the TOLC score is calculated.
1. Deductions from premises (all, some, none)
Premises: all pharmacists are graduates; some graduates are athletes. Which conclusion follows necessarily? (A) Some pharmacists are athletes — (B) All graduates are pharmacists — (C) Anyone who is not a graduate is not a pharmacist — (D) No athlete is a pharmacist — (E) Some athletes are pharmacists
Draw a “pharmacists” circle inside a “graduates” circle. The athletes who are graduates are a part of the graduates, but we do not know where that part falls: it can lie outside the pharmacists, inside them, or straddle the edge. So (A) and (E) may be true but do not follow for certain, (D) might be false, and (B) reverses the premise. That leaves (C): if the pharmacists all sit inside the graduates, anyone outside the graduates is outside the pharmacists too.
The typical mistake is reversing an “all”: from “all A are B” it does not follow that “all B are A”.
2. Necessary and sufficient conditions
If it rains, the road is wet. Knowing that the road is wet, what can you deduce with certainty? (A) It rained — (B) It did not rain — (C) The road got wet for other reasons — (D) Nothing can be deduced about the rain — (E) The rain has just stopped
“If it rains then wet” cannot be read backwards: the road may be wet because of a water tanker or a fountain. The answer is (D). What you can deduce with certainty is the contrapositive: if the road were dry, then it did not rain. The rule is simple: from “if A then B” you get “if not B then not A” and nothing else.
This is by far the most frequent type, and the distractor is always the reversed reading. For the twin case “all the enrolled students handed in the form” and its negation there is an example in the TOLC-SU guide.
3. Negating a statement
What is the negation of “if a candidate has studied, they pass the exam”? (A) If they have not studied, they do not pass — (B) No candidate who has studied passes — (C) At least one candidate has studied and has not passed — (D) If they do not pass, they have not studied — (E) No candidate has studied
An “if… then” is refuted by a single case in which the first part is true and the second false. So the negation is “there is a candidate who has studied and did not pass”: (C). (B) is much stronger and is not needed to negate the statement, while (A) and (D) are other implications, not negations.
Typical mistake: negating an “if… then” with another “if… then”. The negation of an implication is never an implication.
4. Number sequences
Complete the sequence: 3, 6, 11, 18, 27, ? (A) 36 — (B) 38 — (C) 39 — (D) 40 — (E) 42
Write down the differences between consecutive terms: 3, 5, 7, 9. They grow by 2 each time, so the next one is 11 and the missing term is : (B). An equivalent reading: the terms are the squares plus 2 ().
The method is always the same: first differences, then second differences, then ratios. If the differences are constant the sequence is linear, if the second differences are constant it is quadratic, if the ratios are constant it is geometric. Almost everything else is two interleaved sequences (take one term, skip one).
5. Orderings and family relationships
Anna is taller than Bruno. Carla is shorter than Bruno. Dario is taller than Anna. Who is the second tallest? (A) Anna — (B) Bruno — (C) Carla — (D) Dario — (E) It cannot be determined
Write a line with the symbol : from “Dario taller than Anna”, “Anna taller than Bruno”, “Bruno taller than Carla” you get . The second tallest is (A) Anna. The trick is not to hold the order in your head: you build it on paper, one piece of information at a time, and the question closes itself.
Family relationships work the same way: draw a small tree. “Luca’s mother is my father’s sister”: Luca’s mother is my aunt, so Luca is my cousin. Never work through these chains in words.
6. Who tells the truth
Three friends say: A “B is lying”, B “C is lying”, C “A and B are lying”. Only one person tells the truth. Who? (A) A — (B) B — (C) C — (D) Nobody — (E) It cannot be determined
You test each hypothesis and look for a contradiction.
- If A tells the truth: then B is lying, so C tells the truth (because B said “C is lying”). Two truth-tellers: impossible.
- If C tells the truth: A and B are lying. But A said “B is lying”, and if A lies, then B tells the truth: contradiction.
- If B tells the truth: A and C are lying. Check: A said “B is lying”, which is false because B is truthful, so A lies, fine. C said “A and B are lying”, which is false because B is truthful, so C lies, fine. Everything fits.
The answer is (B). This type is slow because you work by trial and error: if it takes you more than two minutes, it is the ideal candidate to leave blank.
To practise against the clock, use the TOLC-I practice test, which has a logic section of 10 questions in 20 minutes, or the TOLC-E practice test, where logic weighs more. The correct answer is free; the guided solution is unlocked with your first lesson.
How to train, in practice
- A set of ten questions a day, timed at twenty minutes, for two weeks. At the end of each set, sort your mistakes by type (one of the six above) and not by “logic”.
- Recognising the type is half the work. After a few days you should be able to write the name of the type next to each question before solving it.
- Redo the questions you got wrong “cold”, the next day, without looking at the solution.
- In the full practice test, look at the average time per question: if logic takes more than two minutes each, you are taking time away from maths, which in the TOLC-I accounts for 40% of the questions.
How to read the result of a practice test, section by section, is explained in TOLC practice test: how to take it and read the result.
If you would like a hand
Logic is learned best with someone who shows you where you fall for the distractor: in our lessons we work on one type at a time, with exercises solved aloud on a graphics tablet and a shared screen. To find out where to start, book a lesson or write to me through the contact page for a free 30-minute introductory call; the list of tests I prepare students for is on the admission tests page, and you will find the maths foundation in the maths tutoring page.
