Quantifiers, conditionals and contrapositives

Why the small words carry the argument

Two of the verbal item types Bocconi describes depend on exact reading: data and statements, where you decide whether each statement is true or false, and verbal critical reasoning, where a short passage is taken as true and a question asks, for example, which condition would best support its thesis. Short arguments are often built from rules ("only", "unless", "no ... without") and quantities ("not all", "none but"). Read loosely, these words can reverse a sentence's meaning. The cure is mechanical: translate each rule into a plain arrow, "if this, then that", before you judge anything.

Translating the awkward phrasings

Phrase in the passage	Plain meaning	Arrow or quantity
Not all A are B	At least one A is not B	some A are not B
None but A may do X	Anyone who does X is an A	$\text{X} \to \text{A}$
Only A do X	The same as "none but"	$\text{X} \to \text{A}$
No X without Y	Every X has Y	$\text{X} \to \text{Y}$
X only if Y	Y is needed for X	$\text{X} \to \text{Y}$
X if Y	Y is enough for X	$\text{Y} \to \text{X}$
X unless Y	Without Y, X happens	$\text{not Y} \to \text{X}$
X if and only if Y	Each guarantees the other	both arrows
It is not impossible that X	X has some chance, however small	possible, not probable

Four rows deserve a second look.

"Not all A are B" says only that there is at least one exception. It does not say that some A are B: "not all of the fund's bonds are investment grade" is still true if none of them is.

"None but", "only" and "no ... without" all name a requirement, and the arrow points to the requirement. "No firm wins the tender without an environmental certificate" means that winning requires the certificate: $\text{wins} \to \text{certificate}$. The certificate is a necessary condition. Holding it is not a guarantee, which would be a sufficient condition.

"Unless" works in one direction only. "The launch will be delayed unless the supplier delivers by May" says that if the supplier does not deliver by May, the launch is delayed. Its contrapositive is just as useful: if the launch is on time, the supplier delivered by May. Strictly, the sentence makes no promise about what happens if the supplier does deliver; the launch could be delayed for some other reason.

"Not impossible" is the weakest positive claim there is. It rules out only a zero chance. It does not mean "likely", and an option that turns it into "probably" says more than the passage.

One arrow, four moves

Write the rule as $P \to Q$. Then:

You learn	You may conclude	Name
$P$	$Q$	following the arrow: valid
not $Q$	not $P$	the contrapositive: valid
$Q$	nothing about $P$	the converse: invalid
not $P$	nothing about $Q$	the inverse: invalid

The contrapositive is valid because it says the same thing as the rule: if $P$ were true, $Q$ would be true; $Q$ is false, so $P$ cannot be. The converse fails because $Q$ may come about in other ways, and the inverse fails for the same reason. The converse and the inverse are contrapositives of each other, so they stand or fall together, and the rule supports neither. An argument that runs an arrow backwards commits the flaw described in flaws and evaluation: it treats a necessary condition as sufficient, or a sufficient one as necessary.

Rules with two conditions

Rules often join two conditions: "any project that is funded and finished on time earns a bonus", or $(F \text{ and } T) \to B$. Its contrapositive is: no bonus, so not both $F$ and $T$. That means at least one of them is false, not both. To say which, you need another fact. If you also know the project was funded, the false one must be $T$: it was not finished on time. The negation of "$F$ and $T$" is "not $F$ or not $T$", never "not $F$ and not $T$".

Matching an argument's blueprint

When a question asks which argument reasons in the same way as another, ignore the subject matter. Replace each claim with a letter, keep the logical words, translate them into arrows, and write the pattern, for example "$(P \text{ and } Q) \to R$; $Q$; not $R$; so not $P$". Do the same for each option and compare. Check three things:

• the conclusion: a negative conclusion needs a negative match, a certain one a certain match;
• the direction of every arrow, after translating "only", "unless" and "without";
• the validity: a valid argument is matched by a valid one, and a flawed argument by one with the same flaw.

The order in which the premises appear does not matter; the pattern does.

Testing a rule: look for the case that could break it

A rule of the form "every $P$ is $Q$" can never be proved by cases that agree with it, but a single $P$ that is not $Q$ refutes it. So when you must check a rule, or decide which evidence would test it, look only where a counterexample could hide:

• cases known to be $P$: check that they are $Q$;
• cases known to be not $Q$: check that they are not $P$.

Cases known to be $Q$, or known to be not $P$, cannot break the rule whatever else is true of them. Checking them feels productive because they tend to agree with the rule, which is exactly why they tell you nothing. The same habit helps with evaluation questions: the most useful fact about a general claim is often whether a counterexample exists.