Revision: logic, deductions and arguments

Where this sits

The standard test has 9 critical thinking questions, in two formats: data and statements, where you choose which statements are true (or false); and a short passage with an assertion to judge true, false or cannot be deduced. Both reward strict deduction, which is most of this sheet. Questions on assumptions, strengthening, weakening and flaws are part of the Law test's verbal reasoning, not of Bocconi's list for the standard test; the last section is here because the same skills help you read arguments in passages.

Four conditional forms

Take the rule "if a good is taxed ($A$), its price rises ($B$)".

Form	Pattern	Status	Example
Modus ponens	$A \to B$; $A$; so $B$	Valid	It was taxed, so its price rose
Contrapositive (modus tollens)	$A \to B$; $\neg B$; so $\neg A$	Valid	The price did not rise, so it was not taxed
Affirming the consequent	$A \to B$; $B$; so $A$	Invalid	The price rose, so it was taxed (other causes exist)
Denying the antecedent	$A \to B$; $\neg A$; so $\neg B$	Invalid	It was not taxed, so its price did not rise

Translating "only if" and "unless"

Sentence	Arrow	Contrapositive
If $P$, then $Q$	$P \to Q$	$\neg Q \to \neg P$
$P$ only if $Q$	$P \to Q$	$\neg Q \to \neg P$
$P$ unless $Q$	$\neg Q \to P$	$\neg P \to Q$

In $P \to Q$, $P$ is sufficient for $Q$ and $Q$ is necessary for $P$. "Only if" introduces the necessary condition, so the arrow points towards the clause that follows "only if".

Quantifiers and Venn relations

Statement	Picture	Guarantees
All $A$ are $B$	Circle $A$ inside circle $B$	Nothing about $B$s outside $A$
No $A$ are $B$	Two separate circles	No $B$ is an $A$ either
Some $A$ are $B$	Overlap with at least one member	Some $B$ are $A$

"Some" means at least one, and possibly all. "Some $A$ are $B$" reverses; "all $A$ are $B$" does not. Two "some" premises ("some $A$ are $B$", "some $B$ are $C$") give no certain link between $A$ and $C$: the $B$s involved may be different ones. The opposite of "all" is "not all", not "none".

Chains and negations

Rule	Form	Example
Transitive chain	$A \to B$ and $B \to C$, so $A \to C$	If inflation rises, rates rise; if rates rise, mortgage lending falls
Contrapositive chain	$A \to B \to C$, so $\neg C \to \neg B \to \neg A$	Lending did not fall, so inflation did not rise
De Morgan	$\neg(A \land B) \equiv \neg A \lor \neg B$ and $\neg(A \lor B) \equiv \neg A \land \neg B$	"Not both taxed and fined" means untaxed or unfined
Disjunctive syllogism	$A \lor B$ and $\neg A$, so $B$	Bonds or shares were issued; no bonds, so shares

A chain runs backwards only through negations: from the end of the chain being false you can walk back, but from the end being true you can conclude nothing about the start.

Sequencing puzzles

Place the fixed facts first (a named slot), then the blocks they pull with them ("immediately before"), then the slots that are left. Write down every arrangement that survives; a "must be true" answer holds in all of them, a "could be true" answer in at least one.

Paradoxes: both facts stand

An apparent contradiction is resolved by a fact that lets both statements be true, never by doubting one of them. A carmaker cuts the price of every model by 8%, sells as many cars as the year before, yet earns 6% more. If buyers moved from cheaper sedans to dearer SUVs, the average price per car can rise even though every price fell. With 50 sedans at €40,000 and 50 SUVs at €80,000, revenue is €6.0 million; after the cuts (€36,800 and €73,600), 27 sedans and 73 SUVs bring in about €6.37 million, roughly 6% more. This is a change in the sales mix (composition), not Simpson's paradox.

Arguments: premise, assumption, conclusion

An argument has stated premises, a conclusion the author wants you to accept, and often an unstated assumption that bridges them. A necessary assumption is one the argument cannot do without; test it by negating it: if the negation wrecks the conclusion, it was necessary. A sufficient assumption is one that, added to the premises, makes the conclusion follow for certain. Look for a term that appears in the conclusion but not in the premises: "fewer fatigue crashes" does not by itself give "lower insurance premiums".

To weaken a causal claim ($X$ caused $Y$): show another cause, show that $Y$ caused $X$, or show $X$ without $Y$.

Flaw	Error	Example
Post hoc	$A$ came before $B$, so $A$ caused $B$	Crime fell after the new mayor arrived, so the mayor cut crime
Survivorship bias	Studies only the survivors	Studying successful start-ups and ignoring failed ones
Division	A trait of the whole is given to every part	National output grew 4%, so every province grew 4%
Composition	A trait of the parts is given to the whole	Every player is excellent, so the team is excellent
False dilemma	Two extremes presented as the only options	Either 50% tariffs or the industry dies

Argument pattern	Unstated assumption it needs
It worked in City A, so it will work in City B	The two cities do not differ in any way that matters
A poll shows 80% support, so the public agrees	The sample is representative
Action X will achieve goal Y	X is feasible and has no side effect that defeats Y
A and B rose together, so A caused B	No third factor drove both