Logic and Scientific Thinking

Formal Fallacies

Bogdan G. Popescu

Tecnologico de Monterrey

Welcome

STORY ARC ACROSS THE FALLACY SESSIONS (the trial). Today = the FORMAL half: the shapes that guarantee a conclusion, the twins that only look like them, and the intuition that answers before it checks. Next = the informal field guide. Do NOT say the “seller is your own mind” line — that framing belongs to the cognitive-biases reading (Kahneman). Cognate today: verdict / veredicto. NO jury language — Mexican courts are judge-decided. TIMING (~40 min): recap 5 / the necessary/sufficient decoder + the twins in the wild 14 / Exercise 1 6 / the ledger slide (“or”/“all” families, for the record) 4 / bat-and-ball Act I 8 / close 3. DOOR SLIP: ONE folded slip per student, “CASE FILE / EXPEDIENTE” — the bat-and-ball question only. Answered silently in the first 5 minutes, collected before Act I; TALLY during the algebra slide.

Learning Outcomes

Catch each formal trap by its valid counterpart.

Use necessary vs. sufficient to see why the traps fail.

See the full shape table — the “or” and “all” families, for the record.

Recognize a valid form — or a confident feeling — is not a checked conclusion.

Recap

Question	Answer from the previous session
Validity vs. truth?	Valid logic can carry false premises
The three valid conditional shapes?	Modus ponens, modus tollens, hypothetical syllogism
The two traps you met?	Affirming the consequent, denying the antecedent

That toolkit was conditional (if–then).
Today: why its traps keep fooling us — plus the “or” and “all” families, in one table.

The Formal Toolkit

Necessary vs. Sufficient Conditions

“If dog, then mammal” points one way:

Dog is sufficient for mammal — dog guarantees mammal.
Mammal is necessary for dog — no mammal, no dog.

The Rule:

forward (dog → mammal)
contrapositive (not mammal → not dog).

Invalid:

reverse (mammal → dog)
negate both (not dog → not mammal).

Necessary vs. Sufficient Conditions

“If dog, then mammal” points one way:

Dog is sufficient for mammal — dog guarantees mammal.
Mammal is necessary for dog — no mammal, no dog.

one direction only

Rule: P → Q, not the reverse (Q → P).

The Traps, in Practice

The previous session’s five shapes, in everyday examples. Name each:

“If the election were rigged, there’d be long lines. Lines were long — so it was rigged.”
- Affirming the consequent — long lines aren’t sufficient to prove rigging.
“If it’s spam, it has typos. This email has none — so it’s safe.”
- Denying the antecedent — typos aren’t necessary for spam; polished scams exist.

Necessary vs. sufficient names why each trap fails: it swaps sufficient for necessary.

Exercise 1 — Tool or Trap?

Valid tool or its trap? Name each.

If rain, then wet. It rained. So it’s wet.
If rain, then wet. It’s wet. So it rained.
If in Paris, then in France. Not in France. So not in Paris.
If in Paris, then in France. Not in Paris. So not in France.

Exercise 1 — Answers

Each pair shares one rule. Only the second premise moves.

Pair A — rule: If rain, then wet.

It rained. So it’s wet. → modus ponens · TOOL
It’s wet. So it rained. → affirming the consequent · TRAP — a sprinkler could have wet it.

Pair B — rule: If in Paris, then in France.

Not in France. So not in Paris. → modus tollens · TOOL
Not in Paris. So not in France. → denying the antecedent · TRAP — Lyon is not Paris, and still in France.

One legal move, one look-alike that breaks.

The Toolkit at a Glance

The five conditional shapes — each valid tool beside the trap that mimics it:

Name	Shape	Verdict
modus ponens	If P, then Q. P. so Q	VALID
affirming the consequent	If P, then Q. Q. so P	INVALID
modus tollens	If P, then Q. Not Q. so Not P	VALID
denying the antecedent	If P, then Q. Not P. so Not Q	INVALID
hypothetical syllogism	If P, then Q. If Q, then R. so If P, then R	VALID

Logicians catalogue many more shapes — the “or” and “all” families, and dozens beyond them. These five are the ones this course uses. After Vaughn (2019).

The whole toolkit on one slide — a designated summary, so let it be dense. Read it as pairs: each invalid trap sits directly under the valid tool it mimics. Hypothetical syllogism is the exception — a chain with no twin. The “all” family stays because students meet these five everywhere: Session 5 drills them, the review session rehearses them, and the exam names them. The “or” and “all” families used to sit here and have been cut — nothing in the course rests on either, and the caption now covers them. If a student asks, one line each does the job. “Pet or predator — it’s a pet, so not a predator” fails because a cat is both. “All dogs are mammals, all cats are mammals, so cats are dogs” fails because sharing a container does not nest the sets — and they have already graded that shape in Session 4, so give them the case rather than the name. The 5am-club version lands well: “successful people wake at 5am; I wake at 5am; so I’m successful.” This is the reference students photograph, so say plainly: these five are what the exam names.

So What? — Knowing the Shapes Isn’t Enough

The recipe:

keep premises true
conclusion false
one case refutes it.

So why do the traps keep working, even on people who can check them?

A wrong move can feel valid — and the feeling often replaces the check.

Act I: that feeling at its purest.

Act I: The Answer Comes First

The Bat and the Ball

No tools here — just your immediate answer.

A bat and a ball cost $1.10 in total.
The bat costs $1.00 more than the ball.
How much does the ball cost?

Correct answer: 5 cents.

You’re in good company:

many people answer 10 cents
the answer arrives before any check runs.

Two Lines of Algebra

Call the ball $x$. Then the bat is $x + \$1.00$.

Together they cost $1.10: $\;x + (x + 1.00) = 1.10$
Subtract the $1.00: $\;2x = 0.10$
Divide by 2: $\;x = 0.05$ — the ball costs 5 cents.
Check: ball $0.05 + bat $1.05 = $1.10 ✓

Most solvers split $1.10 into $1.00 + 10¢ without noticing.
Answering an easier question than the one asked.

The same lesson as the counterexample test: check, don’t feel.

So What? — Feeling Right Isn’t Being Right

A conclusion can feel right before reasoning.

Owning the checking tools doesn’t mean we use them.

THE BRIDGE — the whole psychology this session needs, in one sentence. Do NOT explain WHY conclusions arrive early (System 1, the inner lawyer) — that machinery is the Kahneman reading’s territory. HEDGE, if a well-read student invokes the two systems: the System 1/System 2 architecture is contested — Mercier & Sperber (2017) argue reasoning is itself an intuitive inference (about reasons), not a separate slow system. The felt-answer phenomenon on this slide survives under BOTH accounts; the course teaches the phenomenon, not the architecture. (Students read both authors.) Today the frame is external: ARGUMENTS do the selling. FRAME for Act II, said once: “Eleven classic ways to win an argument without being right. You will see them used ON you. Definitions and drills are exam material — learn them clean.”

Conclusion

A valid form is not a true conclusion — and a confident feeling is not a checked one. Today: the formal half.

Four traps in the table, each paired with a valid shape — the conditional pair drilled today; the “or”/“all” pair for the record.
Hypothetical syllogism stands alone — a chain, no matching trap.
The test: a counterexample — premises true, conclusion false.
The bat and the ball: the felt answer ($0.10) wasn’t right ($0.05).

References

Vaughn, L. (2019). The power of critical thinking: Effective reasoning about ordinary and extraordinary claims (6th ed.). Oxford University Press.