Logic and Scientific Thinking

Deductive Tools, Validity, and Truth

Bogdan G. Popescu

bgpopescu@tec.mx

Tecnologico de Monterrey

Learning Outcomes

By the end, you can:

Separate a valid argument from a true one

Use three reasoning shapes you can trust

Spot and name the two traps

NARRATION: Welcome. In the previous session you built the grading machine: classify by claimed support, check the structure, check the premises, issue the verdict. Today we use it twice. First on a real, contested argument — the deepfake claim this course has been circling since day one — to watch conditional form split what is valid from what is actually true. Then we learn five conditional patterns that make grading fast: three reliable tools and their two invalid twins — affirming the consequent and denying the antecedent. A future session will set these two inside the full catalogue of fallacies. Timing plan (90-min half): Outcomes + warm-up ~10 / Part 1 (the trial + deepfake clip) ~28 / Part 2 (five shapes, with flash votes) ~30 / Exercise ~8 (4 work + 4 cold-calls) / Close (bridge + takeaways) ~9. Total ~85 with real slack — spend it on the trap slides, where the misconceptions live.

Recap Exercise

“This region’s economy has grown in every quarter for the past six years. So it will probably grow next quarter too.”

Answer the following questions:

Deductive or inductive? (Does it claim a guarantee, or just a good bet?)
Does the link hold? (deductive → valid? · inductive → strong?)
Are the premises true?
Your one-word verdict? (sound · cogent · or neither)

NARRATION: Prove the previous session stuck — three minutes, no notes, four questions in order. One: is it deductive or inductive — does it claim a guarantee, or just a good bet? Two: does the link hold — valid if deductive, strong if inductive? Three: are the premises actually true? Four: your one-word verdict. The family you pick in question one decides which checks you run. SOLUTION (quick key): inductive — strong — COGENT (probability, never proof). ANSWER KEY: “probably” claims a good bet — inductive, so validity never arises. Support: twenty-four straight quarters with no contraction, a long unbroken run — strong (though economies can turn, which is exactly why it is a bet, not a guarantee). Premise: empirical and, as stated, true. Verdict: COGENT — and per Hume, cogent is the ceiling: probability, never proof. Collect two or three answers aloud; the reliable stumble is students reaching for “valid,” which is the word the previous session spent an hour fencing off.

Part 1: Grading a Real Argument

The Deepfake Argument

Earlier sessions discussed this claim; today we isolate it and grade it.

P1.	If AI can generate realistic fake videos and people cannot tell fake from real, then democratic discourse is undermined.
P2.	AI can generate realistic fake videos.
P3.	People cannot tell fake from real.
C.	Democratic discourse is undermined.

Step 1 done.
Step 2: the claim is a guarantee
Could P1–P3 be true while C is false? (the grid’s validity test)

NARRATION: Here is the deepfake argument, reconstructed — the argument sessions gave us the tools; now step one of the grading procedure is done on the slide. P1 is the bridge: a conditional whose if-part is a conjunction — if AI can generate realistic fakes AND people cannot tell fake from real, then democratic discourse is undermined. P2 and P3 then assert the two facts that bridge asks for: AI can fake, and people cannot tell. Conclusion: discourse is undermined. Formal-detail note for sharp students: with the two facts on separate lines, this is modus ponens plus one trivial step — combining P2 and P3 into the conjunction P1’s if-part names; bundle them into a single premise instead and it is a literal three-line modus ponens. Either way it is valid. Now step two: the wording claims a guarantee, so this is deductive — could P1, P2, and P3 all be true while C is false? Same validity test as the previous session’s grid — look at the shape before you answer.

Validity Is Instant

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flowchart LR
  P1["P1: IF both facts hold,<br/>THEN undermined"] --> C["C: discourse IS<br/>undermined"]
  P2["P2 + P3: both<br/>facts hold"] --> C
  C -.-|"the shape"| MP["MODUS PONENS<br/>if the premises are true,<br/>the conclusion must be true"]
  style P1 fill:#1e293b,color:#f9fafb,stroke:#1e293b
  style P2 fill:#1e293b,color:#f9fafb,stroke:#1e293b
  style C fill:#4a7c6f,color:#f9fafb,stroke:#4a7c6f
  style MP fill:#b7943a,color:#1e293b,stroke:#b7943a

Modus ponens is a fundamental rule of logical inference.

It translates from Latin to “method of affirming”

It follows a simple three-step structure: If P, then Q.
P is true. Therefore, Q must also be true.

Deduction or Induction

Every argument makes one of two promises. Next: what’s inside deduction.

Inside Deduction

Modus ponens is one shape in here — the one you just used on the deepfake. Four more coming; two are invalid.

Testing Soundness

Two premises are empirical facts — test each on its own:

P2 (AI can generate realistic fakes) — true.
P3 (people cannot tell fake from real) — Is that always true?
P1 (the conditional): is the link true?

P1.	If AI can generate realistic fake videos and people cannot tell fake from real, then democratic discourse is undermined.
P2.	AI can generate realistic fake videos.
P3.	People cannot tell fake from real.
C.	Democratic discourse is undermined.

What Does “Undermined Discourse” Mean?

The phrase hides two vague terms:

what does it mean for democratic discourse?
- public discussion?
- forums for political communication, such as journalism
- speech that explicitly promotes democracy

what does it mean for a discourse to be undermined?
- (weak) it just carries some convincing falsehoods?
- are some people censored?
- do citizens stop regarding these forums as credible?
- (strong) the shared basis for what is real collapses?

“Undermined,” Read Weakly

Weak reading: public debate now carries some convincing falsehoods.

P1.	If AI can generate realistic fake videos and people cannot tell fake from real, then public debate carries some convincing falsehoods.
P2.	AI can generate realistic fake videos.
P3.	People cannot tell fake from real.
C.	Public debate carries some convincing falsehoods.

P1 is now plausibly true.
C is too mild to justify alarm.
Falsehoods always circulated.

“Undermined,” Read Strongly

Strong reading: the shared basis for what is real collapses.

P1.	If AI can generate realistic fake videos and people cannot tell fake from real, then the shared basis for what is real collapses.
P2.	AI can generate realistic fake videos.
P3.	People cannot tell fake from real.
C.	The shared basis for what is real collapses.

C is the alarming claim unproven leap.
C isn’t even about democracy anymore:
- “what is real” widens “discourse” into all of reality.

It persuades only because the weak reading is true.
This is equivocation.

NARRATION: Now take “undermined” in its strong sense — the shared basis for what is real collapses. Read that way, the conclusion is exactly the alarming claim that motivates regulation. But now premise one is a huge, unproven leap: it asserts that fakes people cannot spot will actually collapse the shared basis for what is real, and nothing here shows that. Alarming, but unproven — and notice it is not even a claim about democracy any more: “what is real” has quietly widened “democratic discourse” into reality itself, a second slippage riding on the first. And here is the trap: the original argument feels compelling only because it borrows truth from the weak reading and alarm from the strong one. Fix the word to a single meaning and the argument can prove something trivial or assert something unproven — never both at once. That slide between meanings is called equivocation, and you will meet it again in a future session.

Before the Data

Real or fake?

How sure are you?

NARRATION: Before we look at what the studies found, test the claim on yourselves. Watch this clip, and before any reveal, commit to a verdict: real or fake? Jot down what tipped you off, and how confident you are from zero to a hundred. Then we count hands — and in most rooms the vote splits badly, and the confident ones are not the ones who turn out right. Hold that feeling, because in a moment the data will tell you it is not just you. This is P3 — that unaided people cannot reliably tell fake from real — meeting you in person before it meets the forest plot. SETUP: Jordan Peele / BuzzFeed “Obama” deepfake (2018). Embedded from Dropbox; OFFLINE BACKUP: media/obama_deepfake.mp4 (H.264 720p) in VLC, which loads the .en/.es subtitles automatically — TEST IT EACH TERM; the Dropbox rlkey will rot eventually. Clean EN/ES transcripts in media/ — print or share as a handout. Play to about the 0:33 mark and PAUSE before “Someone like Jordan Peele” (the reveal lands at ~0:36): collect real/fake and confidence by show of hands, then play the reveal and re-vote. The payoff is the next slide, where the meta-analysis shows the room’s split is the global pattern.

The Detection Claim

Overall accuracy 55.5%.

NARRATION: P3, the detection claim, on the stand. A 2024 meta-analysis pooled 56 studies — over 86,000 participants asked to tell real from fake. Overall accuracy: 55.5 percent, and the 95 percent confidence interval crosses 50 — statistically, we cannot rule out that people perform at chance. Point at the modalities: video, 57; images, 53; text, 52. Audio looks better at 62 but with eight studies the interval is enormous. Note the discipline before you cheer for the premise: 55.5 percent does not show people categorically CANNOT tell — it shows unaided individuals, on average, across these settings, do only slightly better than a coin flip, with wide variation. Whether that narrower fact is enough for the argument is exactly the next slide’s question. This is what the earlier E-label buys you: an empirical premise is a testable premise, and this one was tested at scale.

The Verdict

Valid?

Yes
if the premises hold, the conclusion can’t be false (modus ponens)

Sound?

That needs true premises:

P3 is absolute: people cannot recognize deepfakes.
But people are right ~55% of the time (Diel et al. 2024).

The Verdict

Valid?

Yes
if the premises hold, the conclusion can’t be false (modus ponens)

Sound?

That needs true premises:

So P3 — “people cannot recognize deepfakes” — is false:

Valid: if P1–P3 are correct, then C must follow (modus ponens).
Unsound: a premise (P3) is false.

NARRATION: Now the verdict — and only after hearing both sides, because rigor means reading the whole literature. The evidence cuts two ways: unaided individuals do only slightly better than a coin flip, but crowds reach 74 to 86 percent, per Groh and colleagues, and training with feedback lifts accuracy above chance, per the meta-analysis. Now grade the argument as written, with the two checks. Valid? Yes — grant the premises and the conclusion cannot be false; that guarantee comes from the form, the modus ponens shape from earlier, so it is secure and not worth attacking. Sound? That needs true premises, and here premise three overshoots. It is an absolute — people cannot tell, full stop. But the evidence shows people are bad, not blind: unaided, they only edge past a coin flip, and, the bigger point, crowds reach seventy-four to eighty-six percent and training lifts accuracy well above chance. So “cannot tell” is too strong; the honest claim is only that unaided, untrained individuals barely beat chance. So P3 as an absolute — people cannot recognize deepfakes — is false. That gives a two-part verdict: valid, because the form guarantees the conclusion if the premises hold; but unsound, because that guarantee runs through a false premise. Good link, bad premise — the trouble is a premise, not the logic.

Repairing the Argument

Replace P3 with the claim the evidence does support:

P1.	If AI can generate realistic fake videos and people spot them only slightly better than chance, then the shared basis for what is real collapses.
P2.	AI can generate realistic fake videos.
P3.	People spot fakes only slightly better than chance (~55%).
C.	The shared basis for what is real collapses.

P1 and P3 had to change: weaken one, you weaken the other.
The new P1 is a fresh empirical claim: harder to defend.

Final verdict:

the form is valid
new premise is unproven, so not sound.

NARRATION: Can we repair it? Try swapping P3 for the claim the evidence does support: unaided people detect fakes only slightly better than chance. That premise is true — but you cannot drop it into the old argument for free, because premise one’s “if” clause was written for the stronger condition. To match, premise one has to be rewritten into a new conditional: if fakes spread and people only barely beat chance, then the shared basis for what is real collapses. And here is the point students miss — that new conditional is not weaker; it is if anything harder to defend, because it claims reality itself collapses even when people can partly tell fakes apart. It is a fresh empirical claim, and the evidence here does not establish it. So the final verdict stands: the form is valid, but the argument is not shown sound — it does not establish its conclusion. Every unresolved weakness lives in the premises — above all, the claimed link between fakery and damaged democratic discourse. Two lessons ride along: soundness verdicts carry a date stamp, because premises are claims about a changing world; and precision about a premise is not pedantry — fixing one premise here forced us to rebuild the whole argument.

Part 2: Five Shapes — Three Tools, Two Traps

Tool 1: The Condition Holds, So the Result Follows

P1: If something is a dog, then it is a mammal.
P2: Fido is a dog.
C: Fido is a mammal.

modus ponens

Shape: If P, then Q. P. so Q

Valid — the if is met, so the then follows.

Tool 2: The Result Is Missing; the Condition Can’t Hold

P1: If something is a dog, then it is a mammal.
P2: Fido is not a mammal.
C: Fido is not a dog.

modus tollens

Shape: If P, then Q. Not Q. so Not P

Valid — the logic behind falsification.

Tool 1 vs. Tool 2

Modus ponens	Modus tollens
The condition is present	The guaranteed result is absent
So the result follows	So the condition is ruled out
If P, then Q; P; so Q	If P, then Q; not Q; so not P
If dog, then mammal; Fido is a dog; so a mammal	If dog, then mammal; Fido not a mammal; so not a dog

Tool 3: Two Rules Chain into One

P1: If dog, then mammal.
P2: If mammal, then animal.
C: If dog, then animal.

hypothetical syllogism

Shape: If P, then Q. If Q, then R. so If P, then R

Valid — but every added link is another premise that can fail.

Dogs Inside Mammals

Inside Dogs → must be inside Mammals: modus ponens.
Outside Mammals → can’t be inside Dogs: modus tollens.
That cat — a mammal that isn’t a dog — is the exception both traps forget.

Trap 1: The Result Happened, So This Condition Did

P1: If something is a dog, then it is a mammal.
P2: Fido is a mammal.
C: Fido is a dog.

But Fido could be a cat — a dog is only one way to be a mammal.

affirming the consequent (invalid)

Shape: If P, then Q. Q. so P

NARRATION: Now the first trap — and the answer to the modus tollens flash vote. If dog then mammal; Fido is a mammal; so Fido is a dog. Invalid: the rule says being a dog is enough to be a mammal, not that it is the only way — Fido could be a cat. In symbols: if P then Q; Q; therefore P — affirming the consequent. It spots the result after “then” and reasons backward to the condition after “if,” and that backward step is not guaranteed. Remember the previous session’s dead battery: a dead battery stops a car, but so does an empty tank or a bad starter — the car’s failure makes “dead battery” plausible, an inductive “probably,” never a deductive proof. The tell: a hunch about what is likely is induction and must say “probably”; this trap states it flatly, claiming a guarantee it has not got.

Trap 2: The Condition Is Absent, So the Result Is Too

P1: If something is a dog, then it is a mammal.
P2: Fido is not a dog.
C: Fido is not a mammal.

But Fido could be a cat — not a dog, still a mammal.

denying the antecedent (invalid)

Shape: If P, then Q. Not P. so Not Q

Tool vs. Trap

Each trap reverses its valid counterpart (all share the rule: If P, then Q):

Valid tool	Invalid counterpart	Where it goes wrong
Modus ponens — P, so Q	Affirming the consequent — Q, so P	reads the rule backward
Modus tollens — Not Q, so Not P	Denying the antecedent — Not P, so Not Q	forgets Q can hold without P (the cat)

Two questions catch most errors: (1) Am I going the direction the conditional states? (2) Was P the only route to Q, or just one?

The Five Shapes

Name	Plain-language movement	Shape	Verdict
Modus ponens	Condition holds, so result follows	If P, then Q. P. so Q	VALID
Modus tollens	Result missing, so condition can’t hold	If P, then Q. Not Q. so Not P	VALID
Hypothetical syllogism	Two rules chain into one	If P, then Q. If Q, then R. so If P, then R	VALID
Affirming the consequent	Result happened, so this condition did	If P, then Q. Q. so P	INVALID
Denying the antecedent	Condition absent, so result absent	If P, then Q. Not P. so Not Q	INVALID

The five shapes — designated summary slide. After Vaughn (2019).

Memorize the two INVALID rows — they are next session’s first two fallacies.

Exercise: Name That Shape

Answer these questions:

1. “If it’s a dog, then it’s a mammal. Rex is a dog. So Rex is a mammal.”

2. “If the video is fake, the detector flags it. The detector flagged it. So the video is fake.”

3. “If the vaccine caused the outbreak, the vaccinated would get sick. They didn’t. So it didn’t.”

4. “If she deleted the files, she’s guilty. She didn’t delete them. So she’s not guilty.”

Answers by cold-call: name the shape, then say valid or trap — and why.

NARRATION: The pattern exercise, four minutes — solo first, then compare with your neighbor. Four arguments; your job is purely pattern recognition now: name the shape, flag the traps, and for each trap say in one sentence why the conclusion escapes. SOLUTION (quick key): (1) modus ponens — valid. (2) affirming the consequent — TRAP. (3) modus tollens — valid. (4) denying the antecedent — TRAP. ANSWER KEY: 1 — modus ponens, valid: the condition is met, so the result follows. 2 — affirming the consequent, TRAP: detectors flag real videos too (false positives), so the flag arriving doesn’t tell you the video is fake. Ties straight to today’s through-line: even an AI detector can’t rescue a broken inference shape. 3 — modus tollens, valid: the predicted result is missing, so the cause is ruled out — also good scientific reasoning. 4 — denying the antecedent, TRAP: guilt could rest on other evidence entirely. Cold-call one student per argument; ask “valid or trap” before “why.” Note the deliberate interleave: valid, trap, valid, trap — so position gives nothing away.

What You Leave With

You walked in able to argue. You walk out able to grade — to take any argument apart and judge it fairly.

Guarantee or good bet? — you can tell deduction from induction.
Valid ≠ true — you can judge the logic and the facts separately.
Three shapes to trust, two traps to avoid — named and spotted.
The deepfake argument is valid — so the real disagreement is about facts.
When the support is weak, claim less — your habit for life.

NARRATION: What you leave with. You walked in able to argue; you walk out able to grade — to take any argument apart and judge it fairly. Five things you now own. One: you can tell a guarantee from a good bet, deduction from induction. Two: you can separate whether the logic holds from whether it’s actually true — the confusion almost everyone lives inside, and you can now see straight through it. Three: you can name three reasoning patterns you can trust and dodge the two traps that mimic them. Four: you took the course’s hardest real case, the deepfake argument, all the way to a verdict — valid, so every honest fight about it is a fight about facts, not logic. And five: you carry one habit for life — when an argument struggles, claim less. That is a genuine skill, and it is yours now. Next session files those two traps into the full catalogue.

References

Diel, A., Lalgi, T., Schröter, I. C., MacDorman, K. F., Teufel, M., & Bäuerle, A. (2024). Human performance in detecting deepfakes: A systematic review and meta-analysis of 56 papers. Computers in Human Behavior Reports, 16, 100538. https://doi.org/10.1016/j.chbr.2024.100538
Vaughn, L. (2019). The power of critical thinking: Effective reasoning about ordinary and extraordinary claims (6th ed.). Oxford University Press. (Ch. 3 — the course PDF; the syllabus lists this reading as Vaughn & MacDonald, the Canadian edition.)