Logic and Scientific Thinking

Deduction and Induction

Bogdan G. Popescu

bgpopescu@tec.mx

Tecnologico de Monterrey

Learning Outcomes

Overview

Classify any argument as deductive or inductive — by the support it claims.

Separate validity from truth — and name the one impossible combination.

Grade arguments with the full vocabulary: valid, sound; strong, cogent.

NARRATION: Welcome. You can already spot an argument, rebuild it in standard form, and label its premises. Today we grade the “therefore” itself. Three outcomes: classify any argument by the support it claims; separate validity from truth; and grade arguments with the full vocabulary — valid and sound for guarantees, strong and cogent for good bets. The next session puts this to work: the deepfake argument, and the five inference patterns. Timing plan (90-min half): Outcomes + warm-up ~6 / Part 1 (two families + two-jobs pair + Hume, deductive coda of 3) ~28 / Part 2 (validity & truth) ~15 / Exercise 1 ~8 (5 work + 3 share-outs) / Part 3 (grading grid + examples) ~14 / Exercise 2 ~9 (5 work + 4 debrief) / Close ~4. Total ~84 — the 3-slide deductive coda is the first block to trim if short; real air for questions after validity and after soundness, the two places students stall.

Warm-Up: Vitamin C

“I caught a cold, so I took vitamin C every day. A week later the cold was gone. Therefore, the vitamin C cured me.”

The parts from earlier sessions, refreshed:

Conclusion: the vitamin C cured me.
Premises:
- P1: took vitamin C daily;
- P2: the cold ended.
Hidden assumption:
- The cold would not have disappeared without Vitamin C

NARRATION: Here is the vitamin C argument from an earlier session — and before anything new, the sixty-second refresher of its parts, because every tool today operates on them. The conclusion is the claim you are asked to accept: the vitamin C cured me. The premises are the reasons offered in support: I took it every day, and a week later the cold was gone. And the hidden assumption is the premise the argument needs but never says aloud: that the vitamin — and nothing else — did the work. Cold-call these three; they should be familiar from the earlier sessions. Today’s question is different and comes before all of that: what kind of support do these premises even claim to give the conclusion? Is the speaker promising a guarantee — or offering a good bet? Hold your answer; we resolve it in ten minutes, once the distinction has a name.

Part 1: Two Ways to Reason

Two Broad Families

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flowchart LR
  P["PREMISES<br/>the reasons offered"] -->|"promise a<br/>GUARANTEE"| D["DEDUCTION<br/>premises true means<br/>conclusion MUST be true"]
  P -->|"promise a<br/>GOOD BET"| I["INDUCTION<br/>premises true means<br/>conclusion PROBABLY true"]
  style P fill:#1e293b,color:#f9fafb,stroke:#1e293b
  style D fill:#4a7c6f,color:#f9fafb,stroke:#4a7c6f
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Induction is about probability.

Deduction is about guarantee.

NARRATION: Here is the map — and let me be precise about its status: this is our working simplification, two broad families. A guarantee: if these premises are true, the conclusion cannot possibly be false — that is deduction. Or a good bet: if these premises are true, the conclusion is probably true — that is induction. Logicians distinguish finer kinds — arguments from analogy, statistical generalizations, inference to the best explanation — and for this course they all live under induction’s umbrella, because they all claim probability rather than necessity. Say that now and the sharp student’s objection in minute five becomes a confirmation. One more subtlety we will need all day: the promise is about intent, not success. An argument that promises a guarantee and fails to deliver is still a deductive argument — just a broken one. Vaughn’s chapter 3 builds everything on this distinction.

Job 1 — Logic: Classify Arguments by Claimed Support

Examples:

“95% of University students own a phone. Marco is a university student. So Marco probably owns one.”

“Probably” claims a bet: induction.

“Ana is taller than Luis. Luis is taller than Sofía. So Ana is taller than Sofía.”

If the premises are true, the conclusion can’t be false: deduction.

Indicator words — must vs. probably — are clues.

NARRATION: The words “deduction” and “induction” do two different jobs in this university, and confusing the jobs is where the trouble starts. Job one, ours, in logic: classifying INDIVIDUAL ARGUMENTS. There the definition is by claimed support — a guarantee makes it deduction, probability makes it induction — and the direction rule you may have met in school, general-to-specific versus specific-to-general, is wrong for this job: it assigns arguments to the wrong family. Watch it fail twice. Marco: ninety-five percent of university students own a phone; Marco is a university student; so Marco probably owns one. The shortcut looks at the direction — general statistic down to one person — and says deduction. But listen to the argument: it says PROBABLY; it claims only a bet. It is induction — logicians call this shape a statistical syllogism — and the shortcut just misfiled it. Ana: Ana is taller than Luis; Luis is taller than Sofía; so Ana is taller than Sofía. Nothing general anywhere, so the shortcut says this cannot be deduction. But grant the premises and the conclusion is inescapable — a full guarantee: deduction. Misfiled again, in the other direction. So classify by ear, not by direction: indicator words help — “necessarily” and “must” versus “probably” and “likely” — but as an earlier session taught, words are clues, not guarantees; if the intended standard is genuinely ambiguous, evaluate both readings. The next slide gives these words their second, fully legitimate job — do not skip it, because half the room has already met that job in a methods course.

Job 2 — Methods: Direction Describes Strategies

In the social sciences, induction and deduction usually describe the direction of a research process.

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flowchart LR
  A1["DEDUCTIVE STRATEGY<br/>theory-testing"] --> B1["THEORY<br/>general"] --> C1["HYPOTHESES<br/>specific, testable"] --> D1["DATA<br/>the test"]
  A2["INDUCTIVE STRATEGY<br/>theory-building"] --> B2["DATA<br/>specific cases"] --> C2["PATTERNS<br/>regularities"] --> D2["THEORY<br/>general"]
  style A1 fill:#1e293b,color:#f9fafb,stroke:#1e293b
  style A2 fill:#1e293b,color:#f9fafb,stroke:#1e293b
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  style C2 fill:#b44527,color:#f9fafb,stroke:#b44527
  style D2 fill:#b44527,color:#f9fafb,stroke:#b44527

Deduction moves from a general rule to a specific conclusion.

Induction moves from specific observations to a broader generalization.

NARRATION: Job two, and this is important because you will meet it in this very course: in research methodology courses, the same words label research STRATEGIES, not single arguments. Top row: the deductive strategy starts from general theory, derives specific, testable hypotheses, and lets the data deliver the verdict — theory-testing. Bottom row: the inductive strategy starts from specific observations, finds patterns, and builds toward general theory — theory-building. For this job, the general-to-specific language is perfectly correct — it describes the direction of the workflow. And the names are earned, not accidental: deriving a prediction from a theory IS deduction in today’s sense, and generalizing from data IS induction in our sense. So do not tell your methods professor the direction talk is wrong — it is right about strategies. It is only wrong as a classifier of individual arguments, which was the previous slide’s job. If a student conflates the two later in the course, send them back to Marco and Ana.

Explanation

The use of the induction vs deduction meaning is related.

Methods textbooks use the terms more loosely:

Context	Deduction	Induction
Logic	Premises guarantee the conclusion	Premises make the conclusion probable
Research design	Theory-testing; theory to hypothesis and data	Theory-building; data to patterns and theory

Flash Vote: Guarantee or Good Bet?

Hands up — one finger = guarantee (deduction), two = good bet (induction). No explaining yet — just classify.

1. “All squares have four sides. This shape is a square. So it has four sides.”

2. “The last ten winters here were cold. So next winter will likely be cold.”

3. “No fish are mammals. A shark is a fish. So a shark is not a mammal.”

Empirical Generalization

Unemployment rose during or soon after each recession.
Therefore, the next recession will raise unemployment
- A bet, not a guarantee.

NARRATION: Now induction in its most familiar setting: empirical economics. Real data — the US unemployed share from 1967 to 2015, with recessions shaded. In each recession shown, joblessness rose during or soon after the downturn. So the economist argues: in the next recession, unemployment will rise too. Notice the difference from a deductive guarantee: even with all seven premises solid, the conclusion could still turn out false. The premises make it probable — highly probable, even — but nothing guarantees it. That is a bet, not a guarantee. To be fair to the economists: inside their formal models they deduce constantly — theorems from axioms. But the moment a claim leaves the model and faces data, the support on offer is inductive. One caveat to have ready if asked, which does not touch the inductive point: the series is unemployed persons as a share of the TOTAL population, not the official unemployment rate that divides by the labor force — say “unemployed share,” as the axis does.

The Missing “Probably”

Honest induction should hedge: “probably,” “likely,” “may.”
In real speech, that hedge is the first thing dropped.
So a bet gets stated as a guarantee.
Why? Audiences reward confidence, not hedging.

“Every past technology ended up creating more jobs than it destroyed.”

Overconfident: “AI will create more jobs than it destroys.”

Honest: “AI will probably create new jobs too.”

The honest move — and the rarer one — is to keep the “probably.”

NARRATION: One habit worth naming before we move on. An inductive argument only earns a probable conclusion, so stated honestly it should carry a hedge — “probably,” “likely,” “may.” Yet in real speech that hedge is the very first word to disappear. It just happened on the last slide: the economist’s conclusion was “the next recession WILL raise unemployment,” not “will probably” — a bet, quietly dressed up as a guarantee. Why does the “probably” vanish? Because we live in a certainty-thirsty culture: audiences take confidence as a proxy for competence and hear hedging as weakness or evasion — the confidence heuristic you met previously (Price & Stone 2004). Take the case on the slide: every past technology, from the loom to the spreadsheet, ended up creating more jobs than it destroyed. The honest induction is “AI will PROBABLY create new jobs too” — but what people actually say is the flat, confident “AI will create more jobs than it destroys.” And notice the sharper risk on a contested, forward-looking claim like this: the past may not decide the future, which is exactly why the hedge matters. So the speaker who wants to be believed strips the qualifier, and the honest, calibrated claim ends up sounding weaker than the overconfident one. The discipline this course asks is the harder, rarer move: keep the “probably” in, even when it costs you persuasive force. It is the same lesson as “claim less” — a conclusion you actually earn beats one that merely sounds strong.

Warm-Up, Resolved

“…A week later the cold was gone. Therefore, the vitamin C cured me.”

First question: what does the speaker claim?

No must, no necessarily — a causal diagnosis from one case.
The claim is a good bet, not a guarantee: inductive.
So we skip validity and ask about strength instead.

Hume: Induction’s Problem

“The bread, which I formerly eat, nourished me … but does it follow, that other bread must also nourish me at another time?”

— An Enquiry Concerning Human Understanding (1748), Section IV, Part II, ¶16

Induction runs from observed cases to unobserved ones.
It assumes observed patterns hold in similar unseen cases.
Deduction can’t prove that — nature could conceivably change.
Experience can’t either — that defense already uses induction (a circle).
Science manages this uncertainty; it never makes it proof.

NARRATION: Before we grade inductions, one hard problem from your reading. Hume, 1748: the bread I ate before nourished me — but does it follow that other bread must nourish me later? Notice the shape of ALL induction, not just claims about the future: it runs from cases we have observed to cases we have not — the next loaf, yes, but also unseen places, past events, a whole population from a sample. Every such step assumes that observed patterns hold in relevantly similar unseen cases; Hume’s own phrase for one version of this is “the future will be conformable to the past.” Now Hume asks how we could justify that assumption, and he closes two doors. Could we prove it by pure logic, demonstratively? No — there is no contradiction in imagining nature changing tomorrow, so its denial is perfectly conceivable. Could we prove it from experience? Only by noting it has held so far — which already assumes the very principle in question, so that defense runs in a circle. Neither route works. This is the famous problem of induction, and philosophy has no agreed solution. What science offers is not a solution but discipline: replication, controls, randomization, and statistics help us detect error, separate explanations, and make our uncertainty explicit — but they do not deductively prove that observed patterns will persist beyond the evidence. That is why “strong,” today’s grade, is always a statement of probability, never of necessity.

So What?

Deduction promises a guarantee — so we can check whether it delivers.
That check has a name: validity.
It is the most misunderstood word in logic — next section fixes that.

Part 2: Validity and Truth

Validity: The Link

A deductive argument is valid when:

IF the premises are true, the conclusion must be true.

Validity evaluates the reasoning from premises to conclusion.
It does not tell us whether the premises are actually true.
Truth applies to statements; validity applies to arguments.

The test: Can all the premises be true while the conclusion is false?

No → valid
Yes → invalid

NARRATION: The definition, slowly. An argument is valid when, if the premises were true, the conclusion could not possibly be false. Everything lives inside that “if.” Validity inspects the link — the earlier sessions’ arrow, the therefore — and nothing else. It does not certify the premises; it does not certify the conclusion. Truth is a property of individual statements; validity is a property of the structure connecting them. And here is the working test, the one you will run all day: could the premises all be true and the conclusion still false? Really try to imagine it. Impossible: valid. Conceivable: invalid — and note carefully what invalid does NOT mean: the argument is still deductive; it promised necessity and broke the promise. The counterexample test never decides deductive versus inductive — that was classification, done by claimed support — it decides valid versus invalid. Keep classification and evaluation apart, and the examples that follow will make sense. ASK FIRST (before revealing the definition): “What would have to be IMPOSSIBLE for an argument to count as valid?” Pause; let them reach “the premises all true, yet the conclusion false” on their own, then confirm it as the test.

Valid Nonsense

P1.	All cats are reptiles.
P2.	All reptiles speak French.
C.	All cats speak French.

Every line is false.
Yet if P1 and P2 were true, C would have to be.
The structure itself is flawless.
Valid argument, false everything. Validity ≠ truth.

The form: All A are B; all B are C; so all A are C.

Same Structure, Sensible Content

P1.	All poodles are dogs.
P2.	All dogs are mammals.
C.	All poodles are mammals.

Premises true, conclusion true — and it had to follow.
Same structure as the French-speaking reptiles.
Different facts, identical logic. Validity ignores the subject.

Change the content, keep the form — validity does not change.

The Forbidden Combination

P1.	All cats are mammals.
P2.	All dogs are mammals.
C.	All cats are dogs.

Both premises true. Conclusion plainly false.
True premises and a false conclusion — so: invalid.
Sharing a category is not being the same thing.

This is the one pattern a valid argument can never show.

A Lucky Conclusion

P1.	All cats are mammals.
P2.	All dogs are mammals.
C.	All whales are mammals.

Every line is true — yet the premises never mention whales.
The conclusion is true independently of the premises.

A true conclusion never proves the reasoning was good.

NARRATION: This one is subtler, because every sentence is true: cats are mammals, dogs are mammals, whales are mammals. But read the premises — they say nothing whatever about whales. The conclusion could stay true even if the premises were about something else entirely; the premises did no work to support it. The conclusion is true by luck, not by link, so the argument is invalid. Keep this example close: students constantly assume that a correct conclusion proves the reasoning was good. It does not. A true conclusion can sit at the end of a completely broken argument. THE EMPIRICAL TWIST (Kahneman 2011, ch. 3), runnable live in 30 seconds: “All roses are flowers. Some flowers fade quickly. Therefore some roses fade quickly.” Vote valid or invalid — a large majority of college students endorse it, and it is INVALID (the quickly-fading flowers may include no roses at all). Belief bias: when the conclusion is believable, people believe the argument — “the conclusion comes first and the arguments follow.” The mirror of this slide: here, true lines with no connection; there, a believable conclusion smuggling in a broken link. That is WHY the counterexample method must be mechanical.

So What?

Validity is a quality check on structure
- it ignores whether the claims are true on their own.
But we argue to be right about the world.
We need words for valid + true — and for good bets.
We also need additional vocabulary: soundness, strength, cogency.

Exercise 1: Sort and Judge

In pairs — 5 minutes. For each: deductive or inductive?

Then — deductive: could the premises be true yet the conclusion false?

(no → valid);
inductive: strong or weak?

Give one sentence.

1. “Everyone who entered the lab signed in. Leo entered the lab. So Leo signed in.”

2. “Every café that opened on this street closed within a year. So this new one will too.”

3. “If the battery is dead, the car won’t start. The car won’t start. So the battery is dead.”

NARRATION: First exercise — five minutes, in pairs; move fast, these are classification calls, not essays. For each argument, first ask what kind of support the premises claim: a guarantee, or only likelihood? Then use the right test — for deduction, could the premises be true while the conclusion is false? for induction, do the premises make the conclusion likely, strongly or weakly? SOLUTION (quick key): (1) deductive, VALID. (2) inductive, reasonably STRONG. (3) deductive, INVALID. ANSWER KEY: 1 — deductive and valid: if everyone who entered signed in, and Leo entered, he must have signed in; the premises cannot be true while the conclusion is false. 2 — inductive and reasonably strong: the run of closures makes another closure likely, but nothing guarantees it — this café could break the streak. 3 — deductive and invalid: a dead battery is only one reason a car won’t start — no fuel, a bad starter, and more — so the premises can be true while the conclusion is false. A possible cause is not the only cause. Circulate; the reliable stumble is #3 — it sounds right because a dead battery really would stop the car, but that is not the same as its being the only thing that could. Its name — affirming the consequent — waits for the next session; here, just collect the verdict.

Part 3: Grading Arguments

Evaluating an Argument: Two Checks

Grading Arguments: Two Checks

First, identify the argument’s claimed support:

Deductive → evaluate validity
Inductive → evaluate strength

Grading Arguments: Two Checks

Check 1: How good is the link?

Deductive argument	Inductive argument
Could the premises be true while the conclusion is false?	How probable do the premises make the conclusion?
No → Valid	Highly probable → Strong
Yes → Invalid	Not very probable → Weak

Grading Arguments: Two Checks

Check 1: How good is the link?

Deductive argument	Inductive argument
Could the premises be true while the conclusion is false?	How probable do the premises make the conclusion?
No → Valid

P1: If it rains, the street gets wet.
P2: It rained.
C: Therefore, the street is wet.

Why? Rain must wet the street.
The premises can’t be true with a false conclusion. No → Valid

Grading Arguments: Two Checks

Check 1: How good is the link?

Deductive argument	Inductive argument
Could the premises be true while the conclusion is false?	How probable do the premises make the conclusion?

Yes → Invalid

P1: If it rains, the street gets wet.
P2: The street is wet.
C: Therefore, it rained.

Why? Wet doesn’t mean it rained.
A sprinkler could have wet it. Yes → Invalid

Grading Arguments: Two Checks

Check 1: How good is the link?

Deductive argument	Inductive argument
Could the premises be true while the conclusion is false?	How probable do the premises make the conclusion?
	Highly probable → Strong

P1: It has rained here every day this month.
C: Therefore, it will probably rain here tomorrow.

Why? A month is strong evidence.
Many observations strongly support the conclusion. Highly probable → Strong

Grading Arguments: Two Checks

Check 1: How good is the link?

Deductive argument	Inductive argument
Could the premises be true while the conclusion is false?	How probable do the premises make the conclusion?

	Not very probable → Weak

P1: It rained here yesterday.
C: Therefore, it will probably rain here tomorrow.

Why? One day proves almost nothing.
A single day barely supports the conclusion. Not very probable → Weak

Grading Arguments: Two Checks

Check 2: Are the premises actually true?

Deductive arguments	Inductive arguments
Valid + true premises = SOUND	Strong + true premises = COGENT

Sound arguments provide a deductive guarantee based on true premises.

Cogent arguments provide strong inductive support based on true premises.

Grading Arguments: Two Checks

Check 2: Are the premises actually true?

Deductive arguments	Inductive arguments
Valid + true premises = SOUND

Example:

P1: If it rains, the street gets wet.
P2: It rained.
C: Therefore, the street is wet.

The conclusion must follow from the premises → Valid
Suppose it did in fact rain → both premises are true

Therefore, the argument is SOUND.

Grading Arguments: Two Checks

Check 2: Are the premises actually true?

Deductive arguments	Inductive arguments
	Strong + true premises = COGENT

Example:

P1: It has rained here every day this month.
C: Therefore, it will probably rain here tomorrow.

The premise makes the conclusion highly probable → Strong
Suppose the month of rain checks out → the premise is true

Therefore, the argument is COGENT.

NARRATION: Here is the entire evaluation scheme on one grid — and there are really only two questions. First, do the premises provide the kind of support the argument claims? For a deductive argument that means validity: if the premises were true, the conclusion could not be false. For an inductive argument that means strength: if the premises were true, they would make the conclusion probable. Second, are all the premises actually true? A deductive argument that passes both checks is sound; an inductive argument that passes both is cogent. Soundness and cogency are the corresponding top grades in the two families — the right inferential support plus true premises. They are parallel, but not identical: a sound argument guarantees a true conclusion, while a cogent one makes its conclusion probable, not certain. Picture validity as a switch — the conclusion either follows necessarily or it does not — and inductive strength as a dimmer, where the evidence can support the conclusion to a greater or lesser degree. ASK FIRST (before the grid): “A valid argument can still be a bad one — what else must we check?” Let them say “whether the premises are actually true” before you reveal check two. ASK (before the cogent cell): “If valid + true premises = sound, then strong + true premises = ?” Let them construct “cogent” themselves rather than reading it off the grid.

Take-Away: Deduction

In plain words:

Good logic + true premises → conclusion must be true.
Good logic, a false premise → not proven (may still be true).
Broken logic → doesn’t follow (may still be true).

A bad argument doesn’t make its conclusion false — it just hasn’t earned it.

Grade Each Argument

Deductive or inductive? Then grade it — valid / invalid, or strong / weak.

1. “All squares have four sides. This shape is a square. So it has four sides.”

2. “All dogs are mammals. This animal is a mammal. So it’s a dog.”

3. “This mountain pass has closed from snow every winter for thirty years. So it will probably close this winter too.”

4. “Two of my friends got sick after eating there. So that restaurant probably has bad hygiene.”

Take-Away: Induction

In plain words:

Strong support + true premises → reasonable to accept (still revisable).
Strong pattern, a false premise → not established (may still be true).
Too little evidence → not enough to accept (may still be true).

A weak argument doesn’t make its conclusion false — it just hasn’t earned it.

What You Leave With

Two families: a guarantee (deduction) or a good bet (induction).
One forbidden cell: valid + true premises + false conclusion.
Four grades: valid, sound; strong, cogent.
Four steps that grade any argument you meet.
One habit: when the support is weak, claim less.

References

Hume, D. (2007). An enquiry concerning human understanding (P. Millican, Ed.). Oxford University Press. (Original work published 1748; Section IV, Part II, ¶16 / SBN 33–34.)
National Bureau of Economic Research. (2023). US business cycle expansions and contractions. https://www.nber.org/research/data/us-business-cycle-expansions-and-contractions