Logic and Scientific Thinking

Building an Argument — Build · Probe · Repair

Bogdan G. Popescu

bgpopescu@tec.mx

Tecnológico de Monterrey

Always Someone Else’s Argument

You separated premises from conclusions.
You made the hidden premise say itself out loud.
You sorted claims: empirical · conceptual · normative.
You checked the link — validity, then truth.
You named the move: fallacy, rhetoric, misinformation.
You asked what a correlation could and could not show.

You did all of that to someone else’s argument.

Today you find out how hard it is to build one.

Six lines, and they are the whole course — read them at a steady pace and let students recognise each one rather than explaining any of it. Line by line that is Session 2, Session 2 again, Session 3, Sessions 4 and 5, the fallacies and rhetoric sessions, and Session 13. If a line draws blank faces, name the session and move on; this is a roll call, not a review. Do not attach specimens to them — the abstraction is the point, because what they are being asked to notice is the shape of the whole term rather than any one example.

Then let the first fragment land on its own. Every skill in that list is a skill for receiving an argument. Five weeks of equipment for taking things apart, and not one hour spent on making something that could survive the same treatment.

Then the framing line of the session, delivered slowly and close to verbatim: “For five weeks you have learned how to critique other people’s arguments. Today you will discover how hard it is to build one yourself.” Say it straight and do not apologise for it; the difficulty is the lesson. Nothing new is being taught today; everything they need they already have.

Opening straight onto this slide is deliberate — no housekeeping first. Someone will nonetheless ask within the first ten minutes whether this is on the exam, so have the answer ready and give it once, briefly: nothing produced today is examinable, but every move made today is a move the exam tests. Then get back to the session. Full exam scope goes at the close, where it will not compete with the framing.

The Toolkit, Recalled

The Four Moves

At every “therefore”: conclusion? premises? unstated assumption?

Two sessions gave you the whole procedure:

Spot the argument — reasons for a disputed claim, or just noise?
Rebuild it — standard form, hidden premises said aloud, charitably.
Sort the premises — empirical, conceptual, normative — catch the hidden ought.
Probe the weak point — often the therefore, not the facts.

Today someone runs all four on your argument. Build · probe · repair.

Sixty seconds, brisk, no re-teaching — this is one slide now, not two, because the room needs to be building sooner than it needs another recap. Open with the three questions; they are the reflex from Session 2 and have been used in every session since. Two terms are worth saying aloud once more because they are load-bearing today. An ENTHYMEME is an argument that leaves a premise unstated — which is almost every argument anyone has ever made, including the ones they are about to write. And CHARITY: attribute the strongest plausible hidden premise, then judge. Charity matters more today than it ever has, because in a few minutes they will be judging a classmate sitting next to them.

Recite the four moves fast; they have seen the slide before, at the close of Session 3. The one worth pausing on is move 3, because today’s arguments are all POLICY arguments — every one of the five topics ends in an “ought”. That means every table’s argument will need at least one normative premise, and Hume’s gap says it cannot be derived from the empirical ones. Students who forget this write three factual premises and a conclusion containing “should”, and their reviewer should catch it. That is the single most common failure they will see today, so plant it now. The closing fragment is the pivot of the session and is worth ten seconds of silence: everything so far has been analysis, and analysis lets you hide behind someone else’s mistakes. Production does not. The checklist their reviewer will use is the same one they have been running on Romney, on advertisers, on the papers — turned around to face them.

Part 1 — The Build

The Six Boxes

Write these by hand on the sheet. They can be rough.

My conclusion
My premises — at least one empirical, at least one normative
For the empirical one: what would decide it
My hidden premise
Where I expect to be attacked — the step I am least sure of
The repair

Facts on their own never get you to a should. If all your premises are empirical, the argument has a hole in it.

Boxes 1–5 now. Box 6 only after someone has probed it.

Hand the sheets out while this slide is up. Stress the physical format: boxes, not prose. Nobody is writing an essay and nobody is being graded on style — a premise is one line. Nothing on this list is new: boxes 1 and 2 are standard form with the E/C/N sort from Session 3 attached, box 3 is the third step of the three-step diagnosis, box 4 is the enthymeme, box 5 is probe turned on yourself, box 6 is the repair they did to the Romney–Obama exchange in Session 2, turned on their own argument this time. Box 5 asks for a PREDICTION, not a verdict: where do you expect to be attacked? The reviewer finds the actual weak point independently, and the two get compared afterwards. Say that when the sheets go out, because it is the point of the exercise — an author who correctly predicts their own vulnerability has done something genuinely hard, and an author who did not has just met the thing this session exists to teach. The instruction that saves the most time: leave box 6 EMPTY. A repair written before anyone has probed the argument is not a repair; it is the conclusion written twice. Box 6 is the only box that requires another person. Warn them about box 4 too, so the worked example lands harder: hunting your own hidden premise usually turns up the one you are most comfortable with, not the one the argument is standing on. Finding the second is the reviewer’s job.

The mixed-premise rule in box 2 is the one to enforce while circulating, because it is the most common way a sheet comes out broken. Every topic today ends in a “should”, and Hume’s gap says no pile of facts delivers one — so a sheet with three empirical premises and a normative conclusion has a hole in it whether or not anyone notices. Requiring one of each also guarantees there is something for Session 13 to bite on: an empirical premise is a premise you can ask “what would decide it?” about, and a sheet made entirely of value claims gives the reviewer nothing checkable to work with.

Nothing Here Is New

Box	Where you learned it
Conclusion · premises	Standard form — Session 2
Empirical premise	E / C / N — Session 3
What would decide it	The three steps — Session 7
Hidden premise	Enthymeme — Session 2
Where I expect to be attacked	Probe the therefore — Session 3
The repair	Repair the Debate — Session 2

The Worked Example

Should class attendance count toward the final grade?

Make It Arguable

Too vague to attack, and therefore impossible to defend:

“Attendance matters.”

Arguable:

Class attendance should count toward the final grade.

The test: could a reasonable person deny it?

Nobody knows what “attendance matters” is asking them to agree to.

A conclusion nobody can attack is not a strong conclusion; it is an unfinished one — and the quickest way to tell the difference is to ask whether a reasonable person could deny it. Nobody denies that attendance matters, which is why it feels safe to write and why it is useless: nobody can tell what they are being asked to agree to, so the reviewer has nothing to get hold of and the author has nothing to defend. A vague box 1 is the most common reason a peer review comes back as taste rather than diagnosis. Do NOT demand scope conditions and exceptions here — “in which courses”, “with what exceptions” — even though a fully specified policy needs them. Those are what the repair adds at the end of the worked example, and asking for them now spends the payoff early. Box 1 needs a claim someone can disagree with; the qualifications are earned later, by surviving the probe.

Student 1 — The Sheet

This is what a student would write:

P1.	Coming to class takes real time and effort.
P2.	Grades should reward the effort students put in.
C.	Attendance should count toward the final grade.

Student 2 — The Critique

The reviewer writes three things:

Name it

“You’re treating showing up as if it is the effort.”

Quote it

“coming to class takes real time and effort”

Say what would decide it

“Someone sits here doing nothing. Someone skips and does all the work. Which one does your rule pay?”

In standard form — the premise nobody put on the page:

P3. (unstated)

Showing up is the effort P2 wants to reward.

The three quotes are deliberately what a student would actually write in the box, rather than what a logic textbook would say. The machinery of the course is a way of writing down ordinary suspicion; it does not replace it. The reviewer did not go hunting for a hidden premise. They noticed that P1 calls attending an effort, P2 asks to reward effort, and nothing on the sheet says the two are the SAME effort — the author has quietly treated showing up as if it were the effort that earns a grade. Only afterwards does the last fragment give that suspicion its formal name and its own line, and the moment it has a line, everyone can see it was never argued for.

Say why the reviewer left the other lines alone, because choosing the target is the judgement they need this afternoon. P2 is a values claim — reasonable people differ about whether grades should track effort at all, and a review that opens there starts an argument nobody settles in three minutes. P1 is not in dispute either — sitting in a room three hours a week plainly costs time and effort, and nobody denies it. What nobody has argued for is that THAT effort is the effort P2 means. The unstated step between them is the one doing all the work, which is the profile of the line worth attacking. Note for yourself that P1 is deliberately NOT a correlation: “students who attend get better grades” would be exactly the claim Session 13 taught them to distrust, and waving it through here to keep the example tidy would undo that session in one slide. Note what the author could not do for themselves: every line they wrote is true, and the premise the argument stands on is still not on the page. This is the three-step diagnosis from Session 7 pointed at a classmate.

Presence Is Not Effort

Ana comes to every class and does nothing.
Beto misses class and does all the work.

The rule pays Ana and punishes Beto.

But the author wanted to reward effort, so the rule does the opposite of its job.

Two students, and the argument is in trouble — no technical vocabulary required. Give the room a moment on each name; everyone has sat next to Ana and everyone has been Beto at least once, and that recognition is what does the teaching here. Then let them say the punchline before you do: ask what the proposed rule actually pays out to each of them.

The objection is not that the author is wrong about anything they wrote down. It is that the thing they propose to grade and the thing they say they value come apart in both directions at once. That is why it bites without being unfair — nothing here calls the author foolish, and the reviewer granted their evidence and their value entirely.

Make the Session 13 connection explicit — it is an exam-relevant check run on a classmate. This is the measurement question — what was ACTUALLY measured? Attendance measures bodies in a room, the way a 1-to-10 survey answer measures “happiness”. A proxy is only as good as the gap between it and the thing you care about, and Ana and Beto are that gap walking around. Then set up the repair with the question the author’s own second premise has already asked: if attendance is not the effort, what is?

Student 1 — The Repair

The author takes the point:

“Fair — showing up isn’t what I want credit for. The work that can only happen in here is.”

	Before	After
P1	Coming to class takes real time and effort.	Some course learning needs real-time activities that cannot be replicated outside class.
P2	Grades should reward the effort students put in.	Students should get credit for completing required activities that serve the course objectives.
P3	(unstated) Showing up is the effort P2 wants to reward.	(now stated) Attendance is only a proxy for completing those activities.
C	Attendance should count toward the final grade.	Credit should attach to completing the in-class work, not to physical presence.

The author’s line — “fair, showing up isn’t the same as doing the work, but I still want effort to count” — is the whole skill: concede the point that was actually made, and hold on to what you actually wanted. Students under criticism do one of two wrong things — defend the original sentence word for word, or abandon the position entirely. This is neither, and it is worth naming as neither.

Then put the table up and read DOWN the right-hand column, not across the rows — it is a complete argument on its own and should be heard as one: some learning needs real-time activity that cannot be replicated outside, students should get credit for completing required activities, attendance is only a proxy for that, therefore credit attaches to completing the work rather than to presence. Only then go back and compare across.

Check the repair against the counterexample out loud, because that is what makes it a repair and not a retreat. The student who sits there doing nothing now earns nothing — they did not do the work. The student who misses class is not punished for being absent; they simply miss the chance to do something worth credit. Both halves of the reviewer’s objection are answered, and no proxy is left anywhere in the argument.

Then the row that matters most: P3. It went from an unstated assumption that attendance IS the effort to a stated admission that attendance is only a PROXY for what the author cares about — and once that is on the page, the conclusion cannot stay where it was. Point at P2 as well. It did not survive untouched; it got sharper. “Reward effort” was vague enough to be bought with the wrong policy, and being probed forced the author to say what they actually wanted credit to track. That is the repair doing its real work.

What the Repair Did

What the repair removed: the proxy. Attendance was standing in for something else.

What it put in its place: the work itself — and a sharper account of what the author actually valued.

So the conclusion changed, because what the author wanted was never attendance.

The repaired argument is still arguable — you can deny that credit should track completion at all. A repair makes an argument better, not finished.

Neither of them won the exchange. They ended up with a better claim than either of them started with.

The most important slide in the session, and it is about collaboration, not logic. It is built as stayed-then-changed on purpose. What students should carry out of the room is that the criticism took nothing away from this author; it showed them their policy did not reach their own goal.

Take the fragments one at a time. What went was the proxy: attendance was never the thing the author cared about, only a stand-in for it, and a premise nobody writes down can stay wrong for years because not even the person who holds it ever looks at it. Give it a line of its own and it has to survive inspection. This one did not. What replaced it was the thing itself — and getting there forced the author to say what they actually wanted credit to track, which turned out to be sharper than “effort”. The conclusion moved with it: the author did not want attendance, they wanted the work attendance stood in for, and they had never separated the two.

The fourth fragment is the one to protect against over-claiming, and it is worth saying even though it costs a little drama. The repaired argument is not beyond dispute — plenty of people think grades should measure mastery rather than completion or effort, and a reviewer could open there tomorrow. That is not a failure of the repair. Fixing one inferential link is what a repair does; it does not hand you an argument nobody can touch. If a student objects to the new P2, they are right, and they have just demonstrated the point of the afternoon.

Nobody won this exchange: the reviewer did not defeat the author, the author did not concede, and between them they produced a claim neither walked in with — which the author now owns. The corollary is worth stating outright: being probed is not losing, and keeping your conclusion is not winning. The measure is whether the argument underneath got better. That is what the peer review this afternoon is for.

Repaired, Not Born

Two ways a conclusion survives criticism:

Narrow it — when the weak point is that you claimed too much.
Rebuild it — when the weak point is the therefore itself.

The second one is harder, and it matters more.

You will usually end up defending a different argument from the one you started with.

Say this now and say it again at the close — it is the durable lesson of the session and the one that outlives the vocabulary. The distinction is the payoff of the worked example: the worked example was the SECOND kind, and that is worth saying out loud, because it is the rarer one. Narrowing would not have saved it: “attendance should count a little, in some courses” still pays the student who sits there doing nothing, so the counterexample survives any qualifier you attach. The argument only recovered once the author stopped defending the policy and asked what the policy was for. Students will reach for narrowing every time, because it costs nothing and feels like conceding something. Push them to the harder question first: is that premise true as written, and is it doing the job I think it is? Connect it back to Session 13: a flaw is not a refutation, and a qualified claim is not a defeated one.

Part 2 — How Today Runs

The Run Sheet

5 min	Build — your own argument, boxes 1–5
5 min	Probe — pass it left; find one weak point, then read their box 5
5 min	Repair — box 6

Ground Rules

By hand. No phones, no laptops — as every session.
Your own argument. Everyone at the table writes one.
One topic per table. Assigned, not chosen.
Exchange within your table, one to one.

The Topics

Five Topics, One Per Table

Should AI tools be allowed on assignments if their use is disclosed?
Should lectures be recorded and shared with all students?
Should final exams be worth more than 50% of the grade?
Should group projects carry one shared grade, or grades differentiated by contribution?
Should students be allowed to resubmit major assignments for a higher grade?

Assign these by table before the slide goes up; do not let students choose, or four tables take the same one. With five topics and six tables, two tables will share one — that is fine, and the debrief is better for it. All five were selected on one criterion: a reasonable person can disagree. That rules out topics where everyone in the room already agrees — “phones distract” is true and therefore useless, because an argument nobody resists is an argument nobody has to defend. Each of these forces a genuine trade-off: integrity against access, transparency against attendance, assessment validity against stress, fairness against measurement, mastery against workload. Topic 3 is worth thirty seconds when you assign it: that table is arguing about exam weighting the day before an exam, so they are the interested party. Put one question to them and leave it unresolved — what would decide this independently of your interest? It is the closest thing in the session to Session 12’s third defence question, and the strongest version of their argument is the one that would persuade someone with the opposite stake.

The Reviewer’s Job

You get one classmate’s sheet and you write one weak point — the same three lines Student 2 wrote:

Name it — what is wrong with that step?
Quote it — which words on the page?
Say what would decide it.

A value premise? Find where it gives an answer they would reject.

Charity first — attack the strongest version of what they wrote.

Then look at their box 5: did they predict the attack you actually found?

Point at the worked example rather than explaining this from scratch: it is the same three lines they watched Student 2 write, on a sheet they are now holding. That is the whole instruction, and it should take under a minute.

Enforce the one-weak-point rule hard. A reviewer who lists six things has made none of them useful, and the author has five minutes to act on it. Insist on the quotation too — the difference between a useful review and a useless one is whether the author can find the words being complained about. “I disagree” and “needs more evidence” are not findings; send those reviewers back to the page. The one principled exception is the one the worked example demonstrated: if the weakness is a premise that is NOT on the page, say so — the absence is the finding.

For reviewers who freeze, three prompts are usually enough: look at the hidden premise, look at the scope of the words, look at the therefore. Do not read out a longer catalogue of fallacy names. A printed list turns the exercise into label-hunting, which the Fallacy Fallacy already warned them about.

The value-premise line matters because roughly half the premises in the room will be normative, and “what would decide it?” has no empirical answer for those — nothing in the world settles whether grades ought to reward effort. The move that does work is a counterexample: a case where the principle delivers a result its own author would refuse. Expect to say this out loud to at least two tables. And say the charity line as a social rule, not a logical one: the strongest version of a classmate’s argument may be better than the one they actually wrote, and pointing that out is the generous move.

The box 5 comparison comes last, and only after the reviewer has committed to a finding of their own — reading box 5 first would just hand them the answer. Two outcomes, both useful. If the author predicted the attack, they have done something genuinely difficult and should be told so. If they did not, they have just felt the session’s thesis from the inside: the step you cannot see is invisible precisely because you are the one who wrote it. Neither outcome is a score, and it is worth saying so before the exchange starts, or box 5 turns into a competition.

Conclusion

Most real-world arguments lean on something the author never wrote down.
That is very hard to see in your own argument, and easy in someone else’s.
If someone shows you the weak step, they have done you a favour.
An argument that has answered a real objection is stronger than one that never faced any.

Good arguments are repaired, not born.

Four claims about arguments in general, not four instructions for this afternoon — they have already done the afternoon. This is the last taught session of the course and the slide is doing the work of a conclusion.

The first two belong together and are the discovery of the session: real arguments almost always lean on something unwritten, and the person who cannot see it is the person who wrote it. Say “most”, not “every” — a fully formalised argument can state all of its premises, and that is precisely why formalisation is worth the trouble. If a student raises it, agree with them. The claim that matters is about arguments made by people under time pressure, which is all of them outside a logic textbook. That is why the answer is another person rather than more care. Say it plainly, because students hear “you missed your own hidden premise” as an accusation unless you tell them how ordinary it is.

The third is the one they will need under pressure: being shown the weak step is not losing. Their instinct will be to defend the sentence they wrote. The fourth is why it is worth resisting that instinct — a claim that has survived a real objection is worth more than one that never met any, and they watched exactly that happen this morning.

The arc in one sentence: the course spent five weeks building a machine for taking arguments apart, and today they pointed it at their own and the argument got better. That is the justification for the whole discipline — not to police other people’s reasoning, but to make your own case survivable.

Housekeeping last, not first: sheets are collected, the final is Thursday and covers Sessions 10 through 13, and the optional session on reading a real study is posted on the site for anyone who wants it.