How to work with this page
This page is the workshop for Lesson 2 of the Logic unit. You work through it alone, in about 45 minutes, with your Lesson 2 handout beside you. It has one job: to make sure that after this lesson you never again say valid when you mean true.
Each part is built the same way:
- Explanation — short, with the lesson’s own examples.
- Exercise — you type or choose, then press Check. Nobody sees your mistakes. The page counts your first try; that number is for you.
Two rules for this page
Retrieve before you look. Where the page says “from memory”, scroll the explanation away first. A wrong guess you then correct sticks better than a right answer you copied.
Every exercise says what goes in the field: a word from a closed list, yes or no, or a choice. Spelling counts as written; capitals do not.
The page has three stages. Stage 1 puts the lesson’s case on trial and asks you to take a side before you have the tools. Stage 2 gives you the tools: validity, soundness, the counterexample, the error-spotting. Stage 3 sends you back to your verdict with the tools in hand. Your sheet tells you when each stage is due.
Recall — Lesson 1 from memory
Before anything new: eight questions on Lesson 1, without your notes. Guess first: how many of the eight will you get? …… of 8.
Exercise 1 — Lesson 1
Write one word or a short phrase in each field, in English, without looking anything up.
- In logic, an argument is a set of propositions in which some, the , are offered as reasons to believe another, the .
- “Therefore”, “so”, “hence” mark the ; “because”, “since”, “given that” mark a .
- An inference whose premises, if true, guarantee the conclusion is ; one whose premises make the conclusion probable is .
- Aristotle’s three means of persuasion: logos (argument), ethos (the speaker’s ), pathos (the audience’s ).
- Socrates’ method of refuting a claim by drawing out its consequences until they contradict is the .
- The Surgisphere papers in The Lancet collapsed not because the inference was bad but because the could not be verified.
Fewer than 5? Read the Lesson 1 key terms on your handout again before you go on — one minute, then come back and do not redo the exercise.
Stage 1 — The case first
Why the case comes first
A tool you have never needed is a tool you will not remember. Take your side now; Stage 2 will give you the words for what you did.
Eric Loomis was sentenced in Wisconsin in 2013 with the help of a COMPAS risk score, a proprietary algorithm that classifies defendants by their statistical likelihood of reoffending. The workings of the algorithm are a trade secret. Loomis argued he had been sentenced as a member of a class, not as an individual, on a premise he could not examine. The Wisconsin Supreme Court ruled against him in 2016. The same year, ProPublica showed that COMPAS falsely labelled Black defendants “high risk” almost twice as often as white defendants; the company replied that the analysis used the wrong fairness criterion, and researchers then proved that the competing fairness criteria cannot all be met at once when base rates differ.
The state’s argument, laid out:
- COMPAS reliably predicts recidivism risk on a population basis.
- Risk of recidivism is a relevant consideration in sentencing.
- Therefore, the COMPAS score was a legitimate input into Loomis’s sentence.
Exercise 2 — Read the argument
Choose one answer per question.
Which line is the conclusion?
Loomis did not say the inference was faulty. He said something about a premise. Which?
Now take your side, in your notebook, before you go on. Two or three sentences: was Loomis sentenced on an argument (with the court: population-level instruments are legitimate inputs if they are not determinative and the judge is warned of their limits), or handed a conclusion without one (with Loomis: a premise he cannot examine is not a premise he has been given)? Name your side and one reason. You will come back to this at Stage 3.
Stage 2 · Part A — Validity
A1. The definition
An argument is valid if the conclusion cannot be false when all the premises are true. Validity is a fact about the relationship between premises and conclusion. It says nothing about whether the premises are true.
This argument is valid:
- All fish can fly.
- Salmon are fish.
- Therefore, salmon can fly.
Both premises are false, the conclusion is false, and the argument is valid: if the premises were true, the conclusion would have to be true. The form carries truth from the premises to the conclusion; it does not put any truth in.
The trap
“Valid” in ordinary English means roughly “good” or “acceptable”. In logic it means one thing only: no counterexample exists. An argument can be valid and worthless (false premises), or invalid and persuasive.
A2. The test: build a counterexample
To show an argument invalid, describe one situation in which all the premises are true and the conclusion is false. If you can, the argument is invalid. If no such situation is possible, it is valid.
| Argument | Counterexample? | Verdict |
|---|---|---|
| All cats are mammals. All dogs are mammals. So all dogs are cats. | Premises true, conclusion false: the world as it is | invalid |
| If it rained, the street is wet. The street is wet. So it rained. | A street-cleaning truck passed: wet street, no rain | invalid |
| If it rained, the street is wet. It rained. So the street is wet. | None possible: rain is the wet street’s guarantee | valid |
The first is the fallacy of the undistributed middle: “mammals” does not pin cats and dogs to the same subset. The second is affirming the consequent: “If P then Q; Q; therefore P.”
A3. Four forms to know by name
| Name | Shape | Valid? |
|---|---|---|
| Modus ponens | If P then Q. P. Therefore Q. | yes |
| Modus tollens | If P then Q. Not Q. Therefore not P. | yes |
| Affirming the consequent | If P then Q. Q. Therefore P. | no |
| Denying the antecedent | If P then Q. Not P. Therefore not Q. | no |
Modus tollens is how falsification works: relativity predicts that light bends around the sun (if the theory is true, the light bends). Had Eddington found in 1919 that the light did not bend, the theory would have been refuted.
Exercise 3 — Valid or invalid?
Scroll the tables away. For each argument, choose valid or invalid by asking one question: could the premises all be true and the conclusion false?
All philosophers are mortal. Socrates is a philosopher. So Socrates is mortal.
If the defendant is guilty, his fingerprints are on the knife. His fingerprints are on the knife. So he is guilty.
If the theory is right, the light bends. The light does not bend. So the theory is not right.
All politicians are untrustworthy. The librarian is untrustworthy. So the librarian is a politician.
If you study, you pass. You did not study. So you will not pass.
All swans are white. This bird is a swan. So this bird is white.
If COMPAS is reliable, its scores should be used. COMPAS is reliable. So its scores should be used.
Every study that used Surgisphere data was retracted. This study was retracted. So this study used Surgisphere data.
Exercise 4 — Name the form
Write exactly one of these five names in each field: modus ponens, modus tollens, affirming the consequent, denying the antecedent, undistributed middle. The first row is done for you.
| Argument | Form |
|---|---|
| If it is a fish, it has gills. It is a fish. So it has gills. | modus ponens |
| If the alarm works, it rings when there is smoke. There is smoke and it did not ring. So the alarm does not work. | |
| If the vaccine works, cases fall. Cases fell. So the vaccine works. | |
| All Swiss cantons have a parliament. Geneva has a parliament. So Geneva is a Swiss canton. | |
| If she was in Bern, she missed the train. She was not in Bern. So she did not miss the train. | |
| If the truck drove under its own power, the video is honest. The truck did not drive under its own power. So the video is not honest. | |
| If the poll is right, Trump wins. Trump won. So the poll was right. |
Exercise 5 — Which one is the counterexample?
An argument: If a study is published in The Lancet, it has been peer-reviewed. This study has been peer-reviewed. So it was published in The Lancet.
Which of these is a counterexample of the same form — premises true, conclusion false?
Part B — Soundness
B1. The definition
An argument is sound if it is (1) valid and (2) all its premises are true. Soundness is what you want in practice: a sound argument guarantees a true conclusion.
| Valid | Invalid | |
|---|---|---|
| All premises true | Sound — the conclusion must be true | Not sound — the conclusion may or may not be true |
| Some premise false | Not sound — the conclusion may be false | Not sound |
Two tests, in a fixed order: first validity (a question about the form, settled without looking at the world), then the premises (a question about the world). Philosophy works a great deal with valid but unsound arguments: thought experiments stipulate their premises instead of establishing them.
The consequence that matters most
A valid argument with a false conclusion must have a false premise. So if you reject a conclusion, you owe the room a premise you reject — and you have to say which.
Exercise 6 — Valid, true, sound
Three fields per argument, each yes or no: is it valid? are all the premises true? is it sound? The first row is done for you.
| Argument | Valid? | Premises true? | Sound? |
|---|---|---|---|
| All fish can fly. Salmon are fish. So salmon can fly. | yes | no | no |
| All whales are mammals. All mammals breathe air. So all whales breathe air. | |||
| All cats are mammals. All dogs are mammals. So all dogs are cats. | |||
| If Bern is in Switzerland, it is in Europe. Bern is in Switzerland. So Bern is in Europe. | |||
| If the moon is made of cheese, it is edible. The moon is made of cheese. So the moon is edible. | |||
| If a study is retracted, its data were fabricated. The Surgisphere study was retracted. So its data were fabricated. |
Row 6: the form is modus ponens, so it is valid. But the first premise is false — studies are retracted for many reasons — so the argument proves nothing, however true its conclusion may happen to be.
Exercise 7 — Which premise do you owe?
Each argument below is valid and you reject its conclusion. You must therefore reject a premise. Which one?
If an instrument predicts well for the population, it predicts well for each member. COMPAS predicts well for the population. So COMPAS predicts well for Loomis. You reject the conclusion. Which premise goes?
Whatever the majority of experts say is true. The majority of experts said hydroxychloroquine was harmful. So it was harmful. Suppose later trials show no effect. Which premise goes?
Anyone who benefits from a fraud is complicit in it. Investors in Nikola benefited while the stock rose. So they were complicit. Which premise goes?
Part C — Spot the error
C1. How arguments go wrong in this lesson
Six things students say, and the one mistake in each. Learn to name them: naming is what lets you find them in your own writing.
Exercise 8 — Name the mistake
Write exactly one of these four labels in each field: validity for truth (calls an argument valid because its premises or conclusion are true), truth for validity (calls an argument invalid because a premise is false), soundness for validity (uses a false premise to attack the logic), consequent affirmed. The first row is done for you.
| The student writes | The mistake |
|---|---|
| “The argument is valid, because both premises are obviously true.” | validity for truth |
| “The argument is invalid: the first premise is simply false.” | |
| “The state’s reasoning is illogical — COMPAS doesn’t predict individual risk.” | |
| “Trump won, so the neighbour poll was clearly a valid argument.” | |
| “The retraction proves the study’s inference was invalid.” | |
| “The vaccine worked because cases fell after the rollout — cases falling is what a working vaccine predicts.” |
Row 3 is the one to watch in your own essays. The objection may be a good one — but it is an objection to a premise, so it belongs at the soundness step, not the validity step. “Illogical” is the wrong word for it.
Exercise 9 — Where does the rule need a judgment call?
Not every application is mechanical. Ordinary-language arguments leave premises unsaid, and whether the argument is valid depends on which premise you supply. Decide for each:
“She is a doctor, so she will know what this rash is.” As it stands, is this valid?
“COMPAS is calibrated across groups, so it is fair.” As it stands?
This is the philosophy
The answer key to Exercise 9 says cannot say yet both times. That is not a dodge. Supplying the missing premise, and seeing that the argument’s fate turns on it, is the work — the drill can only tell you where it is.
Stage 3 — Back to the case
Exercise 10 — The retrieval checks
Two questions from your handout, from memory. Write your answer in the box in full sentences; the bar fills as your words match the model answer’s key words. Then compare.
State the two conditions an argument must meet to be sound. Then: are arguments from analogy, or inference to the best explanation, valid in the formal sense?
Mathematics consists of valid deductions from axioms. What does a valid deduction guarantee about its conclusion, and what does it leave unestablished?
Your verdict, with the tools
Take out what you wrote at Stage 1. Now rewrite it in three sentences:
- Your side (with the court, or with Loomis).
- Your reason — and say at which step it lives: validity or soundness.
- The price your side pays. Court: the point at which an algorithmic premise becomes too opaque to count as a premise at all. Loomis: what replaces risk-scoring — unaided judicial discretion, with its own documented disparities, or something else.
Then check your own paragraph:
- Did you keep validity and soundness apart? “The argument was valid, so the sentence was fine” is the exact confusion this lesson exists to end.
- Did your reason name a premise, not the outcome (the sentence, the ProPublica numbers)?
- Did you say the price in your own words, not the sheet’s?
Only now: compare with the three model paragraphs — two good ones, one bad one.
Done?
You can now:
- say what valid means without using the word true;
- build a counterexample to show an argument invalid;
- name modus ponens, modus tollens, and the two look-alikes that fail;
- run the two tests in order: form first, premises second;
- say, when you reject a conclusion, which premise you reject;
- catch “illogical” being used for “false premise” — in your own essays first.
Go back to your sheet.