Test a definition: Logic, Round 1 | SizzlinShred Curriculum

1 Learn

You will be able to

State a definition by genus and differentia, not by example.
Find a counterexample and say whether the definition is too wide or too narrow.
Tell a verbal disagreement from a real one by fixing the key term.
Rewrite a vague word as an operational definition.

LabThe ElenchusThe six-step cross-examination of a belief, with a worked example.
LabDefinitionsWhat a definition is and is not, plus the entries on elenchus and aporia.
LibraryObserve, Define, Represent: The First Three Moves · Definition: what exactly do we mean?Genus and differentia, operational definitions and verbal disagreements.2 min glance · 6 min
LibraryProof and Precise Reasoning: From Arguments to Theorems · What a proof is, and is notWhy examples and computations can refute a universal claim but not establish one.2 min glance · 6 min
LibraryProof and Precise Reasoning: From Arguments to Theorems · Counterexample and falsificationThe single-counterexample asymmetry and its link to the elenchus.2 min glance · 6 min
LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium · Three teachers of methodSocrates in the Euthyphro: ask what the word means before arguing about the thing.2 min glance · 7 min

2 Practise

Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.

recall
What are the six steps of the Academy's elenchus, the Socratic cross-examination you run on a belief of your own?

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State the claim so it could be wrong; define its terms; give your grounds and what kind of evidence they are; state the strongest objection; decide, before running it, the test that could refute it; give a verdict for now.
recall
What does the Greek ὅρος, the root of ὁρισμός ("definition"), literally mean, and what does that suggest about defining a word?

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It means a boundary-stone, the marker set at the edge of a field. To define is to say where a word stops.
recall
What definition of knowledge does Plato consider at Theaetetus 201d?

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True belief together with an account of why it is true, that is, belief plus logos. Plato himself refutes this definition later in the same dialogue.
recall
What is an operational definition, and how would it sharpen the word "slow"?

Show answer
Following Bridgman, it defines a concept by the operations used to measure it. For "slow", it asks: slower than what, measured where, for which users?
explain
How does William James's squirrel example show that some disputes are verbal?

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Both sides agree on the facts: the man passed north, east, south and west of the squirrel, but never in front of it, to its right, behind it and to its left, since it kept turning to face him. They differ only on what "going around" means, so once that is fixed, nothing is left to dispute.
connect
This page says the counterexample habit underlies the testing habit in 'Hypotheses, Predictions and Tests'. What asymmetry drives that habit, and how does it shape the way you test an explanation?

Show answer
One valid counterexample refutes a universal claim, while checking some cases cannot prove it. So you ask "what would refute this?" and design tests whose result could show the explanation wrong, writing the prediction down before you look.
recall
What is the elenchus, and what habit does the page say to take from it?

Show answer
The elenchus is Socrates's method of testing a claim by asking for definitions and drawing out consequences. The habit to take is to ask what the word means before arguing about the thing.
apply
A student finds that a formula gives a prime for every whole number from 0 to 20 and is ready to claim it always does. What does the page advise, and what does its Euler example show?

Show answer
Passing cases is not proof: testing refutes quickly but never proves, so keep testing and seek a proof. Euler's n squared plus n plus 41 gives a prime for every n from 0 to 39, yet n = 40 gives 1681, which is 41 squared.

3 Prove it: the mastery check

5 questions drawn from a pool of 12. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.

4 Prove it: the performance task

Cross-examine a definition

Pick an everyday abstract word that people argue about, such as 'fair', 'safe', 'healthy', 'stable' or 'skill'. Choose one that needs no personal details. (1) Write your definition in one sentence, with a genus and a differentia. (2) Write six test cases, three that should count and three that should not, each with a one-line reason for belonging on its side. (3) Mark each case as admitted or excluded by your definition. (4) Find the case that does the most damage, say whether the definition is too wide or too narrow, and rewrite the definition. (5) Run the same six cases against the new wording. (6) Finish with one paragraph naming a case you still cannot settle, and turn one vague word in your final definition into an operational one. Allow 30 to 45 minutes.

What to hand in: A one-page table of the six cases showing both versions of the definition and how each case fared, plus the closing paragraph. Save it on the page or download it.

Check your work against the rubric The first definition is one sentence that names a genus and a differentia. It is not an example, a list or a description of typical features. There are at least six test cases, three that should count and three that should not, and each has a one-line reason for belonging on its side. Each case is marked admitted or excluded, and at least one failure is classified as too wide or too narrow. The revised definition changes its wording in response to the named failure, and the same six cases are run against it again. The write-up names one case still unsettled, and one vague word in the final definition is restated as an operational definition (what is measured, where, and what result counts).

Your work Or a link to it (optional)

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5 Discuss: the seminar

Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.

Logic or dialectic · Round 1

Plato, Euthyphro, 2a-16a (the whole dialogue)Read the whole dialogue in one sitting. Give most attention to 5d-6e (Socrates asks for the one form that makes every pious act pious), 10a-11b (is the pious loved by the gods because it is pious?) and 15c-16a (the ending). The link is H. N. Fowler's translation on Perseus, which renders the key word (hosion) as 'holy'; many other translations say 'pious'. Benjamin Jowett's translation is also free at Project Gutenberg (ebook 1642).

Euthyphro is certain he knows what piety is, yet each answer he gives fails under Socrates' questions. What does the questioning accomplish, and would you call the ending a failure?

At 6d-e Socrates asks for the one form that makes every pious act pious, not a list of pious acts. Why is a list of examples not a definition?
At 10a Socrates asks whether the gods love the pious because it is pious, or whether it is pious because they love it. Why does the order of the explanation matter for the definition?
At 11b-d Euthyphro complains that whatever they propose moves about and will not stay where they put it, and each speaker suggests the other is the Daedalus who sets it moving. Whose fault is it: the definitions', the speaker's or the questioner's?

Keep notes on the seminars page →

Test a definition