Round A uses P → Q and P to conclude Q; the full four-row table has no counterexample. Round B uses P → Q and Q to conclude P; FT refutes it. The HTML checks a selected row and separately computes the complete table under the material-conditional convention.
Work through the example
Select Round B, enter FT, and Check row. P is false, Q true, and P → Q true. Both premises are true while conclusion P is false. TF does not qualify because the conditional premise is false. Review all rows reports Round B invalid from FT; it does not infer validity from one successful row.
Start with the valuation cards before revealing the complete table. For each failed counterexample, ask whether at least one premise is false or the conclusion is true. In Round A, TT fails because the conclusion is true even though both premises are true. Validity concerns every permitted valuation; it does not establish that a real card has a star or is blue. Review written reasons separately from the fixed Boolean checks so a symbolic result is not mistaken for a mastery judgment.
State the propositions and the convention
Use two original propositions about a hypothetical card:
- P: “The fictional card has a star.”
- Q: “The fictional card is blue.”
The game uses the material conditional P → Q. Under that convention, the conditional is false only when P is true and Q is false. It is not a claim that a star causes blue coloring, a promise about a future card, or an observation of physical tokens.
The Oklahoma State hosted truth-table chapter explains complete truth-value checking and argument validity: a counterexample has all premises true and the conclusion false. We use those concepts with our own star/blue fixture, not its diagrams or worked stories. Truth tables and validity
Prepare four hypothetical valuation cards, with each ID encoding P then Q:
| Card ID | P | Q | P → Q |
|---|---|---|---|
| TT | True | True | True |
| TF | True | False | False |
| FT | False | True | True |
| FF | False | False | True |
The two letters represent truth values, not a learner's initials. No card is an actual observation. All four rows are available because the fixture imposes no additional relation between star and color.
Round A: Keep both premises visible
Round A has premises P → Q and P, with conclusion Q.
Read it as: suppose the conditional holds, and suppose the card has a star. Does Q follow in every case where both premises hold?
| Valuation | Conditional premise | P premise | Both premises true? | Q conclusion | Counterexample? |
|---|---|---|---|---|---|
| TT | True | True | Yes | True | No |
| TF | False | True | No | False | No |
| FT | True | False | No | True | No |
| FF | True | False | No | False | No |
TT is the only row where both premises are true, and Q is true there. None of the four rows has true premises and a false conclusion. Round A is valid under the stated truth-functional convention.
TF does have a false Q, but its conditional premise is also false. It therefore cannot refute Round A. FF has a false Q too, but P is false, so it also fails the counterexample test.
Ask the learner to identify why each attempted counterexample fails: a false premise or a true conclusion. TT illustrates the second reason. That explanation is more precise than saying “the row is wrong.”
Round B: The reverse inference needs its own test
Round B keeps premise P → Q but uses Q as its second premise and P as the conclusion.
Recreate this example
Build one self-contained responsive HTML example titled “Logic classroom game: find a counterexample” for readers completing the stated task. Return the complete HTML with embedded CSS and JavaScript. Use no remote fonts, libraries, images, requests, account connections, or saved-state claims.
Use P=fictional card has star, Q=blue. Conditional !P||Q. Round A premises conditional,P conclusion Q. B premises conditional,Q conclusion P. Round A/B buttons change formulas and clear feedback. Valuation input accepts TT/TF/FT/FF and starts blank. Check row reports premise/conclusion booleans and counterexample condition; missing/invalid rows prompt selection. Review all rows calculates four unique rows and reports A valid, B invalid with FT counterexample. Reset returns A and blank row. Explain that a row fails as a counterexample if a premise is false or the conclusion is true; Round A TT has true premises and a true conclusion. Do not grade arbitrary prose.
Use a truth-table laboratory with formulas and complete evaluated row ledger design with a compact header. Keep the actual sample data, controls, and useful default result visible early on a 360px-wide screen. Use readable text at least 16px, generous paragraph spacing, strong contrast, semantic labels, visible keyboard focus, and wrapping source blocks. Stack records on narrow screens rather than forcing a wide table.
Wire each action to an explicit type=button click handler using element selectors; the host permits scripts but blocks form submission. Update the visible result when the described action occurs. Show specific error feedback for invalid or missing input and clear stale feedback after dependent inputs change. Use plain text when displaying user values.
This bounded symbolic checker does not establish real-card observations or learner mastery. Keep the source fixture and worked teacher or developer example distinct from actual user records. Include the domain-specific explanation and next check beside the result. Provide no fake download, send, deployment, approval, or persistence button. Everything operates locally for the current page session.
Review the prepared copy in NOS
Use FT as the debrief question: how can both premises remain true when P is false? A prepared HTML explanation in NOS can keep that row beside the formulas. Revise eligible static teaching text visually, then retain the learner’s own rationale in the worksheet.