Build a counterexample model
Test whether a conclusion is forced by premises by trying to build one valid counterexample.
Adult planning suggestion: about 9–12 · Logic & rule checking · Language & storytelling · Cooperation & communication
- Players
- Groups of 3
- Time
- 15–20 minutes
- Materials and quantities
- Three object rows, P/Q yes–no property cards, three premise/conclusion cards and a model table.
Thirty pupils: ten trios; builder and the two separate checkers rotate each card.
Prepare
The universe is exactly three fictional objects numbered 1–3. Each may have property P, property Q, both or neither. All P are Q means every P object also has Q; it does not assert that any P exists. Some means at least one of these three, and no P are Q means no object has both. Do not use pupil traits, gender, ability or social groups as the properties.
Play
For each card, first read the premises and proposed conclusion aloud. To show the conclusion is not forced, assign P/Q values to the three objects so every premise is true and the conclusion is false. Check the premises separately before announcing a counterexample. To say it is forced requires a reason that no such counterexample can satisfy the stated premises, not merely failing to find one quickly.
Worked example
Card 1 gives All P are Q and Some Q exist, proposing All Q are P. Object 1 can have P and Q, object 2 Q only and object 3 neither. The premises hold but object 2 refutes the conclusion. Card 2 gives No P are Q and Some P exist, proposing Some P are not Q; that conclusion is forced because an existing P must lack Q. Card 3 gives only All P are Q, proposing Some P are Q; three objects with neither property form a counterexample.
Finish and check
Finish after recording a valid model or a short necessity explanation for each of the three cards. The teacher key checks all sixty-four P/Q assignments for this three-object universe. Learners do not have to enumerate them, and the finite check is not a proof of every statement over arbitrary infinite universes. Keep the difference between false in one model and impossible under the premises explicit.
Adaptation and practical limits
Thirty pupils: ten trios as model builder, premise checker and conclusion checker. Adult planning suggestion is about 9–12 with substantial language support; the philosophy reference is formal background, not a child-tested lesson. Use the supplied concrete three-row table and allow oral explanations. Never assess a child’s real-life honesty, personality or diagnostic ability from a symbolic counterexample.
Aim: Test whether a conclusion is forced by premises by trying to build one valid counterexample.
Practice within this activity: Keep all premises true while testing a conclusion rather than changing the rules.
Adult observation: Ask which object refutes which universal claim and whether existence was stated.
Mechanism and adaptation: Our version uses finite-model logical entailment and quantified counterexamples. Examples and sheets are separately written; the earliest inventor is not established here.
Reference checked 25 October 2026: Stanford Encyclopedia of Philosophy · Quantifiers and Quantification — Formal quantifier and satisfaction background. Our three-object tasks and finite model enumeration are separate; the source is not a child-validated activity.
Goals describe observable activity choices, not measured long-term gains or institutional endorsement.
Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Player logical task cards
Exactly3 fictional objects. Some=at least1. All does not by itself assert existence.
Player three model tables
Card 1
| Object | P? | Q? |
|---|---|---|
| 1 | ____ | ____ |
| 2 | ____ | ____ |
| 3 | ____ | ____ |
Card 2
| Object | P? | Q? |
|---|---|---|
| 1 | ____ | ____ |
| 2 | ____ | ____ |
| 3 | ____ | ____ |
Card 3
| Object | P? | Q? |
|---|---|---|
| 1 | ____ | ____ |
| 2 | ____ | ____ |
| 3 | ____ | ____ |
Host / teacher checked example · separate
Thirty pupils: ten trios; builder and the two separate checkers rotate each card.
Three object rows, P/Q yes–no property cards, three premise/conclusion cards and a model table.
Keep this answer page separate. Read the full web rules and preview materials before leading; allow pauses, passing and supported writing.
Card1 is not forced: 1 both,2 Q-only,3 neither. Card2 is forced: an existing P must lack Q. Card3 is not forced: all3 neither. Check premises before checking the conclusion.
| Card | Premise-satisfying models /64 | Of these, conclusion false |
|---|---|---|
| 1 | 26 | 19 |
| 2 | 19 | 0 |
| 3 | 27 | 8 |
New premise audits in the same three-object universe · separate player exercise
| Card | Object1 | Object2 | Object3 | Forced? |
|---|---|---|---|---|
| ________ | ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ | ________ |
New premise audits in the same three-object universe · host check
Card1 forced: any P would be Q, contradicting no Q. Card2 not forced: object1 has P/Q, object2 P only, object3 neither. Card3 not forced: object1 Q only and the others neither. Finite64-model checks confirm these exact statements, not arbitrary domains.
Related activities
New premise audits in the same three-object universe
The original three cards remain unchanged. For each separate card below, check premises before the conclusion; use exactly three fictional objects and the same P/Q meanings. A proposed counterexample outside the premises is not valid evidence.
Build a Counterexample Without Breaking the Premises
Write the target conclusion separately
A model that makes a conclusion false is only useful if it still obeys every premise. In Build a counterexample model, keep one box for the premises and another for the proposed conclusion. The universe contains exactly three fictional objects. Each object can have P, Q, both or neither. Read “some” as at least one and read “all P are Q” as excluding any P-only object. These conventions matter more than inventing a persuasive story. Stay with the fictional labels instead of assigning properties to classmates or real social groups.
Start with the object that would refute the claim
For the first printed card, the proposed conclusion says that every Q object has P. To make that false, you need an object with Q but without P. Give object 1 Q only. Leave objects 2 and 3 with neither property. Now examine the two premises rather than assuming the work is finished. A single violating object refutes a universal conclusion, but the other constraints still decide whether that object is permitted in the model. Finding a counterexample is a constrained construction, not permission to make every statement in the exercise false.
Check both premises in the fresh model
The first premise, all P are Q, holds because there are no P objects that lack Q. The second premise, some Q exist, holds because object 1 has Q. The conclusion, all Q are P, fails at object 1. This is another valid counterexample for the supplied card, different from the host example that includes an object with both properties. The ANU logic glossary explains the importance of true premises and a false conclusion when testing an argument form. Our three-row assignment is a separate concrete exercise.
Reject a model that breaks an assumption
Suppose someone instead gives object 1 P only and gives the other two objects neither property. The target conclusion is true in that model because there are no Q objects to violate it. More importantly, both printed premises fail: the P-only object violates all P are Q, and no Q exists. This is not a near-success that needs a more confident explanation. It is a model outside the permitted cases. Write which premise fails before repairing it. Keeping separate truth checks prevents a dramatic-looking exception from being mistaken for evidence about the original question.
Distinguish an example from a necessity claim
A model where all three objects have both P and Q makes the first card's premises and conclusion true. That establishes that the conclusion can hold, not that it must hold. The Q-only model already shows it is not forced. For the second printed card, however, an existing P object must lack Q because the premises forbid overlap. That gives a reason for the proposed conclusion. Simply trying several arrangements and failing to find a counterexample would be weaker. The distinction resembles checking existence versus a stronger puzzle claim, but here the premises define which models count.
Announce any extra premise before using it
Consider an optional new question: all P are Q, and some P exist; does some Q exist follow? Yes: choose the P object whose existence was promised, and the first premise gives it Q. This question adds an existence assumption that the third printed card does not contain. Do not quietly import it when judging the original card. If you want to explore the variation, write it on a separate slip and label the new premise. Keeping changes visible lets participants see exactly why adding information can change what follows.
End with an inspectable assignment
Ask the builder to name each object's properties, the premise checker to test every assumption, and the conclusion checker to identify the decisive object or necessity reason. Rotate those jobs for the next card. A table with one valid counterexample is a complete rejection of “forced” for that card. A finite enumeration can check the supplied three-object exercise, but does not establish every result over unrestricted domains. Record that boundary beside any shared answer. The finish is an explanation another group can verify, rather than an assessment of someone's honesty, personality or general reasoning ability.