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evaluate a universal claim in a finite declared domain
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will evaluate a universal claim in a finite declared domain, recording the intermediate model values and the precise reason each conclusion follows.
Recall the truth conditions for not, and, or and if-then. Those rules still govern compound formulas here, but atomic truth now comes from objects and predicate extensions or from a world's valuation. Identify which new structure this lesson introduces before using a familiar propositional rule.
| Term | What it means |
|---|---|
| Interpretation | A declared domain and meanings for the nonlogical symbols; a modal interpretation also specifies worlds, accessibility and valuations. |
| Assignment | A choice of domain object for a free variable during an evaluation; it is not itself another domain object. |
| Witness | An eligible object or accessible world satisfying the property required by an existential or possibility claim. |
| Counterexample | An eligible case where the required condition fails; a countermodel to an inference additionally makes every premise true. |
| Validity | Truth in every interpretation of the specified kind, a stronger claim than truth in one supplied model. |
The sentence forall x P(x) requires P of every object in the domain. In a finite domain named a, b and c, its truth condition can be displayed as P(a) and P(b) and P(c), provided the list really covers every object. One false conjunct makes the universal false. A true instance by itself is insufficient; it tells us about one object rather than the whole range of the quantifier.
Universal statements can express restricted claims. If the domain contains people and S means student, 'Every student submitted' is represented by forall x (S(x) -> U(x)). A nonstudent makes the conditional true through its false antecedent. This does not claim that everybody is a student or that everybody submitted. By contrast, forall x (S(x) and U(x)) would say both of those things about every domain member.
The distinction explains vacuous truth. If no object is a student, there is no counterexample to the claim that every student submitted. The conditional universal is true in that interpretation, but it does not establish that a student exists. First-order logic normally assumes a nonempty domain; an empty extension for one predicate is entirely compatible with that assumption. Do not confuse an empty class inside the domain with an empty domain.
Another way: An explicit audit sheet
Keep four parts on the page: the declared objects or worlds, the meaning of each symbol, the intermediate values, and the conclusion. A changed interpretation belongs on a new sheet so the premises and conclusion are never checked in different models.
The domain consists exactly of a, b; these names denote distinct objects. P is true exactly of a, b. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
For object a, P(a) is T. The extension explicitly includes a, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
For object b, P(b) is T. The extension explicitly includes b, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
Instances checked: 2. A universal quantifier ranges over every member of the declared domain. Give each object a turn as the variable's value; do not count only the objects that already satisfy the predicate. Checking the extension alone can hide the objects outside it, which are exactly the objects capable of refuting this universal sentence. The finite domain lets us finish this complete inspection.
False instances: 0. Compare the full domain with the predicate's extension. Each domain member outside the extension makes its corresponding instance false. These objects are counterexamples to the universal claim, not exceptions permitted by the quantifier. The statement says every object, so even one failure is enough. Several failures still refute one sentence; they do not make it a different kind of quantifier.
forall x P(x): T. The universal is true exactly when the false-instance count is zero. This condition is stricter than most instances being true and different from at least one being true. In a nonempty domain, a true universal supplies a true instance for any declared name. The calculation establishes truth in this model; it does not establish validity across interpretations with different extensions.
Add an object to the domain without adding it to P's extension. The expanded model now has a false instance, so a previously true universal becomes false. Adding a P-object instead preserves truth. This is why the domain must remain explicit when somebody says that all objects have a property. A claim about all records in today's checked file is not automatically a claim about future records or objects outside that file.
The domain consists exactly of a, b, c; these names denote distinct objects. P is true exactly of a. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
For object a, P(a) is T. The extension explicitly includes a, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
For object b, P(b) is F. The complete extension omits b, so this named object does not satisfy P. Omission here means false because the interpretation is complete; an incomplete real-world record would not license the same assumption.
For object c, P(c) is F. The complete extension omits c, so this named object does not satisfy P. Omission here means false because the interpretation is complete; an incomplete real-world record would not license the same assumption.
Instances checked: 3. A universal quantifier ranges over every member of the declared domain.
False instances: 2. Compare the full domain with the predicate's extension.
forall x P(x): F. The universal is true exactly when the false-instance count is zero.
Add an object to the domain without adding it to P's extension. The expanded model now has a false instance, so a previously true universal becomes false. Adding a P-object instead preserves truth. This is why the domain must remain explicit when somebody says that all objects have a property. A claim about all records in today's checked file is not automatically a claim about future records or objects outside that file.
The domain consists exactly of a, b, c, d; these names denote distinct objects. P is true exactly of a, b, c. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
For object a, P(a) is T. The extension explicitly includes a, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
For object b, P(b) is T. The extension explicitly includes b, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
For object c, P(c) is T. The extension explicitly includes c, so this named object satisfies P. This is a membership fact about one object; it does not by itself say that another object satisfies P.
For object d, P(d) is F. The complete extension omits d, so this named object does not satisfy P. Omission here means false because the interpretation is complete; an incomplete real-world record would not license the same assumption.
Instances checked: 4. A universal quantifier ranges over every member of the declared domain.
False instances: 1. Compare the full domain with the predicate's extension.
forall x P(x): F. The universal is true exactly when the false-instance count is zero.
Add an object to the domain without adding it to P's extension. The expanded model now has a false instance, so a previously true universal becomes false. Adding a P-object instead preserves truth. This is why the domain must remain explicit when somebody says that all objects have a property. A claim about all records in today's checked file is not automatically a claim about future records or objects outside that file.
A complete finite inspection can establish a universal for its declared model. If the domain has exactly three objects and each satisfies P, there is no remaining object in that domain waiting to be checked. The qualification exactly is doing important work. If the three objects are only a sample from a larger collection, checking them establishes only their instances.
The number of successes is not the logical source of the universal conclusion. Completeness of coverage is. Checking three of three objects establishes the finite universal; checking a thousand of a million does not establish it by enumeration. Other evidence or a general argument may support a broader claim, but it is a different method with a different justification.
A proof using an arbitrary object can avoid enumeration. Suppose an argument takes an unspecified member of the domain and establishes P without relying on special information about that member. Under appropriate proof rules, the result can be generalized. By contrast, selecting Ada because she has a special badge cannot justify a conclusion about everyone. Arbitrary does not mean a person chosen at random and then treated as typical.
When reading a universal conditional, inspect the qualifying class as well as the property. All P-objects are Q-objects is true exactly when P's extension is contained in Q's extension. The P-extension may be empty, equal to Q, or a proper subset. None of these possibilities is determined by the words all P are Q alone. A set picture shows the permitted arrangements while keeping existential information separate from the inclusion claim.
A depot audit has a declared domain of six parcels currently on one trolley. Let L mean has a readable label. Five parcels have readable labels and one has a torn, unreadable label. The claim forall x L(x) is false. The five successes are useful operational information, but they do not rescue a claim that covers every parcel. The unlabelled parcel is a counterexample within the declared domain.
An accurate report can name the counterexample and request a replacement label. Once it is replaced and rechecked, the same universal may become true in the updated model. The earlier report was not wrong about the earlier state; the interpretation has changed over time. Record the time of the audit rather than treating a universal result as permanent.
Suppose the rule instead says that every refrigerated parcel has a temperature label. Define R for refrigerated and T for temperature-labeled. The formula is forall x (R(x) -> T(x)). A non-refrigerated parcel without a temperature label is not a counterexample, because it does not satisfy the restriction. A refrigerated parcel lacking that label is a counterexample because its antecedent is true and its consequent false.
This distinction prevents two practical mistakes. Inspecting only the labeled parcels would hide failures. Requiring the special label on every parcel would expand the rule beyond its stated class. The method is to list the full domain, identify the restricted objects when there are any, and check the relevant condition for each of them. The audit establishes the bounded rule's status for this trolley, not for every parcel at every depot.
A correct evaluation answers the stated question for its stated interpretation. Do not turn a true instance into a universal rule or a successful example into a proof of validity. When the task is a proof audit, keep local assumptions and fresh parameters within their declared scope.
Record the interpretation and the question.
The domain consists exactly of a, b, c; these names denote distinct objects. P is true exactly of a, b. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Use the declared objects and meanings throughout this calculation: Instances checked is the first requested result.
Determine the requested value: Instances checked.
3
A universal quantifier ranges over every member of the declared domain.
Determine the requested value: False instances.
1
Compare the full domain with the predicate's extension.
Determine the requested value: forall x P(x).
F
The universal is true exactly when the false-instance count is zero.
Collect the results in the requested order.
3 / 1 / F
Each result belongs to its own entry: Instances checked; False instances; forall x P(x).
Record the interpretation and the question.
The domain consists exactly of a, b, c, d; these names denote distinct objects. P is true exactly of a, b, c, d. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Use the declared objects and meanings throughout this calculation: Instances checked is the first requested result.
Determine the requested value: Instances checked.
4
A universal quantifier ranges over every member of the declared domain.
Determine the requested value: False instances.
0
Compare the full domain with the predicate's extension.
Determine the requested value: forall x P(x).
T
The universal is true exactly when the false-instance count is zero.
Collect the results in the requested order.
4 / 0 / T
Each result belongs to its own entry: Instances checked; False instances; forall x P(x).
Record the interpretation and the question.
The domain consists exactly of a, b, c, d, e; these names denote distinct objects. P is true exactly of none. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Use the declared objects and meanings throughout this calculation: Instances checked is the first requested result.
Determine the requested value: Instances checked.
5
A universal quantifier ranges over every member of the declared domain.
Determine the requested value: False instances.
5
Compare the full domain with the predicate's extension.
Determine the requested value: forall x P(x).
F
The universal is true exactly when the false-instance count is zero.
Collect the results in the requested order.
5 / 5 / F
Each result belongs to its own entry: Instances checked; False instances; forall x P(x).
Test which alteration would change the conclusion.
Add an object to the domain without adding it to P's extension. The expanded model now has a false instance, so a previously true universal becomes false. Adding a P-object instead preserves truth. This is why the domain must remain explicit when somebody says that all objects have a property. A claim about all records in today's checked file is not automatically a claim about future records or objects outside that file.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: Instances checked.
4
A universal quantifier ranges over every member of the declared domain.
Determine the requested value: False instances.
Determine the requested value: forall x P(x).
The domain consists exactly of a, b, c, d, e; these names denote distinct objects. P is true exactly of a, b, c, d. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
| Computed result | |
|---|---|
| Instances checked | |
| False instances | |
| forall x P(x) |
The domain consists exactly of a, b, c, d, e, f, g; these names denote distinct objects. P is true exactly of a, b, c, d, e, f. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Count the entire declared domain.
b0
The requested entry concerns instances checked; retain its stated scope.
Count objects outside the extension.
b1
The requested entry concerns false instances; retain its stated scope.
Ask whether there is any failing instance.
b2
The requested entry concerns forall x p(x); retain its stated scope.
The domain consists exactly of a, b, c, d, e, f; these names denote distinct objects. P is true exactly of a, b, c, d, e, f. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Instances checked: b0
False instances: b1
forall x P(x): b2
The domain consists exactly of a, b, c, d, e; these names denote distinct objects. P is true exactly of a, b, c, d. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Instances checked: b0
False instances: b1
forall x P(x): b2
The domain consists exactly of a, b, c, d, e, f; these names denote distinct objects. P is true exactly of a, b, c, d, e. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Instances checked: b0
False instances: b1
forall x P(x): b2
In a parcel audit, objects are parcels and P means has a readable label. The data below form the complete invented audit. The domain consists exactly of a, b, c, d, e, f; these names denote distinct objects. P is true exactly of a, b, c, d, e, f. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Instances checked: b0
False instances: b1
forall x P(x): b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The domain consists exactly of a, b, c, d, e, f, g; these names denote distinct objects. P is true exactly of a, b, c, d, e, f. Evaluate forall x P(x). Record the number of instances checked, the number of false instances, and the universal sentence's truth value (T or F).
Instances checked: b0
False instances: b1
forall x P(x): b2
You can evaluate a universal claim in a finite declared domain. Reconstruct the three audit entries from a fresh model without consulting the examples; explain what change to the interpretation would change one answer.
14. Complete the next model audit, step 2
1
Compare the full domain with the predicate's extension.
14. Complete the next model audit, step 3
F
The universal is true exactly when the false-instance count is zero.