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evaluate an existential 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 an existential 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 exists x P(x) is true when at least one domain object satisfies P. A witness need not be unique. With a finite domain a, b and c, the truth condition is P(a) or P(b) or P(c). One true disjunct suffices. To establish falsity, each candidate must fail; an unsuccessful check of only the first object is not a complete search.
Restricted existence uses conjunction. 'Some student submitted' becomes exists x (S(x) and U(x)), requiring one and the same object to be both a student and a submitter. An implication inside that existential would be too weak: a nonstudent would make S(x) -> U(x) true and could serve as a witness without any student submitting.
The phrases some and at least one do not mean some but not all in this formal setting. A universal can make the corresponding existential true in a nonempty domain. Everyday conversation sometimes suggests a stronger contrast, but that suggestion is not part of the existential quantifier's truth condition. If a problem needs both a witness and a non-witness, it must state both requirements.
Do not confuse a witness with proof that a familiar name denotes it. An existence statement may give no information identifying which object works. Later proof rules preserve that uncertainty through fresh local parameters rather than turning existence into a claim about an arbitrary favorite object.
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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (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.
Witnesses: 2. A witness is an object in the domain satisfying the formula inside the existential quantifier. Count the members of P's extension, since these and only these satisfy the property in this model. The witness is the object, not merely the printed name used to identify it. Several witnesses can support the same existential sentence; the quantifier does not say exactly one.
exists x P(x): T. The existential is true when the witness set is nonempty. Once a witness is found, no other object's failure can undo that existence claim in the fixed model. To show the existential false, however, every domain member must fail. This difference explains why a successful search can stop early while an unsuccessful search over an infinite domain cannot simply announce that nothing exists.
forall x P(x): T. Check all members separately before making a universal claim. Existential truth by itself does not settle this second question. A model with one successful object and one failing object satisfies the existential but refutes the universal. A model in which every object succeeds satisfies both. Record the actual model result rather than applying an invalid rule from some to all.
Remove one witness while leaving another in the domain and extension. Existence remains true, because it requires at least one rather than the original number. Removing the last witness makes the existential false. Conversely, adding a non-witness does not destroy an existing witness. This behavior is the reverse of a common mistake: people sometimes reject an existence claim by pointing to an object that lacks the property, which only addresses a universal claim.
The domain consists exactly of a, b, c; these names denote distinct objects. P is true exactly of a. Evaluate exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (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.
Witnesses: 1. A witness is an object in the domain satisfying the formula inside the existential quantifier.
exists x P(x): T. The existential is true when the witness set is nonempty.
forall x P(x): F. Check all members separately before making a universal claim.
Remove one witness while leaving another in the domain and extension. Existence remains true, because it requires at least one rather than the original number. Removing the last witness makes the existential false. Conversely, adding a non-witness does not destroy an existing witness. This behavior is the reverse of a common mistake: people sometimes reject an existence claim by pointing to an object that lacks the property, which only addresses a universal claim.
The domain consists exactly of a, b, c, d; these names denote distinct objects. P is true exactly of a, b, c. Evaluate exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (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.
Witnesses: 3. A witness is an object in the domain satisfying the formula inside the existential quantifier.
exists x P(x): T. The existential is true when the witness set is nonempty.
forall x P(x): F. Check all members separately before making a universal claim.
Remove one witness while leaving another in the domain and extension. Existence remains true, because it requires at least one rather than the original number. Removing the last witness makes the existential false. Conversely, adding a non-witness does not destroy an existing witness. This behavior is the reverse of a common mistake: people sometimes reject an existence claim by pointing to an object that lacks the property, which only addresses a universal claim.
An existential sentence records a lower bound of one witness. To say exactly one object has P, existence must be combined with uniqueness. One useful form says there is an x with P, and every y with P is identical to x. This allows the domain to contain many non-P objects while preventing a second distinct P-object.
To say at least two objects have P, require witnesses x and y, require P of both, and require x and y to be distinct. Without the final condition, both variables can take the same value. Variables are positions in an assignment, not tickets reserving different objects. A two-variable formula can therefore be true in a one-object domain.
The difference can be seen with a shelf containing three books. If only one book is borrowed, some book is borrowed is true and exactly one is borrowed is true. If a second distinct book is borrowed, the first sentence remains true while the exact-one sentence becomes false. At least two becomes true. The sentence's numerical force comes from its logical conditions, not from how many names happen to appear in its written form.
Existential introduction is correspondingly modest. Once P(a) is established, existence follows even if a is an alias for an object with another name. The step does not count a new object into being. It removes the need to specify which object witnesses the property. This is why a named fact can support an existential conclusion while an existential fact generally cannot identify a previously designated name as its witness.
A study room has four sockets in the declared search area. Let A mean available for the visiting group's equipment. A checked inspection finds that the first two are occupied, the third is available and the fourth is occupied. The existence claim that an available socket exists is true, with the third socket as a witness. The three occupied sockets do not count against that particular existential claim.
They do count against the stronger universal that every socket is available. Keeping the two claims separate helps the group decide what information it has. One available socket may be enough for one device, but it does not establish that four devices can all be connected. That practical requirement would need a different count or a formula expressing several distinct available objects.
If the third socket has not actually been checked, the first two failures and the fourth failure do not settle whether an available socket exists. They narrow the search without finishing it. A formal exercise supplies a complete extension so that the answer is determined; an incomplete inspection should be recorded as incomplete rather than silently treated as a complete model.
Finally, change the domain to sockets in a different room. The original witness no longer belongs to the new domain, so it cannot establish existence there. This does not make the original observation false. It shows why existence always has a range: some object in the declared collection satisfies the stated property. Writing both the range and the property makes the result useful to somebody who needs to act on it.
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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Use the declared objects and meanings throughout this calculation: Witnesses is the first requested result.
Determine the requested value: Witnesses.
2
A witness is an object in the domain satisfying the formula inside the existential quantifier.
Determine the requested value: exists x P(x).
T
The existential is true when the witness set is nonempty.
Determine the requested value: forall x P(x).
F
Check all members separately before making a universal claim.
Collect the results in the requested order.
2 / T / F
Each result belongs to its own entry: Witnesses; exists x P(x); 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Use the declared objects and meanings throughout this calculation: Witnesses is the first requested result.
Determine the requested value: Witnesses.
4
A witness is an object in the domain satisfying the formula inside the existential quantifier.
Determine the requested value: exists x P(x).
T
The existential is true when the witness set is nonempty.
Determine the requested value: forall x P(x).
T
Check all members separately before making a universal claim.
Collect the results in the requested order.
4 / T / T
Each result belongs to its own entry: Witnesses; exists x P(x); 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Use the declared objects and meanings throughout this calculation: Witnesses is the first requested result.
Determine the requested value: Witnesses.
0
A witness is an object in the domain satisfying the formula inside the existential quantifier.
Determine the requested value: exists x P(x).
F
The existential is true when the witness set is nonempty.
Determine the requested value: forall x P(x).
F
Check all members separately before making a universal claim.
Collect the results in the requested order.
0 / F / F
Each result belongs to its own entry: Witnesses; exists x P(x); forall x P(x).
Test which alteration would change the conclusion.
Remove one witness while leaving another in the domain and extension. Existence remains true, because it requires at least one rather than the original number. Removing the last witness makes the existential false. Conversely, adding a non-witness does not destroy an existing witness. This behavior is the reverse of a common mistake: people sometimes reject an existence claim by pointing to an object that lacks the property, which only addresses a universal claim.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: Witnesses.
3
A witness is an object in the domain satisfying the formula inside the existential quantifier.
Determine the requested value: exists x P(x).
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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
| Computed result | |
|---|---|
| Witnesses | |
| exists x P(x) | |
| 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Count the property witnesses.
b0
The requested entry concerns witnesses; retain its stated scope.
Ask whether the witness set is nonempty.
b1
The requested entry concerns exists x p(x); retain its stated scope.
Check whether the extension exhausts the domain.
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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Witnesses: b0
exists x P(x): 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Witnesses: b0
exists x P(x): 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Witnesses: b0
exists x P(x): b1
forall x P(x): b2
In a study-room search, objects are sockets and P means available for the group's equipment. 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Witnesses: b0
exists x P(x): 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 exists x P(x). Record the number of witnesses, the existential sentence's truth value, and whether forall x P(x) is also true (T or F).
Witnesses: b0
exists x P(x): b1
forall x P(x): b2
You can evaluate an existential 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
T
The existential is true when the witness set is nonempty.
14. Complete the next model audit, step 3
F
Check all members separately before making a universal claim.