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identify a permitted existential witness move
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 identify a permitted existential witness move, 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. |
Existential elimination reasons from an unspecified witness without pretending to identify it. From exists x P(x), open a local assumption P(c) with a fresh parameter c. Derive a conclusion that does not contain c, then close the subproof and export that conclusion. The parameter must not appear in the relevant outside assumptions or in the exported result.
Freshness prevents extra facts about an old name from being attached to the witness. If a already denotes a particular person known to lack Q, choosing a as the witness to exists x P(x) could create a false contradiction. The existential only says somebody has P; it never said that particular person does. A new local parameter avoids that unsupported identification.
Combine this rule with a universal conditional. Assume P(c) locally, instantiate forall x (P(x) -> Q(x)) at c, and derive Q(c). Existential introduction then gives exists x Q(x). That last sentence has no c, so it can be exported when the witness subproof closes. Q(c) itself cannot be exported merely because it appeared inside the subproof.
The course's proof audits require the witness assumption, local result and parameter-free conclusion in canonical notation. They assess the permission conditions explicitly rather than using a propositional proof widget to pretend to verify arbitrary quantified proofs. Draw a box around local lines on paper and inspect which names remain in the result after the box is closed.
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.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name c; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
The first premise commits the proof to a nonempty P-extension, but it leaves the identities of its members unspecified. Naming a convenient object from an earlier problem would add an unsupported commitment. The fresh parameter stands for a witness under the local assumption rather than for a specially selected favorite object.
The universal conditional constrains every P-object to have Q. Consequently any permitted witness to P also witnesses Q. This semantic picture explains why the proof can export an existential conclusion even though it cannot export the fresh named instance. The conclusion captures what is shared by every admissible choice of witness.
Check the undischarged assumptions at the point the witness subproof is opened. If the supposedly fresh name already appears in an outside assumption, its use may smuggle in information about a particular object. A correct-looking local sequence does not repair a violated freshness condition; scope is part of the rule, not a decorative indentation convention.
Local witness assumption: P(c). Open a temporary subproof using a fresh parameter for an unspecified witness. The existential premise guarantees that some object has P; it does not tell us that a previously designated object has P. Freshness prevents the witness from inheriting facts already attached to an old name. The assumption remains local and cannot simply be copied outside the subproof as a named conclusion.
Local consequence: Q(c). Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens. This derives Q for that same unspecified witness. No additional property of the witness has been assumed. The argument would work for whichever object fulfilled the existential premise, which is the reason the witness's particular identity does not matter to the eventual exported conclusion.
Exported conclusion: ∃x Q(x). Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination. The exported sentence contains no occurrence of the fresh parameter. This is essential: the result must not depend on knowing which object the witness is. The global conclusion states only that a Q-object exists, not that an independently named object has Q or that every object does.
Try to export Q of the fresh name instead. That proposed conclusion violates the restriction because it identifies the result with the local parameter. To see the problem, take a model with one P-object and another object lacking P and Q. The existential premise and universal conditional can both be true while the independently named second object lacks Q. Exporting the existential avoids that illicit identification.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name d; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
The first premise commits the proof to a nonempty P-extension, but it leaves the identities of its members unspecified. Naming a convenient object from an earlier problem would add an unsupported commitment. The fresh parameter stands for a witness under the local assumption rather than for a specially selected favorite object.
The universal conditional constrains every P-object to have Q. Consequently any permitted witness to P also witnesses Q. This semantic picture explains why the proof can export an existential conclusion even though it cannot export the fresh named instance. The conclusion captures what is shared by every admissible choice of witness.
Check the undischarged assumptions at the point the witness subproof is opened. If the supposedly fresh name already appears in an outside assumption, its use may smuggle in information about a particular object. A correct-looking local sequence does not repair a violated freshness condition; scope is part of the rule, not a decorative indentation convention.
Local witness assumption: P(d). Open a temporary subproof using a fresh parameter for an unspecified witness.
Local consequence: Q(d). Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
Exported conclusion: ∃x Q(x). Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.
Try to export Q of the fresh name instead. That proposed conclusion violates the restriction because it identifies the result with the local parameter. To see the problem, take a model with one P-object and another object lacking P and Q. The existential premise and universal conditional can both be true while the independently named second object lacks Q. Exporting the existential avoids that illicit identification.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name e; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
The first premise commits the proof to a nonempty P-extension, but it leaves the identities of its members unspecified. Naming a convenient object from an earlier problem would add an unsupported commitment. The fresh parameter stands for a witness under the local assumption rather than for a specially selected favorite object.
The universal conditional constrains every P-object to have Q. Consequently any permitted witness to P also witnesses Q. This semantic picture explains why the proof can export an existential conclusion even though it cannot export the fresh named instance. The conclusion captures what is shared by every admissible choice of witness.
Check the undischarged assumptions at the point the witness subproof is opened. If the supposedly fresh name already appears in an outside assumption, its use may smuggle in information about a particular object. A correct-looking local sequence does not repair a violated freshness condition; scope is part of the rule, not a decorative indentation convention.
Local witness assumption: P(e). Open a temporary subproof using a fresh parameter for an unspecified witness.
Local consequence: Q(e). Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
Exported conclusion: ∃x Q(x). Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.
Try to export Q of the fresh name instead. That proposed conclusion violates the restriction because it identifies the result with the local parameter. To see the problem, take a model with one P-object and another object lacking P and Q. The existential premise and universal conditional can both be true while the independently named second object lacks Q. Exporting the existential avoids that illicit identification.
Consider premises exists x Packed(x) and forall x (Packed(x) -> Labeled(x)). The goal is exists x Labeled(x). Open a temporary witness scope with Packed(c), using c as a fresh name for a witness whose identity is otherwise unspecified. Instantiate the universal premise to obtain Packed(c) -> Labeled(c), then derive Labeled(c). Existential introduction now yields exists x Labeled(x), a conclusion that no longer names c. The witness scope can close with that identity-independent result.
The proof did not establish that a particular previously named parcel was packed. If d already names the parcel on the desk, replacing c by d without justification would assume the existential witness is that parcel. A model with a packed parcel in a cupboard and an unpacked desk parcel shows why this step fails. Existence supplies some qualifying object, not whichever object is convenient for the conclusion.
Nor may Labeled(c) simply escape the temporary scope as a permanent fact about an independently identified object. The fresh name was a device for reasoning about an unspecified witness. The conclusion carried outside must respect the rule's restrictions, including independence from that fresh name and from the temporary witness assumption. An existential conclusion is often the natural way to express exactly what survived without pretending to know who the witness was.
Fresh names also do not automatically assert distinctness. Two existential premises, exists x Packed(x) and exists x Insured(x), might concern the same parcel or different parcels. Giving their temporary witnesses different names does not add an inequality premise. To establish that two distinct objects exist, the reasoning must include an appropriate inequality condition or other evidence forcing distinctness.
Audit an existential argument by tracking both properties and dependencies. Which premise supplied existence? Which fresh assumption represented the witness? Which statements depended on that scope? What conclusion remained after the name disappeared? This record separates a legitimate use of an unspecified witness from an unsupported identification or a leaked assumption.
An event's premises say that at least one person is registered and that every registered person has a badge. The conclusion sought is that at least one person has a badge. We do not know the registrant's name, and the argument does not need it.
Introduce a fresh local label c for an unspecified registrant. Under that local assumption, the universal rule gives a conditional from c's registration to c's badge. Registration then yields the badge fact for c. From that local badge fact, introduce the existential statement that somebody has a badge. Close the local witness argument and retain only this parameter-free statement.
Now compare an invalid shortcut: 'Someone is registered; therefore the receptionist is registered.' Nothing in the premise identifies its witness with the receptionist. Adding a familiar role name has introduced information that was never supplied. Even if the receptionist happens to be registered in the actual event, the proposed inference does not guarantee it across models of the premise.
The same restriction applies to exporting c. The local label was a reasoning device for whatever witness exists, not a new globally identified person. A record written after the proof should say 'A badge-holder exists', not 'The person named c in our permanent register has a badge'. The proof's discipline is useful precisely because it preserves what remains unknown while still deriving what the premises guarantee. Existence can support a meaningful conclusion without naming, locating or counting every witness.
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.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name f; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Use the declared objects and meanings throughout this calculation: Local witness assumption is the first requested result.
Determine the requested value: Local witness assumption.
P(f)
Open a temporary subproof using a fresh parameter for an unspecified witness.
Determine the requested value: Local consequence.
Q(f)
Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
Determine the requested value: Exported conclusion.
∃x Q(x)
Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.
Collect the results in the requested order.
P(f) / Q(f) / ∃x Q(x)
Each result belongs to its own entry: Local witness assumption; Local consequence; Exported conclusion.
Record the interpretation and the question.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name g; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Use the declared objects and meanings throughout this calculation: Local witness assumption is the first requested result.
Determine the requested value: Local witness assumption.
P(g)
Open a temporary subproof using a fresh parameter for an unspecified witness.
Determine the requested value: Local consequence.
Q(g)
Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
Determine the requested value: Exported conclusion.
∃x Q(x)
Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.
Collect the results in the requested order.
P(g) / Q(g) / ∃x Q(x)
Each result belongs to its own entry: Local witness assumption; Local consequence; Exported conclusion.
Record the interpretation and the question.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name h; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Use the declared objects and meanings throughout this calculation: Local witness assumption is the first requested result.
Determine the requested value: Local witness assumption.
P(h)
Open a temporary subproof using a fresh parameter for an unspecified witness.
Determine the requested value: Local consequence.
Q(h)
Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
Determine the requested value: Exported conclusion.
∃x Q(x)
Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.
Collect the results in the requested order.
P(h) / Q(h) / ∃x Q(x)
Each result belongs to its own entry: Local witness assumption; Local consequence; Exported conclusion.
Test which alteration would change the conclusion.
Try to export Q of the fresh name instead. That proposed conclusion violates the restriction because it identifies the result with the local parameter. To see the problem, take a model with one P-object and another object lacking P and Q. The existential premise and universal conditional can both be true while the independently named second object lacks Q. Exporting the existential avoids that illicit identification.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: Local witness assumption.
P(i)
Open a temporary subproof using a fresh parameter for an unspecified witness.
Determine the requested value: Local consequence.
Determine the requested value: Exported conclusion.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name j; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
| Computed result | |
|---|---|
| Local witness assumption | |
| Local consequence | |
| Exported conclusion |
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name n; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Use the fresh parameter only inside its local assumption.
b0
The requested entry concerns local witness assumption; retain its stated scope.
Apply the universal conditional to the local witness.
b1
The requested entry concerns local consequence; retain its stated scope.
Export a sentence without the fresh parameter.
b2
The requested entry concerns exported conclusion; retain its stated scope.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name k; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Local witness assumption: b0
Local consequence: b1
Exported conclusion: b2
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name l; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Local witness assumption: b0
Local consequence: b1
Exported conclusion: b2
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name m; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Local witness assumption: b0
Local consequence: b1
Exported conclusion: b2
For an event register, P means registered and Q means has a badge; the argument must preserve the registrant's unknown identity. The data below form the complete invented audit. Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name o; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Local witness assumption: b0
Local consequence: b1
Exported conclusion: b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Premises: exists x P(x); forall x (P(x) -> Q(x)). In an existential-elimination subproof choose the fresh name p; it occurs in no premise or undischarged assumption. Fill the witness assumption, the named Q-result inside the subproof, and the existential sentence exported after closing it. Use P(c), Q(c), ∃x Q(x) style notation, substituting the stated fresh name. The symbol ∃ is the existential quantifier: there is at least one. Enter ∃x Q(x) for the exported formula.
Local witness assumption: b0
Local consequence: b1
Exported conclusion: b2
You can identify a permitted existential witness move. 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
Q(i)
Instantiate the universal conditional at the fresh parameter and use the local P-assumption with modus ponens.
14. Complete the next model audit, step 3
∃x Q(x)
Introduce an existential conclusion from the local Q-result, then close the witness subproof with existential elimination.