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translate a negated quantified claim
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 translate a negated quantified claim, 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. |
Not every object has P exactly when some object lacks P. In symbols, not(forall x P(x)) and exists x not P(x) have the same truth conditions. The first denies a universal success; the second supplies its counterexample. A mixed domain is the best diagnostic case: some objects have P and others do not. Both expressions are true there.
No object has P exactly when every object lacks P. Thus not(exists x P(x)) is equivalent to forall x not P(x). This second pair is false in a mixed domain because at least one object does have P. Comparing the two pairs prevents the common jump from not all to none. A single counterexample refutes all; it does not refute every possible witness.
Negation applies to the formula within its scope. In not(forall x (P(x) -> Q(x))), pushing the negation inward first changes forall to exists, then negates the conditional. The result is exists x (P(x) and not Q(x)). It names an object with the antecedent property but without the consequent property. This is exactly what a counterexample to an all-P-are-Q claim requires.
Work from the outside in, one operation at a time. Do not swap quantifiers and erase negations by appearance alone. Check the transformed statement on an all-success model, a mixed model and an all-failure model. The correct equivalence agrees on every model; these three simple patterns expose many mistaken rewrites.
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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use 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.
not(forall x P(x)): F. First evaluate the entire universal sentence, then negate that result. The outer negation denies that every object has the property. It does not separately deny the property of every object. Parentheses make the scope visible: the negation applies to a quantified sentence. A single domain member outside P is sufficient to make this outer denial true.
exists x not P(x): F. Evaluate the inner negated predicate for each object. Its witnesses are precisely the domain members outside P's extension. There is such a witness exactly when the original universal fails. This gives the same truth value as the first expression for a semantic reason, not a typographical trick: both assertions require at least one counterexample to P holding everywhere.
forall x not P(x): F. This third expression requires every object to be outside P's extension. It therefore says that no object has P, which is stronger than saying that not every object has P. A mixed model, with at least one P-object and at least one non-P-object, makes the first two expressions true and this one false. The placement of the quantifier matters.
Test the analogous pair not(exists x P(x)) and forall x not P(x). Both require the witness set for P to be empty. Compare this with exists x not P(x), which needs only one object outside P. Writing the domain and extension as two explicit sets makes the distinction visible. Do not replace a denied universal by a universal denial simply because the same words all and not appear in both.
The domain consists exactly of a, b, c; these names denote distinct objects. P is true exactly of a. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use 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.
not(forall x P(x)): T. First evaluate the entire universal sentence, then negate that result.
exists x not P(x): T. Evaluate the inner negated predicate for each object.
forall x not P(x): F. This third expression requires every object to be outside P's extension.
Test the analogous pair not(exists x P(x)) and forall x not P(x). Both require the witness set for P to be empty. Compare this with exists x not P(x), which needs only one object outside P. Writing the domain and extension as two explicit sets makes the distinction visible. Do not replace a denied universal by a universal denial simply because the same words all and not appear in both.
The domain consists exactly of a, b, c, d; these names denote distinct objects. P is true exactly of a, b, c. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use 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.
not(forall x P(x)): T. First evaluate the entire universal sentence, then negate that result.
exists x not P(x): T. Evaluate the inner negated predicate for each object.
forall x not P(x): F. This third expression requires every object to be outside P's extension.
Test the analogous pair not(exists x P(x)) and forall x not P(x). Both require the witness set for P to be empty. Compare this with exists x not P(x), which needs only one object outside P. Writing the domain and extension as two explicit sets makes the distinction visible. Do not replace a denied universal by a universal denial simply because the same words all and not appear in both.
Take the claim 'Every returned book has a receipt.' With R for returned and C for has a receipt, the formula is forall x (R(x) -> C(x)). Its denial says that some returned book lacks a receipt. Both parts are essential: the witness must be returned, and it must fail the receipt condition.
A book that has not been returned and has no receipt is not a counterexample. Its failure to have a receipt does not violate a condition that applies only upon return. Likewise, a returned book with a receipt supports one successful instance rather than the denial. Constructing a counterexample requires respecting the antecedent and falsifying the consequent in the same object.
Now negate the denial. 'There is no returned book without a receipt' returns to the original universal rule. The inner conjunction R(x) and not C(x) describes precisely the forbidden case. Negating its existence rules that case out everywhere. This is a useful way to read regulations: write the violation condition, then state that no violation occurs.
English negatives can be less tidy than symbolic scope. 'All the books were not returned' may be used to mean none were returned or merely that not all were returned. Do not assign one formalization without checking the intended reading. A paraphrase such as 'At least one was not returned' removes the ambiguity. Formal notation helps only after its relation to the intended sentence is clear. Once the reading is fixed, testing a mixed model is a quick way to distinguish the denial of all from the assertion of none.
An inspection checks five display lights. Four work and one does not. The report says, 'Not all five lights work.' This is a denial of universal success. It is equivalent to saying that at least one of the five does not work. The failed light is a witness to that negative property, and the four working lights do not undo the report.
Rewriting the report as 'All five lights do not work', meaning every light is broken, changes the claim. That stronger statement is false in the supplied case. A repair team relying on it might replace working lights unnecessarily. Precise quantifier scope therefore matters even in a simple maintenance note.
Suppose the rule is more specific: every emergency light must work. A counterexample must be an emergency light that does not work. A broken decorative light does not by itself refute that restricted rule. In the logical form, the failed universal is a universal conditional, and its negation requires both membership in the emergency class and failure to work.
Write a report that names the scope and preserves the evidence: 'One of the five checked lights failed; it is an emergency light, so the emergency-light rule was not satisfied at inspection time.' This supplies a bounded witness and the rule it challenges. It does not say that no emergency light worked or that every light in the building was checked. A logical negation is strongest when it is exact, rather than when it sounds most dramatic.
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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
Use the declared objects and meanings throughout this calculation: not(forall x P(x)) is the first requested result.
Determine the requested value: not(forall x P(x)).
T
First evaluate the entire universal sentence, then negate that result.
Determine the requested value: exists x not P(x).
T
Evaluate the inner negated predicate for each object.
Determine the requested value: forall x not P(x).
F
This third expression requires every object to be outside P's extension.
Collect the results in the requested order.
T / T / F
Each result belongs to its own entry: not(forall x P(x)); exists x not P(x); forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
Use the declared objects and meanings throughout this calculation: not(forall x P(x)) is the first requested result.
Determine the requested value: not(forall x P(x)).
F
First evaluate the entire universal sentence, then negate that result.
Determine the requested value: exists x not P(x).
F
Evaluate the inner negated predicate for each object.
Determine the requested value: forall x not P(x).
F
This third expression requires every object to be outside P's extension.
Collect the results in the requested order.
F / F / F
Each result belongs to its own entry: not(forall x P(x)); exists x not P(x); forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
Use the declared objects and meanings throughout this calculation: not(forall x P(x)) is the first requested result.
Determine the requested value: not(forall x P(x)).
T
First evaluate the entire universal sentence, then negate that result.
Determine the requested value: exists x not P(x).
T
Evaluate the inner negated predicate for each object.
Determine the requested value: forall x not P(x).
T
This third expression requires every object to be outside P's extension.
Collect the results in the requested order.
T / T / T
Each result belongs to its own entry: not(forall x P(x)); exists x not P(x); forall x not P(x).
Test which alteration would change the conclusion.
Test the analogous pair not(exists x P(x)) and forall x not P(x). Both require the witness set for P to be empty. Compare this with exists x not P(x), which needs only one object outside P. Writing the domain and extension as two explicit sets makes the distinction visible. Do not replace a denied universal by a universal denial simply because the same words all and not appear in both.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: not(forall x P(x)).
T
First evaluate the entire universal sentence, then negate that result.
Determine the requested value: exists x not P(x).
Determine the requested value: forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
| Computed result | |
|---|---|
| not(forall x P(x)) | |
| exists x not P(x) | |
| forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
Negate the result of the universal check.
b0
The requested entry concerns not(forall x p(x)); retain its stated scope.
Search for an object outside the predicate extension.
b1
The requested entry concerns exists x not p(x); retain its stated scope.
Check whether every object is outside the predicate extension.
b2
The requested entry concerns forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
not(forall x P(x)): b0
exists x not P(x): b1
forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
not(forall x P(x)): b0
exists x not P(x): b1
forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
not(forall x P(x)): b0
exists x not P(x): b1
forall x not P(x): b2
In a display-light inspection, objects are lights and P means working at inspection time. 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
not(forall x P(x)): b0
exists x not P(x): b1
forall x not 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. Record the truth values of not(forall x P(x)), exists x not P(x), and forall x not P(x), in that order. Use T or F.
not(forall x P(x)): b0
exists x not P(x): b1
forall x not P(x): b2
You can translate a negated quantified claim. 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
Evaluate the inner negated predicate for each object.
14. Complete the next model audit, step 3
F
This third expression requires every object to be outside P's extension.