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distinguish the scope of two quantifiers
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 distinguish the scope of two quantifiers, 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. |
In forall x exists y R(x,y), a y may be chosen after x is fixed. Different x-values may therefore have different witnesses. In exists y forall x R(x,y), one y must be chosen before the universal check and work for every x. The order determines a dependency, not merely the order in which words are read aloud.
Picture a relation table with sources as rows and targets as columns. The first formula asks whether every row contains at least one true cell. The second asks whether some column consists entirely of true cells. A diagonal table over two distinct objects has a true cell in each row but no all-true column. It therefore separates the two readings using the smallest useful counterexample pattern.
The stronger common-target claim entails the row-witness claim: if one target works for everyone, each individual has at least that target. The reverse inference fails, as the diagonal model shows. This is a semantic argument about the two conditions, not a rule saying quantifiers may always be reversed when one direction happens to work.
The variables themselves can be renamed consistently without changing meaning. Swapping the roles or scopes of their quantifiers is different. Parentheses help identify the formula governed by each quantifier. In these exercises, both quantifiers range over the same declared domain and self-relations are permitted unless expressly excluded. If an application separates students and books, additional predicates or a stated many-sorted convention must enforce those roles.
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.
Domain: a, b, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Fix the first argument at a. Its row has targets a, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets b, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Rows with a witness: 2. Hold the outer variable fixed while evaluating the inner existential. Each row gets its own search for a target. Different rows may choose different witnesses, because the existential choice occurs inside the universal's scope. A witness chosen after seeing the row is not a single object chosen in advance for the entire model. Record the row successes before combining them.
forall x exists y R(x,y): T. The outer universal requires the row-level existence check to succeed for every first argument. It does not require the chosen targets to match. The row count must equal the domain size. This makes the order of operations explicit: choose a source, search its row, then repeat for every source. Reading the symbols backwards would impose a different dependency.
exists y forall x R(x,y): F. Now choose one target first and test whether every source relates to that very target. Such a witness appears as an entirely true column. A different successful target in every row is insufficient. The existential is outside the universal here, so its choice must work before the sources are considered individually. Compare the column test with the earlier row test rather than guessing from English word order.
For this interpretation, the number of targets shared by every source is 0. Compare that with the 2 successful rows. If each source has a target but no target is shared by all sources, the two quantifier orders differ in truth value. Adding one common target to every row repairs the stronger column claim. Deleting every entry in one row refutes the row claim. These changes explain which structural feature each formula measures.
Domain: a, b, c, all distinct. R holds exactly for these ordered pairs: (a,c), (b,c), (c,c). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Fix the first argument at a. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at c. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(c,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Rows with a witness: 3. Hold the outer variable fixed while evaluating the inner existential.
forall x exists y R(x,y): T. The outer universal requires the row-level existence check to succeed for every first argument.
exists y forall x R(x,y): T. Now choose one target first and test whether every source relates to that very target.
For this interpretation, the number of targets shared by every source is 1. Compare that with the 3 successful rows. If each source has a target but no target is shared by all sources, the two quantifier orders differ in truth value. Adding one common target to every row repairs the stronger column claim. Deleting every entry in one row refutes the row claim. These changes explain which structural feature each formula measures.
Domain: a, b, c, d, all distinct. R holds exactly for these ordered pairs: (a,b), (b,c), (c,d), (d,a). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Fix the first argument at a. Its row has targets b, so there are 1 true entries in this row. The statement exists y R(a,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at b. Its row has targets c, so there are 1 true entries in this row. The statement exists y R(b,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at c. Its row has targets d, so there are 1 true entries in this row. The statement exists y R(c,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Fix the first argument at d. Its row has targets a, so there are 1 true entries in this row. The statement exists y R(d,y) is T. The inner existential can choose from this row only; it cannot borrow a target from a row belonging to a different first argument.
Rows with a witness: 4. Hold the outer variable fixed while evaluating the inner existential.
forall x exists y R(x,y): T. The outer universal requires the row-level existence check to succeed for every first argument.
exists y forall x R(x,y): F. Now choose one target first and test whether every source relates to that very target.
For this interpretation, the number of targets shared by every source is 0. Compare that with the 4 successful rows. If each source has a target but no target is shared by all sources, the two quantifier orders differ in truth value. Adding one common target to every row repairs the stronger column claim. Deleting every entry in one row refutes the row claim. These changes explain which structural feature each formula measures.
A useful reading game makes the dependency explicit. For forall x exists y R(x,y), imagine one person choosing any x and another responding with a suitable y after seeing that choice. The responder may use a different y on a later round with a different x. Success requires a response to every possible x, not a fixed response chosen before the game starts.
For exists y forall x R(x,y), the responder must commit to y first. The challenger may then choose any x, and the same y must work. This is a stronger demand. The game is a way of reading the quantifiers, not a claim that literal players or strategies are objects in the first-order domain.
Use a two-object equality relation as a separating model. Each object relates to itself and not to the other object. When x is a, choose y as a; when x is b, choose y as b. The first game succeeds. No single y works for both x-values, so the second game fails. This countermodel pinpoints dependence instead of relying on an ambiguous English slogan.
By contrast, exchanging two adjacent universal quantifiers preserves the requirement that every ordered pair satisfies the inner formula. Exchanging two adjacent existential quantifiers preserves the requirement that some ordered pair satisfies it. The mixed case is where a dependency can change. Even then, inspect the actual formula and premises: special facts about a particular relation may make both readings true. Equality of truth in one convenient model is not a general permission to interchange the quantifiers.
Three volunteers each need a helper for a task. Ari is helped by Bo, Bo by Cy, and Cy by Ari. Let H(x,y) mean y helps x, so the first argument is the person receiving help. In this supplied arrangement, every volunteer has a helper. The formula forall x exists y H(x,y) is true. The witnesses are allowed to depend on the volunteer being considered.
There is no single volunteer helping everybody. Each potential helper appears in only one of the listed helper roles, so exists y forall x H(x,y) is false. Announcing 'Someone helps everyone' would change an arrangement of individual help into a claim about one common helper.
To repair the stronger claim, suppose Bo also helps Bo and Cy, as well as Ari. Under the explicitly permitted self-help interpretation, the Bo column then has a true entry for every recipient. Bo becomes a common witness. If the task forbids self-help, that restriction must be included instead, and this particular repair would not work for Bo's own row.
The table therefore asks a real planning question: is the requirement that every task has somebody assigned, or that one coordinator covers every task? Those requirements may need different arrangements. A short phrase such as everyone has someone can conceal the distinction. Declaring the relation's argument order and writing both formulas makes the intended staffing claim checkable without assuming that individual coverage implies a common coordinator.
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.
Domain: a, b, c, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (b,a), (b,b), (b,c). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Use the declared objects and meanings throughout this calculation: Rows with a witness is the first requested result.
Determine the requested value: Rows with a witness.
2
Hold the outer variable fixed while evaluating the inner existential.
Determine the requested value: forall x exists y R(x,y).
F
The outer universal requires the row-level existence check to succeed for every first argument.
Determine the requested value: exists y forall x R(x,y).
F
Now choose one target first and test whether every source relates to that very target.
Collect the results in the requested order.
2 / F / F
Each result belongs to its own entry: Rows with a witness; forall x exists y R(x,y); exists y forall x R(x,y).
Record the interpretation and the question.
Domain: a, b, c, d, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b), (c,c), (d,d). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Use the declared objects and meanings throughout this calculation: Rows with a witness is the first requested result.
Determine the requested value: Rows with a witness.
4
Hold the outer variable fixed while evaluating the inner existential.
Determine the requested value: forall x exists y R(x,y).
T
The outer universal requires the row-level existence check to succeed for every first argument.
Determine the requested value: exists y forall x R(x,y).
F
Now choose one target first and test whether every source relates to that very target.
Collect the results in the requested order.
4 / T / F
Each result belongs to its own entry: Rows with a witness; forall x exists y R(x,y); exists y forall x R(x,y).
Record the interpretation and the question.
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,e), (b,e), (c,e), (d,e), (e,e). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Use the declared objects and meanings throughout this calculation: Rows with a witness is the first requested result.
Determine the requested value: Rows with a witness.
5
Hold the outer variable fixed while evaluating the inner existential.
Determine the requested value: forall x exists y R(x,y).
T
The outer universal requires the row-level existence check to succeed for every first argument.
Determine the requested value: exists y forall x R(x,y).
T
Now choose one target first and test whether every source relates to that very target.
Collect the results in the requested order.
5 / T / T
Each result belongs to its own entry: Rows with a witness; forall x exists y R(x,y); exists y forall x R(x,y).
Test which alteration would change the conclusion.
For this interpretation, the number of targets shared by every source is 1. Compare that with the 5 successful rows. If each source has a target but no target is shared by all sources, the two quantifier orders differ in truth value. Adding one common target to every row repairs the stronger column claim. Deleting every entry in one row refutes the row claim. These changes explain which structural feature each formula measures.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: Rows with a witness.
4
Hold the outer variable fixed while evaluating the inner existential.
Determine the requested value: forall x exists y R(x,y).
Determine the requested value: exists y forall x R(x,y).
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (a,d), (a,e), (b,a), (b,b), (b,c), (b,d), (b,e), (c,a), (c,b), (c,c), (c,d), (c,e), (d,a), (d,b), (d,c), (d,d), (d,e). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
| Computed result | |
|---|---|
| Rows with a witness | |
| forall x exists y R(x,y) | |
| exists y forall x R(x,y) |
Domain: a, b, c, d, e, f, g, all distinct. R holds exactly for these ordered pairs: (a,a), (a,b), (a,c), (a,d), (a,e), (a,f), (a,g), (b,a), (b,b), (b,c), (b,d), (b,e), (b,f), (b,g), (c,a), (c,b), (c,c), (c,d), (c,e), (c,f), (c,g), (d,a), (d,b), (d,c), (d,d), (d,e), (d,f), (d,g), (e,a), (e,b), (e,c), (e,d), (e,e), (e,f), (e,g), (f,a), (f,b), (f,c), (f,d), (f,e), (f,f), (f,g). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Search each row for an inner witness.
b0
The requested entry concerns rows with a witness; retain its stated scope.
Combine the successful rows using the outer universal.
b1
The requested entry concerns forall x exists y r(x,y); retain its stated scope.
Search for one column succeeding for every row.
b2
The requested entry concerns exists y forall x r(x,y); retain its stated scope.
Domain: a, b, c, d, e, f, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b), (c,c), (d,d), (e,e), (f,f). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Rows with a witness: b0
forall x exists y R(x,y): b1
exists y forall x R(x,y): b2
Domain: a, b, c, d, e, all distinct. R holds exactly for these ordered pairs: (a,e), (b,e), (c,e), (d,e), (e,e). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Rows with a witness: b0
forall x exists y R(x,y): b1
exists y forall x R(x,y): b2
Domain: a, b, c, d, e, f, all distinct. R holds exactly for these ordered pairs: (a,b), (b,c), (c,d), (d,e), (e,f), (f,a). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Rows with a witness: b0
forall x exists y R(x,y): b1
exists y forall x R(x,y): b2
For a volunteer plan, R(x,y) means y helps x; self-help is permitted in this planning exercise. The data below form the complete invented audit. Domain: a, b, c, d, e, f, all distinct. R holds exactly for these ordered pairs: (a,a), (b,b), (c,c), (d,d), (e,e), (f,f). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Rows with a witness: b0
forall x exists y R(x,y): b1
exists y forall x R(x,y): b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Domain: a, b, c, d, e, f, g, all distinct. R holds exactly for these ordered pairs: (a,g), (b,g), (c,g), (d,g), (e,g), (f,g), (g,g). Every unlisted pair is false, and self-pairs are permitted. Complete the three quantified-model entries named below. Use T or F for sentence truth.
Rows with a witness: b0
forall x exists y R(x,y): b1
exists y forall x R(x,y): b2
You can distinguish the scope of two quantifiers. 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 outer universal requires the row-level existence check to succeed for every first argument.
14. Complete the next model audit, step 3
F
Now choose one target first and test whether every source relates to that very target.