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use identity to state that two names pick out one object
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 use identity to state that two names pick out one object, 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. |
Identity is sameness of denotation. The statement a=b is true when both names designate the very same domain object. Different spelling does not imply different objects, and sharing several properties does not imply identity. In first-order logic with identity, equality has this fixed interpretation; it is not a freely assigned relation alongside the other predicates.
From a=b and P(a), substitution of identicals permits P(b). The reason is semantic: there is one object, so membership in an extensional predicate cannot depend on which co-referring name is used. The same applies to a term position in a relation. If R(a,c) holds and a=b, then R(b,c) holds. Substitution preserves argument position and does not reverse the relation.
An existential formula with two variables does not guarantee two objects. The formula exists x exists y (P(x) and P(y)) can be satisfied by assigning the same P-object to both variables. To require at least two distinct P-objects, add not(x=y). Exactly one P-object can be expressed by requiring a P-witness and requiring every P-object to be identical to it.
The model may contain unnamed objects, and several names may denote a single object. Count the domain, the names and the denotations separately. Our worksheets state an explicit mapping from names to numbered objects so that these three counts cannot be confused. Quotations and belief reports are outside the extensional substitution task taught here.
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 is the distinct objects [1, 2]. Name a denotes object 1; name b denotes object 1. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Object 1 is inside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 2 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
a=b: T. Compare denotations, not the printed shapes of the names. Two different names may pick out one object, so distinct spelling does not establish inequality. Conversely, objects sharing a property need not be identical. Identity is interpreted as actual sameness within the model. It is not an ordinary relation that the author can assign freely while leaving the objects distinct.
P(b): T. Follow b to the object it denotes and inspect that object's membership in the predicate extension. If a and b co-refer, this agrees with P(a), as substitution of identicals requires. If they refer to different objects, P(a) alone supplies no answer about P(b). The explicit extension gives the answer in this model without using an invalid resemblance-to-identity inference.
Objects named by a and b: 1. Count distinct denoted objects rather than occurrences of names. Two names may contribute one object to this count. This is why two existential variables do not by themselves express the existence of two different objects: a distinctness condition is required. Keep the number of named objects separate from the size of the complete domain, which may also contain unnamed objects.
Add a new name c for the object denoted by a. The vocabulary has gained a name, but the domain has gained no object. Every extensional predicate applied to c must agree with its application to a. This substitution principle does not license careless replacement inside quotations or reports of what somebody believes; those contexts require additional analysis beyond the first-order extensional language used here.
The domain is the distinct objects [1, 2, 3]. Name a denotes object 1; name b denotes object 2. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Object 1 is inside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 2 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 3 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
a=b: F. Compare denotations, not the printed shapes of the names.
P(b): F. Follow b to the object it denotes and inspect that object's membership in the predicate extension.
Objects named by a and b: 2. Count distinct denoted objects rather than occurrences of names.
Add a new name c for the object denoted by a. The vocabulary has gained a name, but the domain has gained no object. Every extensional predicate applied to c must agree with its application to a. This substitution principle does not license careless replacement inside quotations or reports of what somebody believes; those contexts require additional analysis beyond the first-order extensional language used here.
The domain is the distinct objects [1, 2, 3, 4]. Name a denotes object 2; name b denotes object 2. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Object 1 is inside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 2 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 3 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
Object 4 is outside P's extension. This fact belongs to the object, so every name denoting this same object must agree about whether it has P. The interpretation cannot make P true of one alias and false of another alias while treating them as identical.
a=b: T. Compare denotations, not the printed shapes of the names.
P(b): F. Follow b to the object it denotes and inspect that object's membership in the predicate extension.
Objects named by a and b: 1. Count distinct denoted objects rather than occurrences of names.
Add a new name c for the object denoted by a. The vocabulary has gained a name, but the domain has gained no object. Every extensional predicate applied to c must agree with its application to a. This substitution principle does not license careless replacement inside quotations or reports of what somebody believes; those contexts require additional analysis beyond the first-order extensional language used here.
Suppose an interpretation has one object and two names for it. The sentences P(a) and P(b) can both be true while there is only one P-object. Counting the written instances as two witnesses would count the vocabulary instead of the domain. An equality statement exposes the aliasing directly: a=b.
To require two different objects, assert not(a=b) when the witnesses are named, or include not(x=y) in a quantified formula. To require three different witnesses, one pairwise inequality is insufficient. Each pair must be constrained: x differs from y, x differs from z, and y differs from z. Otherwise two of the variables can still coincide while the third is different.
Identity also supports a useful inconsistency check. If a=b and P(a) are true, not P(b) cannot be true in the same extensional interpretation. The contradiction does not come from a resemblance between names; it comes from assigning incompatible membership to one object. This makes identity useful when reviewing records that may contain aliases.
Be careful about descriptions rather than direct names. 'The current chair' and a person's name can pick out the same person now without having done so last year. A time-sensitive interpretation must say which time is being modeled. Likewise, quoting the string 'Ada' concerns a name's spelling, not Ada's height or membership in a club. Substitution of identical objects preserves extensional claims about those objects; it does not erase distinctions between the objects and words used to describe them. The finite exercises deliberately keep that boundary simple and explicit.
A phrase such as 'the chair of the committee' seems to identify an object, but its successful use can depend on there being exactly one chair. Russell's analysis represents the descriptive sentence 'The chair is registered' by three conditions: a chair exists, every chair is identical to that witness, and the witness is registered. With C for chair and R for registered, this is exists x (C(x) and forall y (C(y) -> y=x) and R(x)).
If the committee has no chair, the existential condition fails. If it has two distinct chairs, uniqueness fails. Either way the analyzed sentence is false, even when every committee member is registered. If there is exactly one chair, that object's registration determines the remaining condition. The analysis therefore does not treat the description as an ordinary name guaranteed to denote.
Negation has two scopes. Externally denying the whole sentence says that the combined existence, uniqueness and registration conditions do not all hold. Asserting instead that the unique chair is not registered keeps existence and uniqueness and negates only the property. When there is no chair, the external denial is true but the latter sentence is false under this analysis. The apparent alternatives do not exhaust the possibilities if their scope differs.
This is one influential analysis, not the only philosophical account of descriptions. Ordinary conversation may instead treat an empty description as a failed presupposition. The graded audit explicitly specifies Russell's analysis so it evaluates a declared formal interpretation rather than marking an alternative theory as a factual mistake.
A library's donation system gives a book the temporary label a. The permanent catalog later gives that same physical book the label b. The inventory identifies both labels with object 17. Therefore a=b is true in the intended model. Two labels have been used, but only one book has been added to the library.
Suppose P means has a damaged cover and object 17 is in P's extension. P(a) and P(b) must agree. The physical cover does not change when the catalog switches labels. This is the practical reason substitution of identicals works for this predicate. If one database entry says damaged and the other says undamaged, the entries need reconciliation; they cannot both be a consistent interpretation of one unchanged object's condition.
A second book may have the same title, author and cover design. Those similarities do not establish identity with object 17. A record showing object 23 for the second copy distinguishes the two physical books even when their descriptive fields match. Counting matching titles would answer a different question from counting copies.
This distinction also matters in a lending relation. If L(u,a) means user u borrowed the book named a, replacing a with its alias b preserves the borrowed object. It does not license reversing the arguments or substituting another copy with the same title. A careful catalog model records object identity, properties and relations separately. The result is a precise account of what is duplicated: sometimes the data label, sometimes a descriptive property, and sometimes an actual additional object.
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 is the distinct objects [1, 2, 3]. Name a denotes object 2; name b denotes object 3. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Use the declared objects and meanings throughout this calculation: a=b is the first requested result.
Determine the requested value: a=b.
F
Compare denotations, not the printed shapes of the names.
Determine the requested value: P(b).
F
Follow b to the object it denotes and inspect that object's membership in the predicate extension.
Determine the requested value: Objects named by a and b.
2
Count distinct denoted objects rather than occurrences of names.
Collect the results in the requested order.
F / F / 2
Each result belongs to its own entry: a=b; P(b); Objects named by a and b.
Record the interpretation and the question.
The domain is the distinct objects [1, 2, 3, 4]. Name a denotes object 3; name b denotes object 3. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Use the declared objects and meanings throughout this calculation: a=b is the first requested result.
Determine the requested value: a=b.
T
Compare denotations, not the printed shapes of the names.
Determine the requested value: P(b).
F
Follow b to the object it denotes and inspect that object's membership in the predicate extension.
Determine the requested value: Objects named by a and b.
1
Count distinct denoted objects rather than occurrences of names.
Collect the results in the requested order.
T / F / 1
Each result belongs to its own entry: a=b; P(b); Objects named by a and b.
Record the interpretation and the question.
The domain is the distinct objects [1, 2, 3, 4, 5]. Name a denotes object 3; name b denotes object 4. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Use the declared objects and meanings throughout this calculation: a=b is the first requested result.
Determine the requested value: a=b.
F
Compare denotations, not the printed shapes of the names.
Determine the requested value: P(b).
F
Follow b to the object it denotes and inspect that object's membership in the predicate extension.
Determine the requested value: Objects named by a and b.
2
Count distinct denoted objects rather than occurrences of names.
Collect the results in the requested order.
F / F / 2
Each result belongs to its own entry: a=b; P(b); Objects named by a and b.
Test which alteration would change the conclusion.
Add a new name c for the object denoted by a. The vocabulary has gained a name, but the domain has gained no object. Every extensional predicate applied to c must agree with its application to a. This substitution principle does not license careless replacement inside quotations or reports of what somebody believes; those contexts require additional analysis beyond the first-order extensional language used here.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: a=b.
T
Compare denotations, not the printed shapes of the names.
Determine the requested value: P(b).
Determine the requested value: Objects named by a and b.
The domain is the distinct objects [1, 2, 3, 4, 5]. Name a denotes object 4; name b denotes object 5. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
| Computed result | |
|---|---|
| a=b | |
| P(b) | |
| Objects named by a and b |
The domain is the distinct objects [1, 2, 3, 4, 5, 6, 7]. Name a denotes object 6; name b denotes object 7. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
Compare the denotations of the names.
b0
The requested entry concerns a=b; retain its stated scope.
Check the property of the object denoted by the second name.
b1
The requested entry concerns p(b); retain its stated scope.
Count distinct denotations rather than names.
b2
The requested entry concerns objects named by a and b; retain its stated scope.
The domain is the distinct objects [1, 2, 3, 4, 5, 6]. Name a denotes object 5; name b denotes object 5. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
a=b: b0
P(b): b1
Objects named by a and b: b2
The domain is the distinct objects [1, 2, 3, 4, 5]. Name a denotes object 5; name b denotes object 1. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
a=b: b0
P(b): b1
Objects named by a and b: b2
Domain: three committee members a, b, c. Nobody is chair; all three are registered. Under Russell's existence-and-uniqueness analysis, evaluate 'The chair is registered'. Enter the number of chairs, that sentence's truth value, and the truth value of its external negation. Use T or F.
Chairs: n. The chair is registered: claim. External denial: denial.
In a library inventory, a and b are catalog labels and P means has a damaged cover. The data below form the complete invented audit. The domain is the distinct objects [1, 2, 3, 4, 5, 6]. Name a denotes object 1; name b denotes object 1. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
a=b: b0
P(b): b1
Objects named by a and b: b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The domain is the distinct objects [1, 2, 3, 4, 5, 6, 7]. Name a denotes object 7; name b denotes object 1. P holds exactly of object 1. Other domain objects need not have names. Give a=b, P(b), and the number of objects denoted by a and b together.
a=b: b0
P(b): b1
Objects named by a and b: b2
You can use identity to state that two names pick out one object. Reconstruct the three audit entries from a fresh model without consulting the examples; explain what change to the interpretation would change one answer.
15. Complete the next model audit, step 2
F
Follow b to the object it denotes and inspect that object's membership in the predicate extension.
15. Complete the next model audit, step 3
1
Count distinct denoted objects rather than occurrences of names.