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identify the stated scope of a basic modal inference
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 the stated scope of a basic modal inference, 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 inference from Box P to P is valid on reflexive frames because the current world belongs to its own accessible set. If P holds at every accessible world, it then holds here. On an arbitrary frame the current world may be absent from that set, so Box P can be true while P is false here. A countermodel needs both the accessibility facts and the valuation.
To evaluate Box P -> P, compute the modal antecedent first, read the actual consequent second, and only then apply the ordinary conditional truth rule. Treating Box P as if it were automatically the same as P would assume the very frame property being tested. A true instance at one world also does not establish validity on every world or frame.
Other modal principles require other conditions. Transitive accessibility supports the familiar step from Box P to Box Box P: successors of successors are already successors of the starting world. Seriality supports moving from Box P to Diamond P, because there is at least one successor. Do not borrow these principles without stating the needed frame restriction.
This lesson's graded audits use unnested formulas and an explicit successor list, so every required value is determined. The discussion of nested boxes explains why a richer problem would need accessibility information from the successor worlds as well. A single successor list is enough for Box P here but not for every nested modal formula one might write.
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.
Worlds: w0, w1. Evaluate at w0. Its accessible worlds are w0, w1. P is true exactly at w0. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
World w0 has P T and is accessible from w0. It therefore contributes to the box and diamond tests at w0. A false accessible instance defeats necessity, while a true accessible instance supplies a possibility witness.
World w1 has P F and is accessible from w0. It therefore contributes to the box and diamond tests at w0. A false accessible instance defeats necessity, while a true accessible instance supplies a possibility witness.
Box P at w0: F. Inspect exactly the accessible worlds and combine their P-values universally. The resulting value depends on both the valuation and the accessibility relation. Declaring a frame reflexive adds a world to its own successor set, which can change this computation. A modal inference cannot be assessed solely from the ordinary truth table of P at the evaluation world.
P at w0: T. Read actual P separately. If w0 is not accessible from itself, the box calculation may have ignored this very value. That separation is the source of a countermodel to Box P implies P in arbitrary frames. In reflexive frames it cannot happen, since every world's own P-value is included among the values its box requires to be true.
Box P -> P at w0: T. Apply the ordinary truth condition for implication only after computing the modal antecedent. The conditional is false exactly when Box P is true and actual P is false. A true conditional at one world does not establish validity on the frame, and validity on one frame does not establish validity on all frames. State the level of the result before generalizing.
Add a reflexive edge from w0 to itself while keeping all truth values fixed. A false P at w0 must now make Box P false, so the proposed inference from necessity to actuality cannot have true premise and false conclusion there. Adding edges makes necessity harder to satisfy and possibility easier to satisfy. Deleting edges has the opposite effect. This checks a modal result by changing the frame rather than changing the proposition.
Worlds: w0, w1, w2. Evaluate at w0. Its accessible worlds are w1, w2. P is true exactly at w1, w2. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
World w0 has P F and is not accessible from w0. Its truth value does not enter the box or diamond calculation at w0. Merely belonging to the model is different from being accessible from the evaluation world.
World w1 has P T and is accessible from w0. It therefore contributes to the box and diamond tests at w0. A false accessible instance defeats necessity, while a true accessible instance supplies a possibility witness.
World w2 has P T and is accessible from w0. It therefore contributes to the box and diamond tests at w0. A false accessible instance defeats necessity, while a true accessible instance supplies a possibility witness.
Box P at w0: T. Inspect exactly the accessible worlds and combine their P-values universally.
P at w0: F. Read actual P separately.
Box P -> P at w0: F. Apply the ordinary truth condition for implication only after computing the modal antecedent.
Add a reflexive edge from w0 to itself while keeping all truth values fixed. A false P at w0 must now make Box P false, so the proposed inference from necessity to actuality cannot have true premise and false conclusion there. Adding edges makes necessity harder to satisfy and possibility easier to satisfy. Deleting edges has the opposite effect. This checks a modal result by changing the frame rather than changing the proposition.
Worlds: w0, w1, w2, w3. Evaluate at w0. Its accessible worlds are w3. P is true exactly at w0, w1, w2. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
World w0 has P T and is not accessible from w0. Its truth value does not enter the box or diamond calculation at w0. Merely belonging to the model is different from being accessible from the evaluation world.
World w1 has P T and is not accessible from w0. Its truth value does not enter the box or diamond calculation at w0. Merely belonging to the model is different from being accessible from the evaluation world.
World w2 has P T and is not accessible from w0. Its truth value does not enter the box or diamond calculation at w0. Merely belonging to the model is different from being accessible from the evaluation world.
World w3 has P F and is accessible from w0. It therefore contributes to the box and diamond tests at w0. A false accessible instance defeats necessity, while a true accessible instance supplies a possibility witness.
Box P at w0: F. Inspect exactly the accessible worlds and combine their P-values universally.
P at w0: T. Read actual P separately.
Box P -> P at w0: T. Apply the ordinary truth condition for implication only after computing the modal antecedent.
Add a reflexive edge from w0 to itself while keeping all truth values fixed. A false P at w0 must now make Box P false, so the proposed inference from necessity to actuality cannot have true premise and false conclusion there. Adding edges makes necessity harder to satisfy and possibility easier to satisfy. Deleting edges has the opposite effect. This checks a modal result by changing the frame rather than changing the proposition.
Suppose w0 accesses w1, and w1 accesses w2. Let P be true at w1 and false at w2, with no other relevant edges. At w0, Box P is true because its immediate successor w1 satisfies P. But Box Box P is false: at w1, Box P is false because w2 is accessible there and falsifies P. The second box adds another evaluation step. It cannot be erased merely because the same symbol repeats.
If the frame also includes an edge directly from w0 to w2, Box P at w0 becomes false as well. That illustrates how a frame condition governing two-step paths can affect relationships between nested and single modalities. Do not infer the extra direct edge from the drawing unless the problem states the corresponding condition or explicitly includes that edge.
Scope also matters with ordinary connectives. Suppose w0 accesses two worlds: P is true only at the first, Q only at the second. Diamond P and Diamond Q are both true at w0, so their conjunction is true. Diamond (P & Q) is false because neither accessible world satisfies both. The two separate possibility claims can use different witnesses, whereas the single diamond over the conjunction requires a shared witness.
Similarly, a true Box (P | Q) need not imply Box P | Box Q. Every accessible world may satisfy the disjunction while different disjuncts do the work at different worlds. Draw the two worlds and evaluate the subformulas separately to expose the change in scope rather than relying on a verbal impression of necessity distributing everywhere.
A reliable nested evaluation proceeds from the inside out but records the world at every stage. Calculate the inner modal formula at each successor, using that successor's own outgoing edges. Then apply the outer operator at the original world. Flattening all reachable worlds into one collection loses the path structure that determines the formula's meaning.
A simple workflow has a current state w0 and two permitted next states. Both next states have an approved document, but the current state does not. Let P mean the document is approved. If accessibility means permitted next step and the current state is not its own next step, Box P is true at w0 while P is false there. The inference from every permitted next state having approval to approval now fails in this model.
The counterexample does not say that approval is impossible. It says the temporal placement of the property matters. A workflow rule about what comes next is not automatically a statement about what is already true. Record the current valuation separately from the successor valuations before drawing conclusions.
Now change the accessibility convention to include the current state among the states considered. Because P is false at w0, Box P becomes false under that reflexive interpretation. The earlier true-premise, false-conclusion combination disappears. The ordinary truth rule for implication did not change; the modal antecedent changed because the frame changed.
In a real design discussion, these two accessibility conventions could model different questions. One asks about immediate next actions; the other includes remaining where one is. Neither should be smuggled into the wording without explanation. A useful modal audit names the worlds, gives the relevant arrows, records truth at each world and states which frame conditions are assumed. Only then can a proposed inference be evaluated rather than merely recognized as a familiar-looking symbol pattern.
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.
Worlds: w0, w1, w2. Evaluate at w0. Its accessible worlds are none. P is true exactly at w1, w2. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Use the declared objects and meanings throughout this calculation: Box P at w0 is the first requested result.
Determine the requested value: Box P at w0.
T
Inspect exactly the accessible worlds and combine their P-values universally.
Determine the requested value: P at w0.
F
Read actual P separately.
Determine the requested value: Box P -> P at w0.
F
Apply the ordinary truth condition for implication only after computing the modal antecedent.
Collect the results in the requested order.
T / F / F
Each result belongs to its own entry: Box P at w0; P at w0; Box P -> P at w0.
Record the interpretation and the question.
Worlds: w0, w1, w2, w3. Evaluate at w0. Its accessible worlds are w0, w1, w2, w3. P is true exactly at w0, w1, w2. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Use the declared objects and meanings throughout this calculation: Box P at w0 is the first requested result.
Determine the requested value: Box P at w0.
F
Inspect exactly the accessible worlds and combine their P-values universally.
Determine the requested value: P at w0.
T
Read actual P separately.
Determine the requested value: Box P -> P at w0.
T
Apply the ordinary truth condition for implication only after computing the modal antecedent.
Collect the results in the requested order.
F / T / T
Each result belongs to its own entry: Box P at w0; P at w0; Box P -> P at w0.
Record the interpretation and the question.
Worlds: w0, w1, w2, w3, w4. Evaluate at w0. Its accessible worlds are w1, w2, w3, w4. P is true exactly at w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Use the declared objects and meanings throughout this calculation: Box P at w0 is the first requested result.
Determine the requested value: Box P at w0.
T
Inspect exactly the accessible worlds and combine their P-values universally.
Determine the requested value: P at w0.
F
Read actual P separately.
Determine the requested value: Box P -> P at w0.
F
Apply the ordinary truth condition for implication only after computing the modal antecedent.
Collect the results in the requested order.
T / F / F
Each result belongs to its own entry: Box P at w0; P at w0; Box P -> P at w0.
Test which alteration would change the conclusion.
Add a reflexive edge from w0 to itself while keeping all truth values fixed. A false P at w0 must now make Box P false, so the proposed inference from necessity to actuality cannot have true premise and false conclusion there. Adding edges makes necessity harder to satisfy and possibility easier to satisfy. Deleting edges has the opposite effect. This checks a modal result by changing the frame rather than changing the proposition.
The altered interpretation checks the dependence of these answers on the stated model, rather than replacing it during the calculation.
Determine the requested value: Box P at w0.
F
Inspect exactly the accessible worlds and combine their P-values universally.
Determine the requested value: P at w0.
Determine the requested value: Box P -> P at w0.
Worlds: w0, w1, w2, w3, w4. Evaluate at w0. Its accessible worlds are none. P is true exactly at w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
| Computed result | |
|---|---|
| Box P at w0 | |
| P at w0 | |
| Box P -> P at w0 |
Worlds: w0, w1, w2, w3, w4, w5, w6. Evaluate at w0. Its accessible worlds are none. P is true exactly at w1, w2, w3, w4, w5, w6. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Compute the modal antecedent from accessible worlds.
b0
The requested entry concerns box p at w0; retain its stated scope.
Read the consequent at the designated world.
b1
The requested entry concerns p at w0; retain its stated scope.
Apply the conditional truth rule to those two results.
b2
The requested entry concerns box p -> p at w0; retain its stated scope.
Worlds: w0, w1, w2, w3, w4, w5. Evaluate at w0. Its accessible worlds are w0, w1, w2, w3, w4, w5. P is true exactly at w0, w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Box P at w0: b0
P at w0: b1
Box P -> P at w0: b2
Worlds: w0, w1, w2, w3, w4. Evaluate at w0. Its accessible worlds are w1, w2, w3, w4. P is true exactly at w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Box P at w0: b0
P at w0: b1
Box P -> P at w0: b2
Worlds: w0, w1, w2, w3, w4, w5. Evaluate at w0. Its accessible worlds are w5. P is true exactly at w0, w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Box P at w0: b0
P at w0: b1
Box P -> P at w0: b2
The worlds represent workflow states, accessibility means permitted next state, and P says the document is approved. The data below form the complete invented audit. Worlds: w0, w1, w2, w3, w4, w5. Evaluate at w0. Its accessible worlds are w0, w1, w2, w3, w4, w5. P is true exactly at w0, w1, w2, w3, w4. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Box P at w0: b0
P at w0: b1
Box P -> P at w0: b2
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Worlds: w0, w1, w2, w3, w4, w5, w6. Evaluate at w0. Its accessible worlds are w1, w2, w3, w4, w5, w6. P is true exactly at w1, w2, w3, w4, w5, w6. These are all the accessibility facts needed for the displayed, unnested formulas. Complete the displayed truth entries using T or F.
Box P at w0: b0
P at w0: b1
Box P -> P at w0: b2
You can identify the stated scope of a basic modal inference. 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
Read actual P separately.
14. Complete the next model audit, step 3
T
Apply the ordinary truth condition for implication only after computing the modal antecedent.