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distinguish necessary, possible and actual claims
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 necessary, possible and actual claims, 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. |
A basic modal model has worlds, an accessibility relation and a truth assignment at each world. At a world w, Box P is true when P is true at every world accessible from w. Diamond P is true when P is true at at least one world accessible from w. P by itself reports its truth at w. The three tests must not be conflated.
Accessibility expresses the relevant alternatives for the chosen interpretation of modality. It is not automatically physical travel, and it need not connect every world to every other. The exercise must say which alternatives count. A true P-world outside the accessible set is no witness to Diamond P at w. A false P-world outside that set is no counterexample to Box P at w.
An empty successor set gives Box P true and Diamond P false. The box has no failing accessible instance; the diamond has no successful accessible witness. If a particular modal reading requires at least one accessible world, this must be expressed by a frame condition. Seriality provides that nonempty-successor requirement, and reflexivity also guarantees a successor by making each world accessible from itself.
The duality Box P equals not Diamond not P follows directly from universal and existential negation over the successor set. It holds even at a world with no successors. These semantics explain the symbols; they do not settle philosophical debates about which worlds or accessibility conditions best model a particular kind of possibility.
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.
P at w0: T. Actual truth in this exercise means truth at the designated evaluation world. Read its valuation directly rather than inspecting other worlds. The fact that a sentence is possible elsewhere does not make it true here. Conversely, without a reflexivity assumption, actual truth need not contribute to possibility from this world, because the actual world may not be accessible from itself.
Box P at w0: F. Necessity requires P at every world accessible from w0. Restrict the inspection to that successor set; worlds elsewhere in the model are irrelevant to this particular modal evaluation. If the set is empty, the universal condition has no counterexample and the box is true. This is a semantic result in a general frame, not a claim that an actual event must occur.
Diamond P at w0: T. Possibility requires at least one accessible world where P is true. An inaccessible true world is not a witness for this evaluation. If the successor set is empty, the diamond is false because there is no witness. This explains why necessity alone need not imply possibility in arbitrary frames: empty successor sets satisfy the box condition without satisfying the diamond condition.
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.
P at w0: F. Actual truth in this exercise means truth at the designated evaluation world.
Box P at w0: T. Necessity requires P at every world accessible from w0.
Diamond P at w0: T. Possibility requires at least one accessible world where P is true.
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.
P at w0: T. Actual truth in this exercise means truth at the designated evaluation world.
Box P at w0: F. Necessity requires P at every world accessible from w0.
Diamond P at w0: F. Possibility requires at least one accessible world where P is true.
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.
Take three worlds w0, w1, and w2. Let P be true at w1 and false at w0 and w2. Suppose the only world accessible from w0 is w1. At w0, Box P is true because every accessible alternative satisfies P, and Diamond P is true because an accessible witness exists. Yet P is false at w0. This is possible because the frame has not required w0 to access itself. The example separates necessity relative to a frame from actuality at the evaluation world.
Now add an edge from w0 to w2 without changing any truth values. Diamond P remains true: w1 is still an accessible witness. Box P becomes false because the new accessible world falsifies P. Adding an alternative can destroy necessity while preserving an existing possibility witness. Removing the w1 edge instead would remove that witness and leave only a false alternative.
Consider a different frame where w0 has no accessible worlds. Box P is true there because no accessible world violates P, while Diamond P is false because no accessible witness exists. These are consequences of the quantified truth conditions over an empty successor set, not contradictory claims that an actual event both must and cannot occur. If a modeling application requires at least one alternative from every world, that frame requirement must be stated.
Accessibility is a relation chosen for the intended modal interpretation. It need not connect every pair, run both ways, or include self-connections unless the frame conditions say so. An edge from w0 to w1 alone does not create an edge back. A drawing should therefore show direction clearly and identify the world at which a formula is being evaluated.
Before calculating a modal value, list the immediate successors, then inspect the proposition at exactly those worlds. Keep frame information and truth assignments separate. Changing either can change the result, but each change has a different explanation that an auditable calculation should preserve.
Consider an invented scheduling model with three allowed schedule configurations. At the current configuration, the art session is indoors. In one alternative it is indoors and in another it is outdoors. Let P mean the art session is indoors, and let the accessible worlds be exactly the alternatives the organizer permits from the current plan.
If both alternatives are accessible, Diamond P is true because one permitted alternative has the session indoors. Box P is false because another permitted alternative has it outdoors. The actual P-value remains true at the current configuration. Thus actual, possible and necessary receive different answers without any contradiction: they inspect different parts of the model.
Suppose the outdoor alternative is removed because its venue is unavailable. If the remaining accessible alternatives all put art indoors, Box P becomes true relative to the revised model. This is a claim about the alternatives declared relevant to this scheduling task, not a claim that art could never be outdoors in any imaginable circumstance.
If no alternatives are listed at all, the formal box and diamond values follow the empty-set rules. A planner may instead intend the current configuration always to remain an allowed alternative. That intention adds a reflexive accessibility edge and changes the model. The point of declaring a frame is to make such assumptions visible. Without the accessible-world list, the phrase must be indoors can slide between a rule of this plan, a practical limitation and an unrestricted claim of necessity.
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: P at w0 is the first requested result.
Determine the requested value: P at w0.
F
Actual truth in this exercise means truth at the designated evaluation world.
Determine the requested value: Box P at w0.
T
Necessity requires P at every world accessible from w0.
Determine the requested value: Diamond P at w0.
F
Possibility requires at least one accessible world where P is true.
Collect the results in the requested order.
F / T / F
Each result belongs to its own entry: P at w0; Box P at w0; Diamond 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: P at w0 is the first requested result.
Determine the requested value: P at w0.
T
Actual truth in this exercise means truth at the designated evaluation world.
Determine the requested value: Box P at w0.
F
Necessity requires P at every world accessible from w0.
Determine the requested value: Diamond P at w0.
T
Possibility requires at least one accessible world where P is true.
Collect the results in the requested order.
T / F / T
Each result belongs to its own entry: P at w0; Box P at w0; Diamond 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: P at w0 is the first requested result.
Determine the requested value: P at w0.
F
Actual truth in this exercise means truth at the designated evaluation world.
Determine the requested value: Box P at w0.
T
Necessity requires P at every world accessible from w0.
Determine the requested value: Diamond P at w0.
T
Possibility requires at least one accessible world where P is true.
Collect the results in the requested order.
F / T / T
Each result belongs to its own entry: P at w0; Box P at w0; Diamond 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: P at w0.
T
Actual truth in this exercise means truth at the designated evaluation world.
Determine the requested value: Box P at w0.
Determine the requested value: Diamond 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 | |
|---|---|
| P at w0 | |
| Box P at w0 | |
| Diamond 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.
Read the valuation at the designated world.
b0
The requested entry concerns p at w0; retain its stated scope.
Check every accessible world.
b1
The requested entry concerns box p at w0; retain its stated scope.
Search the accessible set for a witness.
b2
The requested entry concerns diamond 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.
P at w0: b0
Box P at w0: b1
Diamond 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.
P at w0: b0
Box P at w0: b1
Diamond 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.
P at w0: b0
Box P at w0: b1
Diamond P at w0: b2
The worlds represent permitted schedule configurations and P says the art session is indoors. 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.
P at w0: b0
Box P at w0: b1
Diamond 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.
P at w0: b0
Box P at w0: b1
Diamond P at w0: b2
You can distinguish necessary, possible and actual claims. 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
F
Necessity requires P at every world accessible from w0.
14. Complete the next model audit, step 3
F
Possibility requires at least one accessible world where P is true.