Quantifier Scope Widening Legality Checker
Restricted scope widening for (Qx P) connective R to Qx(P connective R), given caller complete free-variable inventory; preserves empty-domain exceptions.
Description
Quantifier Scope Widening Legality Checker: return passes declared scope conditions, nonempty domain needed.
Moving a quantifier across a connective can change which variables it binds. A variable free in the independent part cannot simply be captured by widening the scope.
This tool returns passes declared scope conditions, nonempty domain needed. Use the stated units and model convention when interpreting these fields.
When to use this tool
Use this tool when the supplied inputs match the specific operation described above and you need a repeatable result under its stated convention.
- Check the supported input and output contract against an existing calculation or observation.
- Use the complete example to confirm units, selector spelling and result interpretation.
How to use the tool and how it works
- Enter every field in the input table using the stated units and the exact supported selector spelling where applicable.
- Run the tool. Correct input errors before interpreting an output.
- Read the result fields below; use accompanying flags and the stated sign, boundary or model convention when interpreting a number.
Check that the bound variable is absent from the declared free-variable list. Universal-with-AND and existential-with-OR also require callerDomainNonempty; report that requirement separately.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| quantifier | Quantifier. | Exact value: "forall", "exists" |
| connective | Connective. | Exact value: "AND", "OR" |
| boundVariable | Bound variable. | Text |
| independentPartFreeVariables | Independent part free variables. | List of text values |
| callerDomainNonempty | Caller domain nonempty. | true or false |
| Output field | Interpretation |
|---|---|
| result | Object containing the following named result fields. |
| result.passesDeclaredScopeConditions | Passes declared scope conditions flag under the exact conditions described above. |
| result.nonemptyDomainNeeded | Nonempty domain needed flag under the exact conditions described above. |
{
"input": {
"quantifier": "forall",
"connective": "AND",
"boundVariable": "x",
"independentPartFreeVariables": [
"y"
],
"callerDomainNonempty": true
},
"output": {
"result": {
"passesDeclaredScopeConditions": true,
"nonemptyDomainNeeded": true
}
}
}Limitations and assumptions
- The tool checks supplied side conditions, not parsed formulas or the completeness of the free-variable declaration.
- Every declared field is required and unexpected fields are rejected. Numeric inputs must be finite; input errors are distinct from a valid false result or an explicitly qualified output.
- Extremely large inputs and nonfinite or excessively large outputs are rejected. List positions reported by the tool use zero-based indices unless stated otherwise.
Alternative or Complementary methods
Check the underlying observations, units and conventions independently before using the result in a larger workflow. The tools listed below address nearby tasks; compare their inputs and scope before substituting one for this operation.
References
Similar or alternative tools
- Quantifier Negation Push-Through Rewriter (¬∀→∃¬)
Classical quantifier-negation duality for one explicitly declared unary atomic predicate only; not arbitrary capture-avoiding formula parsing.
- a proposition classifier
Categorical propositions use A for universal affirmative, E for universal negative, I for particular affirmative and O for particular negative.