Quantifier Negation Push-Through Rewriter (¬∀→∃¬)
Classical quantifier-negation duality for one explicitly declared unary atomic predicate only; not arbitrary capture-avoiding formula parsing.
Description
Quantifier Negation Push-Through Rewriter (¬∀→∃¬): return rewritten formula.
Negating a universal statement changes it to an existential statement with a negated predicate, and conversely. This tool represents only one unary atomic predicate.
This tool returns rewritten formula. 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.
Accept forall or exists, validate the variable and predicate identifiers, swap the quantifier and construct the exact string with a negated predicate, such as exists x.!P(x).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| negatedQuantifier | Negated quantifier. | Exact value: "forall", "exists" |
| variableName | Variable name. | Text |
| predicateName | Predicate name. | Text |
| Output field | Interpretation |
|---|---|
| result | Object containing the following named result fields. |
| result.rewrittenFormula | Rewritten formula; use the stated units and sign convention. |
{
"input": {
"negatedQuantifier": "forall",
"variableName": "x",
"predicateName": "P"
},
"output": {
"result": {
"rewrittenFormula": "exists x.!P(x)"
}
}
}Limitations and assumptions
- It does not parse arbitrary quantified formulas or perform general capture-avoiding substitution.
- 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
- double negation elimination validator
Double-negation elimination permits an explicitly double-negated atom to be affirmed in classical logic.
- material implication rewriter
Material implication is false only when its antecedent is true and consequent false.