Involution Double Complement Cancel Checker
Cancel pairs of nested negations around an identifier atom using classical involution.
Description
Classical double negation cancels in pairs.
Classical double negation cancels in pairs. An even number of nested negations leaves an atom unchanged; an odd number leaves one negation.1
The tool returns simplified, remainingNegations. Interpret the result under the rule and assumptions below; an encoded premise’s truth is not independently established.
When to use Involution Double Complement Cancel Checker
Use this for the following task: Cancel pairs of nested negations around an identifier atom using classical involution. Prepare the specified logical encoding before running the check or transformation.
How to use Involution Double Complement Cancel Checker and understand its results
- Prepare the requested Boolean values, complete truth-table rows, or identifier atoms. An atom is a name such as p or premise_1, not a natural-language sentence.
- Enter atom, negationCount according to the field table, then run the tool.
- Read simplified, remainingNegations. Use the matching rule, assignment order or nonempty-class assumptions below to interpret the result.
- For a false check, inspect the supplied encoding and matching rule. A well-formed false result is different from an input error.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| atom | Identifier atom | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
| negationCount | Nested negation count | integer; minimum 0; maximum 9007199254740991 | Required |
Supply an atom identifier and nonnegative safe-integer count. Calculate parity and prepend one ! only when the remainder is one.
remaining negations = negationCount mod 2 The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| simplified | Resulting propositional formula | string (propositional-formula) |
| remainingNegations | Remaining negation count | integer |
{
"input": {
"atom": "p",
"negationCount": 4
},
"output": {
"simplified": "p",
"remainingNegations": 0
}
}Limitations and assumptions
Operands are case-sensitive identifier atoms, not complete expressions. The tool does not parse a proof or determine whether the premises are true. No nested input expression is accepted.
The supplied encodings must represent the intended propositions or classes. No natural-language parsing or assessment of empirical premise truth is performed. Malformed input gives a tool error rather than a logical false result.
Alternative or Complementary methods
Choose Double Negation Elimination Validator (¬¬P ⊢ P) when you need its operation: Validate double-negation elimination from an explicitly double-negated identifier to a claimed atom. Choose Complement Law Contradiction Tautology Pair Checker for the complementary operation: Check for complementary signed identifier literals and apply their conjunction or disjunction law.
Background reference: Additional rules for TFL.
References
-
Additional rules for TFL — forallx Calgary, Open Logic Project
Similar or alternative tools
- Double Negation Elimination Validator (¬¬P ⊢ P)
Validate double-negation elimination from an explicitly double-negated identifier to a claimed atom.
- Complement Law Contradiction Tautology Pair Checker
Check for complementary signed identifier literals and apply their conjunction or disjunction law.