Complement Law Contradiction Tautology Pair Checker

Check for complementary signed identifier literals and apply their conjunction or disjunction law.

Description

Complementary literals name the same atom with opposite signs.

Complementary literals name the same atom with opposite signs. Their conjunction is always false and their disjunction always true.^[1]

The tool returns complementary, simplified. Interpret the result under the rule and assumptions below; an encoded premise’s truth is not independently established.

When to use Complement Law Contradiction Tautology Pair Checker

Use this for the following task: Check for complementary signed identifier literals and apply their conjunction or disjunction law. Prepare the specified logical encoding before running the check or transformation.

How to use Complement Law Contradiction Tautology Pair 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 operator, left, right, leftNegated, rightNegated according to the field table, then run the tool.
Read complementary, simplified. 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
operator	Boolean operator	string; choices "and", "or"	Required
left	Left	string; maximum text length 100; pattern ^(?!(?:true\|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S])	Required
right	Right	string; maximum text length 100; pattern ^(?!(?:true\|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S])	Required
leftNegated	Left literal negated	boolean	Required
rightNegated	Right literal negated	boolean	Required

Supply an operation, two atom identifiers and separate negation flags. Recognize a complement pair when identifiers agree and flags differ; otherwise return the rendered unsimplified expression.

P AND NOT P = false; P OR NOT P = true

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
complementary	Complementary literals	boolean
simplified	Resulting propositional formula	string (propositional-formula)

json

{
  "input": {
    "operator": "and",
    "left": "p",
    "right": "p",
    "leftNegated": false,
    "rightNegated": true
  },
  "output": {
    "complementary": true,
    "simplified": "false"
  }
}

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.

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 Idempotence Duplicate Term Collapser Checker (P∧P=P) when you need its operation: Collapse equal identifier operands under conjunction or disjunction using idempotence. Choose Domination Law Annihilator Checker (P∨T=T, P∧F=F) for the complementary operation: Check whether a Boolean constant is the annihilator of conjunction or disjunction.

Background reference: Additional rules for TFL.

References

Additional rules for TFL — forallx Calgary, Open Logic Project

Similar or alternative tools

Idempotence Duplicate Term Collapser Checker (P∧P=P)
Collapse equal identifier operands under conjunction or disjunction using idempotence.
Domination Law Annihilator Checker (P∨T=T, P∧F=F)
Check whether a Boolean constant is the annihilator of conjunction or disjunction.