Material Equivalence Expansion Rewriter (P↔Q two forms)
Expand material equivalence between identifier atoms into the disjunction of their matching truth states.
Description
A biconditional is true when its two propositions share a truth value.
A biconditional is true when its two propositions share a truth value. It expands into the disjunction of both-true and both-false cases.1
The tool returns original, rewritten. Interpret the result under the rule and assumptions below; an encoded premise’s truth is not independently established.
When to use Material Equivalence Expansion Rewriter (P↔Q two forms)
Use this for the following task: Expand material equivalence between identifier atoms into the disjunction of their matching truth states. Prepare the specified logical encoding before running the check or transformation.
How to use Material Equivalence Expansion Rewriter (P↔Q two forms) 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 p, q according to the field table, then run the tool.
- Read original, rewritten. 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 |
|---|---|---|---|
| p | P | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
| q | Q | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
Supply p and q as identifier atoms. Return the original biconditional and the fixed expansion rendered with <->, &&, || and !.
P↔Q ≡ (P AND Q) OR (NOT P AND NOT Q) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| original | Original propositional formula | string (propositional-formula) |
| rewritten | Equivalent propositional formula | string (propositional-formula) |
{
"input": {
"p": "p",
"q": "q"
},
"output": {
"original": "(p <-> q)",
"rewritten": "((p && q) || (!p && !q))"
}
}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 XNOR Equality Gate Definition Checker when you need its operation: Evaluate XNOR equality and its classical matching-state expansion on two Boolean inputs. Choose Material Implication to Disjunction Rewriter (P→Q ≡ ¬P∨Q) for the complementary operation: Rewrite material implication between identifier atoms as a disjunction with a negated antecedent.
Background reference: Additional rules for TFL.
References
-
Additional rules for TFL — forallx Calgary, Open Logic Project
Similar or alternative tools
- XNOR Equality Gate Definition Checker
Evaluate XNOR equality and its classical matching-state expansion on two Boolean inputs.
- Material Implication to Disjunction Rewriter (P→Q ≡ ¬P∨Q)
Rewrite material implication between identifier atoms as a disjunction with a negated antecedent.