Material Implication to Disjunction Rewriter (P→Q ≡ ¬P∨Q)
Rewrite material implication between identifier atoms as a disjunction with a negated antecedent.
Description
Material implication is false only when its antecedent is true and consequent false.
Material implication is false only when its antecedent is true and consequent false. A negated antecedent joined by inclusive OR reproduces that truth condition.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 Implication to Disjunction Rewriter (P→Q ≡ ¬P∨Q)
Use this for the following task: Rewrite material implication between identifier atoms as a disjunction with a negated antecedent. Prepare the specified logical encoding before running the check or transformation.
How to use Material Implication to Disjunction Rewriter (P→Q ≡ ¬P∨Q) 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 antecedent, consequent 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 |
|---|---|---|---|
| antecedent | Antecedent | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
| consequent | Consequent | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
Supply two atom identifiers. Render both the original conditional and its disjunctive equivalent using ->, ! and ||.
P→Q ≡ NOT P OR 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": {
"antecedent": "p",
"consequent": "q"
},
"output": {
"original": "(p -> q)",
"rewritten": "(!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 Contrapositive Form Generator (P→Q ≡ ¬Q→¬P) when you need its operation: Generate the equivalent contrapositive of a conditional between identifier atoms. Choose Material Equivalence Expansion Rewriter (P↔Q two forms) for the complementary operation: Expand material equivalence between identifier atoms into the disjunction of their matching truth states.
Background reference: Additional rules for TFL.
References
-
Additional rules for TFL — forallx Calgary, Open Logic Project
Similar or alternative tools
- Contrapositive Form Generator (P→Q ≡ ¬Q→¬P)
Generate the equivalent contrapositive of a conditional between identifier atoms.
- Material Equivalence Expansion Rewriter (P↔Q two forms)
Expand material equivalence between identifier atoms into the disjunction of their matching truth states.