Distributivity AND-over-OR Expansion Applier
Distribute conjunction over disjunction for three identifier atoms.
Description
Boolean distributivity expands AND over OR, preserving truth for every assignment.
Boolean distributivity expands AND over OR, preserving truth for every assignment. The same leading atom appears in both resulting factors.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 Distributivity AND-over-OR Expansion Applier
Use this for the following task: Distribute conjunction over disjunction for three identifier atoms. Prepare the specified logical encoding before running the check or transformation.
How to use Distributivity AND-over-OR Expansion Applier 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, r 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 |
| r | R | string; maximum text length 100; pattern ^(?!(?:true|false)$)[A-Za-z_][A-Za-z0-9_]*$(?![\s\S]) | Required |
Supply p, q and r as identifier atoms. Render the fixed original form and its distributed expansion using && and ||.
P AND (Q OR R) ≡ (P AND Q) OR (P AND R) 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",
"r": "r"
},
"output": {
"original": "(p && (q || r))",
"rewritten": "((p && q) || (p && r))"
}
}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. This applies a chosen form rather than searching an expression for applicable rewrites.
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 Distributivity OR-over-AND Expansion Applier when you need its operation: Distribute disjunction over conjunction for three identifier atoms. Choose Absorption Law Simplifier (P∨(P∧Q)=P) for the complementary operation: Simplify either classical absorption-law form for two identifier atoms.
Background reference: Additional rules for TFL.
References
-
Additional rules for TFL — forallx Calgary, Open Logic Project
Similar or alternative tools
- Distributivity OR-over-AND Expansion Applier
Distribute disjunction over conjunction for three identifier atoms.
- Absorption Law Simplifier (P∨(P∧Q)=P)
Simplify either classical absorption-law form for two identifier atoms.