Semantic Tableau Calculator for Propositional Logic

Semantic Tableau Result

Classification: Satisfiable Open Branches: 3 Closed Branches: 0 Stopped Branches: 0 Expanded Nodes: 3

Download CSV Download PDF

Parsed Formula: (((P → Q) ∧ P) → Q)

Variables: P, Q

Mode: Direct satisfiability analysis.

Interpretation: At least one tableau branch remains open.

Calculator

Branch Expansion Steps

Branch B1 used Implication on (((P → Q) ∧ P) → Q).
Created B2 with: ¬(((P → Q) ∧ P))

Created B3 with: Q
Branch B2 used Negated Conjunction on ¬(((P → Q) ∧ P)).
Created B4 with: ¬((P → Q))

Created B5 with: ¬P
Branch B4 used Negated Implication on ¬((P → Q)).
Created B6 with: P, ¬Q

Final Branches

Branch	Status	Depth	Reason	Remaining Formulas
B3	Open	1	All remaining formulas are literals.	Q
B5	Open	2	All remaining formulas are literals.	¬P
B6	Open	3	All remaining formulas are literals.	P, ¬Q

Open Branch Assignments

Open Branch	P	Q
Open 1	Undetermined	T
Open 2	F	Undetermined
Open 3	T	F

Plotly Graph

This graph compares branch depth across open, closed, and stopped outcomes.

Generated Truth Data

P	Q	Formula Value
F	F	T
F	T	T
T	F	T
T	T	T

Example Data Table

Formula	Goal	Expected Result	Reason
((P -> Q) & P) -> Q	Validity test	Unsatisfiable negation	Modus ponens is valid.
P & !P	Direct test	Unsatisfiable	One branch closes immediately.
P \| Q	Direct test	Satisfiable	At least one open branch exists.
(P <-> Q) + R	Direct test	Usually satisfiable	Branching depends on assignments.

Formula Used

Semantic tableaux do not rely on one numeric formula. They rely on decomposition rules that preserve logical meaning while splitting formulas into branch conditions.

Conjunction: A ∧ B adds both formulas to one branch.
Disjunction: A ∨ B splits into one branch with A and another with B.
Implication: A → B becomes branches ¬A or B.
Negated implication: ¬(A → B) becomes one branch containing A and ¬B.
Biconditional: A ↔ B becomes branches A, B or ¬A, ¬B.
Closure test: a branch closes when both P and ¬P appear.

If every branch closes, the tested set is unsatisfiable. If one branch stays open, the set is satisfiable.

How to Use This Calculator

Enter a propositional logic formula using the supported operators.
Select direct satisfiability or validity testing by negation.
Click Build Tableau to expand branches automatically.
Review the result summary above the form.
Read the expansion steps to understand each decomposition rule.
Inspect final branches for contradictions and open assignments.
Use the truth data table for verification.
Download CSV or PDF reports for documentation.

Frequently Asked Questions

1. What does this calculator solve?

It analyzes propositional logic formulas with semantic tableau rules. It shows whether a formula set is satisfiable, unsatisfiable, or limited by the expansion cap.

2. What operators can I enter?

You can use not, and, or, implication, biconditional, and xor. Supported symbols include !, &, |, ->, <->, and +.

3. What is an open branch?

An open branch contains no contradiction. It represents a possible assignment that keeps the tested formulas true together.

4. What is a closed branch?

A closed branch contains a contradiction such as P and not P. That branch cannot represent any valid assignment.

5. Why include a truth table section?

The truth data verifies tableau outcomes. It also helps compare symbolic branch reasoning with direct valuation results for each assignment.

6. What does validity mode do?

Validity mode negates the entered formula first. If the negated formula closes on every branch, the original formula is logically valid.

7. Why might a branch show undetermined values?

Open branches may not force every variable. Some propositions remain free because the branch already stays consistent without assigning them.

8. Can this handle predicate logic quantifiers?

No. This version is designed for propositional logic only. Predicate logic needs quantifier rules, instantiation control, and more advanced parsing.