fuzzy4

A Python library for 4-valued fuzzy logic based on Belnap logic with product t-norm operations.

Overview

Traditional boolean logic has two values: True and False. Real-world knowledge often involves uncertainty (we don't know) and contradiction (conflicting evidence). Belnap's 4-valued logic addresses this by representing propositions as vectors (T, F):

State	Vector	Meaning
TRUE	(1, 0)	Confirmed, not refuted
FALSE	(0, 1)	Refuted, not confirmed
UNKNOWN	(0, 0)	Neither confirmed nor refuted
CONFLICT	(1, 1)	Both confirmed and refuted

This library extends Belnap logic with fuzzy values, where T and F are continuous values in [0, 1].

Installation

pip install fuzzy4

Quick Start

from fuzzy4 import FuzzyBool, TRUE, FALSE, UNKNOWN, CONFLICT

# Create fuzzy values
x = FuzzyBool(0.8, 0.2)  # 80% confirmed, 20% refuted
y = FuzzyBool(0.6, 0.3)  # 60% confirmed, 30% refuted

# Logical operations
result = x & y           # conjunction (AND)
result = x | y           # disjunction (OR)
result = ~x              # negation (NOT)
result = x >> y          # implication (x -> y)
result = x.iff(y)        # bi-implication (x <-> y)

# Accumulate evidence
result = x + y           # combines evidence from multiple sources
result += FuzzyBool(0.5, 0.1)           # in-place accumulation
result = FuzzyBool.accumulate(x, y, z)  # accumulate many values

# Check dominant state
print(x.dominant_state())  # "true", "false", "unknown", or "conflict"
print(x.is_true())         # True if predominantly true

Operations

Negation: ~x

Swaps confirmation and refutation:

x	~x
TRUE	FALSE
UNKNOWN	UNKNOWN
CONFLICT	CONFLICT
FALSE	TRUE

~FuzzyBool(0.8, 0.3)  # -> FuzzyBool(0.3, 0.8)

Conjunction: x & y

Both must confirm, any refutation propagates:

&	TRUE	UNKNOWN	CONFLICT	FALSE
TRUE	TRUE	UNKNOWN	CONFLICT	FALSE
UNKNOWN	UNKNOWN	UNKNOWN	FALSE	FALSE
CONFLICT	CONFLICT	FALSE	CONFLICT	FALSE
FALSE	FALSE	FALSE	FALSE	FALSE

Uses product t-norm for T and probabilistic s-norm for F:

• T = Tx * Ty
• F = Fx + Fy - Fx * Fy

Disjunction: x | y

Any confirmation suffices, both must refute:

|	TRUE	UNKNOWN	CONFLICT	FALSE
TRUE	TRUE	TRUE	TRUE	TRUE
UNKNOWN	TRUE	UNKNOWN	TRUE	UNKNOWN
CONFLICT	TRUE	TRUE	CONFLICT	CONFLICT
FALSE	TRUE	UNKNOWN	CONFLICT	FALSE

Uses probabilistic s-norm for T and product t-norm for F:

• T = Tx + Ty - Tx * Ty
• F = Fx * Fy

Implication: x >> y

Material implication: x -> y = ~x | y

>>	TRUE	UNKNOWN	CONFLICT	FALSE
TRUE	TRUE	UNKNOWN	CONFLICT	FALSE
UNKNOWN	TRUE	UNKNOWN	TRUE	UNKNOWN
CONFLICT	TRUE	TRUE	CONFLICT	CONFLICT
FALSE	TRUE	TRUE	TRUE	TRUE

Bi-implication: x.iff(y)

Equivalence: x <-> y = (x -> y) & (y -> x)

iff	TRUE	UNKNOWN	CONFLICT	FALSE
TRUE	TRUE	UNKNOWN	CONFLICT	FALSE
UNKNOWN	UNKNOWN	UNKNOWN	TRUE	UNKNOWN
CONFLICT	CONFLICT	TRUE	CONFLICT	CONFLICT
FALSE	FALSE	UNKNOWN	CONFLICT	TRUE

Evidence Accumulation: x + y

Combines evidence using probabilistic sum — useful for aggregating information from multiple sources:

+	TRUE	UNKNOWN	CONFLICT	FALSE
TRUE	TRUE	TRUE	CONFLICT	CONFLICT
UNKNOWN	TRUE	UNKNOWN	CONFLICT	FALSE
CONFLICT	CONFLICT	CONFLICT	CONFLICT	CONFLICT
FALSE	CONFLICT	FALSE	CONFLICT	FALSE

weak = FuzzyBool(0.3, 0.0)     # weak confirmation
result = weak + weak + weak    # -> FuzzyBool(0.657, 0.0)

# In-place accumulation
total = UNKNOWN
for source in sources:
    total += source            # accumulates evidence iteratively

# Accumulate a list of values
FuzzyBool.accumulate([source1, source2, source3])

Properties:

• More confirmations → T approaches 1
• More refutations → F approaches 1
• UNKNOWN is the neutral element

Scalar Operations

x = FuzzyBool(0.8, 0.6)
x * 0.5              # -> FuzzyBool(0.4, 0.3)
x / 2.0              # -> FuzzyBool(0.4, 0.3)

Component Extraction

Each FuzzyBool has four derived properties representing the degree of each logical state:

x = FuzzyBool(0.7, 0.4)

x.truth     # degree of pure truth (confirmed AND not refuted)
x.falsity   # degree of pure falsity (refuted AND not confirmed)
x.unknown   # degree of uncertainty (neither confirmed nor refuted)
x.conflict  # degree of contradiction (both confirmed and refuted)

x.as_dict()  # {"truth": ..., "falsity": ..., "unknown": ..., "conflict": ...}

Use Cases

• Knowledge bases: Handle incomplete or contradictory information
• Sensor fusion: Combine readings from multiple sensors with varying reliability
• Expert systems: Model uncertainty in rule-based reasoning
• Database queries: Represent null/unknown values with more nuance than SQL's 3-valued logic

Mathematical Background

This implementation uses:

• Product t-norm for conjunction: a ⊗ b = a × b
• Probabilistic s-norm for disjunction: a ⊕ b = a + b - a × b
• Standard negation: ¬a = 1 - a

De Morgan's laws hold: ~(x & y) = ~x | ~y and ~(x | y) = ~x & ~y

Development

# Clone the repository
git clone https://github.com/mihail-gribov/fuzzy4.git
cd fuzzy4

# Install with dev dependencies
pip install -e ".[dev]"

# Run tests
pytest

# Run tests with coverage
pytest --cov=fuzzy4

License

MIT License — see LICENSE for details.

References

• Belnap, N. D. (1977). "A useful four-valued logic"
• Fitting, M. (1994). "Kleene's Three Valued Logics and Their Children"