Real Logic: A Framework for Real-Valued Inference

• Real Logic is a formal system that assigns real-valued truth degrees to formulas, allowing reasoning about uncertainty beyond binary true/false.
• It uses multidimensional sentences and weighted formulas to express joint constraints and interdependencies in neuro-symbolic and fuzzy systems.
• Decision procedures based on LP/MILP and a strongly complete axiomatic framework enable rigorous and explainable inference in continuous settings.

Real Logic refers to a class of formal systems that extend the truth-functional framework of classical logic to the real numbers, typically the unit interval $[0,1]$, enabling reasoning about degrees of truth and uncertainty. By generalizing Boolean semantics into real-valued (often many-valued or fuzzy) semantics, Real Logic is foundational for neuro-symbolic AI, fuzzy systems, and rigorous reasoning with uncertainty. Its mathematical treatment now encompasses axiomatizations, decision procedures, modal and first-order extensions, and integration with learning systems.

1. Core Semantic Framework: Many-Valued and Real-Valued Logics

In Real Logic, formulas are interpreted not as strictly true or false, but assigned a real value in a given interval, typically $[0,1]$, capturing the “degree of truth” or satisfaction. Formally, for a logic $\mathcal{L}$:

• Language: $\mathcal{L}$ may have propositional, first-order, or modal formulas.
• Truth Values: Interpretations are given by functions $\mu: \text{Form} \rightarrow T_*$, where $T_*\subseteq [0,1]$ is a subalgebra of a BL-algebra (bounded, lattice-ordered algebra based on a continuous t-norm).
• Connectives: Logical connectives are interpreted as real functions (e.g., $\odot$ for Łukasiewicz, $\min$ for Gödel, $\times$ for Product logic).

For example, in Łukasiewicz logic: $$x \odot y = \max\{0, x + y - 1\}, \quad x \to_\odot y = \min\{1, 1 - x + y\}.$$ In Gödel logic: $$x \min y = \min\{x, y\}, \quad x \to_{\min} y = \begin{cases} 1, & x \leq y\ y, & x > y \end{cases}$$ General logics are parametrized by the choice of t-norm, residuum, and $T_*$.

2. Expressive Sentences: Multidimensional and Weighted Constraints

A central advancement is the introduction of multidimensional sentences (MD-sentences), which allow one to express constraints not only on individual formulas but also on their joint distributions of truth values. An MD-sentence has the form: $$(\sigma_1, \ldots, \sigma_k; S), \quad S \subseteq [0,1]^k$$ where each $\sigma_i$ is a formula, $k$ is arbitrary, and $S$ is any (potentially complex or undecidable) subset of $[0,1]^k$. The semantics are: $$M \models (\sigma_1, ..., \sigma_k; S) \iff (\mu(\sigma_1), ..., \mu(\sigma_k)) \in S$$ This enables encoding dependencies and joint constraints among real-valued logical formulas beyond the reach of traditional (scalar) approaches.

Weighted real logics, relevant for systems such as Logical Neural Networks, further generalize formulas to include learnable weights, extending every connective to a function of both truth values and weights: $s_m = f_\alpha(s_i, s_j, w_1, w_2)$ Weights can thus be learned jointly with logical inference in a unified neuro-symbolic setting.

3. Parametrized Axiomatization and Strong Completeness

A distinguishing feature of Real Logic is the existence of a strongly complete, parametrized proof system covering all major real-valued logics by selecting the connective interpretations:

• Axiom: $(\sigma; [0,1])$ for all formulas $\sigma$ (any value is possible initially).
• Inference rules:
  • permutation of tuple components,
  • augmentation (adding components via $S \times [0,1]^{m-k}$),
  • intersection (conjunction of constraints),
  • projection (variable elimination),
  • weakening (set-wise relaxation),
  • operator rule: restrict $S$ to tuples that respect the algebraic operations imposed by connectives.

Strong completeness for finite premise sets states: $$\Gamma \models \gamma \implies \Gamma \vdash \gamma$$ where $\Gamma$ and $\gamma$ are finite sets of MD-sentences. This supports rigorous, sound reasoning about which combinations of real values are entailed by others, exceeding the granularity offered by classical, fuzzy, or Pavelka-style systems.

4. Decision Procedures and Computational Aspects

For broad classes of real logics—particularly those where connectives admit piecewise-linear or min/max semantics (e.g., Łukasiewicz, Gödel)—the system enables decision procedures for entailment and satisfiability grounded in linear programming (LP) or mixed-integer linear programming (MILP):

• Interval-based sentences (MD-sentences constrained to $S = S_1 \times ... \times S_k$ with $S_i$ finite unions of rational intervals) are especially amenable.
• Logical entailment reduces to the feasibility of a system of real or piecewise-linear constraints embodying both the semantic constraints and operator rules.
• Complexity scales exponentially in the number of formulas involved, but is polynomial for fixed formula sets and variable numbers of intervals.
• Practical implementations (e.g., SoCRAtic logic, IBM CPLEX backend) validate the tractability of this approach for nontrivial logics.

5. Comparison and Characterization of Real Logic Systems

A robust theoretical analysis, particularly in “Two principles in many-valued logic” (Aguzzoli et al., 2013), leads to the following characterization:

• Principle 1 (Semantic equivalence): Two formulas are logically equivalent iff they are true in exactly the same degree-1 (fully true) models. Satisfied iff the logic extends Gödel logic.
• Principle 2 (Model separation): Any two distinct valuations are separated by some formula (i.e., there exists a formula true in one and false in the other). Satisfied iff the logic extends Łukasiewicz logic.
• Product logic is characterized (under Euclidean topological closure of $T_*$) as the only proper real-valued logic failing both principles.

This establishes precise boundaries between various flavors of real-valued logics within the BL-algebraic framework and dictates their semantic properties, redundancy, and distinguishability.

6. Applications: Neuro-Symbolic Reasoning, Uncertainty, and Explainability

The Real Logic paradigm underpins a wide range of practical and theoretical applications:

• Neuro-symbolic systems: It furnishes the logical infrastructure for architectures like Logic Tensor Networks, enabling the integration of real-valued reasoning, learning, and background knowledge in tensorized neural architectures.
• Explainable AI: Formal reasoning about partial truth assignments supports robust explainability and trust, as every semantically valid constraint derivable by the system is also provable.
• Automation and Deductive Decision Making: Real Logic’s decision procedures automate fuzzy, probabilistic, and constraint-based reasoning, extending classical deductive tools to the continuous setting.

The parametric, completeness-guaranteeing framework readily adapts to the needs of probabilistic soft logic, logical neural networks, probabilistic programming, and approximate reasoning in uncertain domains.

7. Performance, Scalability, and Limitations

Algorithmic reasoning in Real Logic is governed by:

• Resource complexity: Exponential in the number of formulas; polynomial for fixed-arity and interval-based reasoning.
• Expressivity vs. decidability trade-off: Full generality (arbitrary $S \subseteq [0,1]^k$) can force undecidability; practical applications generally employ tractable subsets (e.g., interval, polyhedral, or piecewise-linear $S$).
• Computational requirements: Efficiency is supported by LP/MILP solvers in many logics of interest, but scaling to large formula collections or deeply nested constraints may require further optimization and approximation methods.

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Summary Table: Core Features of Real Logic

Aspect	Formalism/Property	Example/Key Formula
Truth values	$[0,1]$ (BL-subalgebra)	$x \odot y = \max\{0, x + y - 1\}$
Sentences	Multidimensional: $(\sigma_1, ..., \sigma_k; S)$	$S \subseteq [0,1]^k$, e.g., $\{(a_1, a_2): a_1^2 = a_2\}$
Axioms/inference	Parametric rules + operator-specific constraints	Operator rule: $s_m = f_\alpha(s_i, s_j)$
Completeness	Strong (finite $\Gamma$): provability = entailment	$$\Gamma \models \gamma \implies \Gamma \vdash \gamma$$
Decision procedure	LP/MILP for interval-based sentences	Feasibility encodes both sentence constraints and operations
Characterization	Two principles (semantic equivalence/model sep.)	Gödel logic $\Leftrightarrow$ P1; Łukasiewicz logic $\Leftrightarrow$ P2
Extensions	Weighted formulas, first-order/modal, etc.	$s_m = f_\alpha(s_i, s_j, w_1, w_2)$

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In conclusion, Real Logic as formalized in contemporary research enables rigorous, expressive, and soundly complete reasoning about real-valued logical formulas, supporting the foundations of uncertainty, explainability, and advanced inference in symbolic and neuro-symbolic AI. It unifies axiomatic, algebraic, and algorithmic perspectives, offering a robust substrate for future development in AI, formal reasoning, and logic-based machine learning systems (Fagin et al., 2020).