Logical Expressiveness

Updated 10 May 2026

Logical expressiveness is a measure of a logic's ability to represent and distinguish diverse structures, properties, and behaviors in formal systems.
It is quantified using model-theoretic, proof-theoretic, and translational frameworks, ensuring back-and-forth translation and preservation of key meta-properties.
Its application spans AI, knowledge representation, and computational learning, although increased expressiveness often challenges decidability and succinctness.

Logical expressiveness is a foundational concept for formal logic, computational logic, knowledge representation, and automated reasoning. At its core, logical expressiveness characterizes the ability of a logic or a computational formalism to capture or define certain classes of structures, properties, or behaviors. This concept is critical not only for theoretical classification of logical systems, but also for understanding the capabilities and limitations of formal languages deployed in AI, machine learning, verification, and databases.

1. Notions and Frameworks of Logical Expressiveness

Logical expressiveness does not admit a single definition but is formalized in several related frameworks, each appropriate for different logical paradigms:

Model-theoretic expressiveness: A logic $L_2$ is at least as expressive as $L_1$ if every property definable by a formula in $L_1$ can be defined by some formula in $L_2$ over the same class of models (Fernandes, 2017). Formally, this is captured as

$L_2 \geq_{EC} L_1 \iff \forall \varphi \in F_1\;\exists \psi \in F_2 : Mod_{L_1}(\varphi) = Mod_{L_2}(\psi)$

Proof-theoretic and Tarskian frameworks: Expressiveness may be defined in terms of formula-mapping translations that preserve (and sometimes reflect) theoremhood or derivability, decoupled from concrete model classes. This is often required for comparing logics with divergent underlying structures (e.g., modal vs. first-order logics) (Fernandes, 2017).
Translational expressiveness: The broadest criterion dispenses with model or proof constraints, requiring merely a translation $T: F_1 \to F_2$ that is “back-and-forth” (preserving and reflecting key meta-properties), preferably defined inductively over the connectives of $L_1$ . This enables comparison of non-classical, substructural, and hybrid logics for which other criteria are inadequate (Fernandes, 2017).

Expressiveness can also be refined to “separating power” (the ability to distinguish between models or structures) and “definability” of classes (characterizing sets of models via formulas). These subtleties are crucial for classification, complexity, and succinctness arguments in modal, temporal, and higher-order logics (0905.4332, Schnoor, 2014).

2. Quantifying and Comparing Expressiveness

Logical expressiveness is typically formalized as a pre-order relation between logics, often under further adequacy criteria:

Back-and-forth condition: A translation $T$ between the formula sets of two logics is required to preserve and reflect theoremhood or derivability (Fernandes, 2017).
Structural fidelity: Translations should be general-recursive, i.e., defined inductively over the structure of formulas and sensitive to connectives. This ensures that logical connectives retain their conceptual integrity under translation.
Meta-property preservation: Preserving meta-properties such as non-triviality, the deduction theorem, and decidability is necessary to preclude degenerate or trivial encodings.

Several formal expressiveness relationships widely used in literature are summarized below:

Criterion Type	Canonical Quantification	Examples/Comments
Model-theoretic (EC)	Definable classes of models	Relational fragment comparisons: FO, MSO, modal logic (Fernandes, 2017, Strass, 2014)
Proof-theoretic	Back-and-forth on derivability	Focused/proof-net adequacy (Chaudhuri, 2010)
Expressiveness_{gg}	Inductive translation	Strictest “translational” condition (Fernandes, 2017)

For example, in argumentation and logic programming, languages such as abstract dialectical frameworks (ADFs), normal logic programs (NLPs), and propositional logic (PL) can be strictly ordered by the model-sets each can realize (see formal hierarchies in (Strass, 2014)). In substructural logics, classical and intuitionistic subexponential calculi have been shown to be equally expressive at the level of focal adequacy despite their apparent semantic distinctions (Chaudhuri, 2010).

3. Logical Expressiveness in Computational and AI Formalisms

Contemporary AI and logic-based learning systems bring logical expressiveness into computational practice:

Neural architectures and GNNs: The expressiveness of GNNs and topological message-passing networks has been precisely matched to logical fragments: e.g., 1-WL ≡ graded modal logic ≡ GNN separation power; TNNs’ power is equated with topological counting logics via pairwise counting quantifiers and higher-order pebble games (Akbari et al., 21 Apr 2026, Soeteman et al., 16 Jun 2025, Benedikt et al., 2024).
Knowledge representation: Expressiveness characterizes the ability of different formalisms (e.g., SLR, MSO, FOC $_2$ ) to specify relational and graph properties (Iosif et al., 2022, Chen et al., 2023, Iosif et al., 2023). The existence of strict separation and collapse results (e.g., SLR $\not\subseteq$ MSO and vice versa on unbounded-treewidth graphs) reveals the fundamental limits of each language.
Learning and reasoning with RL: Logical expressiveness has empirical impacts: in reinforcement learning of LLM-based reasoners, the scaling exponent for training steps required to master inference increases strictly with the expressiveness of the logic involved. More expressive logics (adding ∧, ∨, ¬, ∀) induce superlinear compute scaling and support better transfer to downstream mathematical and reasoning tasks (Wang et al., 7 May 2026).

Modal and temporal logics are notable for the intricate relationships among their expressiveness, succinctness, and definability:

Relative expressiveness and complexity: For modal logics, expressiveness and succinctness hierarchies can be as complex as any countable partial order, as shown via constructions exploiting successor-selection functions and alternation languages (Schnoor, 2014). Decidability and invariance properties (e.g., class bisimulation) precisely delineate definability in modal logic (0905.4332).
Temporal and hybrid logics: Temporal GNNs’ expressivity is matched exactly to two-dimensional product logics (PTL × K); more constrained neural architectures only reach restricted fragments depending on the allowed spatio-temporal interaction (Sälzer et al., 17 May 2025).
Higher-order foundations: Systems such as HOL and Church’s STT can uniformly embed both modal and non-classical logics, with “discernment” (primitive equality/difference) ensuring closure under quantification and the definability of all classical connectives (Fuenmayor, 23 Feb 2026).

5. Expressiveness Hierarchies and Succinctness

Expressiveness does not imply succinctness: even when two logics are equally expressive (i.e., all formulas in one have equivalents in the other), the size of representing formulas may differ exponentially. Results from modal logic show expressiveness/succinctness hierarchies of arbitrary complexity (Schnoor, 2014). In knowledge representation, frameworks such as ADFs, BADFs, AFs, and logic programs yield fine-grained stratifications according to which model-sets can be represented, with tight connections to succinctness and computational complexity (Strass, 2014, Zhang et al., 2014).

6. Decidability and Practical Implications

Logical expressiveness acts as a double-edged sword: increased expressiveness frequently breaches decidability frontiers. Restricting quantifiers or connectives (e.g., counting quantifiers, unary conjunctive views in databases, separating conjunction-depth in SLR) balances expressive power and decidability (0803.2559, Chen et al., 2023). In computational learning and verification, knowing the expressiveness–decidability landscape is essential for system design and formal methods.

7. Advancements, Limitations, and Open Problems

Translational expressiveness, with its requirement of inductively structured, back-and-forth translations, is now the most robust general criterion encapsulating the intuitive content of the phrase “everything said in $L_1$ 0 can be said in $L_1$ 1” (Fernandes, 2017). However, no universally optimal metric is possible: certain aspects (e.g., uniform definability of real-valued functions, logical fragments resisting shallow embeddings, interaction of expressiveness with complexity theory) remain open. Success in logical compositionality, meta-property conservation, and the translation of hybrid, modal, and higher-order features remains a target for both proof-theoretical and computational approaches.

In summary, logical expressiveness organizes the comparative power of formalisms at the heart of logic, AI, and computation. Its careful measurement, matched to adequacy criteria and structural fidelity, remains a crucial tool for understanding the limits and possibilities of reasoning systems across theoretical and applied domains.