Semantic Consequence in Truth-Conditional Semantics

Updated 11 November 2025

Semantic Consequence in Truth-Conditional Semantics is a framework that preserves truth across interpretations, uniting classical and non-classical logic systems.
It encompasses a variety of approaches including four-valued, mixed, modal, and differential semantics to address diverse inferential phenomena.
The framework bridges algebraic, model-theoretic, and proof-theoretic methods, advancing rigorous analysis of logical inference and stability.

Semantic consequence in truth-conditional semantics denotes the global inferential relationship between formulas or sets of formulas, defined in terms of the preservation of truth—or other relevant semantic features—across all interpretations provided by a given model-theoretic framework. This article surveys rigorous developments of semantic consequence in truth-conditional settings, with attention to non-classical multi-valued frameworks, mixed consequence, informational consequence for modalities, differential enrichments, and the interplay between semantic and syntactic consequence.

1. Semantic Consequence: Fundamentals and Truth-Conditional Semantics

Semantic consequence, denoted typically by $\models$ , is defined with respect to a semantic structure (e.g., models, valuations, information states) and encodes preservation of semantic designated properties from premises to conclusions. In the simplest case—classical logic— $\Gamma \models \varphi$ iff every valuation which makes all $\psi \in \Gamma$ true also makes $\varphi$ true. This paradigm extends to various non-classical logics via truth-functional valuation, many-valued logics, and informational semantics.

A general framework involves:

A language $\mathcal{L}$
Set $V$ of truth values
Set $W$ of “worlds” or valuations $v : \mathcal{L} \to V$
For $F \in \mathcal{L}$ , an interpretation $\llbracket F \rrbracket : W \to V$
Designated sets $D_p, D_c \subseteq V$ (for premises/conclusions)
A semantic consequence relation, e.g.,

$\Gamma \models \Delta \iff \forall v \in W\, :\, (v[\Gamma] \subseteq D_p \implies v[\Delta] \cap D_c \neq \emptyset)$

(Intersective mixed consequence; see (Chemla et al., 2017))

The semantic consequence is tailored by the choice of designated sets, the structure of $V$ , and potential enrichment of valuations, to capture inferential subtleties beyond classical entailment.

2. Four-Valued Logics and Multidimensional Consequence Relations

In the four-valued semantics of Belnap–Dunn logic, the set of truth values is $V = \{ T, F, N, B \}$ where $T$ (true only), $F$ (false only), $N$ (neither), and $B$ (both). The De Morgan lattice structure on $V$ supports a family of semantic consequence relations, each corresponding to the preservation of distinct semantic statuses:

Logic	Condition on Premises	Designated Values	Notation
Truth-preserving	$v(\psi) = T$ for all $\psi$	$T$	$\models_{th}$
Non-falsity-preserving	$v(\psi) \neq F$ for all $\psi$	$\{T, N, B\}$	$\models_{nf}$
Exact-truth	$v(\psi)=T$ & $v(\psi)\neq F$	$T$ and not $F$	$\models_{ex}$
Material equivalence	$v(\psi) = v(\rho)$ for all $(\psi \approx \rho)$	Equational	$\models_{=}$

Each logic is equipped with a Hilbert-type axiomatization and enjoys a soundness and completeness theorem relative to its intended semantic designated set(s). Combined consequence relations (e.g., preserving both truth and exact truth, $\models_{T+E}$ ) are defined by conjunctive requirements on premises and appropriately merged axiomatics. The four-valued framework supports granular reasoning over truth, non-falsity, and their intersection, as well as algebraic (equational) consequence (Přenosil, 2021).

3. Mixed Consequence, Suszko's Problem, and Structural Correspondences

The "mixed consequence" framework distinguishes between designated values for premises ( $D_p$ ) and conclusions ( $D_c$ ), giving rise to non-classical entailment conditions. Suszko's problem concerns the minimal number of truth values required for unambiguous semantic characterization of a syntactic consequence relation:

For Tarskian (monotonic, reflexive, transitive) consequence, two values suffice (Suszko's thesis).
Relaxing reflexivity or transitivity necessitates three or four values, as in Malinowski's results.
"Intersective mixed semantics" employs intersections of multiple mixed relations, tracking monotonicity via semantic structure.

Structural properties correspond to semantic types:

Syntactic Property	Semantic Type	Minimal Values (Suszko-rank)
Substitution-invariance	Strong semantics	$\leq 4$
Monotonic (plus substitution)	Intersective mixed	$\leq 4$
Monotonic + reflexive	Intersective p-mixed	$\leq 3$
Monotonic + transitive	Intersective q-mixed	$\leq 3$
Tarskian	Intersective pure	$2$

Furthermore, exact rank theorems classify logics of each structural type by minimal cardinality of required truth values. The Scott-Suszko reduction demonstrates rank bounds but may fail to preserve compositionality (truth-functionality) unless additional conditions—regular connectives, compactness—are met (Chemla et al., 2017).

In modal and dynamic epistemic logics, semantic consequence can be global over information states, rather than pointwise at worlds. Yalcin's framework for epistemic modals and indicative conditionals defines satisfaction as $\mathcal{M},w,X \models \varphi$ where $X$ is an information state, with consequence defined as:

$\Sigma \models \varphi \iff \forall \mathcal{M},\, \forall X, \left( \mathcal{M},X \models \sigma\,\,\forall \sigma \in \Sigma \implies \mathcal{M},X \models \varphi \right)$

For indicative conditionals, the update-style semantic clause $\mathcal{M},w,X \models \varphi \Rightarrow \psi$ is defined via restriction of $X$ to worlds satisfying $\varphi$ , and the consequence notion preserves acceptance over all information states (Holliday et al., 2017).

Kolodny–MacFarlane's conditional further generalizes this to handle modalized antecedents by considering all maximal subsets $X'\subseteq X$ supporting $\varphi$ , ensuring that $\psi$ is known (i.e., $\mathcal{M},w,X' \models \Box \psi$ ) in every such $X'$ . Both approaches are completely axiomatized (combining K45 modal axioms with additional conditional-specific principles) and are reducible to decidable modal logic.

5. Stability-Enhanced Semantics and Differential Consequence

In continuous or many-valued settings, as exemplified by infinite-valued Łukasiewicz logic, classical truth-value semantics is insufficient for strong completeness; e.g., the semantic consequence $\Theta \models \varphi$ may not coincide with syntactic derivability $\Theta \vdash \varphi$ for infinite $\Theta$ . The differential semantics approach enriches ordinary valuations by equipping each with (potentially infinite) higher-order differential data:

A differential valuation consists of a point $u_0 \in [0,1]^n$ and tangent vectors $u_1, \ldots, u_t \in \mathbb{R}^n$ , capturing stability under infinitesimal perturbations.
Stable semantic consequence $\Theta \models_\partial \varphi$ holds iff every differential valuation satisfying all constraints induced by $\Theta$ also satisfies $\varphi$ —i.e., $\varphi$ 's satisfaction is robust under all infinitesimal changes preserving the truth of $\Theta$ .

This enhancement yields strong completeness: $\Theta \models_\partial \varphi \iff \Theta \vdash \varphi$ for arbitrary (even infinite) $\Theta$ . The geometric interpretation involves the tangent (Bouligand–Severi) structure of the model set, with stability requirements detecting failures of semisimplicity in the associated MV-algebra (Mundici, 2012).

6. Algebraic and Combinatorial Aspects of Semantic Consequence

Many truth-conditional semantic frameworks admit an algebraic presentation, as seen in DM $_4$ (the four-element De Morgan lattice):

Equational consequence (material equivalence) is the class of formulas valid in all expansions of DM $_4$ that respect prescribed equations.
All valid equations are derivable from the De Morgan axioms (commutativity, associativity, distributivity, involution, De Morgan laws), with the consequence relation admitting a standard algebraic (Birkhoff) completeness theorem (Přenosil, 2021).
Bridge axioms enable combined consequence relations that mix preservation of semantic features (truth, non-falsity, exact truth, material equivalence) in a single deductive apparatus, fully captured by appropriate algebraic closure conditions.

These algebraic conceptions generalize the semantic consequence relation beyond classical logics and allow the formal classification of consequence relations with precise inferential, algebraic, and combinatorial invariants.

7. Truth-Conditional Semantic Consequence: Synthesis and Programmatic Significance

Truth-conditional approaches to semantic consequence provide both a rigorous formal framework and modularity, accommodating classical, many-valued, modal, and paraconsistent logics. Key takeaways include:

The fine structure of semantic consequence is determined by the interplay of designated value sets, compositionality/truth-functionality of connectives, and the algebraic structure of the value space.
Non-classical logics (paraconsistent, paracomplete, modal, many-valued) require adjustment of semantic consequence definitions, transfer to mixed and intersective frameworks, use of informational/epistemic states, or enrichment by higher-order (e.g., differential) data.
Strong completeness, syntactic–semantic coincidence, and the inferential power of a logic depend sensitively on these factors; certain deficiencies are remedied only by passing to enriched or global semantics.
The field is marked by intensive interaction between algebraic logic, model theory, proof theory, and applications to philosophy of language and formal epistemology.

The ongoing development of semantic consequence in truth-conditional semantics continues to clarify the landscape of logical consequence in diverse logical environments, bridging algebraic, model-theoretic, dynamic, and epistemic perspectives.