Seven-Valued Contextual Logic Overview

• Seven-valued contextual logic is a framework that assigns seven distinct truth patterns based on combinations of true, false, and indeterminate in different contexts.
• It resolves quantum paradoxes by encoding context-dependent predications, thereby handling contradictory states without collapsing into classical logic.
• Its robust algebraic and categorical foundations not only unify logical traditions but also offer practical applications in physics, cognitive science, and decision theory.

A seven-valued contextual logic is a formally precise logical framework that generalizes the traditional binary and ternary logics to capture context-dependence, complementarity, and partial indeterminacy, central to quantum theory and other domains involving incompatible perspectives. Originating in quantum foundational studies, logical pluralism, and knowledge representation, this system organizes all nonempty combinations of “true” (T), “false” (F), and “indeterminate/unsayable” (U) across mutually incompatible contexts, yielding exactly seven distinct truth patterns. Its semantic encoding resolves paradoxes such as measurement problems and ambiguous predications by relativizing truth to experimental or observational setups, with algebraic, lattice-theoretic, and categorical foundations unifying various philosophical and formal traditions (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025, Greco et al., 2023, Ghose, 13 Dec 2025).

1. Formal Definition and Structural Properties

The seven-valued logic is rigorously constructed by extending the set of possible truth assignments for a proposition $P$ to encompass its evaluation across one, two, or three pairwise incompatible contexts $c_1, c_2, c_3$. Each context-specific atomic assertion—$c_i{:}T$ (“true under $c_i$”), $c_i{:}F$ (“false under $c_i$”), $c_i{:}U$ (“indeterminate under $c_i$”)—can be combined into compound predications:

Compound Predication	Interpretation	Formulaic Formulation
$c_1{:}T$	True in $c_1$, nowhere else	$\forall x[\phi_1(x)\to p(x)]$
$c_1{:}F$	False in $c_1$	$\forall x[\phi_1(x)\to\neg p(x)]$
$c_1{:}U$	Indeterminate in $c_1$	$\forall x[\phi_1(x)\to q(x)]$
$c_1{:}T \wedge c_2{:}F$	True in $c_1$, false in $c_2$	...$\wedge$... under $\neg(\phi_1\leftrightarrow \phi_2)$
$c_1{:}T \wedge c_2{:}U$	True in $c_1$, indeterminate in $c_2$	...
$c_1{:}F \wedge c_2{:}U$	False in $c_1$, indeterminate in $c_2$	...
$c_1{:}T \wedge c_2{:}F \wedge c_3{:}U$	True, false, indeterminate across three contexts	...

The seven values correspond to the seven nonempty subsets of $\{T,F,U\}$ when viewed through the lens of context separation.

Under this construction, the logic forms a seven-element algebraic lattice (Boolean algebra on context-indexed triples, or Heyting algebra in categorical settings), with meet ($\wedge$) and join ($\vee$) inherited from coordinatewise operations:

$$V(P,c) = (t, f, u) \in \{0,1\}^3 \setminus \{(0,0,0)\}$$

Negation and implication are context-local, blocking explosion and retaining paraconsistency even under contradictory assertions within a single context (Ghose, 13 Dec 2025, Ghose et al., 14 May 2025, Greco et al., 2023).

2. Contextuality, Complementarity, and Quantum Logic Origins

The logic formalizes two pivotal quantum principles:

• Contextuality: The truth of any statement about a quantum system depends on the specific measurement context—a proposition about a system cannot be globally assigned a non-contextual truth-value. Logical syntax encodes this via context-predicates (e.g., $\phi(x)$) and quantification, so all predications are context-indexed (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025).
• Complementarity: Mutually exclusive properties (e.g., wave vs. particle behavior) are required for a full quantum account, but cannot be jointly asserted in a single context. The seven-valued logic enables simultaneous, non-explosive representation of complementary aspects via compound predications—each holding in its distinct context, preventing logical paradox.

These features render the logic robust to canonical quantum paradoxes (Schrödinger’s cat, Wigner’s friend, EPR) by distributing “contradictory” or “indeterminate” statements safely across incompatible modes of observation (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025, Ghose, 13 Dec 2025).

3. Algebraic and Model-Theoretic Foundations

An equivalent algebraic presentation arises from the Pawlak–Brouwer–Zadeh lattice. Each proposition’s state is encoded by evidence-bits $(p,q,r)$, partitioning the universe into seven blocks:

Label	Tuple $(p,q,r)$	Meaning
T	(1,0,0)	Definitely true
sT	(1,0,1)	Sometimes true
U	(0,0,1)	Unknown
K	(1,1,0)	Contradictory
fK	(1,1,1)	Fully contradictory
sF	(0,1,1)	Sometimes false
F	(0,1,0)	Definitely false

Lattice operations, De Morgan–BZ algebra identities, and the preservation of paraconsistency (involution $x''=x$, interconnection $x^\sim\le x'$) are satisfied, providing a mathematically rigorous base for inference and reasoning. Classical two-valued, three-valued, and Belnap’s four-valued logics appear as quotient and sub-logics within this structure (Greco et al., 2023).

In categorical/topos-theoretic context, the logic emerges as the finite Heyting algebra of global elements—a presheaf of context-dependent truth-assignments over a base category of measurement contexts. Contextuality coincides with the obstruction to gluing local sections, measured by Čech cohomology (Ghose, 13 Dec 2025).

4. Semantics: Connectives, Truth-Tables, and Quantifiers

Logical connectives are interpreted contextually. Within a fixed context, connectives mirror standard three-valued logic (Kleene-style):

• Negation: $\neg T = F$, $\neg F = T$, $\neg U = U$
• Conjunction, disjunction: componentwise operations on triples

For formulas under different contexts, $\alpha$ in $c_i$ and $\beta$ in $c_j$, $\alpha \wedge \beta$ becomes the compound predication $(c_i{:}\mathrm{val}_1 \wedge c_j{:}\mathrm{val}_2)$, forming a new element of the seven-valued algebra.

Quantifiers internalize contextuality: $\forall x[\phi_i(x)\to p(x)]$, $\exists x[\phi_i(x)\wedge p(x)]$, always relativized to explicit context predicates, fully aligning with Bohr’s and Wittgenstein's language-game perspectives (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025).

5. Resolution of Paradoxes and Illustrative Applications

The logic provides uniformly consistent resolutions to classical quantum paradoxes:

• Double-Slit: Assigns truth (detection) in which-way context, indeterminacy in interference context, blocking paradoxes.
• Schrödinger’s Cat: “Alive & dead” arises only from crossing contexts ($c_\mathrm{open}$ and $c_\mathrm{closed}$); each context is consistent alone.
• Wigner’s Friend: Collapse and superposition described simultaneously, each within its observer’s context.
• EPR/Bell: Replaces the problematic notion of non-local “real factual situations” with context-indexed predications.

Beyond quantum mechanics, the framework models phenomena in perceptual psychology (contextual thresholds), cognitive science (order effects, contextuality-by-default), and decision theory (ambiguity aversion as contextually indeterminate, dissolving classical Bayesian paradoxes) (Ghose et al., 14 May 2025).

6. Comparative Perspectives and Conceptual Synthesis

Reichenbach’s three-valued logic models quantum indeterminacy but collapses indeterminacy and contextuality into a single undifferentiated value. The seven-valued system generalizes and refines the account by explicit context-indexing and context-separated compound truths (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025).

In comparison to the Jaina saptabhaṅgī­naya, sevenfold predication is mechanized by quantifiers rather than epistemic language, enabling compositional, algebraic, paraconsistent semantics suitable for predicate logic and topos theory (Ghose, 1 Oct 2025).

From the categorical viewpoint, the algebra of seven values corresponds to global elements of the subobject classifier in a presheaf topos over measurement contexts. Classical truth is recovered by sheafification, i.e., forcing gluing of local data—so “measurement” is mathematically the functor $F\mapsto F^\sharp$ collapsing many-valued contextual logic to Boolean (Ghose, 13 Dec 2025).

7. Theoretical Impact and Future Applications

Seven-valued contextual logic thus establishes:

• A general logical foundation for quantum complementarity and contextuality, dissolving all major quantum paradoxes without appealing to nonlocality or ill-defined collapse.
• Robust paraconsistent and algebraic reasoning underpinning pluralistic epistemologies and data-driven inference in non-quantum domains.
• A categorical framework linking logical context-dependence to topological and cohomological obstructions in gluing local perspectives, suggesting further mathematical research and applications in logic, probability, and information theory.
• A plausible implication is the broadened applicability of contextual logics in cognitive modeling, social science, and complex systems, wherever incompatible perspectives and partial truth are empirically manifest.

References: (Ghose et al., 14 May 2025, Ghose, 1 Oct 2025, Greco et al., 2023, Ghose, 13 Dec 2025).