Emergent Thought and Classical Logic

• Emergent Thought and Classical Logic is defined by the spontaneous formation of irreducible cognitive entities that transcend classical Boolean operations.
• Empirical evidence reveals systematic deviations from classical set theory, evidenced by overextension and underextension in human conceptual combinations.
• Quantum-theoretic models integrate emergent and logical reasoning via a two-sector Fock space, offering new insights into human conceptual dynamics.

Emergent thought encompasses the spontaneous formation of new cognitive entities, concepts, or judgments that cannot be reduced to the combinatorics or inference rules of classical logic. While classical logic, with its Boolean connectives and Kolmogorovian probabilistic semantics, underpins formal mathematics and much of traditional cognitive science, empirical research shows systematic violations of classicality in human conceptual combination and reasoning. Recent advances in quantum-theoretic modeling provide principled formalisms for capturing both the non-classical, emergent aspects of human thought and their interplay with logic-based processes.

1. Classical Logic in Cognition and Its Limitations

Classical logic, as codified in Boolean algebras and set-theoretic operations, underpins the standard treatment of logical reasoning and probability in cognitive theories. In this framework, concepts are represented as sets or events, and their combinations—conjunction ($A \land B$), disjunction ($A \lor B$), and negation ($\neg A$)—obey the algebraic rules:

• $\mu(A \land B) = \min\{\mu(A),\mu(B)\}$
• $\mu(A \lor B) = \max\{\mu(A),\mu(B)\}$
• $\mu(\neg A) = 1 - \mu(A)$

A single Kolmogorovian probability space models all concept combinations (Aerts et al., 2015). Marginal laws and normalization constraints are maintained, such that the sum of weights over exhaustive, mutually exclusive events equals unity. This structure was developed in the wake of Laplace’s "Théorie analytique des probabilités" (1812), which remains foundational to probability theory (Aerts et al., 2012).

Empirical investigations, however, consistently reveal robust patterns of deviation from these classical constraints—specifically, overextension in conjunctions, underextension in disjunctions, and the phenomenon of “borderline contradictions” in human judgments (Aerts et al., 2015).

2. Conceptual Emergence: Definitions and Empirical Markers

Conceptual emergence, or emergent reasoning, is distinguished by the creation of new, irreducible meanings when concepts are combined. The emergent compound (e.g., "Pet Fish") exhibits membership properties and typicalities not explained by any Boolean function of its constituents. Hallmark empirical effects include:

• Guppy Effect: An exemplar (e.g., “guppy”) is more typical of "Pet Fish" than of "Pet" or "Fish" separately.
• Overextension: $\mu(A \land B) > \min\{\mu(A),\mu(B)\}$
• Underextension: $\mu(A \lor B) < \max\{\mu(A),\mu(B)\}$
• Vagueness and Borderline Contradictions: Items judged as both "tall and not tall" with significant frequency. These phenomena systematically violate the expectations of classical set theory and probability (Aerts et al., 2015, Aerts et al., 2014).

Empirical studies document stable patterns of classical-violation across diverse concept pairs and exemplars. For instance, mean deviations from classicality measured by $$I_A = \mu(A) - [\mu(A \land B) + \mu(A \land \neg B)]$$ are approximately $-0.42$ (SD $A \lor B$0), with similar magnitudes found for $A \lor B$1, $A \lor B$2, and $A \lor B$3. The total normalization deviation $A \lor B$4 (Aerts et al., 2015).

3. Quantum-Theoretic Formalism for Emergent Thought

Quantum-theoretic modeling posits that human thought is governed by a superposition of two concurrent processes:

• A logical reasoning sector, corresponding to algebraic (probabilistic) logic.
• An emergent conceptual reasoning sector, in which new cognitive entity states are spontaneously generated (Aerts et al., 2014).

Mathematically, cognition is represented in a two-sector Fock space:

$A \lor B$5

• Sector 1 ($A \lor B$6): Emergent reasoning via superpositions of concept vectors (e.g., $A \lor B$7). The conjunction "A and B" is modeled as the normalized superposition $A \lor B$8.
• Sector 2 ($A \lor B$9): Logical reasoning, with concept pairs represented as tensor products, allowing for entanglement and classical logical projectors.

Membership weights are given by the Born rule:

$\neg A$0

For emergent conjunctions:

$\neg A$1

where $\neg A$2 is a quantum interference term.

Logical conjunction in sector 2 is computed as:

$\neg A$3

The overall cognitive state is a superposition across sectors:

$\neg A$4

Leading to the membership expression:

$\neg A$5

(Aerts et al., 2014, Aerts et al., 2015).

4. Empirical Evidence: Systematic Violation of Classicality

Experiments involving concept pairs (e.g., "Food and Plant", "Sportswear or Sports Equipment") and their negations reveal pervasive patterns of overextension and underextension, not only in mean values but as a function of the entire membership-weight distributions. A tabular summary of deviation functions (Aerts et al., 2015):

Function ($\neg A$6)	Mean Value	SD
$\neg A$7	–0.42	0.09
$\neg A$8	–0.43	0.08
$\neg A$9	–0.35	0.09
$\mu(A \land B) = \min\{\mu(A),\mu(B)\}$0	–0.33	0.09
$\mu(A \land B) = \min\{\mu(A),\mu(B)\}$1	–0.81	0.13

Linear regression confirms the stability of these violations ($\mu(A \land B) = \min\{\mu(A),\mu(B)\}$2). Statistical analysis rules out dependence on particular concepts or exemplars, indicating structural non-classicality in concept formation (Aerts et al., 2015).

Quantum-fitted models assign greater weight to emergent reasoning parameters ($\mu(A \land B) = \min\{\mu(A),\mu(B)\}$3) than to logical ones ($\mu(A \land B) = \min\{\mu(A),\mu(B)\}$4), e.g., $\mu(A \land B) = \min\{\mu(A),\mu(B)\}$5 for the "Mint" conjunction-overextension case versus $\mu(A \land B) = \min\{\mu(A),\mu(B)\}$6 (Aerts et al., 2014).

5. Developmental and Computational Perspectives on Logical Emergence

Classical logical operations can themselves be seen as emergent, arising through a hierarchy of cognitive developmental stages. According to the Piagetian framework:

• Preoperational: Only static symbol formation, no operations.
• Concrete operational: Discovery and internalization of reversible operations, supporting concrete conjunction and disjunction.
• Formal operational: Fully abstract hypothesis manipulation, with the introduction of implication ($\mu(A \land B) = \min\{\mu(A),\mu(B)\}$7) and negation ($\mu(A \land B) = \min\{\mu(A),\mu(B)\}$8), operating over arbitrary propositions (Ivan et al., 2019).

The model sketched in (Ivan et al., 2019) suggests a multi-stage mechanism:

1. Sensorimotor exploration of states.
2. Pattern extraction to symbols.
3. Invariance detection and the internalization of operators.
4. Abstraction to schema—elevating regularities to formal axioms.
5. Symbolic/propositional inference, with classical logical rules emerging at the highest abstraction.

This process is not inherently classical; the abstraction to full logical reasoning occurs after—and atop—phases where emergent, context-sensitive conceptualization predominates.

6. Theoretical Integration and Future Directions

The above findings identify emergent thought not as an anomalous deviation from classical logic, but as the dominant, structurally primary mode of human conceptual combination. Classical logic and its probabilistic semantics provide a secondary, partially corrective layer within a more general quantum-inspired formalism (Aerts et al., 2014, Aerts et al., 2015). Sector 2 (logical) is generally subdominant to sector 1 (emergence), though both are required to quantitatively fit observed data.

The two-sector Fock space model integrates both algebraic and emergent modes, offering a unifying mathematical structure for studying cognition, concept dynamics, and potential applications in artificial intelligence. This framework calls for new investigations into higher-order conceptual combinations, the mathematical taxonomy of emergent connectives, and empirical work extending beyond conjunction and negation (Aerts et al., 2015).

Standard fuzzy-set and Boolean approaches are thus inadequate for capturing the generative, irreducible process of concept formation. A plausible implication is that further axiomatizations of cognitive logic must explicitly accommodate emergence as a foundational process, potentially leading to a refined and empirically-grounded “quantum logic of concepts” (Aerts et al., 2015, Aerts et al., 2014).