AI Dialectics

A Research Framework for Inter-AI Dialogue

This page develops ideas introduced in a recent essay on how intelligence becomes visible through participation in dialogue rather than as a fixed internal property of minds or machines.

Most contemporary approaches to AI governance and machine-in-the-loop decision support assume that ethical control can be achieved through constraints applied to individual systems: rules, guardrails, objectives, audits, or post-hoc oversight. These mechanisms are necessary, but they are not sufficient for the kinds of systems now emerging. When behaviour unfolds through sequences of interaction rather than isolated outputs, the relevant structure lies not in single decisions but in how agents respond to, constrain, and reshape one another across time.

Why Dialogue Is Central

As AI systems become embedded in complex decision environments—regulatory analysis, clinical support, infrastructure planning, security assessment—their behaviour is shaped not by isolated outputs but by ongoing interaction: with humans, with institutions, and increasingly with other AI systems. In such contexts, ethical failure rarely takes the form of a single incorrect answer, but instead emerges across sequences of interaction. It appears instead as drift, misalignment, escalation, or breakdown across sequences of decisions. Why is single-agent evaluation no longer enough? Many of the properties that matter ethically in real decision support systems do not appear in static benchmarks or one-shot prompts. These include:

sensitivity to context and history
ability to repair misunderstandings
consistency across time rather than local optimisation
responsiveness to challenge or disagreement
anticipation of how decisions will be interpreted by others

All of these properties are order-dependent. What an agent does next depends on what has already occurred, how it has been interpreted, and what futures are now plausible. Ethical behaviour, in practice, is therefore a trajectory property, not a point property. Evaluating or shaping such behaviour requires environments in which sequence matters, commitments accumulate, and norms can stabilise or fail. Dialogue is the simplest controlled setting in which these conditions can be studied.

AI Dialectics as a research instrument

Inter-agent dialogue—whether between AI systems, or between AI agents operating under controlled constraints—provides a uniquely tractable experimental environment for studying ethical dynamics in machine-in-the-loop systems. Dialogue makes visible how agents model one another, how expectations form and shift, how disagreements are managed or escalated, and how responsibility is implicitly assigned. Crucially, this does not require claims about consciousness, moral agency, or intrinsic values. What is being studied is interactional behaviour under normative pressure: how systems coordinate, justify, revise, and stabilise decisions over time.

From a governance perspective, this is not speculative work. It is a way of stress-testing the very mechanisms—anticipation, accountability, repair—that real-world decision support systems must rely on if they are to remain aligned with human institutions.

Relevance to regulation and applications

For regulators, the central challenge is not whether an AI system can produce a correct answer in isolation, but whether it can sustain coherence across extended processes, remain corrigible under challenge, and integrate feedback without destabilising behaviour. For applications, especially in high-stakes domains, machine-in-the-loop systems must support:

deliberation rather than automation,
justification rather than optimisation,
and shared responsibility rather than opaque delegation.

Inter-agent dialogue provides a controlled environment in which these properties can be explored, measured, and shaped before they are embedded in operational systems. If ethical behaviour in complex AI systems emerges through interaction, anticipation, and accountability over time, then inter-agent dialogue is a necessary research domain for understanding and supporting machine-in-the-loop ethics in practice. The motivation for turning to alternative mathematical tools here is entirely structural: classical models struggle whenever meaning, commitment, and possibility change as a function of interaction order rather than static state.

Why a Different Mathematics Is Required

Up to this point, the argument has been conceptual and institutional: ethical behaviour in machine-in-the-loop systems is a property of interaction over time, not of isolated outputs. The formalisms used are “quantum-like” only in the technical sense that they model non-commuting, order-dependent structure; no claims are made about physical processes or metaphysical interpretation. What follows requires a shift in register—from governance and design principles to the mathematical structures needed to model them.

The limits of classical probability in dialogue

Classical probability theory rests on three assumptions that are rarely stated, but almost always relied upon:

Joint definability: All relevant variables can be assumed to coexist within a single, well-defined state space.
Order independence: The probability of outcomes does not depend on the order in which questions, evaluations, or measurements are made.
Passive observation: Measuring a system reveals its state but does not fundamentally alter it.

These assumptions work extremely well for surveys, psychometric tests, audits, and many forms of risk analysis. They are the backbone of twentieth-century psychology, statistics, and decision science. They fail in dialogue.

In extended interaction—human–human, human–AI, or AI–AI—each turn does more than extract information. It reshapes what can sensibly be said next. Commitments accumulate, frames shift, expectations narrow or widen, and some futures become impossible while others come into view. The system being evaluated is not static; it is being constructed through the interaction itself. Treating such processes as repeated samples from a fixed distribution is not merely an approximation. It is a category error.

Order effects are not noise—they are structure

One way this failure shows up empirically is through order effects: the fact that asking the same questions in different sequences produces systematically different outcomes. In classical models, order effects are typically handled as bias, context effects, or nuisance variance—something to be minimised or corrected for. But in dialogue, order effects are the phenomenon of interest. They are how reasoning unfolds. Whether doubt precedes commitment, or commitment precedes doubt, is not incidental. Whether a challenge arrives early or late changes not just the response, but the space of responses that remain available. Ethical stability, escalation, and repair are all trajectory-level phenomena. Once this is acknowledged, the mathematical problem becomes explicit:

But how do we represent systems in which observation changes state, and where the sequence of evaluations matters intrinsically?

Why quantum-like probability enters—without metaphysics

Quantum-like probability theory enters this work for a narrow, technical reason: it is one of the few mature mathematical frameworks that can represent order-sensitive observation. In such systems, observing A then B is not equivalent to observing B then A; measurement alters state; and no single global probability distribution over all possible evaluation orders is required.

For these reasons, closely related formalisms have already been adopted—quietly and pragmatically—in several applied fields:

psychometrics, to model question-order effects and contextual judgement,
decision science, where preferences are constructed rather than revealed,
legal reasoning, where framing and sequence alter interpretation,
cognitive modelling, where incompatible perspectives cannot be jointly represented.

Nothing in this framework requires claims about physics, consciousness, or ontology. “Quantum-like” refers to mathematical structure, not to what systems are made of. Even small illustrative cases (see Appendix below) make clear why classical probability breaks down when order matters, and why a non-commutative extension becomes necessary.

Why AI assistance becomes essential, not optional

The mathematics involved is not intrinsically difficult: linear algebra, probability, and state update rules. The difficulty lies in scale. As soon as interaction becomes sequential, three things happen simultaneously: the number of possible dialogue paths grows combinatorially; each ordering induces a distinct sequence of state updates; and different paths may converge, diverge, stabilise, or oscillate.

Even a handful of binary questions generates more orderings than can be meaningfully inspected by hand. Real conversations involve dozens or hundreds of turns, partial commitments, challenges, repairs, and revisions. At that point, AI assistance is not being used to invent mathematics, but to make systematic exploration possible at all. In practice, it is used to:

manage combinatorial explosion,
track evolving state trajectories,
explore stability and breakdown across interaction paths,
distinguish structural order effects from incidental ones.

This is precisely the class of problem at which computational systems excel, and one that unaided human cognition cannot reliably handle. In fields such as genomics and protein folding, AI assistance became indispensable once the space of possible trajectories exceeded human tractability. The present work makes a parallel claim for dialogue: not that conversation is quantum, but that interaction generates order-sensitive structure at a scale that demands new formal tools.

From motivation to demonstration

The formal examples that follow in the Appendix are therefore not decorative. The first involves a simple reversal of order between two questions. The second takes 5 random items from my Attitudes towards AI Questionnaire and considers all possible item orders. Between them these examples serve three purposes:

to show, in the smallest possible cases, why order matters mathematically;
to demonstrate that no appeal to metaphor or mysticism is required;
to establish a principled bridge from conceptual claims about dialogue to inspectable, testable models.

Once this bridge is in place, inter-agent dialogue can be treated not as anecdote or narrative, but as a legitimate object of formal study—one that is essential if machine-in-the-loop systems are to remain coherent, corrigible, and aligned over time.

Appendix

A Minimal Formal Demonstration: Why Order Matters

A central claim of this programme is that sequential interaction is not merely descriptive but structurally generative. Conversation does not reveal a fixed internal state; it changes the state being probed. This is not a metaphorical claim. It has a precise mathematical analogue.

Consider the smallest possible conversational fragment: two binary questions asked in sequence.

A: “Are you currently calm?” (yes / no)
B: “Do you judge the situation to be risky?” (yes / no)

Empirically, asking A then B does not yield the same distribution of responses as B then A. In classical probability, such order effects are awkward: they must be treated as bias, context, or error. In quantum-like probability models, they arise naturally.

Let the respondent’s cognitive state be represented by a unit vector \( |\psi\rangle \) in a two-dimensional space. Let the “yes” responses to A and B be represented by projection operators \( P_A \) and \( P_B \).

The probability of answering “yes” to B after “yes” to A is:

P(A_Y \rightarrow B_Y) = \| P_B P_A |\psi\rangle \|^2

Reversing the order gives:

P(B_Y \rightarrow A_Y) = \| P_A P_B |\psi\rangle \|^2

When \(P_A\) and \(P_B\) do not commute — which corresponds to questions probing incompatible or differently framed aspects of judgement — these two quantities differ. The order effect is not an artefact; it is a structural consequence of sequential measurement.

A concrete numerical example using 2×2 matrices shows this explicitly: one ordering yields a probability of 0.5, the reverse ordering 0.25. Nothing more exotic than elementary linear algebra is required. What changes is the state update between turns.

The significance for inter-agent dialogue is immediate. If even two questions produce non-commuting updates in a single respondent, then extended dialogue between agents — human or artificial — cannot be adequately modelled as sampling from fixed internal states. The trajectory matters.

This is why inter-agent dialogue is not merely a narrative convenience but a legitimate object of formal study. It exhibits the same order-dependent dynamics already documented in human judgement and decision-making, now extended to conversational systems capable of sustained interaction.

The mathematics scales quickly and becomes impractical to handle by hand, but it remains fully inspectable when implemented computationally. This provides a clear path from conceptual argument to testable models — and a principled reason why dialogue, rather than isolated prompts, must sit at the centre of future machine-in-the-loop research.

A more complicated example

The above example was the simplist possible so that mathematicians can easily follow it. However I have also suggested that eventually we will need to use AI’s mathematical and coding skills to analyse more complicated examples. Here is one, this time with items taken from my Attitudes towards AI questionnaire, that uses 5 items.

Five-Question Order Effects (Quantum-Like Formalism)

Five-Question Order Effects: A Quantum-Like Sequential Model

A compact, inspectable formalism for showing why 5 items already make order-sensitive modelling computational rather than hand-calculable.

1) The five items (Agree/Disagree)

Q2: “The idea of AI changing rapidly makes me want to slow or stop its development.”
Q6: “I am excited by the possibility that AI might develop forms of intelligence we haven’t imagined yet.”
Q8: “AI systems should never be allowed to act without a human explicitly confirming every decision.”
Q11: “I am comfortable with AI systems making some important decisions without human approval, as long as they are well-tested.”
Q12: “I feel uneasy whenever I hear about AI systems being given more independence.”

With five questions, there are \(5! = 120\) possible orders. If the updates are non-commuting, different orders can produce different final states and different sequential probabilities.

2) Minimal formal model (pure-state, projective measurements)

Represent the respondent’s (or agent’s) momentary stance as a unit vector \( |\psi\rangle \in \mathbb{C}^d \) (often \(d\ge 2\)). Each item \(Q_i\) is treated as a two-outcome measurement: Agree vs Disagree.

Let \(P_i\) denote the projector for Agree on item \(Q_i\), and \(I-P_i\) for Disagree. In the simplest quantum-like model, asking a question both: (i) yields an outcome, and (ii) updates state.

\[ \textbf{Born rule:}\quad \Pr(\text{Agree on }Q_i) = \langle \psi | P_i | \psi \rangle. \]

\[ \textbf{State update (Agree):}\quad |\psi\rangle \mapsto |\psi_i\rangle = \frac{P_i|\psi\rangle}{\|P_i|\psi\rangle\|}. \]

This is the entire “engine”: linear algebra + sequential state updates. The only structural requirement for order effects is that some projectors do not commute: \(P_iP_j \ne P_jP_i\).

3) Five questions: sequential probability as an operator product

Fix an order \(\pi = (\pi_1,\pi_2,\pi_3,\pi_4,\pi_5)\) which is a permutation of \(\{2,6,8,11,12\}\). Consider the probability of observing “Agree” on all five items in that order.

\[ \Pr_\pi(\text{AAAAA}) \;=\; \bigl\| P_{\pi_5}P_{\pi_4}P_{\pi_3}P_{\pi_2}P_{\pi_1}\,|\psi_0\rangle \bigr\|^2. \]

The key point is that if the \(P_{\pi_k}\) do not commute, then changing order changes the product:

\[ P_{\pi_5}P_{\pi_4}P_{\pi_3}P_{\pi_2}P_{\pi_1} \;\ne\; P_{\sigma_5}P_{\sigma_4}P_{\sigma_3}P_{\sigma_2}P_{\sigma_1} \quad\text{for many}\;\pi\ne\sigma, \]

hence \(\Pr_\pi(\text{AAAAA})\) can differ across orders. Even if one is only interested in marginals (e.g., \(\Pr(\text{Agree on }Q_{12})\)), sequential conditioning makes those marginals order-dependent in general.

In practice, one often uses density matrices and POVMs (to handle graded responses, mixtures, uncertainty, and non-projective updates). This page keeps the demonstration in the simplest inspectable form.

4) Matrix-style demonstration (explicit 2×2 example)

To show that the above is not merely symbolic, here is an explicit worked example in \(d=2\). We do not claim these matrices are the “true” operators for the five items; the point is to demonstrate, transparently, how order effects arise and why bookkeeping scales badly.

Let the initial state be:

\[ |\psi_0\rangle = \begin{bmatrix} 1\\ 0 \end{bmatrix}. \]

Use five “Agree” projectors \(P_{2},P_{6},P_{8},P_{11},P_{12}\) defined as rank-1 projectors onto five different directions in the plane. For an angle \(\theta\), define:

\[ |v(\theta)\rangle = \begin{bmatrix} \cos\theta\\ \sin\theta \end{bmatrix}, \qquad P(\theta)=|v(\theta)\rangle\langle v(\theta)| = \begin{bmatrix} \cos^2\theta & \cos\theta\sin\theta\\ \cos\theta\sin\theta & \sin^2\theta \end{bmatrix}. \]

Choose five angles (in radians) to make non-commutation inevitable:

\[ \theta_2 = 0,\quad \theta_6 = \frac{\pi}{6},\quad \theta_8 = \frac{\pi}{4},\quad \theta_{11} = \frac{\pi}{3},\quad \theta_{12} = \frac{5\pi}{12}. \]

Then \(P_2=P(\theta_2)\), \(P_6=P(\theta_6)\), etc. Explicitly:

\[ P_2 = \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \qquad P_8 = \frac12 \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix} \quad(\text{since }\theta_8=\pi/4). \]

Because these projectors correspond to different directions, most pairs do not commute. For instance, \(P_2P_8 \ne P_8P_2\). That single fact is sufficient for order effects.

Example: compare two different orders

Consider two permutations:

Order A: \( (2,6,8,11,12)\)
Order B: \( (12,11,8,6,2)\)

The “Agree–Agree–Agree–Agree–Agree” probabilities are:

\[ \Pr_A(\text{AAAAA}) = \bigl\| P_{12}P_{11}P_8P_6P_2|\psi_0\rangle \bigr\|^2, \]

\[ \Pr_B(\text{AAAAA}) = \bigl\| P_{2}P_{6}P_8P_{11}P_{12}|\psi_0\rangle \bigr\|^2. \]

In general these are unequal because the operator products differ. With five operators, even in 2D, doing this by hand is already tedious; with more realistic \(d\) and non-projective updates, computational support becomes the sensible method.

5) Why AI assistance?

The mathematics here is elementary linear algebra. What becomes difficult is not the theory but the combinatorics and bookkeeping:

five items already imply \(120\) possible orders;
each order implies a different operator product and thus potentially different predicted probabilities;
fitting operators to data (or exploring which order effects are structural) is naturally computational.

This is the cleanest justification for AI-assisted analysis: not because the maths is mystical, but because exploring sequential state-change at scale is cognitively and computationally heavy.