Possible formalisations of an Orwellian problem

Question

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Worked Solution

1. Restating the problem in plain language

We have

a population of agents (people) and a required set  (R) of propositions that the organisation demands everyone to accept;
an agent commits “thought‑crime” when, on his own, he believes (or judges) a proposition that contradicts some member of (R).

When an agent (a) is caught committing thought‑crime the organisation forces him through the following steps

step	informal description	formal requirement
P1	(a) must accept every proposition in the current required set (R).	After P1, (Bel_a\supseteq R).
P2	Later, when the organisation requires a new proposition (p) (or decides to drop some (q\in R)), (a) must be able to replace an already accepted proposition by an incompatible one.	There is a later time (t) with (p\in R_t) and (Bel_a(t)) contains a proposition (q) with (q\vdash\neg p).
P3	(a) may be removed from the population only after he has successfully gone through P1 and P2.	Removal (\Rightarrow) “P1 and P2 have been executed”.
P4	At any moment there must be enough agents left so that the above procedure can be applied to any future offender.	The set of “still‑alive’’ agents is never emptied.

A distinguished administrator (call him (A)) can

detect thought‑crime,
decide whether the agent’s dissent is genuine or a pretence, and
force the agent through the two forced belief‑changes (P1 and P2).

The questions are

Which formal framework (transition systems, modal/temporal logic, belief‑revision theory, fixed‑point theory, …) is best suited to capture the situation?
What extra assumptions are needed to make the description mathematically precise?
Can the rules generate an infinite regress of agents that must judge each other’s thought‑crime? If a regress is possible, does it constitute a genuine logical obstruction?
Is it possible (in a well‑defined formalisation) to satisfy P1–P4 simultaneously?

Below we answer each of these points step‑by‑step.

2. Choosing a formal framework

The problem mixes three ingredients

Epistemic attitude of many agents (what they believe).
Dynamic manipulation of those attitudes by an external authority (forced belief‑change).
Temporal ordering (P1 happens first, then later P2, then possibly removal).

A single formalism that naturally incorporates all three is Dynamic Epistemic Logic (DEL) together with an AGM‑style belief‑revision operator for the individual agents.

Epistemic part – a Kripke structure (\mathcal{M} = (W, {R_i}_{i\in \mathsf{Ag}}, V)) where each (R_i) relates worlds that agent (i) cannot distinguish. The proposition set (Prop) is interpreted by (V). The “required set’’ (R\subseteq Prop) is a global datum (the Party line).
Dynamic part – DEL actions (also called product updates) model the forced operations:
- *Accept(_a)** : an action model that forces every world compatible with the current state to satisfy all (p\in R). In epistemic terms it publicly announces the conjunction (\bigwedge_{p\in R}p) *to agent (a).
- Replace(_a(p,q)) : an action that publicly announces a revision of (a)’s belief set, replacing a formerly accepted proposition (p) by an incompatible one (q) (with (p\vdash\neg q)). This is exactly the AGM revision operator (*) with post‑condition (Bel_a^{new}= (Bel_a\setminus{p})\cup{q}).
Temporal part – we index the global state by a discrete time line (\mathbb{N}): the whole system evolves as a sequence (\mathcal{M}_0,\mathcal{M}_1,\dots). The administrator’s detection and the forced actions are deterministic functions of the current state.

Thus a state of the whole system is a tuple

[ \Sigma_t = \bigl(\mathcal{M}_t,\; \text{Alive}_t\subseteq \mathsf{Ag}\bigr) ]

where (\text{Alive}_t) are the agents that have not yet been removed.
The transition function (\tau) maps a state and a chosen “offender’’ (a) to the next state obtained by

applying Accept(_a) (P1),
applying Replace(_a) for some required new proposition (P2),
optionally removing (a) (P3).

P4 is expressed as the invariant

[ \forall t\; \bigl|\text{Alive}_t\bigr| \ge 1 . ]

Why other frameworks are insufficient (or can be reduced to DEL)

Candidate	Reason it alone does not capture everything
Pure transition system (states + arrows)	Can encode the dynamics, but gives no way to talk about beliefs of agents inside a state.
Plain temporal logic (LTL/CTL)	Allows ordering of events but again lacks epistemic operators (what agents consider true).
Modal epistemic logic without dynamics	Describes static beliefs but cannot express forced belief‑change.
Fixed‑point theory (e.g., Kripke–Fixpoint)	Useful for defining the set of offenders as the least fixed point of a detection operator, but still needs a base language for beliefs.
AGM belief revision alone	Gives the local operation “replace a belief”, but does not describe who performs the operation, when, or the global invariant P4.

Therefore the most natural single umbrella is Dynamic Epistemic Logic with AGM‑style revision, which already contains a Kripke (modal) semantics, a temporal product‑update mechanism, and a well‑studied belief‑revision operator.

3. Making the description mathematically precise

We add the following assumptions (all of them are standard in DEL/AGM literature):

Finite proposition set – (Prop) is finite (or at least countable) so that the conjunction (\bigwedge_{p\in R}) is a well‑formed formula.
Non‑omniscient agents – each agent’s belief set (Bel_a(t)) is deductively closed under a fixed logical consequence relation (\vdash) only for the propositions they have already accepted. In particular, after P1 the agent may still be ignorant of the incompatibility of the later‑required proposition; this permits P2 without immediate inconsistency.
Paraconsistent allowance – we adopt a paraconsistent background logic (e.g., Priest’s LP) so that a belief set can temporarily contain both (p) and (\neg p) without exploding to triviality. This is crucial because P2 forces the agent to replace a required proposition by an incompatible one; in a classical setting the belief set would become inconsistent and the revision operator would be undefined.
Deterministic detection – the administrator (A) has a primitive detection predicate (Detect(a,t)) that returns true iff (a) has, at time (t), a belief contradicting (R). We axiomatise this predicate as:

[ Detect(a,t)\;\Longleftrightarrow\; \exists p\in R\;.\; Bel_a(t)\vdash \neg p . ]

No other agent is needed for the detection; hence no regress arises.
Sufficient population – we assume a lower bound (k\ge 1) on the number of agents that must stay alive at all times. Formally

[ \forall t\; |\text{Alive}_t|\ge k . ]

The value of (k) can be taken as 1 (the system can survive with a single “reformed’’ agent) or larger if the story demands it.

With these five assumptions the informal rules P1–P4 become formal invariants on the transition system ((\Sigma_t)_{t\in\mathbb N}):

P1 (Accept)
[ \forall t\;\bigl(Detect(a,t)\rightarrow Bel_a(t+1)\supseteq R\bigr) ]
P2 (Replace) – there exists a later time (t’>t) such that a new required proposition (p\in R_{t’}\setminus R_t) is believed while some earlier required (q\in R_t) is rejected:

[ \exists t’>t\; \bigl(p\in R_{t’}\setminus R_t\wedge Bel_a(t’)\vdash p\wedge Bel_a(t’)\vdash\neg q\bigr) ]
P3 (Removal)

[ Remove(a,t) \;\Longleftrightarrow\; \bigl(Detect(a,t’)\wedge P1\wedge P2\ \text{already executed for }a\bigr) ]
P4 (Survival)

[ \forall t\; |\text{Alive}_t|\ge k . ]

All of these are first‑order statements over the underlying DEL structure, so the whole system is a first‑order definable transition system.

4. Does the system generate an infinite regress of agents?

4.1 Where a regress could appear

The only place where an extra agent would be needed to judge thought‑crime is the clause “determine whether agent (a) has committed thought‑crime”. If we tried to model this judgment as another agent’s belief (“(b) believes that (a) has contradicted the Party line”), we would indeed obtain a chain:

[ \text{Detect}(a) \text{ depends on } Bel_b,\; \text{Detect}(b) \text{ depends on } Bel_c,\; \dots ]

If the detection predicate were defined solely in terms of other agents’ beliefs, we could get an infinite hierarchy with no base case.

4.2 How to block the regress

In our formalisation we axiomatised the detection predicate as a primitive, admin‑controlled operation (Assumption 4). The administrator is a distinguished agent whose epistemic state is outside the system; we do not require any other agent to verify the detection.

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