PRH | Essay | 7.27 • Blurred Boolean Algebra

Blurred Boolean Algebra Logic at Positive Blur and the Limit b→0 Aleksandar Perišić December 2025 Abstract The blur program insists that most hard mathematical questions should be first asked at positive blur: with finite resolution, finite budgets, and explicit control of what is being averaged away. In practice, this is exactly how we analysed the Collatz problem, Navier– Stokes, and the Church–Turing thesis: we first found clean, finite, blur–level statements, and only later asked which of them can be upgraded to the sharp b= 0 world. In this paper we make the corresponding logical layer explicit. We introduce a blurred Boolean algebra (or Bloorean algebra): a family of three-valued valuations vb∈ {T, F, U} indexed by blur levels b≥ 0, together with an availability predicate Ab ( P )which says when a proposition P is actually decidable at blur b . On the available fragment at each positive blur the logic is fully classical: the law of excluded middle holds as Ab(P)⇒Pb∨ ¬Pb, whereas globally we only have the blurred law Ab(P)∧(Pb∨ ¬Pb)∨ ¬Ab(P), which matches our everyday practice of saying “either it is decidable and classical, or we simply do not know yet at this resolution”. The value b = 0 plays a special role: truth at blur zero is not assumed to be primitive. Instead we show how it arises only when a blur–stable pattern ( vb ( P )) b↓0 with uniform guards allows us to send b→ 0without changing the logical value. This explains why Collatz could be settled at a discrete blur level long before we had a safe passage to b = 0, and why for Navier–Stokes the blur–only analysis was correctly agnostic about the Platonic problem. We end by positioning blurred Boolean algebra among other non-classical logics (three-valued, intuitionistic, partial) and arguing that it captures exactly the epistemic structure of “reasoning under blur” used throughout the program. 1 Why blur needs its own logic The blur viewpoint starts from a simple operational fact: in any serious problem we never have access to the Platonic configuration. We can only probe it through blurred measurements, approximations, or certificates at some finite resolution b > 0and with a finite budget of computation or observation. In the Collatz program, for example, we worked with residue classes modulo 2 k and Lyapunov drift budgets. In the Navier–Stokes analysis we examined Gaussian-blurred avatars of flows and derived energy inequalities at each blur level. In the Church–Turing story we insisted that “effective” means “able to extract more information from blur with finite cost”. At each of these blur levels we ended up making logical statements: •“For every odd residue rmod 213 we have negative drift” (Collatz). • “For each finite blur scale b > 0solutions of Navier–Stokes stay bounded in the appropriate norm” (blur–only NS). 1 • “This model of computation can always take a finite blurred description and refine it, at finite cost, into a sharper one” (Church–Turing under blur). The puzzle is that these statements behave logically as if they were classical on their own domain, yet we cannot treat the Platonic b = 0 statements in the same way without extra work. For Collatz the missing piece was to formalise what it means to move smoothly from a discrete blur level to b = 0 without changing the logical value. For Navier–Stokes we explicitly stopped at blur and said: from blur alone the sharp problem is undecidable; we need additional structure (the Platonic universes pattern) to make the bridge. This paper is simply a precise account of that practice. We do not propose a radical new logic. Instead we introduce a light–weight, but very explicit, layer on top of ordinary Boolean logic: •a three-valued semantics at each blur b:true,false,orundetermined, • an availability predicate Ab ( P ), saying when the blur protocol makes P actually decidable at level b, • classical behaviour (including the law of excluded middle) on the available part at every positive blur, •and a derived notion of truth at blur zero, obtained only from blur–stable patterns. We call this structure a blurred Boolean algebra (or tongue-in-cheek, a Bloorean algebra): Boolean in spirit, but indexed by blur and honest about what we can and cannot decide at each resolution. 2 Blur levels and three-valued semantics We start with the semantic side: blur levels and valuations. The syntax can be the usual propositional or first-order language; nothing essentially new is needed there. 2.1 Blur levels Definition 2.1 (Blur levels).Ablur scale is a parameter b in B := (0 ,∞ ) ∪ { 0 } . Intuitively, b > 0measures how coarse our current resolution is (larger b means more blur), while b = 0 denotes the idealised sharp world. In concrete problems bmay stand for: •the width of a Gaussian kernel in a heat-blur of a measure or flow, •the scale of a residue modulus 2kor pk, •a numerical grid size in a discretisation, •a tolerance or error budget in a computation. 2.2 Three-valued valuations at each blur At each blur level we want to allow for the possibility that some propositions are not decidable yet. This is different from being false; it simply means “we do not know at this resolution / budget”. Definition 2.2 (Blurred valuation).Let L be a fixed propositional language with connectives ∧,∨,¬. A blurred valuation on Lis a family vb:L −→ {T, F, U}, b ∈B, assigning to each formula P and each blur level b one of three epistemic values: true,false, or undetermined. 2 We write Tb(P)for vb(P) = T, Fb(P)for vb(P) = F, Ub(P)for vb(P) = U. The value U is not a third truth value in the Platonic sense; it is an honest admission that the blur protocol does not (yet) decide P at that resolution. This is a key difference from, say, Kleene’s strong three-valued logic where U is a genuine third truth value. Here, U is epistemic, tied to band the available budgets. 2.3 Availability at blur b We single out the formulas which are actually decidable at a given blur. Definition 2.3 (Availability).For each blur level bdefine the availability predicate Ab(P) :⇐⇒ vb(P)∈ {T, F }. The available fragment at blur bis Avb:= {P∈L:Ab(P)}. Intuitively, Ab ( P )reads: “with blur b and the agreed protocol (numerical precision, Lyapunov certificate, energy inequality, . . . ), we can actually decide P ”. In practice Ab will depend not just on b but also on resource budgets: how far we are allowed to iterate, which certificates we have, which a priori bounds are available, and so on. For the present paper we keep that dependence implicit. 2.4 Why blurred Boolean algebra is not superfluous One might object that a blurred Boolean algebra is unnecessary: why add an explicit third truth value when, in practice, mathematicians already say “we do not know yet” and move on? From the blur perspective, the point is more subtle. The extra value Ub ( P )is not a metaphysical new truth status; it is a piece of bookkeeping about missing information at a given blur. Concretely, the blurred evaluation Tb(P), Fb(P), Ub(P) does two things that classical T/F does not: • It gives us a controlled way to move between blur levels. When Ub ( P )holds, we know that the current blur is too coarse to decide P , but we can still track how P behaves at finer blurs b′< b . This lets us say “unknown at this scale, but compatible with the following sharper behaviours”, instead of a featureless “unknown”. • It forces us to account for the missing bits. In the transition from Ub ( P )to a sharp T0 ( P ) or F0 ( P ), the stability conditions spell out exactly which arguments must be upgraded: which error terms must be controlled, which parameters must be fixed, which invariants must be added. In other words, Ub ( P )carries a structured “unknown yet because”: it points to the gap, rather than merely reporting ignorance. In this way, a proposition at blur b can “communicate” with its more precise avatars: we can compare Ub ( P )with Tb′ ( P )or Fb′ ( P )for b′< b and see which line of attack has the best chance of stabilising at b = 0. This is exactly what happens in the blur treatment of Collatz or Navier–Stokes: we first obtain T¯ b ( P )at a coarse blur ¯ b , then show that P is blur-stable, and only then promote this to a sharp T0(Q)for the underlying Platonic statement Q. 3 Equally importantly, the blurred algebra helps us recognise when we must exit the current theory. If Ub ( P )persists across all b > 0without any coherent pattern of stabilisation, the framework tells us that the existing model lacks the invariants needed to decide P at all. In that sense, blurred Boolean logic is not about decorating proofs with an extra symbol; it is about organising the flow of information between scales, and about making the difference between “unknown in principle” and “unknown at this blur, for a very specific reason” mathematically explicit. 3 Blurred Boolean algebra: axioms We now impose axioms on the family ( vb ) b∈B which capture what we actually use in the blur program. These axioms are deliberately minimal; they are satisfied by the concrete valuations underlying our Collatz, Navier–Stokes, and Church–Turing analyses. 3.1 Definition and basic structure Definition 3.1 (Blurred Boolean algebra).Ablurred Boolean algebra on a propositional language Lis a family of blurred valuations (vb)b∈Bsuch that the following conditions hold. (B1) Classical behaviour on the available fragment. For each b > 0, the set Avb is closed under ∧,∨,¬, and the induced two-valued map b vb:Avb−→ {T, F},b vb(P) := vb(P) is a classical Boolean valuation. In particular, for all P, Q ∈Avbwe have Tb(P∧Q)⇐⇒ Tb(P)and Tb(Q), Tb(P∨Q)⇐⇒ Tb(P)or Tb(Q), Tb(¬P)⇐⇒ Fb(P). (B2) Blurred law of excluded middle. For every P∈ L and every b > 0, Ab(P)∧(Pb∨ ¬Pb)∨ ¬Ab(P)(1) is valid, where Pbis the assertion Tb(P)and ¬Pbis Fb(P). (B3) Monotonicity across scales (semantics). The family ( vb )is monotone in the blur scale in the following sense. Suppose 0< b′≤b(i.e. b′is a finer blur than b). Then: •if Tb(P), then Tb′(P); •if Fb(P), then Fb′(P). In particular, positive knowledge does not disappear as we increase resolution. (B4) Monotonicity of availability. Availability can only increase as blur decreases: if 0< b′≤band Ab(P), then Ab′(P). (B5) Blur-zero as a limit object. The valuation v0 is not arbitrary: T0 ( P )and F0 ( P )may only be declared for formulas Pwhich satisfy a stability condition across (vb)b↓0; see §4. Axiom (B1) says that once a formula is available at blur b , its truth behaviour is exactly classical. Axiom (B2) is the blurred law of excluded middle: either the formula is not available at this blur, or it is available and then behaves classically. Axioms (B3) and (B4) formalise the idea that acquiring more resolution never invalidates correct positive statements; it can only make more formulas available. Axiom (B5) is the crucial special treatment of blur zero, to which we now turn. 4 4 Blur-zero truth and stability In all substantive uses of blur there is a sharp distinction between positive blur b > 0and the Platonic limit b= 0: • At b > 0we can typically derive strong existence, uniqueness, and stability statements, because the blur suppresses singularities and bad configurations. • At b = 0 we are dealing with the full Platonic configuration space, in which all possible pathologies can in principle appear. The logic should reflect this: b = 0 should not be treated as just another point of B . Instead, truth at blur zero should be a derived notion, licensed only when the behaviour at positive blur is stable enough. 4.1 Blur-stable formulas Definition 4.1 (Blur-stable pattern).Let P∈ L . We say that P has a blur-stable pattern if there exists b0>0and a function ε: (0, b0]→[0,∞)with ε(b)↓0as b↓0such that: 1. for all b∈(0, b0], the formula Pis available at blur b:Ab(P)holds; 2. for all b∈ (0 , b0 ], the truth value vb ( P )is determined by a quantitative estimate with error bounded by ε(b); and 3. the dependence of P on the underlying blurred data is monotone or Lipschitz in the blur parameter in a way that guarantees no oscillation beyond what ε(b)allows. This definition is intentionally schematic: in concrete settings P will be a statement about Lyapunov drift, energy bounds, or spectral gaps, and the guard function ε ( b )will be a specific error term one can follow analytically. The key point is that P must be decidable at all sufficiently fine blur levels, and its truth must be protected by a quantitative guard which shrinks to zero. Definition 4.2 (Truth at blur zero).Let ( vb )be a blurred Boolean algebra. We say that P is true at blur zero, and write T0 ( P ), if P has a blur-stable pattern and Tb ( P )for all sufficiently small b > 0. Similarly, we say F0 ( P )if P has a blur-stable pattern and Fb ( P )for all sufficiently small b > 0. If P does not have a blur-stable pattern, or if the truth value oscillates as b↓ 0, we set U0 ( P ). Thus there is no primitive axiom of the form ∀b > 0Tb(P)⇒T0(P). The implication must always be mediated by a blur-stability argument: we must show explicitly that the estimates we used at positive blur continue to hold, with shrinking error, all the way down to the limit. 4.2 Why we found Collatz at blur before blur zero In the Collatz program the key statement P is: there exists a Lyapunov function ϕ on residues modulo 2 k (with k = 13 in the final certificate) which enforces negative drift for the accelerated map, up to a small budget ρ absorbed in the Lyapunov inequality. Semantically, this is a statement at a particular discrete blur b=¯ b > 0. In the language of blurred Boolean algebra, we can describe the situation as follows. Let L contain the propositional variable PCol := “There exists a residue Lyapunov certificate of level kenforcing negative drift.” Then: 5 • At blur b = ¯ b corresponding to the chosen modulus and budget, the Collatz certificate is available and verified: A¯ b(PCol), T¯ b(PCol). On the available fragment at that blur, everything is classical. • At the Platonic level b = 0, the statement “the accelerated Collatz map has no cycles and no divergent trajectories” is a different formula QCol . To show T0 ( QCol )one must relate it to PCol via a blur-stable pattern, which requires a careful telescoping argument and a drift-to-global-convergence lemma. The early blur-only results in the Collatz work amounted to T¯ b ( PCol ). They did not yet license T0 ( QCol ). Once the stability lemmas were in place, we could show that PCol is blur-stable and that its implications propagate to b = 0. Only then did T0 ( QCol )become justified. Notice that we could reduce the certificate to k = 3, which is equivalent to decreasing the blur, yet this still required a few manual steps to complete the argument, in particular checking that we enter the 1→4→2→1cycle and verifying the remaining local inequalities. 4.3 Why blur-alone left Navier–Stokes undecided For Navier–Stokes, the blur-only analysis produced statements of the form PNS(b) := “Every NS solution at blur bobeys a uniform energy bound and no finite-time blow-up at scale b.” For each fixed b > 0one can derive such a statement with appropriate guards: AbPNS(b), TbPNS(b). However, the guards depend on b in a way that does not immediately give a blur-stable pattern as b↓ 0. The possibility that a Platonic solution might concentrate energy in finer and finer scales cannot be ruled out by the blur inequalities alone. In the language of this paper: •For each b>0, the formula PNS(b)lies in Avband is true. • The corresponding sharp statement QNS (“no blow-up for smooth data at blur zero”) does not yet have a blur-stable pattern provable from the blur inequalities alone. • Therefore v0 ( QNS ) = U : the Platonic NS problem is undetermined by blur-only information, exactly as we reported in the dedicated paper. Later work, which decomposes the Platonic NS world into 16 universes and isolates the physically relevant one, can be seen as a way to build the missing stability patterns for certain NS statements, but that construction sits strictly on top of the blurred Boolean layer described here. 5 Blurred LEM and its role We can now state explicitly the two forms of excluded middle in a blurred Boolean algebra. 5.1 Internal LEM on the available fragment Proposition 5.1 (Internal law of excluded middle).Let ( vb )be a blurred Boolean algebra. For each b > 0and each formula P∈Avbwe have Tb(P)⊕Fb(P), where ⊕denotes exclusive or. Equivalently, Ab(P)⇒Pb∨ ¬Pb, where Pbis the assertion Tb(P)and ¬Pbis the assertion Fb(P). 6 Proof. This is immediate from (B1): on Avb the induced valuation b vb is classical Boolean. In particular, for each P∈Avb, classical excluded middle holds: b vb(P) = Tor b vb(P) = F. Thus we preserve the classical flavour of logic wherever the blur machinery has the strength to decide a proposition. There is no weakening of LEM inside the decidable fragment. 5.2 Blurred global LEM Globally, however, we must account for the possibility that P is simply not available at blur b . This yields the blurred version already stated in (B2): Ab(P)∧(Pb∨ ¬Pb)∨ ¬Ab(P). This matches precisely our actual reasoning in blur-based arguments: • Either the certificate, energy inequality, or spectral computation we have chosen is strong enough to decide Pat blur b; then we treat Pin a fully classical way at that blur. • Or the current blur and budget are insufficient to make the question meaningful (for instance, we have not yet fixed a modulus in a residue computation, or we lack guards for a limit); in that case we simply refrain from ascribing any truth value. 6 Position among other non-classical logics Blurred Boolean algebra sits in a well-populated landscape of logics which go beyond pure classical two-valued truth. Here we briefly indicate how it relates to three standard families. 6.1 Three-valued and partial logics In Kleene’s strong three-valued logic K3 and in Priest’s logic of paradox LP one works with three truth values, typically interpreted as true,false, and both or neither, with truth tables for ∧,∨,¬ designed to capture vagueness or inconsistency. In partial logics one interprets U as “undefined” and often treats logical connectives as partial operations. Blurred Boolean algebra borrows the three-valued vocabulary but interprets U as epistemic and blur-dependent. The key differences are: • The truth tables on Avb are exactly classical; there is no modification of Boolean algebra inside the available fragment. • The U value is not a third ontological truth state; it is “we have not spent enough blur or budget to decide this question at this scale”. • The monotonicity axioms (B3)–(B4) link the valuations across blur levels, which has no analogue in static many-valued logics. In this sense blurred Boolean algebra is closer to a family of partial classical logics linked by a scale parameter than to a single three-valued system. 6.2 Intuitionistic logic Intuitionistic logic rejects the global law of excluded middle: it is not valid in general that P∨ ¬P. Instead, proofs of existence and disjunctions must be constructive. Kripke semantics for intuitionistic logic uses a partial order of worlds with monotone truth: once a proposition becomes true at some world, it remains true in all stronger worlds. Blurred Boolean algebra shares some surface features with this picture: 7 • There is a parameter ( b ) along which truth is monotone: once a proposition is true at coarse blur it stays true at finer blur. •Globally we cannot assert P∨ ¬P; only the blurred version involving Ab(P)is valid. However, the reason for this is different. In blur we do not reject LEM as a law of thought; instead we relativise it to the available fragment: “excluded middle holds once the blur protocol has made P decidable”. In addition, blur-zero truth is derived from positive blur via stability, whereas in intuitionistic logic the semantics is usually given directly on the Kripke frame. One could develop an intuitionistic version of blurred Boolean algebra by requiring that the available fragments Avb themselves carry an intuitionistic valuation, but this lies beyond the present paper. 6.3 Modal and epistemic viewpoints Blurred truth also has a modal reading. For each blur level b we can form a modal operator □b meaning “provable or certified at blur b”. Then □bPstands for Tb(P), and availability becomes □bP∨□b¬P . The monotonicity □bP⇒□b′P for b′≤b says that proofs at coarse blur remain valid at finer blur. From this perspective, blurred Boolean algebra is a very specific multi-modal logic in which the blur parameter indexes a family of epistemic modalities, and b = 0 corresponds to an ideal modality that is only accessible along stable paths. 6.4 Inevitable undecided sharp statements The special role of blur zero would be misleading if it suggested that, in principle, every formula could ultimately be assigned a sharp truth value. In any realistic setting where the b= 0 layer reflects a concrete axiomatic theory of arithmetic or computation, there must exist formulas whose blur-zero value remains undetermined in our sense. Proposition 6.1 (Undecided b = 0 statements are unavoidable).Let T be a consistent, recursively axiomatizable theory extending a fragment of arithmetic strong enough to formalise Turing machines (for example, Peano Arithmetic or ZFC). Consider a blurred Boolean algebra (vb)b∈Bin which blur-zero truth is tied to Tin the following sense: T0(P) =⇒T⊢P, F0(P) =⇒T⊢ ¬P. Then there exists a formula P∈ L such that U0(P), i.e. v0(P)is neither Tnor F. Proof sketch. By Gödel’s first incompleteness theorem, there is a sentence G in the language of T such that T proves neither G nor ¬G . Interpreting G as a formula of L , the assumptions give: T0(G)⇒T⊢G, F0(G)⇒T⊢ ¬G. Both right-hand sides fail, so neither T0(G)nor F0(G)can hold. Hence U0(G). Equivalently, one can take P to be the halting statement for a specific Turing machine M whose behaviour is independent of T : “ M halts on empty input”. By Turing’s analysis of the halting problem, there exist such M for any T as above. Again, T proves neither P nor ¬P , so v0(P)cannot be Tor F. 8 Remark 6.2 (Alignment with halting and undecidability).Proposition 6.1 says that in any blur model whose b = 0 layer respects our standard axiomatic universe, the set of formulas with U0 ( P )is non-empty; undecided sharp statements are not a pathology but a structural necessity. This matches both: • Gödel-style incompleteness: some arithmetic sentences can never be assigned T0 or F0 without extending the axioms. • The halting problem: for some concrete programs M , the proposition “ M halts” remains U0relative to a fixed theory Tand its blur-zero semantics. From the blur perspective, these U0 formulas are precisely those whose behaviour cannot be captured by a blur-stable pattern as b↓0: no amount of finite blur refinement, within the given axiomatic universe, will force a stable truth value. In practice we experience this as the boundary between “problems that yield to blur” (such as the Collatz Lyapunov certificate) and “problems that remain formally undecided” at the current epistemic level. 7 Outlook and relation to the blur program The main purpose of this paper is not to create a new logical fashion, but to make explicit the layer that we have been using implicitly in the blur program from the very beginning: • In Collatz we first found a blur-level certificate in a purely discrete setting (residues modulo 2 k ), then proved that its consequences are stable as we let blur go to zero. The blurred Boolean algebra explains why it was consistent to “know Collatz” at blur before blur zero, and why additional work was needed to bridge the gap. • In Navier–Stokes we derived strong statements for each finite blur but correctly refused to claim anything at b = 0 from blur-only data. In the language of this paper, the relevant formulas simply did not have blur-stable patterns yet, so v0(P) = U. • In the Church–Turing analysis we characterised effectivity in terms of the ability to extract more structured information from blur with finite cost. Blurred Boolean algebra provides the propositional skeleton of that story: availability is exactly the domain on which a given model of computation can make Pdecidable at a fixed blur. In that sense, blurred Boolean algebra is a bookkeeping device for epistemology under blur. It says: • where we are allowed to think classically (on the available fragment at fixed positive blur), •where we must remain agnostic (undetermined formulas at each blur, and at blur zero), • and under which conditions a blur-level truth can legitimately be promoted to a Platonic truth at b= 0. The broader lesson is that what we can prove is tightly linked to how we are allowed to ask the question. Blur does not weaken logic; it refines the epistemic context in which classical logic operates. If nothing else, it explains how we were even able to look for Collatz at simpler blur while leaving b = 0 undecided, or why a blur-only treatment of Navier–Stokes had to stop short of the full Clay question. The blurred Boolean algebra introduced here is the thin logical skeleton behind those choices. 9