Cooper’s final move in Twin Peaks: The Return is the attempt to “fix” Laura by going back and pulling her out of the night. You can say, in an ordinary psychological way, that it is grief, guilt, saviour fantasy, trauma, a refusal to accept loss. All of that may be true. But it does not touch the more unnerving feature of the series, the sense that the reasoning itself, the “if I do this, then that will be restored” structure, becomes strangely unreliable, as if the very act of making the plan precise introduces a kind of loop, a silent snapping-back to the beginning. Fine sees what a loop in reasoning is without having to psychologise it. An argument is circular when it assumes what it is trying to prove. That sounds obvious until you notice that any valid argument has that structure, because if the premisses are true then the conclusion must be true, and so the conclusion is, in a sense, already “contained” in them. If you want a real distinction between harmless validity and question-begging circularity, you need more than the idea of assuming what you want to prove. Fine takes the argument from p to p. If anything is circular, that is. It is like the blatant contradiction p and not p. You might hope that every more complex circularity could, one way or another, be reduced to this blatant form, the way Lynch reduces elaborate horror to a simple fixed stare or a single word repeated until it becomes wrong. Cooper’s project is something like: Laura is missing, if I bring her back, then the horror that followed will not happen. It sounds like progress. But is this a disguised p to p, a motion that feels like forward movement but returns you to the same place. Fine’s logic is built to tell us when a step is genuinely progressive and when it is only the appearance of progress. There are three ideas that each look plausible, but together cannot all be true. The first is transitivity, which is the idea that progress composes. If you have a noncircular argument from p to q, and you have an argument from q to r, then you ought to be able to stitch them and get a noncircular argument from p to r. If logic could not compose steps, it would be useless as a model of any extended reasoning at all. The second is reversibility, which is the idea that sometimes reasoning can go back. Not always, but sometimes. If you can get from p to q, then in some cases you can also get from q back to p, perhaps by a different route, perhaps in a way that feels less progressive, but still as an argument. In Twin Peaks this is the seductive thought that if Cooper can get from “Laura died” to “Twin Peaks became what it is”, perhaps he can reverse, from “Twin Peaks became what it is” back to “Laura died”, and then reverse again, undoing the death. The show itself behaves as if reversal is available, the Lodge seems built out of reversals and returns. The third is noncircularity, the requirement that there is no genuinely noncircular argument from p to p. Otherwise the whole distinction collapses. If a p to p step could count as progress, then every loop could be called progress, and you have given away the very thing you were trying to define. Fine points out you cannot keep all three. If you allow transitivity and you allow even occasional reversibility, then you can start at p, go to q in a noncircular way, go back from q to p by some argument, and transitivity will then hand you a noncircular route from p back to p, which is exactly what you wanted to forbid. It is like Cooper finding a route out of the Lodge that is supposed to be a real escape, and yet because the structure permits reversals, he ends up back at the same red curtains, only shifted, only stranger. The logic has looped. So Fine says: different theories of noncircular reasoning can be classified by how they respond to this dilemma. You can keep transitivity, letting noncircular reasoning build and build, but then you must treat noncircular arguments as irreversible. This is the progressive approach. Or you can keep reversibility, allowing back-and-forth between equivalent claims, but then you must give up the idea that noncircularity behaves cumulatively under composition. That is the noncumulative approach. Fine chooses the progressive path, partly because it fits the idea of reasoning as making genuine headway, and partly because it is not obvious you can make it coherent. Can you have a system where steps compose, where progress can be tracked, and yet you never get a strong p to p? He takes a very austere logical system as a testing ground, a system built around implication, the “if … then …” connective. That is already very Twin Peaks, because so much of the series is powered by “if”. If Cooper follows the Fireman’s clues, then something will happen. If he crosses at 430, then he will reach a certain place. If he finds Laura, then the world can be restored. These are all implication-shaped thoughts, and the show tests what happens when implication starts to behave like a trap. Fine wants to distinguish two kinds of derivation. There are strong theorems, marked with an s, which are supposed to correspond to genuinely noncircular arguments, and there are weak theorems, unmarked, which are simply valid, whether or not their proofs smuggle the conclusion in through the back door. Think of the difference between a real clue and a false clue. A real clue moves the investigation forward. It changes what is live. It gives you something you did not have. A false clue can still be consistent with everything, it might even be “valid” in the sense that it fits, but it does not move you forward. It is the kind of thing that leads you back to where you started. Fine wants logic to mark this difference inside the proof itself, not by external commentary. He then has to confront conjunction, “and”. In ordinary logic, from “A and B” you can infer A. That is simplification. If you know “Cooper is in the Dougie state and he is in Las Vegas”, you can infer “Cooper is in Las Vegas”. But Fine says: if you are trying to formalise noncircularity, this kind of move is suspicious because the conclusion is literally sitting inside the premiss. If the only reason you can “infer” A is that A is already present as a component of what you assumed, then that is dangerously close to p to p. It is like saying, the reason I can assert “Laura is gone” is that I already asserted “Laura is gone and beautiful”. Nothing has happened. No progress has occurred. It is question begging in miniature. So Fine refuses to automatically treat these simplification patterns as strong. They remain derivable, but they do not count as progress by default. Only certain proofs of a simplification-shaped statement may be strong, namely those where the conclusion is reached by a route that does not rely on the conclusion being merely contained in the premiss. You can sometimes get from “A and B” to B in a way that really uses A to justify B, not by simply picking B out of the conjunction. If A strongly implies B, then “A and B implies B” can be strong, not because B is present, but because A is doing genuine work. This is exactly the kind of distinction Twin Peaks keeps forcing. Sometimes a return to a familiar place is a true return with new information, and sometimes it is an empty loop. To make all of this precise, Fine shifts from a standard style of proof system, where assumptions are treated as an unstructured list, to where the left side, the assumptions, are structured terms. These terms do not just record which assumptions you have, they record how they are arranged and combined. Think about how The Return treats scenes as structured composites. A scene is not just a set of elements. It is a specific arrangement, Dougie with coffee, Dougie with electricity, Dougie with Janey-E, Dougie with the office. You cannot freely reorder these without changing what the scene is doing. Fine’s terms play a similar role. They keep track of the fine structure of an argument, not just its ingredients. Fine then introduces structural transformations, rules that tell you when you can rearrange the antecedent structure. Some of these transformations are weak, they preserve validity. Some are strong, they count as genuine progress regardless of whether the input argument was already strong. This is the formal analogue of a move that forces the narrative forward, a move that, once made, cannot be undone without losing the status of progress. In Twin Peaks terms, some crossings, like leaving the Lodge in a certain way, feel like strong transformations. They change what is possible for the character. Other movements feel like weak transformations, they reshuffle the furniture but do not change the underlying situation. Fine abstracts away from any specific set of transformations and isolates general conditions that any progressive system’s transformations must satisfy. Monotonicity says you can extend contexts in a predictable way. In Twin Peaks terms, if a certain rearrangement is permitted in a small scene, it should still be permitted when you embed that scene inside a larger one. Transitivity says transformations compose. If you can go from one arrangement to a second, and the second to a third, you can go from the first to the third. Termination is the crucial “no strong looping” condition. It rules out strong transformations that would let you take a genuinely progressive step and come right back to where you started, which would be the formal version of a strong p to p. The mates conditions say that if you transform one structured antecedent into another, you should be able to track occurrences of formulas across the transformation. Nothing should appear from nowhere, nothing should vanish without trace, and the “left” occurrences, the ones playing a certain role in the structure, should remain on the left where they belong. This is the demand that when the narrative shifts, you can still trace what carried over. The Lodge is frightening partly because it violates this. People appear with no causal bridge. Names change. Faces swap. The mates discipline is an attempt to keep the logic from becoming Lodge-like in its basic bookkeeping, even while the logic is designed to diagnose loops. The conversion condition says that if you have a certain lefted structure, a chain of “if … then … then …”, then under transformation you can factor the result in a way that isolates the genuinely progressive component from the mere rearrangement components. This is like insisting that when a scene turns into another, you can recognise the main driver of the change, and separate it from incidental shifts. Once the system is set up, Fine proves two meta-theorems. One is inversion, which says that certain rules can be run backwards safely. If you have proved an implication, you can recover a way to use it. If you have proved a conjunction, you can recover the conjuncts. So if you have a proof that “if Dougie goes near electricity then Cooper returns”, you can actually exploit it in the intended way. The second is a cut theorem, which shows you can splice proofs together in a controlled way without introducing new strong circularities. Cut is the formal place where hidden loops like to hide, the moment where you take something proved elsewhere and insert it as a lemma. Fine’s cut theorem ensures that the system’s notion of strong progress is stable under this kind of splicing. If you allow the proof to roam freely, it can keep re-expressing the same content in slightly different shapes, the way Lynch keeps giving you Laura as Laura, Laura as an absence, Laura as Carrie Page, the same “content” in different guises. To prevent an infinite chase, Fine defines a way of measuring complexity so that every time you reduce a formula into a certain normal form, you strictly decrease complexity. That guarantees that you cannot keep deferring the problem forever, you cannot keep running around the lodge corridors of notation. The normal form is basically a conjunction of lefted implications, a bundle of “if … then …” chains ending in atomic propositions. The point is that once everything is in this form, you can force the narrative into a set of simple motifs that you can count and track, rather than letting it remain a fog of overlapping scenes. Fine says that if a normal form formula A strongly yields one of its own component implication chains B, then one of three things must be happening. Either the supposed progress relies on something smaller, which will then be ruled out by the induction hypothesis. Or the “progress” is really happening elsewhere, and you can chase that back to a basic component. Or you are in a degenerate case involving permutation of antecedents, where the derivation is effectively just reshuffling, and that again cannot create strong progress without contradicting the termination condition. An analogue in Twin Peaks is the way attempts at repair reveal hidden dependencies. You think you are trying to restore Laura, but to do so you must already have, somewhere, a simpler stabilising fact that makes the restoration coherent. When you look for it, it is not there. Or you discover that what is driving the move is not Laura at all but another motif, another conjunct, the Lodge, Judy, the Fireman. Or you realise you have done nothing but permute the pieces, change the order of the signs without changing what they are. Fine then adds a combinatorial fact which says that you cannot have a finite closed circuit of genuine progress where each step is strongly supported by another, because that would amount to a loop that secretly contains an identity. There must be a starting point, a basic element, otherwise the alleged progress is an illusion. Fine's proof runs as an induction. If there were a strong A implies A, you could reduce A to normal form, chase the strong derivation into a simpler intermediate or into a cycle among conjuncts, and each route contradicts the induction hypothesis or the cycling lemma. So strong identity is impossible. That is what makes the logic progressive. It allows reasoning to compose, it keeps transitivity in the strong sense, but it forces strong reasoning to have a direction, not a direction you can point to like “simpler” or “more complex” in every case, but a direction embodied in the permitted strong transformations and the impossibility of looping back to your starting point while still calling it progress. The dread in The Return is often the feeling of an attempted reversal that becomes a loop, the feeling of forward motion that returns you to the same words and the same question, but now stripped of ordinary meaning. Fine’s progressive logic is a way of saying that a certain kind of loop should not count as progress, and if you want progress to compose, you must build the system so that this kind of return is impossible at the level of strong proof. That is exactly the pressure Cooper runs into. The show lets him make the move anyway, but then it shows what it feels like when the world refuses to treat the loop as a repair.
Twin Peaks: The Return seems staged inside Fine’s problem space, an extended meditation on what happens when a loop is mistaken for progress. This is something the show itself repeatedly performs and tests, as if it were asking, again and again, whether a return can ever count as a genuine step forward. Fine’s basic worry is that reasoning feels like it should move somewhere. If you start from one claim and reason correctly, you should end up somewhere else. If you always end up back where you started, then however elaborate the route, nothing has really happened. The difficulty is that many forms of reasoning look progressive while secretly looping. They appear to add detail, sophistication, structure, but when examined carefully they amount to p to p, the conclusion smuggled back in under another name. Fine’s entire apparatus is designed to separate real movement from disguised return. The Return dramatizes exactly this distinction. At the surface level, the series is full of motion. Characters travel, timelines split, identities shift, scenes are repeated with variations, portals are crossed, years change. It looks hyper dynamic. But the dominant affect is stasis and dread, the sense that nothing is being resolved, that motion is not carrying anyone out of the problem space. The system permits weak validity everywhere but blocks strong progress. Take Cooper himself. His defining project is framed as an implication. If Laura can be saved, then the horror can be undone. If I go back, then the future will change. This is exactly the kind of reasoning Fine wants to scrutinise. Is this a genuinely progressive inference, or is it a loop dressed up as a plan? In Fine’s terms, Cooper is attempting to treat a reversible move as if it were progressive. He assumes that because he can reason from Laura’s death to the state of the world, he can reason back from the state of the world to Laura’s death, and then forward again to a repaired world. But Fine’s dilemma tells us that if you allow both reversibility and transitivity, you generate a forbidden identity. You get back to p, but you insist it is not the same p. The Return shows the cost of insisting on that distinction when the structure does not support it. This is why Lynch is so careful to make repetition feel wrong. Scenes recur, phrases recur, faces recur, but never in a way that stabilises. These are weak derivations. They preserve something recognisable, but they do not count as progress. Saying “this looks like before” is not the same as being back before. The show forces the viewer to feel the gap between formal resemblance and genuine return, which is exactly the gap Fine is formalising when he says that validity alone is not enough, you must look at the structure of the proof. The Lodge spaces make this especially clear. Inside them, reversibility is everywhere. You can walk one way and appear somewhere else, speak backwards and be understood forwards, meet someone who is and is not the person you knew. The Lodge is a space where reversibility has been maximised and progress eliminated. Nothing there can be strong, because everything can be undone, mirrored, permuted. This is why the Lodge cannot resolve anything. It is too permissive. It allows too many weak equivalences. A logic built entirely on such moves collapses into circularity. The figure of Dougie Jones is the clearest illustration of weak versus strong reasoning made visible. Dougie’s actions are valid in a minimal sense. He responds to stimuli, he produces effects, he even succeeds in certain tasks. But nothing he does counts as progress in the strong sense. He cannot initiate, cannot infer, cannot direct himself. He is trapped in a world of shallow transitions. He inhabits a space where only weak theorems are available. Everything is derivable, nothing is explanatory. This is why his eventual “return” to Cooper feels like a release from a suspended loop. The series’ obsession with electricity works in the same way. Electricity is pure transmission without content. It carries signals but does not interpret them. It connects points without understanding direction. It is the perfect image of transitivity without progress. Signals move, but nothing is settled. Progressive logic insists that transitivity alone is not enough. You must also know that the steps you are chaining are not reversible in the wrong way. Electricity is transitivity stripped of direction, which is why it becomes uncanny. When Cooper finally “succeeds” and reaches a world where Laura did not die, it produces the most disturbing loop of all. Laura becomes Carrie Page, memory fractures, recognition fails, and the final question, “What year is this?” detonates the entire project. This is the moment where the attempted strong derivation collapses into a blatant identity. Cooper has reached p again, but the system refuses to mark it as progress. The scream is the affective signal of noncircularity being violated. Something has gone wrong at the level of possibility itself. This is why The Return is so resistant to psychological or sociological closure. Those explanations operate at the level of weak validity. They tell us why a person might reason this way, why a culture might repeat its traumas, why memory might distort itself. All of that can be true, but it leaves untouched the deeper structure. The problem is not that Cooper reasons badly, it is that he attempts to treat a loop as if it were a path. If you want reasoning, or action, or repair, to count as progress, you must accept limits on reversibility. You must accept that some returns are not allowed if progress is to be preserved. The show refuses that acceptance. It lets Cooper make the move anyway, and then shows us what a world looks like when the distinction between strong and weak reasoning collapses. Everything still “works”, scenes still follow scenes, causes still have effects, but nothing means restoration. Circularity is something you can inhabit. It has a feel. It is the feel of endless implication without arrival, of motion without direction, of explanation without grounding. Lynch builds a world in which that prevention fails, and lets us live inside the consequences.
