The Vampires of Paradox, Is Anybody Out There?

Although I have no hard data on the matter, I believe I know more about paradoxes than any assistant professor north of Battery Park and south of Herald Square. For the past twenty years I’ve sought the holy grail of absurdity in every nook and cranny of Western civilization. Thus far no dragons have crossed my path, no ogres, trolls, or griffins, and yet my quest has often proved perilous. Over the past decade three initially auspicious marriages have crumbled before my eyes, largely because the corresponding wives couldn’t be bothered to understand my burning passion to definitively deconstruct Zeno’s disproof of common sense before some upstart Columbia grad student beat me to it. My department head, the reprehensible Dr. Virginia Sayles, has likewise regarded my specialty with scorn. From the instant I got on the tenure track she began playing political games with my pet project, the Bertrand Russell Institute of Paradox Studies, first moving it out of the philosophy building and then off the NYU campus entirely, so that today I pursue my research in a subterranean office in NoHo.

A paradox may be resolved in one of three ways. First, we can embrace the seemingly unacceptable conclusion. In the case of our suicidal traveler, we might argue that he is simultaneously hanged and sent away unharmed, but such a solution is beneath the dignity of rational minds. Second, we may find fault with the reasoning that led to the contradiction. Concerning our lie-detecting guard, we might insist that his quandary—do I deliver the traveler to the executioner or not?—only appears to be genuine, and an obvious course of action lies before him, but such a demonstration would be beyond my powers. Third, we may attack the paradox’s underlying premises. That is, we might attempt to establish that the notion of stating one’s business truly or falsely is incoherent on first principles, though I cannot imagine a sane person taking such a position.

On the day the desperate abbot entered my office in need of an unassailable absurdity, it took me a while to realize that he’d slipped past Mrs. Graham and padded softly up to my desk, so intently were my thoughts fixed on that desert traveler and his equivocal death wish. Peering through my grimy basement window at the cavalcade of feet and paws marching down Bleecker Street, I decided that the guard’s dilemma owed its structure to the classic Liar Paradox, to wit, “What I am saying now is false.” I tell my students that anyone who desires a more concrete version of the Liar Paradox should take an index card and write on the front, “What’s on the other side is true,” then turn the card over and write, “What’s on the other side is false.” So venerable is this contradiction that, when Bertrand Russell devised his notorious antinomy concerning the set of all sets that are not members of themselves, he was happy to frame it as a version of the Liar Paradox—though the problem he identified is rather more profound than that.

“Dr. Kreigar, I presume?”

I rotated my beloved swivel chair. My visitor, a towering figure in a black cassock, introduced himself as Abbot Articulis, the head of “a small and dwindling monastic community that traces its origins to that magnificent heretic, Quintus Septimius Florens Tertullianus, also known as Tertullian.”

“How dwindling are you?”

The abbot heaved a sigh. He was a handsome man with anthracite eyes and a lantern jaw. “There are only six of us left—me, three monks, and two nuns, maintaining our Adirondack monastery with the help of a small staff.”

“If Tertullian was a heretic, why do you follow him?”

“Because we are heretics, too. So you see, our Order is free of contradiction. Well, not entirely free, for it was Tertullian himself who famously asserted, concerning the Resurrection, ‘Credo quia absurdum,’ ‘I believe because it is absurd,’ though what he actually said was, ‘Certum est, quia impossibile’—‘it is certain, because impossible. ’ Forgive me for interrupting your meditations, Dr. Kreigar. I shall state my business quickly, after which you may return to staring out the window.”

“Call me Donald.”

“Bartholomew.”

“Are you sure you came to the right university?” I asked. “Drop by Fordham, and you’ll be up to your eyebrows in fellow Catholics. Me, I’m a lapsed logical positivist who now believes in the rarefied God of Paul Tillich, which I suspect won’t do you much good.”

“For all I care, you could be a raging atheist,” Articulis said, “as long as you can supply our monastery with what it needs: a robust paradox. Having read your book—”

“Which one?” My personal favorite is The Title of This Book Contains Threee Erors, but I’m better known for Adventures in Self-Reference and A Taxonomy of Nonsense.

“Adventures in Self-Reference. It convinced me you’re the man of the hour. Five potent paradoxes are already in our possession, but unless we acquire a sixth—I know this sounds ridiculous—without the sixth, the world will be laid waste by a voracious metaphysical menace.”

“I see.” Thus far my visitor’s résumé failed to match the profile of the typical East Village lunatic, but that didn’t mean he was a harmless Hudson Valley monk. “Metaphysical menace—fascinating. If it’s a theological conundrum you seek, I’m afraid most of them aren’t particularly confounding. Can God make a stone too heavy for Him to lift? If you think about it, the Omnipotence Paradox—”

“It has an answer,” Articulis interrupted, studying the Escher print beside my bookcase, Ascending and Descending : thirteen hooded figures climbing a rectangular stairway clockwise and getting nowhere, even as another thirteen descended counterclockwise with the same result. “No, God can’t do that, and by the way it doesn’t matter, for He nevertheless remains all-powerful with respect to making and lifting stones. Specify any coherent weight—a hundred trillion tons, a thousand trillion tons—and He will easily fashion the expected object and drop-kick it across the universe. No, Donald, we are not interested in trivial riddles involving God’s presumed limitations.”

Evidently I’d misjudged the man. Articulis knew something about paradoxes, and perhaps about voracious metaphysical menaces as well. “I immediately think of Bertrand Russell’s great and bewildering insight concerning categories,” I noted. “Might that one turn the trick for you?”

“Alas, we’re already running Russell’s Antinomy at full capacity. The fact is, you won’t grasp the nature of our plight without visiting our monastery and observing the Order in person. Might you come on Sunday afternoon? You could stay for dinner, spend the night, and leave the next morning. I believe you’ll get a journal article out of it, perhaps a whole book.”

I gazed inwardly, scanning the dreary parameters of my insipid life. My four-week summer school class, Philosophy 412: Varieties of Infinity, met every afternoon at 3:00 PM, which meant I could take my sweet time returning from Rhinebeck on Monday morning. “Fine. How long is the drive?”

Articulis shrugged and pulled a slip of paper from his cassock. “By train the journey takes slightly more than an hour. I’ve presumed to write out the directions”—he set the paper on my desk, anchoring it with my pewter sculpture of Escher’s Band van Möbius—“and we’re prepared to advance you fifty dollars toward gasoline.”

I waved away his offer. “Keep your money. Academia pays better than Jesus.”

“True, Donald, though our winery does make a small profit. Château Pelagius, have you heard of it?”

I was astonished to learn that Articulis’ heretics were responsible for the best bargain red to be found in any Greenwich Village bodega. “Not only have I heard of it, I’ve got six bottles on my shelf.”

“Driving toward Rhinebeck on Route 9, you may find yourself revisiting Achilles’ fruitless attempt to outrun the tortoise—or do you agree with Charles Peirce that such paradoxes present no difficulty to a mind adequately trained in logic?”

“I’m with Aristotle. Every serious thinker should hone his intellect on Zeno’s whetstone.”

“We’ll expect you by three o’clock,” Articulis said, backing out of the room in a series of steps so short and smooth they constituted a credible demonstration of why the swift-heeled Achilles, tangled in the laws of geometry, would never win the great footrace. “Bring a thirst for merlot and an appetite for perplexity.”

Crossing the bridge, I could not help imagining that, as in the Gallows Paradox, a guard would now appear and ask me to state my business. But instead I was greeted by a dour, bespectacled, clubfooted dwarf who used a croquet mallet as a cane. He introduced himself as Constantine, “major-domo of the monastery, and minor-domo as well,” then guided me through the portal into the lush and labyrinthine vineyard on which the community’s finances depended.

On all sides the trellises held great skeins of vines, each as dense and convoluted as the Gordian knot. Clusters of purple grapes peeked out from among the leaves, soaking up the sun like the naked lymph nodes of a flayed giant. Here and there a student in a Vassar, Bard, or rock band T-shirt stood poised on a stepladder, picking the harvest with an eye to meeting the fall semester’s tuition bill.

“Don’t let this go to your head, Professor, but I believe God has sent you to us,” Constantine said as we reached the center of the maze, where a wrought-iron bench stood flanked by marble angels. “A plague has come to our community, and Abbot Articulis has taken the burden entirely on himself. Now he has you to share his troubles.”

Uncertain how to respond, I made a paradoxical remark, pointing to a Vassar student and saying, “That elephant is about to charge.”

We continued our journey, always making left turns, the strategy for solving any connected maze except one designed by Escher, until at last we walked free of the vineyard. Before us stood the abbey, a dilapidated two-story affair, stricken with creepers. As Constantine sidled back into the labyrinth, Articulis emerged from the building and descended the stone steps, clutching to his chest an antique, leather-bound volume that I assumed was a Bible. We exchanged innocuous and anodyne pleasantries. His handshake was earthy and vigorous. He flashed the cover of his book—not Holy Writ after all but rather a Confessions of Tertullian—then gestured toward the adjacent chapel, its windows displaying as much empty leading as stained glass, the bell tower faced with exfoliating stucco.

“Upon your arrival, you doubtless noticed that our community occupies an island.” Taking my arm, the abbot led me across the monastery grounds and into the cool air of the chapel. “One thousand years ago, on the blackest of Black Fridays, the world suffered a momentous rupture—a crack in the Teilhardian lithosphere.”

“You don’t say.”

“In the primeval reaches of the Hudson Valley, a basin then populated largely by Algonquins and Mohawks, the divine pipework sprang a leak.”

“My goodness.”

A spiral staircase presented itself. We ventured upward through the bore of the tower, round and round the moldering bell rope, eventually disembarking into an open-air turret dominated by an enormous bronze bell laden with spiderwebs and lacking a clapper. Articulis guided me to the nearest Gothic window. Cirrus clouds hung in the sky like cuneiform characters. Below us spread the malevolent lake, coiled around the monastery like an immense anaconda.

“Left unchecked, the tarn will ooze across the entire lithosphere, menacing the biosphere with poison, pestilence, and pandemonium,” Articulis said, indicating the moat with a rigid finger. “At the moment, only a few denizens of the noosphere, my brave band of ordained monks and nuns, stand in opposition to the rift. Through their religious devotions they keep the tarn at bay.”

“That doesn’t sound very plausible,” I noted, the dust of NYU rationality still clinging to my shoes.

Articulis laughed and said, “‘Certum est, quia impossibile. ’ ” He brushed the cover of the Confessions. “No, I’m being needlessly clever. It is certain because of what I find in these pages. Tertullian not only foresaw the rift, he founded a religious community and charged it with containing the exudate when it appeared fifteen hundred years later. The man was a devotee of riddles. Indeed, it was he who gave Christianity its counterintuitive Trinity, ‘tres Personae, una Substantia.’ He believed that when the fateful year 1700 arrived, his monks and nuns were duty-bound to travel to the New World, make their way to the Adirondacks, build a monastery on the site of the fault, and begin to cultivate the Five Primal Paradoxes. If they held these contradictions in their minds night and day, decade after decade, century after century, then the leak would never grow so copious as to threaten the planet.”

I’d say one thing for the discourse available in Hudson Valley monasteries—it effortlessly eclipsed the polysyllabic chatter that occurred at NYU Philosophy Department meetings.

“Did Tertullian specify the five problems?” I asked, instantly receiving the abbot’s nod. “Let me guess. I’ll bet the Liar Paradox made the cut.”

“Not only did our founder mandate that one, he anticipated all the flourishes Bertrand Russell would add many years later. The Confessions also prescribes the Ruby Paradox, the Nonsense Paradox, the Lawyer Paradox, and the Sorites Paradox.”

“All good choices.”

“Every one is Greek in origin, a fact that gave Tertullian pause—‘What has Athens to do with Jerusalem?’ he once remarked—but for the sake of the greater good he set aside his anti-pagan prejudices. Through fifty generations, our Order has pondered these fundamental puzzles. Hour by hour, minute by minute, heartbeat by heartbeat, we have eaten the paradox, drunk the paradox, breathed it, smelled it, sweated it, voided it, dreamt it.”

“A recipe for lunacy, I’d say.”

Articulis hummed in corroboration. “No one understood better than Tertullian that this great commission would blur the distinction between a monastery and a madhouse. And yet the Order cleaved to its task, inspired by our founder’s revelation that, come the dawn of the third millennium, the fault would finally begin to heal. By the year 2015, or 2020, or perhaps 2025—Tertullian was never clear on this point—it would be as if the crack had never occurred.”

“So you’ve beaten the Devil?” I asked, gazing through the Gothic window. A flock of vultures wheeled above the tarn in apparent prelude to feasting on the corpse of the Teilhardian biosphere.

“Not quite,” Articulis said. “For it happens that in the final days of our project Lucifer has acquired an ally—five allies, actually, one for each paradox.”

“Allies?”

“They are difficult to describe, but happily I needn’t try, as the quintet is available for your inspection. With characteristic prescience, Tertullian foretold their advent. In the Confessions he called them cacodaemons.”

It occurred to me that, to enjoy a conversation of this caliber, a person would normally have to organize a seminar among the dozen most accomplished schizophrenics in the five boroughs. “Are they in fact from Hell?”

“Why would you ask such a question, Donald? You don’t even believe in Hell.”

“True—but you do.”

“Not really, no. Concerning the source of the cacos, Tertullian is most explicit. He calls them the children of Vorg—Vorg being their creator-god, an elusive deity inhabiting an inaccessible netherworld on another plane of reality. The Confessions predicted that when the five cacos spewed forth from the fault, they would arrive singing hymns to Vorg.”

I leaned through the window. A blast of hot summer air pulsed against my brow and cheeks. The tarn’s iniquity ascended to my nostrils in spiraling filaments of stench. I gagged and jerked back inside.

“A crack in the lithosphere,” I muttered. “You know, Bartholomew, I almost believe you.”

“Will you help us weld the fractured world whole?”

“What I’m about to say is true,” I told the abbot. “What I just said is false.”

Articulis tapped the monk on the shoulder. “Brother Francis, I would like you to meet a visitor, Dr. Donald Kreigar. He has come to help us seal the tarn.”

Brother Francis rose but, instead of facing me, remained fixed on Line 1 and Line 2.

“Ah, the infamous Nonsense Paradox,” I said, attempting to strike up a conversation. “In assailing the self-referentiality of Line 1, the author of Line 2 has offered a cogent and credible definition of nonsense—and yet Line 2 is the very sentence it so justly criticizes.”

Francis said nothing.

“Is he under a vow of silence?” I asked.

“A pledge of paradox,” Articulis corrected me. “Speaking would compromise his ability to nurture the contradiction.” The abbot stepped forward, interposing himself between the monk and his conundrum. “Dr. Kreigar would like to see your caco.”

Without saying a word, Francis slid back his cowl. His face, a pale, wizened, bulbous- nosed affair, seemed almost attractive compared with the slimy, legless, roach-brown vermin attached to his left temple. In shape the parasite suggested a horseshoe crab, in size it evoked a spatula, and in spirit it echoed a cancerous tumor. My immediate impulse was to locate a pair of canvas gloves, pull the thing off the monk, and mash it beneath my shoe—but when I proposed this remedy to Articulis, he countered with Tertullian.

“The Confessions claims that a cacodaemon cannot be forcibly extracted without killing its host. We have no reason to doubt this assertion. Were Brother Francis to die, not only would the Order lose a beloved child of God, the Nonsense Paradox would lose its strongest avatar on Earth. And yet, as things stand, Francis will soon be unable to serve the lithosphere, for the caco is sapping his strength and impeding his powers of concentration.”

“If I were ever colonized by one of these fiends,” I told Articulis, “my powers of concentration would be the least of the casualties. I would also lose my ability to control my bowels, retain my breakfast, and sustain my reason.”

Among the strangest of the caco’s aspects was the noises it made, a series of staccato gasps testifying to the satisfaction it took in absorbing the Nonsense Paradox, every seventh exhalation accompanied by a sound that fell on my astonished ears as “Vorg.”

“It seems to be praising its god,” I said.

“Exactly,” Articulis said. “Vorg, Vorg, Vorg—just as Tertullian foresaw.”

“This plague must be combated,” I said.

“Combated, exactly—with all our philosophical resources. We don’t know whether the cacos are immortal, or what their weaknesses might be. We know only that they feed on paradox the way a butterfly feeds on nectar or a tick on blood. Last week I hit upon a strategy—tentative, untested, but a strategy all the same. I shall give you the details presently, but first you must witness the full scope of the infestation.”

We bid farewell to Brother Francis, ascended to the first floor, and entered the library, sphere of Sister Margaret, whose calm demeanor and placid apple face belied the apocalyptic quality of her devotions. Two ornately carved jewel boxes rested before her on the reading table. I recognized the accoutrements of the Ruby Paradox, sometimes called Newcomb’s Paradox after the refinements wrought in modern times by William Newcomb of the Lawrence Livermore Laboratory. By the rules of the game, Sister Margaret was permitted to open both Box A and Box B, or else just Box B. She could not open only Box A. Whatever resided inside any box Margaret opened was hers to keep, but she could not retain the contents of any box she declined to open.

“Sister Margaret, please remove your wimple,” Articulis commanded.

Silently she complied, never lifting her eyes from the jewel boxes. Her caco jutted from her occipital area like a hair bun, emitting moans of psychosexual ecstasy and panegyrics to Vorg.

The abbot proceeded to expound on the Christianized iteration of the Ruby Paradox practiced at the Monastery of Tertullian. Heaven’s canniest angel, Gabriel, whose prophecies are almost always accurate, has placed in Box A a single gold coin, valuable enough to pay for refacing the monastery’s bell tower. If Gabriel has predicted that Sister Margaret will open only Box B, he has also deposited an enormous ruby, worth a thousand gold coins, in Box B. With such a sum, Sister Margaret could fulfill her lifelong dream of building an orphanage in Kabul. However, if Gabriel has predicted that Margaret will open both boxes, then he has put nothing in Box B.

Paradoxically, there was a decisive case for Margaret opening both receptacles, and an equally decisive case for her opening only Box B. Think about it. As I explained in chapter two of Adventures in Self-Reference, the classic Prisoner’s Dilemma turns on the same schizoid logic. The recently apprehended criminal will find himself constructing a sound argument for confessing and an equally sound argument for remaining silent.

Next Articulis took me upstairs to Brother Jonathan’s cell, a clean and ill-lighted place, spare as a crypt. His caco sat atop his cranium like a hideous brown yarmulke. Articulis explained that Brother Jonathan passed every waking hour meditating on the Liar Paradox, which, owing to its Scriptural heritage, enjoyed particular prestige within the Order. Speaking of the Jewish Cretans in his brief Epistle to Titus, Saint Paul warns his proxy, “One of themselves, even a prophet of their own, said, ‘The Cretans are always liars, evil beasts, slow bellies.’ This witness is true. Wherefore rebuke them sharply.” Beyond Paul’s usual anti-Semitism—not without reason have I become a Tillichean dialectical humanist—we find in these verses a cogent though evidently unconscious formulation of the Liar Paradox.

Not only was Jonathan engaged in an intense inward consideration of Russell’s Antinomy, he’d also written the basic formulation in an unbroken hundred-character line along the vertical surfaces of his cell, twenty-five characters per wall.

Standing in the center of the room, the monk pivoted slowly on his heel, repeatedly pondering the paradox, reifying the vicious cycle by dint of his own continuous rotations. A buzz of sensual contentment arose from his caco. According to legend, puzzles of this sort had caused the death of at least one ancient logician, Philetas of Cos—a story I now believed: Philetas had indeed succumbed to his own thoughts.

“Alas, Russell’s discovery of set paradoxes did not mark the end of his troubles,” I noted.

“Gödel’s Incompleteness Theorem,” Articulis said with a knowing nod. “In any formal logical system predicated on the natural numbers, there exists at least one statement that can be neither proved nor disproved within the given structure.”

We exited the abbey and started across the courtyard, casting long El Greco shadows on the weed-choked grounds. A late afternoon breeze enveloped us, filling my nostrils with the heady scent of ripe grapes. Our destination, Articulis informed me, was the cloister, where we would meet Sister Ruth, keeper of the Lawyer Paradox.

Consider sly Protagoras, who trains attorneys and each year makes an unusual contract with his pupils. “You may attend all my lectures without any money changing hands,” he tells the class, “but you must pay me ten drachmas if you win your first case.” One particular pupil, Euathlus, takes Protagoras to court with the aim of getting free tuition, arguing that if he wins the case, then the teacher’s fee must be waived, since that is the object of the suit. If Euathlus loses, however, Protagoras’s fee must still be waived, since this is the pupil’s first case. So far, so good.

Reaching the arcade, we beheld a diminutive nun circumnavigating the quadrangle, east to west, north to south, west to east, south to north, her glassy eyes contemplating some unseen and ever receding focal point. Articulis told Sister Ruth to remove her wimple. Without breaking stride, she did as instructed, revealing a parasite suckling obscenely on her left parietal. The beast, an albino, emitted the usual sounds of cacodaemonic satisfaction.

As Ruth passed me for the third time, I realized that she was muttering, over and over, now and forever, Protagoras’s argument to the judge.

“If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case, but if you resolve the dispute to my advantage, Euathlus still owes me ten drachmas, since that is the content of your decision . . . If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case, but if you resolve the dispute to my advantage, Euathlus still owes me ten drachmas, since that is the content of your decision . . . If you find in favor of Euathlus . . .”

Our final stop was the vineyard. The student pickers had gone home for the day, leaving the labyrinth to Brother Thomas, a blowsy, roly-poly man, his enormous eyes resting in their sockets like cupped eggs. The monk wandered aimlessly through the maze, enacting his assigned conundrum, a version of the Sorites Paradox, sorites being Greek for “heap.” Stained with red grape juice and affixed to Thomas’s right temple like an ancillary brain, his parasite feasted greedily.

Consult chapter six of Adventures in Self-Reference, and you’ll learn that the Sorites Paradox belongs to a class of antinomies predicated on the pervasiveness of vagueness in the world. Consider a sand dune. Take away one grain. Is what remains still a heap of sand? Almost certainly. As a general rule, we may assert that if two sand heaps differ in number by just one grain, then both are dunes—or else neither is a dune. Alas, this seemingly innocuous principle leads to the unacceptable conclusion that all collections of sand, even single-grain collections, are heaps.

Not surprisingly, Brother Thomas employed the omnipresent grapes in contemplating the paradox. When is a bunch not a bunch? he wondered aloud. An arbor not an arbor? A vineyard not a vineyard? A harvest not a harvest?

“No doubt my philosopher has deduced the strategy by which I hope to defeat the cacos,” Articulis said as, abandoning Thomas, we meandered among the trellises.

“You intend to turn their appetites against them,” I said.

“Exactly.”

“This will require a baited trap.”

“Quite so,” Articulis said. “The snare in question is now walking beside you.” Reaching the core of the maze, we eased ourselves onto the wrought- iron bench. “Give me an impregnable paradox, Donald, something so tempting from a cacodaemonic perspective that they’ll all desert their hosts for me.”

My temples throbbed as if a pair of parasites had attached themselves to my head. “An impregnable paradox.”

“Impregnable, delectable, ironclad, and unique.”

A wave of nausea rolled through the whole of my digestive system. “I really want to help, but, you see—what am I trying to say?—I’ve never faced a challenge of this magnitude. I’m an academic. This smacks of the real world. I don’t like it.”

“And yet you’re clearly the man for the job.”

I reached toward a low-hanging cluster, plucked a grape, and popped it in my mouth. “Do you think they’d be lured by Zeno’s quarrels with space, time, and motion?”

Quitting the bench, Articulis rose to full height and presented me with a tortuous smile. “The Confessions explicitly states, quote, ‘Zeno the Pagan will prove useless in healing the crack,’ and therefore I must assume a caco would find those puzzles equally unexciting. Tertullian is likewise unimpressed by the Grid Paradox, the Lottery Paradox, and the Paradox of the Ravens. He also anticipated and dismissed Moore’s Paradox—‘It’s raining outside but I don’t believe it is’—as lacking genuine eschatological muscle.”

“I won’t even ask about the Barber Paradox.”

In a remote Sicilian village, the barber shaves every man who does not shave himself. Who shaves the barber? Although the problem has some features in common with Russell’s Antinomy, it’s not especially difficult to resolve. The barber doesn’t shave. The barber is a woman. My favorite answer: the barber doesn’t exist.

“Tertullian says nothing about the Barber Paradox, but I can’t imagine it would beguile our parasites,” Articulis said, starting toward the abbey. “Dinner will be served at eight o’clock. If the cacos continue to feed for another month, or perhaps just another fortnight, I fear our community will die of depletion”—my client vanished, enshrouded by the dusk—“which means the Devil wins after all.”

It came to me that I’d entered into a kind of contest with Tertullian. The two of us had become long-distance rivals, facing one another across the valley of the shadow of death. Whatever venerable absurdity I might put forward by way of helping Abbot Articulis and, by extension, the rest of homo sapiens, Tertullian had almost certainly anticipated the problem and ruled it out. What I needed was a conundrum that couldn’t possibly have occurred to even the smartest third-century theologian—a wholly contemporary paradox, notorious for having tried the patience and taxed the sanity of modernity’s shrewdest thinkers.

By some poetic coincidence, stars bloomed over Rhinebeck just as the required riddle took shape in my brain. Stepping into the quadrangle I scanned the celestial dome with its countless suns, each twinkling point a piece of my epiphany. The answer to our needs, in every possible meaning of that perplexing predicate, was heaven sent.

I repaired to the library, seeking to verify my revelation. The cosmology section was larger than I expected, featuring not only eccentric Christians like Teilhard and Bergson but the secular likes of Michael Hart, Stephen Hawking, Nikolai Kardashev, Freeman Dyson, and M. D. Papagiannis. I perused each relevant paragraph with the frustrated frenzy of Achilles attempting to catch the tortoise, madly scribbling notes on the flyleaf of Shklovskii and Sagan’s Intelligent Life in the Universe. Fixed on her jewel boxes, Sister Margaret ignored my presence. Even after the nun left for dinner, I continued my self-incarceration, until I finally had enough material for a credible presentation to Articulis.

“Shazam!” I shouted, striding into the gloomy, tenebrous, candle-lit refectory.

The abbot sat at the head of the table, monks to his left, nuns to his right. Not a caco was in sight, the infestation being entirely hidden beneath wimples and cowls. At that moment the parasites’ murmurings were redolent of discontent—a sensible enough reaction: although the five hosts were still cultivating paradoxes, they had to devote a modicum of mentation to eating their stew and drinking their wine, with a concomitant reduction in the quality of the cacos’ nourishment.

“The Fermi Paradox!” I cried.

“Tell me more,” Articulis said.

“Why is it that, in such a vast cosmos, with two hundred and fifty billion stars in our galaxy alone, of which one hundred billion may be orbited by Earth-like planets, we have found no evidence of intelligent alien life? Why this Great Silence?”

“Continue,” Articulis said.

Consulting my jottings on the flyleaf of Intelligent Life in the Universe, I told the abbot that astronomers had to date detected over 300 exoplanets—worlds circling sun-like stars outside our solar system—and the tally was increasing every year. Many astrobiologists believed that somewhere between one thousand and one million advanced civilizations may have already arisen in the Milky Way, for an average of 500,500. And yet: no extraterrestrial radio signals, no signs of galactic colonization, no plausible UFO narratives, no credible archaeological evidence of past visitations, no von Neumann probes, no Bracewell probes, no Dyson spheres, no Matrioshka brains. Nothing, nada, zero, zilch, bupkis.

“It’s a genuine paradox,” I continued, “complete with an absurd conclusion: five hundred thousand five hundred advanced civilizations have thus far eluded our senses because we should have detected them by now. As Enrico Fermi famously put it, ‘Where is everybody?’ ”

Dressed in a splotched white apron, a hulking member of Constantine’s staff appeared at my side holding a fissured ceramic tureen. He introduced himself as Jeremiah, then proceeded to serve me a steaming glob of boiled venison using a ladle the size of a caco.

“I’m impressed,” Articulis said.

“I’ve been serving you this sludge for the past twenty years, and tonight you’re impressed?” the cook said.

Articulis scowled and squeezed my hand. “Give me your indulgence while I play the contrarian. Like Tertullian, I subscribe to the doctrine of Original Sin. Perhaps the answer to the Fermi Paradox is simply that intelligent life, upon discovering thermonuclear weapons, inevitably destroys itself. Or maybe all those hypothetical advanced civilizations, in their mad pursuit of absolute security, have succeeded in annihilating one another.”

“Line 1: The sentence written on Line 1 is nonsense,” Brother Francis said.

“Like many renegade Catholics, I’m a Darwinist,” Articulis continued. “A Teilhardian and teleological Darwinist to be sure, convinced that evolution is drawing us toward an Omega Point, but still a Darwinist. We can dispense with the Fermi Paradox simply by assuming that the biological processes found on Earth do not obtain elsewhere. Perhaps the odds are formidably stacked against the transition from prokaryotic cells to eukaryotic cells. Or maybe the move from single-celled to multicellular life is difficult in the extreme.”

“I wonder if I should open Box B only?” Sister Margaret said.

“Perhaps Earth has been deliberately sealed off as a kind of wildlife refuge,” Articulis persisted. “Perhaps we’re under a moral quarantine, doomed to isolation until we eliminate poverty or outlaw warfare. Perhaps the aliens are deliberately concealing themselves, knowing from bitter experience that extraterrestrial contact inevitably leads to disaster.”

“The class of all classes that are not members of themselves is a member of itself if and only if it is not a member of itself,” Brother Jonathan said.

“Maybe the aliens are so intelligent they have no more reason to communicate with us than they do with petunias,” Articulis went on. “Maybe their minds defy our notions of conscious rationality. Maybe they’re philosophically and spiritually advanced but don’t make machines and never will.”

“If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case,” Sister Ruth said.

The abbot was up to speed now. With uncanny perspicacity he reeled off a litany of possible resolutions, most of which I’d already scrawled on the Intelligent Life flyleaf. The aliens are already here, but they’re keeping out of sight. We haven’t been searching long enough. We aren’t listening properly. Advanced civilizations broadcast radio signals for only brief intervals in their histories. Such societies are too distant in space and time to contact us using their existing technologies.

Now Jeremiah got into the act. “You know what I think? I think they’ve uploaded themselves into computers and don’t give a fig about space travel.”

“If there are sixty grapes in a bunch,” Brother Thomas said, “does that mean fifty-nine grapes likewise constitute a bunch?”

“My favorite refutation is pure science fiction,” I said. “Our universe is a simulation created by aliens who deliberately left it devoid of extraterrestrial intelligence.”

“There, you see—even you don’t believe it’s a serious contradiction,” Articulis said.

“No, I merely believe that the only substantive threat to Fermi’s Paradox is the simulated-universe hypothesis, an argument for which the burden of proof clearly lies with its devotees. The remaining challenges all leave the conundrum in place. You see, Bartholomew, for any given advanced alien civilization C₁, that is, an extraterrestrial society whose unavailability to our empiricism is easily explained, there is always C₂, a civilization that by any rational measure should have become manifest by now. If in our perversity we choose to dismiss C₂ with a plausible unavailability scenario, we must still deal with C₃, a different civilization that should have revealed itself already. If we take the trouble to build a case against C₃, we have to cope with C₄, not to mention C₅, C₆, C₇, C₈, C₉, C₁₀—all the way up to C_500,500. I don’t know about you, but somewhere around C_250,000 I would begin to admit there’s a problem here. It takes just one community of celestial eager beavers to break the Great Silence—just one, that’s all. What’s going on, Bartholomew? Are we truly alone? How could that be?”

The abbot knitted his brow, smacked his lips, and took a long swallow of Château Pelagius. “All right, Donald, let’s give it a try. You’ve not quite won me over, but for the moment you’ve allayed my doubts.”

“Excuse me, Dr. Kreigar, but aren’t you overlooking the most obvious solution of all?” Jeremiah asked. “Isn’t the best answer staring us in the face?”

The instant I heard the word “obvious,” a razor-toothed chill coursed along my spine. I winced internally. The cook’s tacit argument was cogent, lucid, and supremely rational—and I had no riposte.

“Poe,” I said.

“What?” Jeremiah said.

“The Purloined Object Effect,” I muttered feebly. “I call it Poe because he wrote a story about the phenomenon. Some secrets hide in plain sight. The tarn didn’t give rise to the cacos—they came from outer space. The parasites themselves have broken the Great Silence.”

“I don’t read much, but that’s what I meant,” Jeremiah said.

An interval of quietude descended upon the refectory—not as profound as the Great Silence, but palpable nevertheless.

“We have nothing to lose by proceeding as if Jeremiah is wrong,” Articulis said at last. “Ergo, I shall brew myself a pot of coffee and spend the night pondering the Fermi Paradox. If the cacos are aliens, the problem will hold no interest for them, since they are its resolution. Otherwise, God willing, our visitors will come after me, determined to draw sustenance from a particularly erotic conundrum.”

“God willing,” Jeremiah echoed.

“God willing,” I said.

“If you resolve the dispute to my advantage, Euathlus still owes me ten drachmas, since that is the content of your decision,” Sister Ruth said.

“If there are fifty-nine grapes in a bunch,” Brother Thomas said, “does that mean fifty-eight also constitute a bunch?”

“All Cretans are liars, the Cretan insisted,” Brother Jonathan said.

Sobered by the Purloined Object Effect, I sought to become intoxicated with Château Pelagius. The first glass failed to meet my criteria for drunkenness, as did the second, but by the third measure Achilles overtook the tortoise, Euathlus received free tuition, the barber got his shave, and nobody gave a damn when a heap stopped being a heap. I was now so rickety that my journey to the dormitory required Jeremiah to stand behind me, grasp my arms, and gingerly walk me up the stairs and along the corridor. I collapsed on the straw pallet, my brain reeling with wine and paradox. A moment later I was dreaming of The Day the Earth Stood Still, that ingenious allegory of solipsistic contact, humankind invading itself with its own crippled conscience, the iridescent angel of our better nature emerging from the gleaming vulva and imploring the world’s leaders to heed his warning, for otherwise humanity will be chastised by the afterbirth, whose destructive power is without limit.

Overflowing with love for the nonexistent occupants of our sterile galaxy, I descended to the library and, approaching the reading table, sat down across from Sister Margaret. She glanced up from her devotions, offered me a nod, then returned to cathecting the Ruby Paradox. Her bare head held wisps of hair as ethereal as corn silk. Her shorn wimple lay beside the jewel boxes, marred now by a gaping hole rimmed with bits of torn thread. I rose and, circling the nun, inspected her occipital. Having broken free of Margaret’s cloth, her caco was now gone, called to a higher paradox.

I lurched out of the library and charged down the basement stairs, praying that Brother Francis’s parasite had also found satisfaction in the Great Silence. The monk sat before his blackboard, eyes fixed on a negation absurdly synonymous with the proposition it negated. Breathlessly I snuck up behind Francis, then gently slid back his hood to reveal a naked left temple where once the Nonsense Paradox caco had feasted.

Upon returning to the main floor, I realized that my respect for Tertullian now rivaled my appreciation for Bertrand Russell. That clever theologian had gotten everything right. Not only had he accurately prophesied the coming of the parasites, he’d intuited their terrestrial provenance. I rushed outside. A soft moan filled the morning air, the mellifluous correlative of cacodaemonic bliss. I followed the sound to its predictable source, Bartholomew Articulis. He sat on the wrought-iron bench in the center of the vineyard, a caco clamped to his left temple, a second to his crown, a third to his occipital. Constantine stood behind the bench, hovering protectively over his employer.

“Not a single von Neumann probe,” Articulis muttered. “No Bracewell probes or Dyson spheres.”

Catching sight of me, the dwarf raised a finger to his lips—a superfluous gesture, for I already knew it would be disastrous to distract the abbot.

“Vorg,” sang the caco trio. “Vorg . . . Vorg . . . Vorg . . .”

“Success,” Constantine whispered.

“No colonizations, visitations, or plausible abductions,” Articulis hissed.

Even as the abbot sustained the Fermi Paradox, the two remaining cacos appeared in the clearing, wriggling across the grass like primeval slugs, thickening the dew with glistening slime. I recognized Brother Thomas’s caco from the grape stains on its dorsal side, Sister Ruth’s from its pallor. Reaching Articulis’s feet, the cacos began their ascent, slithering up his legs and across his torso.

“No radio contact,” said the unflinching abbot.

At last the Sorites Paradox caco came to rest on Articulis’s left parietal, then set about nourishing itself. Seconds later the Lawyer Paradox caco reached the abbot’s right temple.

“How long will it be, I wonder, before he goes mad?” I asked, sotto voce.

“This is a monastery,” Constantine replied. “Providence is on our side. I know in my heart that Abbot Articulis will remain healthy until the rift is healed. God didn’t bring the Order this far only to abandon it on the shores of the tarn.”

“Vorg,” trilled the happy cacos. “Vorg . . . Vorg . . . Vorg . . .”

“Perhaps I should stay here today,” I said. “Shall I cancel my three o’clock class?”

Constantine shook his head and said, “If I need you, I’ll send for you, but right now I’m abrim with hope. I must thank you for helping us defeat these malicious imps.”

At a loss for words, I merely said, “This sentence is not about itself, but about whether it is about itself.”

“I must admit, I’ve never cared for you Tillichean ground-of-being types,” the dwarf said. “It all seems to me like atheism by another name. But God works in mysterious ways. He may even work through assistant professors. Tonight I shall remember you in my prayers.”

I even tortured the class with the notorious Examination Paradox, claiming that I would give them a surprise multiple-choice test sometime the following week. The catch, I sadistically insisted, is that any teacher who wields such a threat cannot possibly make good on it. Friday will not bring the alleged surprise, for if the previous four days have been examination- free, then the test will inevitably fill the one remaining slot. The same reasoning applies to Thursday. Since the previous three days have been examination-free, and because Friday has already been ruled out, springing the test on day four would hardly qualify as unexpected. Through this chain of logic we can likewise assert that neither Wednesday nor Tuesday nor Monday allows the teacher to fulfill his desire.

Given my irrational condition, I was actually relieved when Constantine phoned and begged me to return to the monastery. Alas, the Fermi Paradox had proven less robust than Articulis had hoped, but he had no intention of surrendering to the cacos. If I could possibly manage it, I must come to Rhinebeck immediately and furnish the Tertullianists with a superior antinomy. I promised the dwarf I would leave posthaste.

At three o’clock I marched into my Philosophy 412 class and announced that the unexpected examination had been canceled, the course itself terminated, then cheerily informed them they were all getting As. Returning to my office, I told Mrs. Graham that I planned to spend the rest of the summer in the Adirondacks, healing a crack in the lithosphere.

“An academic conference?” she asked.

“Something like the opposite,” I replied, wondering what I meant. “I am the thought you are thinking now.”

“The day you stop talking crazy,” Mrs. Graham said, “I’ll know you’ve gone insane.”

I passed through the gates and entered the vineyard, soon reaching the center. Brother Thomas sat on the bench, considering his heaps, oblivious to everything save the problem of ill-defined boundaries. Absently he rolled back his hood, no doubt with the unconscious intention of savoring the evening breeze. For me the gesture proved impossibly distressing. The monk’s grape-stained caco had returned, once again drawing sustenance through his right temple.

I broke free of the maze, then dashed toward the cloister, drawn by a glimpse of Sister Ruth. She negotiated the quadrangle with measured strides, all the while cultivating the Lawyer Paradox. “If you find in favor of Euathlus, he owes me ten drachmas, as this is his first case,” she muttered. A malign bulge protruded from the left parietal area of her wimple, testament to a recrudescent caco.

Constantine came hobbling across the grounds, his pointed hat and lederhosen giving him the appearance of a garden gnome. Fighting tears, suppressing sobs, he guided me into the abbey and up the stairs to the dormitory floor. An instant later I found myself in Articulis’s cell, his recumbent form stretched out along his pallet. The abbot’s head was free of cacos, but his face betrayed a pervasive anguish. Sweat speckled his forehead like condensation on a glass of iced tea.

“Did you bring another paradox?” he inquired through clenched teeth.

“Yes,” I said, lying.

“A good one?”

“Tremendous.”

“Thank God.” Articulis released a rasping cough, the sort of wheezing hack that was less its own event than the symptom of a dire condition. “Last night a sixth parasite came among us. Tertullian neglected to foretell its advent, but I cannot deny the evidence of my senses. By cacodaemonic standards this new creature is apparently a great sage—though its appetites are not exotic. Even as we speak, it feeds contentedly on the Liar Paradox.”

“Before attaching itself to Brother Jonathan, our Übercaco made a presentation to its fellows,” Constantine said, wiping Articulis’s fevered brow with a cold damp cloth. “In honor of Tertullian, the sage spoke in Latin. There is no Vorg, it revealed. The creator-god, like all creator-gods, is a fiction. Our galaxy is bereft of transcendent beings, but it does contain billions of spacefaring vagabonds, the category to which every caco belongs.”

“Evidently they believed their sage,” I said.

“Every word,” Articulis said.

“And so the Fermi Paradox has collapsed.”

“Never to rise again.”

Closing his eyes, Articulis rolled onto his chest and pressed his moist face into the naked pillow. The shaft of a goose feather poked through the ticking. The abbot groaned, abducted from the moment by fever, pain, and weariness, his torments now bearing him to some muzzy borderland between sentience and sleep.

Constantine took me aside and explained that, on learning the truth of their origins, the cacos became furious with Articulis, convinced that he’d deliberately deluded them. Before abandoning the abbot and wriggling away to their accustomed hosts, the invaders had avenged themselves, planting their stingers in his flesh and filling his veins with venom.

Articulis, awakening, cried out for water. I filled his glass from a clay pitcher and placed it to his lips. “Tell me, Bartholomew, do you believe the sage’s argument? Is every creator-god a fiction?”

Articulis took a protracted swallow. “For me, God the Father will always be real. I know my Redeemer liveth. The Omega Point beckons.” His sentences came slowly, each syllable purchased at the price of a spasm. “But until humanity attains that celestial apex, our Order must continue its great commission.” A seismic shudder possessed the abbot, a fleshquake rolling from his cranium to his knees. “My monks and nuns are all still at their posts, fighting the good fight, and as soon as I’m rid of this poison, I’ll tutor myself in your new paradox, then proceed to ponder—”

Ponder. An appropriate last word for so thoughtful a man. We applied the usual tests, feeling for his pulse, pricking his heel, holding a mirror to his nostrils. Constantine drew a sheet over the abbot’s imposing frame, whereupon the two of us, Tertullianist and Tillichean, prayed for the soul of our departed friend.

For the next four hours I sat in the refectory, cultivating the conundrum in the garden of my brain. Sister Margaret was the first to arrive. No sooner had she taken her place before a steaming bowl of venison stew than the caco detached itself from her occipital and started in the direction of its new Omega Point, myself, inching along the dining table like some ghastly casserole come horribly to life, leaving lambent threads of slime behind. Slithering up my abdomen, chest, cheek, brow, the parasite at last came to rest atop my head. The alien was predictably cold-blooded, assuming it had a circulatory system, but this sensation was easily offset by the hot flush of conquest I now experienced. I’d lured the beast, by God. I’d seduced it with my genius for absurdity.

Brother Thomas appeared next, followed by Brother Francis, then Sister Ruth. In a matter of minutes my occipital region, right parietal, and left temple had each received a caco. Now Brother Jonathan entered the refectory, bearing his customary parasite on his crown. Sucking on the Liar Paradox, the Übercaco sprouted from his right temple, a lewd glistening invader, fatter than its fellows. I was not surprised when the lesser alien abandoned its host, twisting and flopping toward me until at last it found a home on my left parietal. The Übercaco remained in place, drawing affirmation from that vast set of antinomies by which a dissembler may dissemble to obscure his dissemblance. But my brain harbored a paradox more pleasing still—or so I hoped.

Imagine a box, large enough to contain a cat plus an outlandish contraption consisting of a hammer, a flask of deadly gas, a nugget of radium, and a Geiger counter. For this particular radium sample it happens that, over the span of one hour, there is a fifty- fifty chance that the nucleus of a single atom will decay. Detected by the Geiger counter, this event will trip the hammer, which will in turn break the flask, thus releasing the poison gas and killing the cat. Note that our Gedanken demonstration perversely couples the macroworld of discernible reality to the microworld of quantum mechanics, that probabilistic plane on which, under certain circumstances, we may legitimately claim that the collapse of the wave function occurs in consequence of the observer’s consciousness. We are thus forced to the unacceptable conclusion that, until the instant the experimenter opens the box and peers inside, the cat is simultaneously alive and dead, trapped in an absurd superposition of eigenstates.

Concentrating ferociously, breathing as deliberately as a mother giving birth, I fixed Schrödinger’s Paradox in my mind, its paraphernalia growing more delicious with each passing second. The radium sample glowed. The flask coruscated. The Geiger counter acquired a silver aura. The hammer became a Platonic archetype. The cat’s pelt transmuted into a sleek numinous rainbow.

After an excruciating interval the Übercaco abandoned Brother Thomas and undertook the journey to my brow, in time coming to rest on my right temple. Jeremiah entered and began to serve everyone Château Pelagius from a one- liter bottle. As he filled my goblet, I decided that my powers of concentration would not suffer in consequence of a few swallows. Thus it came to pass that, as the Tertullianists savored their wine, ate their stew, and battled the crack, I drank a toast to Schrödinger’s impossible cat.

The victory turned on our willingness to nurture the six absurdities or die in the attempt. We persisted, and we won. After a mere three weeks of contemplation, we beheld the tarn evaporate completely and the parent rift seal itself. The cacos hung around for another ten days, then finally took leave of my cranium, bound for some brighter star with better puzzles. And so it happens that, held to a standard of tangible evidence, the strange events related herein might as well never have occurred—a judgment that, the more you think about it, the faster it flips its facets: a verbal Necker cube.

Upon my return to the university, I made a point of keeping silent. Any attempt to relate these adventures to my colleagues might have greased the wheels of Dr. Virginia Sayles’s hostility, and before long she would have required me to undergo a psychiatric evaluation, subsequently manipulating the results so that they seemed to demand my institutionalization. Appearances to the contrary, becoming a mental patient is rarely an astute career move in academia.

A question hovers in the air. Why should a man like myself, an aesthete who would rather achieve tenure than a state of grace, have risked his sanity in appeasing a voracious band of extraterrestrial parasites? Why should he place the common good before his self-interest? I really don’t know. The mystery stumps me. Call it a paradox.

There remains only the matter of the boxed cat. You will be pleased to hear that Snowball lived. She recently gave birth. The kittens’ names are Bartholomew, Francis, Margaret, Jonathan, Ruth, and Thomas. How can I pronounce with such confidence on Snowball’s fate? Simple. Every word in the narrative you have just read is false, as befits a composition by yours truly, Donald Kreigar, assistant professor and competent philosopher, who never opens his mouth or picks up his pen except to tell a lie.