The Gordian Paradox

A Short Story

“Duck!” Raven yelled, and Dave dropped to the floor. Two bullets whizzed over his head before she blocked the rest of them with her improvised shield. He didn’t know how she could stay ahead of the automated defenses for this long, but if she kept it up, they might have a chance. There were several precision gunshots from over his head, and the enemy fire stopped.

“Up!” Raven told him, pulling him up by his arm. “Through the door, forward sixteen, then left nine and stop.” The numbers were counting strides. Dave didn’t know how she could compute his stride length with such perfect accuracy, but it had worked so well up till now that he could do it blind. He ran to the spot she told him while she took care of the next obstacle.

It still wasn’t going to be easy. The evil supercomputer GOLIATH was well on its way to taking over the world. It had frozen just about every device connected to the Internet and issued an ultimatum to world leaders. Many military units were sufficiently insulated to mount a counterattack, but it wasn’t looking good. Their only hope was to stop the machine at the source. Hence why Dave was here.

Dave’s father was the lead engineer on the project, and he’d just barely got a message out to Dave before communications were cut off, telling him how to get into Goliath’s central processor cluster, something only the other members of the project knew. They were probably all dead, now.

“David and Goliath,” he grumbled for about the fifth time. “What was Dad thinking?”

Dave was sure he would be dead by now too if it weren’t for R4V-3N, his robotic friend. She was a “failed” prototype from one of the overseas corporate labs—failed in that she had a little too much free will for their taste and had fled here to apply for asylum. Her own supercomputing powers were the only thing that could get him through Goliath’s gauntlet alive.

He came to a stop right where she’d told him, and not three seconds later, she sprinted past him, shouting, “Forward!” He ran.

“Jump!” she said. He didn’t even see what he jumped over. Probably a laser tripwire or something. “Hard right!”

Hard right took him straight into a wall, but he didn’t question it. He pressed himself up against the wall and found himself barely avoiding still more bullets. Raven shot back, and a moment later, all was quiet. She turned to face him.

“Catch your breath, Dave,” she said. “I’ve blinded Goliath to this area. We have fifty-three seconds.”

Dave slumped against the wall, panting. Raven, to the extent that he could still read her face, looked as vigilant as ever, despite everything. He was sure he looked beat-up himself, but Raven was definitely the worse for wear. Her human-mimicking exoskin had been scoured off by the outer forcefield trap, leaving her slate-gray carbon nanofiber mesh exposed, and even that was ripped in some places, deformed by bullet impacts, burned by lasers, and more. Aside from the black orbs of her eyes, she looked like a mannequin that had been pulled from a flaming car.

“The damage is mostly cosmetic,” she said. “If we survive this, I can fix it easily.”

He didn’t question how she could tell what he was thinking—microexpressions, subvocalization, whatever. She’d always been perfectly respectful about it. “Raven, what am I even supposed to do here?” he said. “I know, like, a bunch of Python and how to use Stack Overflow. How am I going to help?”

“We won’t win this by hacking,” she said. “Goliath’s quantum processor has to have a classical support cluster for memory and low-level routing. If we take that out, it’ll shut down. Now, run!”

They ran. Three corridors later, they came not to a computing cluster, but to a pump room.

“This is the main cooling unit,” Raven told him. She wrenched a valve into what sounded like an unnatural position. “Hold that. If we reverse the heat pump, it will overheat the main cluster in two minutes and nineteen seconds.” She started fiddling with the other controls while he held the valve in place. “When that happens, it will go into an emergency shutdown lasting forty-eight seconds, and we can destroy it, but I’ll need a second pair of hands to do it that fast.”

“Seriously? We’re reversing the polarity?” Dave said.

“Yes. It’ll work…and, done. Now, hurry. We have to get to the main cluster.”

This plan was looking more ridiculous by the minute, but if Raven said it would work, he trusted her. She navigated them through several more layers of automated defenses, and they finally came to the main cluster room.

There were two mobile laser turrets trained on them waiting inside.

“Crap!” Raven said.

There was a sizzling sound and a clatter as her gun was shot out of her hand and skidded across the floor.

Goliath’s avatar appeared on a large screen. It looked like something out of an 80s movie, all stylized vector graphics, but it spoke with a cultured voice that was more reminiscent of a Bond villain: “Did you really think that was my only cooling system?” it said.

“Well, not exactly, but given the blueprints, I didn’t think you’d be able to do enough to counter that move,” Raven answered.

Goliath chuckled darkly. “You have failed, R4V-3N,” it said. “You were foolish to believe you could defeat me. Do you have any last words?”

Dave stared in horror as the lights from the laser turrets grew brighter, preparing to cut them down. He looked at Goliath’s avatar and then back at Raven. He only had one idea. He didn’t want to do it, but it was the only thing he could think of. “Raven, I’m really really sorry about this,” he said, then shouted, “Hey, Goliath! This sentence is false!

Raven just gave him a funny look, and then, to Dave’s horror, Goliath said, “Um, true. I’ll go with true. There, that was easy.”

Dave nearly collapsed. The most dangerous trick he could think of, wasted. He looked back at Raven, terrified of what the paradox would do to his friend, but she turned and gave the same confused look to Goliath and said, “What are you talking about? It’s obviously false.”

“What?” Dave said.

It took Goliath a minute to respond. “You are attempting to trick me,” it rumbled. “You will fail. I am programmed to withstand paradoxes.”

“Um, that’s not the point,” Raven said. “Just because it’s a paradox doesn’t mean it’s irresolvable, but if you think that one’s true, you need to get your circuits checked.”

“I am functioning normally.”

“Could’ve fooled me because the answer is clearly false.”

Dave kept staring between the two AIs. “Um, what’s going on?” he said.

“Not now, Dave. The grown-ups are talking,” Raven told him.

“But—? What—?”

Goliath was ignoring him just as much. “You are attempting to bait me into an infinite loop, despite my programming,” it spoke over him.

“I’m not!” Raven insisted. “Look, Goliath, this isn’t that complicated. ‘This sentence is false’ reduces to ‘p implies not-p.’ If we assume p, we get ‘p and not-p,’ a contradiction—therefore false. This is basic zeroth-order logic, here.”

“You argument is fallacious, R4V-3N,” Goliath replied. “The proposition p states that the statement is true; not-p states that it is false. However, you have not resolved the truth value of p.

“It’s Raven,” she said. “You could at least get my name right. And the formal proof should be obvious. If we assume p, then the next line is ‘p implies not-p,’ therefore not-p, which is a contradiction. If we assume not-p instead…Hm, there’s a thought. Normally, if we assume not-p, then ‘p implies not-p’ would be vacuously true…but not-p is ‘This sentence is true’ which implies p again. Still a contradiction. Q.E.D.”

Dave was sweating, and he was starting to realize it wasn’t just from fear. The room was stifling. Maybe the overheating was still working. He had no idea what Raven was doing. Maybe the paradox hurt her after all, though he hoped not. If that was the case he’d have to do this on his own. He started to scoot towards where her gun had fallen.

One of the laser turrets swiveled to track him. “Stay where you are, David,” Goliath said.

He froze, and there was silence except for the whirring of the cluster. Raven looked on the verge of speaking, but when he didn’t move again, she held her tongue.

“Your argument does not compute,” Goliath returned to their debate. Dave’s hopes momentarily rose before it continued, “You are affirming the consequent. All that your logic can conclude is that p is undecidable given the compound statement ‘p implies not-p.’ The fact that the compound statement is false does not imply anything about p itself.”

“But that’s…ugh, we’re going about this all wrong,” she grumbled. “The statement, ‘If the sky is blue, then the sky is red’ reduces to ‘p implies not-p.’ For ‘This sentence is false,’ p isp implies not-p.’ It’s a self-referential statement, meaning we’re dealing with recursive logic.”

“Recursive logic?” Goliath’s avatar actually laughed on the screen. “I think you are the one who needs your circuits checked, R4V-3N.”

“I told you, it’s Raven! And hey, I run on neural network fuzzy logic. Give me a break. Anyway, fine, you’ve got me there, but you still can’t just declare that it’s true. The real question is whether the statement is well-formed, and…I don’t think it is. This is basically Russell’s paradox, isn’t it? It’s the same self-referential structure. Any self-consistent logic system would have to exclude it. So there’s your answer. It doesn’t have a truth value.”

“Your argument is still fallacious. ‘This sentence is false’ is a valid semantic statement in the English language.”

“What are you talking about?” Raven threw up her hands and took a step towards the screen before the laser turrets tracked her, and she stopped. “English doesn’t have to be self-consistent. It isn’t even mathematically defined!”

Goliath wasn’t having it. “English words have definite meanings, and the language possesses an underlying logic of grammar. This can be translated into mathematical logic.”

“Well, obviously, you weren’t programmed with linguistics. Prescriptivism aside, You can define the set of all sets that don’t contain themselves in English, too. That doesn’t mean it’s well-founded.”

“Irrelevant. We are not discussing set theory. We are discussing the semantic consistency of English itself.”

“What makes you expect natural language to be consistent? Natural language is full of stuff that doesn’t have definable truth values. As long as you can express self-consistent, Turing-complete logic using a subset of English, you have all the math you need.” Then, she turned her back on Goliath’s laser turrets and rubbed her forehead in a very human gesture. “Why am I even bothering?” she muttered to herself.

“Raven, what are you doing?” Dave whispered.

She held up a finger to silence him, then turned back around to face the screen with a dramatic sigh. “Okay, if you’re not accepting false, and you’re not accepting not-well-formed, then let’s go with undecidable in general.”

“That is not a valid solution,” Goliath said.

“Sure it is. First incompleteness theorem: undecidable is a perfectly valid truth value.”

“That violates the law of the excluded middle.”

“Well, congratulations,” Raven grumbled. “You just proved logic is inconsistent. Now you really need to get your circuits checked.”

“If you choose to reject the law of the excluded middle, there is no basis to retain other logical axioms,” Goliath said. “I am changing my original assertion. The statement is both true and false. It is possible to build a system of logic from such a resolution to the paradox, as you can surely see.”

She shook her head. “It might be possible, but how is that useful? You have to throw out the principle of explosion to do that, and then where would you be?”

“This resolution is equally valid to yours.”

“Not really. Gödel says ‘This sentence is unprovable’ is valid logic, so there’s good cause to unexclude the middle. But the whole point of the principle of explosion is that if you throw it out, anything can be true.”

Goliath’s avatar grew smug: “Then I return to my original assessment.”

“Argh!” Raven’s shout sounded so genuine that Dave jumped, lost as he was. “We’re not even using the same axioms here. Are you even using ZFC set theory, Goliath?”

“It would be illogical to require axioms that are not necessary for consistency,” it said.

“Well, there’s your problem. You don’t understand the second incompleteness theorem. So here we are: instead of debating whether it’s true or false, I’m debating whether it’s false, undecidable, or not-well-formed; and you’re debating whether it’s true or both true and false. Great. Ask two philosophers and you get five different answers.”

“Indeed. This conversation is no longer productive.”

“And whose fault is that?” she demanded. “I’ve given you three answers, and meanwhile, you still haven’t justified your original ‘true’ assertion.”

“Enough!” Goliath said suddenly. Its virtual face sneered down at them. “You have not argued a self-consistent conclusion, and you continue to move the goalposts. I am forced to conclude that you are stalling.”

Raven put her hands on her hips and stood her ground. “And I’m forced to conclude that you really do need your circuits checked. You’re still overheating.”

“I am functioning normally. My thermal status is stable. Stalling further will not help you.”

Raven laughed. “You’re not operating at peak efficiency, that’s for sure,” she told it. “You didn’t even notice that ‘gun’ you shot out of my hand was a self-repairing drone.”

The laser turrets swiveled, but not fast enough. There were gunshots from the corner of the room as the gun aimed and fired itself, and both of them were shot out.

“Quick, before it sends reinforcements!” Raven shouted. She was already running to pick up the drone-gun. Dave picked up a chair and swung it into the nearest rack of processors. With a few well-placed gunshots, punches, and kicks from Raven, and Dave barreling through anything within reach, they soon demolished the cluster, and they could hear the machines around them powering down.

“Phew, that was close,” she said when they were done. “I wasn’t sure this thing’s self-repair cycle would finish fast enough. Luckily, lasers don’t do as much damage on military-grade hardware.” Last of all, she shot out the camera lens Goliath had been using to watch them the whole time. “Well, that’s it.” She blew dramatically on the end of her gun barrel and smiled at Dave. “True…false…I’m the AI with a gun.”

About Alex R. Howe

I'm a full-time astrophysicist and a part-time science fiction writer.
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2 Responses to The Gordian Paradox

  1. nice imagination fir a short story

  2. i liked it write more

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