What Snout the Bear Said to Natalie: A Playful Puzzle of Logic

On what may or may not have been Opposite Day — determining such things is never easy — Natalie and Snout the Bear lounged in their hammocks, basking in the late summer sun.

“Snout,” Natalie began, “I’ve been pondering a most perplexing puzzle.”

“Oh?” Snout’s ears perked up. “Do enlighten me, dear Natalie.”

“It’s about Opposite Day,” she continued. “Suppose someone says, ‘Today is not Opposite Day.’ How do we know if that’s true?”

Snout adjusted his posture, as bears do when faced with such profound matters. “Ah, a delightful conundrum! Let us be systematic. We shall call the statement ‘Today is not Opposite Day’ our Proposition A.”

“Splendid!” Natalie exclaimed, sitting up straighter.

“Now,” Snout continued, “let us consider two scenarios. Scenario 1: It is, in fact, Opposite Day.”

Natalie nodded eagerly.

“In this case,” Snout said, “Proposition A must be false, as it is indeed Opposite Day. But!” — he raised a finger — “On Opposite Day, one is compelled to declare the opposite of the truth. Therefore, they would still say, ‘Today is not Opposite Day.’”

“Ah!” Natalie’s eyes gleamed with understanding. “But of course! They’re speaking opposite-truths, or as some might say… falsehoods. And what of Scenario 2?”

“Ah, yes. Scenario 2: It is not Opposite Day,” Snout replied. “In this case, Proposition A is perfectly true, and our hypothetical speaker would again say, ‘Today is not Opposite Day.’”

Natalie’s eyes widened as the paradox unraveled and re-knotted. “But that means…”

“Precisely!” Snout interjected, “Whether it is or isn’t Opposite Day, the statement remains the same: ‘Today is not Opposite Day.’ It’s a statement that refutes itself by refusing to refute itself.”

Natalie paused. “So… we’re left with a statement that tells us nothing at all!”

Snout leaned back, a contented smile playing on his face. “Not nothing, dear Natalie. It tells us that certain questions love to tangle themselves into knots.”

“But surely there must be a solution,” Natalie persisted.

Snout sat up slightly. “A second proposition? Ah, now we’re on to something. “Let us introduce Proposition B: ‘This statement is false if and only if today is Opposite Day, and I am not a bear well-versed in paradoxes.’” 

Natalie’s brow furrowed. “But Snout, you are indeed a bear well-versed in paradoxes.” 

Snout glanced down at his paws in mock surprise. “Am I? Or is that just what Opposite Day would have me believe? It’s rather hard to tell.” 

Natalie’s face lit up. “But wait, if Proposition B is true, then it’s not Opposite Day, and you are a bear well-versed in paradoxes, so the statement holds! But if it holds, then it could still be false… and we’re back where we started. It’s a loop we can’t escape!”

“A kind of bear trap for the mind,” Snout added, grinning.

Natalie giggled. “And if Proposition B is false, then it’s Opposite Day, but then you wouldn’t be paradox-savvy, which would mean… you are paradox-savvy, which would mean it’s not Opposite Day, which would mean you are paradox-savvy, which would mean…”

Snout raised a paw. “Precisely! We’d be stuck in a paradoxical pretzel, all twisted up and deliciously unsolvable.” 

“So,” Natalie sighed, “we cannot resolve this puzzle through logic alone?”

“No puzzle of worth ever can be,” Snout replied sagely. “The trick is realizing that some questions aren’t meant to be solved; they’re meant to be wrestled with.”

Natalie grinned. “It’s rather like a carousel, isn’t it? You circle around and around, only to find yourself right back where you began.”

“Precisely,” Snout nodded, a twinkle in his eye. “And therein lies the fun!”

As the day — Opposite or otherwise, though, who could say for sure? — wore on, Natalie and Snout continued their musings, each lap offering fresh insights. Whether they were moving forward or backward, nobody could say for sure — least of all themselves.