Logic Without Sense

Logic Without Sense

Achilles has reached the Turtle and has sat comfortably on his back.

"So you've reached the end of the course of our career?" Said the Tortoise. "Even though it consists of an infinite series of distances? Had not some wise man proved that such a thing could not be achieved? "

"It can be achieved," said Achilles. "It has been achieved! Solvitur Ambulando. You see, the distances were constantly decreasing: and so ... "

"But if they were constantly increasing," the Turtle interrupted. "How is it then?"

Logic Without Sense

"Then I should not be here," Achilles replied modestly; "And at this moment you should have already circled the earth a lot!"

"You beat me ... you flatten me, I mean" said the Turtle; "You are a heavyweight, that is indisputable! Now, would you like to know about a race track that most people imagine they can complete in two or three steps, when in fact it consists of an infinite number of distances, each one longer than the previous one? - Rior? "

"Very much, indeed!" Said the Greek warrior, while taking out of his helmet (Few Greek warriors had pockets in those days) a huge notebook and a pencil. "Proceeds! And speak slowly, please! The shorthand has not been invented yet! "

"That beautiful First Proposition of Euclid!" The Turtle murmured as if between dreams. "Do you admire Euclid?"

"Passionately! So much, at least, as one can admire a treatise that will not be published but in a few more centuries! "

"You flatter me" in the original, which can be understood as arroyar categorically. (N. Of T.)
"Well, let's take a bit of the argument in that First Proposition, just two steps, and the conclusion drawn from them. Be so kind to enter them in your Notebook2. And in order to refer them conveniently, let's call them A, B, and Z:

The things that are equal to the same, are equal to each other.
The two sides of the triangle are the same.
The two sides of the triangle are equal to each other.

"The readers of Euclid will grant, I suppose, that Z follows logically from A and B, so who accepts A and B as true. Will he have to accept Z as true?"

"Undoubtedly! The youngest child in a high school - as soon as the high schools are invented, which will not happen until about two thousand years later - will grant it "

"And some reader who has not yet accepted A and B as true Can you still accept the sequence as valid? I suppose"

"No doubt such a reader can exist. He can say 'I accept as true the Hypothetical Proposition that, if A and B are true, Z must be true; but I do not accept A and B as true. A reader like that should leave Euclid judiciously and dedicate himself to football. "

"And could not there also be a reader who said 'I accept A and B as true, but I do not accept the Hypothetical3'?"

"There certainly can be. He would also do better to dedicate himself to football. "

"And none of those readers," continued the Tortoise, "are you under the logical necessity of accepting Z as true?"

"Right," Achilles nodded.

"Notepad" has been used as a translation of "Note-book", which literally means "Book of notes". At this moment, Carroll proposes a game of words between "Note-book" and "Not-Book", referring the Turtle to the notebook of Achilles as "Not-Book": No-book, no-libreta. (N. Of T.)

It refers in these terms to the Hypothetical Proposition; to the inference connection between A, B and Z. (N. Of T.)
"Well, now, I want you to consider me as a reader of the second type, and force me, logically, to accept Z as true."

"A Turtle playing soccer would be ..." Achilles began to say.

"An abnormality, by the way" interrupted the turtle hastily. "Do not deviate from the point Let's see Z first and football later!"

"Do I have to force you to accept Z?" Said Achilles meditatively. "And your present position is that you accept A and B, but you do not accept the Hypothetical ..."

"Let's say everything is C," said the Tortoise. "... but you do not accept:

If A and B are true, Z must be true: "That is my current position," said the Tortoise.
"Then I must ask you to accept C."

"I will," said the Tortoise, "as soon as you put it in that notebook of yours
What else do you have in it? "

"Just a few notes," said Achilles, letting the sheets run nervously: "a few notes of ... the battles in which I have distinguished myself!"

"Full of white leaves, I see!" The Tortoise remarked cheerfully. "We will need them all! Now write what you dictate ":

The things that are equal to the same, are equal to each other.
The two sides of the triangle are the same.
If A and B are true, Z must be true.
The two sides of the triangle are equal to each other. "

"You should call it D, and not Z," said Achilles "comes after the other three. If you accept A and B and C, you must accept Z. "

"And why should I?"
"Because it follows logically from them. If A and B and C are true, Z must be true. You will not dispute that? I imagine"

"If A and B and C are true, Z must be true," the Tortoise repeated meditatively. "That's another hypothetical, is not it? And, if I have failed to see its truth, I can accept A and B and C, and still not accept Z, can I not? "

"You can," admitted the candid hero; "Even though such foolishness is phenomenal. If it is possible. So I must ask you to grant a more hypothetical. "

"Very well, I am quite ready to grant it, as soon as you have written it. We will call it:

If A and B and C are true, Z must be true.

Have you put it in your notebook? "

"I have it!" Achilles Joyfully exclaimed, as he put the pencil in its case. "And finally we have reached the end of this ideal racetrack! Now that you accept A and B and C and D, do you accept Z, of course? "

"Really?" Said the Turtle innocently. "Let's make it completely clear. I accept A and B and C and D. Suppose I still refuse to accept Z "

"Then Logic would take you by the throat and force you to it!" Replied Achilles. "Logic would tell you 'You can not avoid yourself. Now that you have accepted A and B and C and D, you must accept Z! 'So you will see that you had no choice! "

"Anyway, Logic is good enough to tell me it's worth noting," said the Turtle. "So put in your book please. We will call it
If A and B and C and D are true, Z must be true.

Do you see that it is a completely necessary step? "

"I see," said Achilles; and there was a touch of sadness in his voice.

Here the narrator, having had urgent business in the bank, was forced to leave the happy pair, and did not notice the point again until a few months later. When he did, Achilles was still sitting on the back of the very tough Turtle, and wrote in his notebook, which seemed to be almost completely full. La Tortuga said "Have you recorded the last step? Unless you have lost the account, they are a thousand one. There are many millions more to come. Would you consider, as a personal favor-taking into account that much of the instruction of this colloquium of ours, will provide the Logicians of the Nineteenth Century-would you mind, I say, adopt a play on words that my cousin, the False Turtle4 will invent then? , allowing yourself to be rechristened as the One Who Teach us?

"As you wish!" Replied the tired warrior, in the hollow tones of despair, as he buried his face in his hands. "As long as you, on your side, adopt a play on words that the False Turtle never did, and allow yourself to be renamed" Easy to Kill! "Reference to Mock-Turtle in Alice's Adventures in Wonderland. (N. Of T.)


Translation by Pablo Oyarzún R.

Which is better, a watch that is on time once a year, or one that is twice a day? "The second," you respond, "unquestionably." Very well, now attend.

I have two clocks: one does not walk at all and the other is delayed one minute a day: which one would you prefer? "He who falls behind," you reply, "without a doubt." Now observe: he who falls behind a minute a day has to lose twelve hours, or seven hundred and twenty minutes, before he is again sharp; consequently, it is only at one time in two years, while the other is evidently so many times as many times as the time indicated by it returns, which occurs twice a day.
So you have contradicted yourself once.

"Ah, but," you say, "what's the use of being punctual twice a day, if I can not know what time I am?" Well, suppose the clock is eight o'clock,
Do not you see that the clock will be on time at eight o'clock? Consequently, when it is eight o'clock, your watch will be on time.
"Yes, I see that," you reply.

Very well, then you have contradicted yourself twice: now get out of the jam as best you know, and do not contradict yourself again if you can avoid it.

It could be that you keep asking, "And how will I know when it's eight o'clock? My watch is not going to tell me. "Be patient: you know that when it is eight o'clock your watch will be on time, very well; So, this is your rule: keep your eyes fixed on the clock, and at the precise moment when it is on time, it will be eight o'clock. "But," you will say. And well, that will be enough; the more you argue, the more you will go away from the point, so we'd better stop.


Translation by Pablo Oyarzún R.

Half of the world, or almost half, is always under the light of the sun: as the world turns, this hemisphere of light also moves round, and passes successively on each part of it.

Suppose a Tuesday is tomorrow in London; the next hour will be Tuesday in the morning to the west of England; if everyone were land, we could go to the follow-up on Tuesday morning around all the time, until at twenty-four hours we would go back to London. But we know that in London twenty-four hours after Tuesday morning is Wednesday morning. Where, in its passage around the Earth, does the name change its day? Where does he lose his identity?

In practice, there is no difficulty in this, because a large part of the trip is by water, and what happens over there nobody can say; and, in addition, there are so many different languages, that it would be hopeless to try to follow the name of each day throughout the year. But is it not possible to conceive that the same Earth and the same language could be prolonged around the world? I do not think it is inconceivable; and in that case6 there would be no difference between one day and the next, and the same with the week, with the month, etc., so we would have to say: "The Battle of Waterloo happened today, about two million years ago. hours". Or you would have to fix a line where the change took place, so that the inhabitants of a house would wake up and say: "Ahum Tuesday morn!", And the inhabitants of a neighboring house (on the other side of the house). the line), a few miles to the west, would wake up a few minutes later and say, "Ahum, Wednesday morning!" It is not for me to say in what desperate confusion would be the people who will be lucky. live on the same line. I would fight every morning about what the name of the day would be. I can not imagine a third case, unless each was allowed to decide for himself, what would be a state of affairs far worse than either of the other two.

It is best to imagine oneself walking with the sun, and asking the inhabitants as they go: "What tomorrow is this?" If you suppose that they live all around, and all speak the same language, the difficulty is obvious. .

This, clearly, is an impossible case, and I only put it as a hypothesis.

The usual exclamation on waking, usually said with a yawn.
I realize that this idea has been presented before, namely, to the unknown author of that beautiful poem that begins: "If everyone were apple pie", etc.
The peculiar result we are discussing here, however, does not seem to have occurred to him, since it is limited to the difficulties in obtaining drink that would surely follow.

"If everyone were apple pie, and ink was all over the sea,
And all the trees bread and cheese,
What would we have to drink? "