続続続アキレスがカメに云ったこと [ノート]
ずいぶん間があいてしまった。書きたいことが無いわけではないのだが、どうにも機が熟さない。で、ここはひとまずキャロルの What the Tortoise said to Achilles をまるごと引いて、あらためて少しばかりコメントを付すことでお茶をにごしておきたい。
Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.
"So you've got to the end of our race-course?" said the Tortoise. "Even though it does consist of an infinite series of distances? I thought some wiseacre or another had proved that the thing couldn't be done?"
"It can be done," said Achilles; "It has been done! Solvitur ambulando. You see, the distances were constantly diminishing; and so—"
"But if they had been constantly increasing?" the Tortoise interrupted. "How then?"
"Then I shouldn't be here," Achilles modestly replied; "and you would have got several times round the world, by this time!"
"You flatter me—flatten, I mean," said the Tortoise; "for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?"
"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. "Proceed! And speak slowly, please. Short-hand isn't invented yet!"
"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily. "You admire Euclid?"
"Passionately! So far, at least, as one can admire a treatise that wo'n't be published for some centuries to come!"
"Well, now, let's take a little bit of the argument in that First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in order to refer to them conveniently, let's call them A, B, and Z:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"
"Undoubtedly! The youngest child in High School—as soon as High Schools are invented, which wlil not be till some two thousand years later—will grant that."
"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"
"No doubt such a reader might exist. He might say, 'I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football."
"And might there not also be some reader who would say, 'I accept A and B as true, but I don't accept the Hypothetical'?"
"Certainly there might. He, also, had better take to football."
"And neither of these readers," the Tortoise continued, "is as yet under any logical necessity to accept Z as true?"
"Quite so," Achilles assented.
"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true."
"A tortoise playing football would be—" Achilles was beginning
"—an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards!"
"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you don't accept the Hypothetical—"
"Let's call it C," said the Tortoise.
"—but you don't accept
(C) If A and B are true, Z must be true."
"That is my present position," said the Tortoise.
"Then I must ask you to accept C."
"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it?"
"Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few memoranda of—of the battles in which I have distinguished myself!"
"Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them all!" (Achilles shuddered.) "Now write as I dictate:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."
"You should call it D, not Z," said Achilles. "It comes next to the other three. If you accept A and B and C, you must accept Z."
"And why must I?"
"Because it follows logically from them. If A and B and C are true, Z must be true. You don't dispute that, I imagine?"
"If A and B and C are true, Z must be true," the Tortoise thoughtfully repeated. "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"
"You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical."
"Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it
(D) If A and B and C are true, Z must be true.
Have you entered that in your note-book?"
"I have!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z."
"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"
"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, 'You ca'n't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see. "
"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your book, please. We will call it
(E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
"I see," said Achilles; and there was a touch of sadness in his tone.
Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying, "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century—would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?"
"As you please!" replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. "Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease!"
このアキレスとカメの C の類の仮言命題を逐次書きつらねる共同作業を彼等のレースと対比してみたい。まずは、そのレースについて前に見たところを補足を加えつつまとめておこう。
● 我々は、アキレスがカメのスタート地点に達するまでの間にカメもまた先へ進む、ということからコース上の地点を逐次規定するためのひとつの再帰的アルゴリズムを感得するとともに、その規定の作業が原理的には際限無く続行可能であることをさとる。ただし、そのアルゴリズムはそれに適う地点の無限列（あるいは上でカメが云っているような両者の間の隔たりの無限列（an infinite series of distances））の存在を保証するものではない。
● 「以下同様」の類の表現はレースの記述の一部なのではなくて記述に係るものであり、それによって当の記述がさらに進捗することはない。ところが、「以下同様」の闖入はそれで記述が一挙に進んだかのごとき錯覚を惹き起こすとともに、件のコース上の地点を逐次規定する作業の際限無さをあらためて照らし出す効果をもつ。
● それやこれやでレースの記述にまつわる際限無さがアキレスの走りに投影されるところに、パラドクスの幻影が現われる。
さて、アキレスとカメの共同作業だが、我々はカメの「それ［A と B と C が真ならば、Z は真であるほかない、という命題］もやっぱり仮言ではないか。そしてその真理を看て取るのをしくじったならば、私は A と B と C を認めつつもなお Z を認めないことだろう」という言葉から C の類の仮言を逐次書きつらねるための再帰的アルゴリズムを感得するとともに、その書きつらねの作業が原理的には際限無く続行可能であることをさとる。
ここでは C、D、E と仮言が並べられて行くくだりがレースの記述に相当すると云っていいだろう。そして、「以下同様」の役目を果たしているのは透明だった語り手が見物人として顕われて以降のくだりだ。面白いのは彼のだしぬけの登場とともに語りのレヴェルがあざやかに遷移し、そして、レースの場合とは違って、そのままもとに還ることなくはなしが進行する点だ。つまり、ここにはパラドクスの幻影の現出を援けるような仕掛けは無い。もっとも、ここでのアキレスは、際限無さをあらためて投影されるまでもなく、既に際限無く続行可能な作業の一端を担っているわけだが、しかし、その仕事にパラドクシカルなところは無い。（あるいは、これには異議が呈されるかもしれない。アキレスは無限のタスクを遂行しようとしているのだからパラドクシカルではないか、と。たしかに、彼等の共同作業が無限のプロセスから成るものであれば、そう云い得るだろう。そのような作業もやはりひとつの作業であるには違いないから。だが、彼等がたずさわっているのは際限無く続行可能な作業だ。それはもっぱら続行される限りにおいてひとつの作業なのであり、その限りにおいてつねに有限のプロセスから成る。際限無さはその続行可能性にあり、そこに遂行やら完遂やらの出る幕は無い。そして、アキレスがその共同作業の一端を担うことによって遂行しようとしているのはカメに Z が真であると認めさせることであり、それは無限の果てにおいて達成されるようなものではない。）
こうしてみると、いわゆる無限背進の演出はここでは飾りに過ぎないとも云い得る。パラドクシカルなのは、前にも指摘したとおり、その共同作業をはじめるに至るカメとアキレスの遣り取りであり、その核には次のような課題がある。
A と B を真だと認めるが C を認めない者をして Z を真だと論理的に認めざるを得なくせしめよ。
いったいこれにどう応じればいいのか？ そのような者を前にしてはそもそも論理的ということが宙に浮いてしまうわけで、しかも、まさに問題の推論こそが、通常は、A と B を真だと認める者をして Z を真だと認めせしめるものなのだから。
ついでに、いかにもキャロルらしいくどさが殺伐としたおかしみをそこはかとなく醸しているところを挙げて終えるとしよう。「The two sides of this Triangle are things that are equal to the same」という命題 B のフォーミュレーションがそれだ。ここには「things that are」など無くていいし無い方がすっきりするのにわざわざ付け加えてあるのは、もちろん、A に揃えるためだろう――そうしないことには A の主語と B の述語が正確には一致せず三段論法の型に微妙に合わなくなってしまうから。
☆ ☆ ☆
WHAT THE TORTOISE SAID TO ACHILLES
Lewis Carroll
Achilles had overtaken the Tortoise, and had seated himself comfortably on its back.
"So you've got to the end of our race-course?" said the Tortoise. "Even though it does consist of an infinite series of distances? I thought some wiseacre or another had proved that the thing couldn't be done?"
"It can be done," said Achilles; "It has been done! Solvitur ambulando. You see, the distances were constantly diminishing; and so—"
"But if they had been constantly increasing?" the Tortoise interrupted. "How then?"
"Then I shouldn't be here," Achilles modestly replied; "and you would have got several times round the world, by this time!"
"You flatter me—flatten, I mean," said the Tortoise; "for you are a heavy weight, and no mistake! Well now, would you like to hear of a race-course, that most people fancy they can get to the end of in two or three steps, while it really consists of an infinite number of distances, each one longer than the previous one?"
"Very much indeed!" said the Grecian warrior, as he drew from his helmet (few Grecian warriors possessed pockets in those days) an enormous note-book and a pencil. "Proceed! And speak slowly, please. Short-hand isn't invented yet!"
"That beautiful First Proposition of Euclid!" the Tortoise murmured dreamily. "You admire Euclid?"
"Passionately! So far, at least, as one can admire a treatise that wo'n't be published for some centuries to come!"
"Well, now, let's take a little bit of the argument in that First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them in your note-book. And in order to refer to them conveniently, let's call them A, B, and Z:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(Z) The two sides of this Triangle are equal to each other.
Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"
"Undoubtedly! The youngest child in High School—as soon as High Schools are invented, which wlil not be till some two thousand years later—will grant that."
"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"
"No doubt such a reader might exist. He might say, 'I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football."
"And might there not also be some reader who would say, 'I accept A and B as true, but I don't accept the Hypothetical'?"
"Certainly there might. He, also, had better take to football."
"And neither of these readers," the Tortoise continued, "is as yet under any logical necessity to accept Z as true?"
"Quite so," Achilles assented.
"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true."
"A tortoise playing football would be—" Achilles was beginning
"—an anomaly, of course," the Tortoise hastily interrupted. "Don't wander from the point. Let's have Z first, and football afterwards!"
"I'm to force you to accept Z, am I?" Achilles said musingly. "And your present position is that you accept A and B, but you don't accept the Hypothetical—"
"Let's call it C," said the Tortoise.
"—but you don't accept
(C) If A and B are true, Z must be true."
"That is my present position," said the Tortoise.
"Then I must ask you to accept C."
"I'll do so," said the Tortoise, "as soon as you've entered it in that note-book of yours. What else have you got in it?"
"Only a few memoranda," said Achilles, nervously fluttering the leaves: "a few memoranda of—of the battles in which I have distinguished myself!"
"Plenty of blank leaves, I see!" the Tortoise cheerily remarked. "We shall need them all!" (Achilles shuddered.) "Now write as I dictate:—
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this Triangle are things that are equal to the same.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."
"You should call it D, not Z," said Achilles. "It comes next to the other three. If you accept A and B and C, you must accept Z."
"And why must I?"
"Because it follows logically from them. If A and B and C are true, Z must be true. You don't dispute that, I imagine?"
"If A and B and C are true, Z must be true," the Tortoise thoughtfully repeated. "That's another Hypothetical, isn't it? And, if I failed to see its truth, I might accept A and B and C, and still not accept Z, mightn't I?"
"You might," the candid hero admitted; "though such obtuseness would certainly be phenomenal. Still, the event is possible. So I must ask you to grant one more Hypothetical."
"Very good. I'm quite willing to grant it, as soon as you've written it down. We will call it
(D) If A and B and C are true, Z must be true.
Have you entered that in your note-book?"
"I have!" Achilles joyfully exclaimed, as he ran the pencil into its sheath. "And at last we've got to the end of this ideal race-course! Now that you accept A and B and C and D, of course you accept Z."
"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"
"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, 'You ca'n't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see. "
"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your book, please. We will call it
(E) If A and B and C and D are true, Z must be true. Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
"I see," said Achilles; and there was a touch of sadness in his tone.
Here the narrator, having pressing business at the Bank, was obliged to leave the happy pair, and did not again pass the spot until some months afterwards. When he did so, Achilles was still seated on the back of the much-enduring Tortoise, and was writing in his note-book, which appeared to be nearly full. The Tortoise was saying, "Have you got that last step written down? Unless I've lost count, that makes a thousand and one. There are several millions more to come. And would you mind, as a personal favour, considering what a lot of instruction this colloquy of ours will provide for the Logicians of the Nineteenth Century—would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?"
"As you please!" replied the weary warrior, in the hollow tones of despair, as he buried his face in his hands. "Provided that you, for your part, will adopt a pun the Mock-Turtle never made, and allow yourself to be re-named A Kill-Ease!"
☆ ☆ ☆
このアキレスとカメの C の類の仮言命題を逐次書きつらねる共同作業を彼等のレースと対比してみたい。まずは、そのレースについて前に見たところを補足を加えつつまとめておこう。
● 我々は、アキレスがカメのスタート地点に達するまでの間にカメもまた先へ進む、ということからコース上の地点を逐次規定するためのひとつの再帰的アルゴリズムを感得するとともに、その規定の作業が原理的には際限無く続行可能であることをさとる。ただし、そのアルゴリズムはそれに適う地点の無限列（あるいは上でカメが云っているような両者の間の隔たりの無限列（an infinite series of distances））の存在を保証するものではない。
● 「以下同様」の類の表現はレースの記述の一部なのではなくて記述に係るものであり、それによって当の記述がさらに進捗することはない。ところが、「以下同様」の闖入はそれで記述が一挙に進んだかのごとき錯覚を惹き起こすとともに、件のコース上の地点を逐次規定する作業の際限無さをあらためて照らし出す効果をもつ。
● それやこれやでレースの記述にまつわる際限無さがアキレスの走りに投影されるところに、パラドクスの幻影が現われる。
さて、アキレスとカメの共同作業だが、我々はカメの「それ［A と B と C が真ならば、Z は真であるほかない、という命題］もやっぱり仮言ではないか。そしてその真理を看て取るのをしくじったならば、私は A と B と C を認めつつもなお Z を認めないことだろう」という言葉から C の類の仮言を逐次書きつらねるための再帰的アルゴリズムを感得するとともに、その書きつらねの作業が原理的には際限無く続行可能であることをさとる。
ここでは C、D、E と仮言が並べられて行くくだりがレースの記述に相当すると云っていいだろう。そして、「以下同様」の役目を果たしているのは透明だった語り手が見物人として顕われて以降のくだりだ。面白いのは彼のだしぬけの登場とともに語りのレヴェルがあざやかに遷移し、そして、レースの場合とは違って、そのままもとに還ることなくはなしが進行する点だ。つまり、ここにはパラドクスの幻影の現出を援けるような仕掛けは無い。もっとも、ここでのアキレスは、際限無さをあらためて投影されるまでもなく、既に際限無く続行可能な作業の一端を担っているわけだが、しかし、その仕事にパラドクシカルなところは無い。（あるいは、これには異議が呈されるかもしれない。アキレスは無限のタスクを遂行しようとしているのだからパラドクシカルではないか、と。たしかに、彼等の共同作業が無限のプロセスから成るものであれば、そう云い得るだろう。そのような作業もやはりひとつの作業であるには違いないから。だが、彼等がたずさわっているのは際限無く続行可能な作業だ。それはもっぱら続行される限りにおいてひとつの作業なのであり、その限りにおいてつねに有限のプロセスから成る。際限無さはその続行可能性にあり、そこに遂行やら完遂やらの出る幕は無い。そして、アキレスがその共同作業の一端を担うことによって遂行しようとしているのはカメに Z が真であると認めさせることであり、それは無限の果てにおいて達成されるようなものではない。）
こうしてみると、いわゆる無限背進の演出はここでは飾りに過ぎないとも云い得る。パラドクシカルなのは、前にも指摘したとおり、その共同作業をはじめるに至るカメとアキレスの遣り取りであり、その核には次のような課題がある。
A と B を真だと認めるが C を認めない者をして Z を真だと論理的に認めざるを得なくせしめよ。
いったいこれにどう応じればいいのか？ そのような者を前にしてはそもそも論理的ということが宙に浮いてしまうわけで、しかも、まさに問題の推論こそが、通常は、A と B を真だと認める者をして Z を真だと認めせしめるものなのだから。
ついでに、いかにもキャロルらしいくどさが殺伐としたおかしみをそこはかとなく醸しているところを挙げて終えるとしよう。「The two sides of this Triangle are things that are equal to the same」という命題 B のフォーミュレーションがそれだ。ここには「things that are」など無くていいし無い方がすっきりするのにわざわざ付け加えてあるのは、もちろん、A に揃えるためだろう――そうしないことには A の主語と B の述語が正確には一致せず三段論法の型に微妙に合わなくなってしまうから。
2019-05-11 19:37
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