What Makes a Good Argument? 什麼是好的論證？

🔊Take the most famous example in the subject. All humans are mortal; Socrates is human; therefore Socrates is mortal. If you grant the two premises you cannot consistently deny the conclusion. An argument with this property is called valid: it is impossible for all of its premises to be true while its conclusion is false. The premises are then said to entail the conclusion.

拿這門學問裡最有名的例子來看。所有人都會死；蘇格拉底是人；所以蘇格拉底會死。如果你承認這兩個前提，就不可能不自相矛盾地否認結論。有這種性質的論證叫作「有效」：它的前提不可能全部為真而結論為假。這時我們說，前提「蘊涵」結論。

🔊Aristotle (384–322 BCE) saw that this necessity has nothing to do with Socrates or with death. It holds by virtue of the pattern alone. Replace the terms with letters, as he was the first to do, and the pattern still compels: if A belongs to every B, and B belongs to every C, then A belongs to every C. Whatever you put in for the letters, true premises will never lead to a false conclusion. With that step, the study of reasoning became a science of forms.

亞里斯多德（西元前 384–322 年）看出，這種必然性跟蘇格拉底無關，也跟死亡無關；它單單憑著那個樣式成立。像他首創的那樣，把詞項換成字母，這個樣式照樣有約束力：如果 A 屬於每一個 B，而 B 屬於每一個 C，那麼 A 屬於每一個 C。不管你把什麼填進字母裡，真的前提永遠不會導出假的結論。跨出這一步，對推理的研究就成了一門關於形式的科學。

🔊The definition has a consequence that surprises almost everyone at first. Validity says nothing about whether the premises are in fact true. “All fish can fly; all whales are fish; therefore all whales can fly” is a perfectly valid argument. Every statement in it is false, and yet the conclusion follows: if the premises were true, the conclusion would have to be.

這個定義有一個後果，幾乎每個人初次聽到都會吃驚：有效性完全不管前提事實上是不是真的。「所有的魚都會飛；所有的鯨都是魚；所以所有的鯨都會飛」是一個完全有效的論證。裡面每一句都是假的，可是結論推得出來：假如前提為真，結論就非真不可。

🔊The reverse also happens. “All poets are human; Li Bai is human; therefore Li Bai is a poet” has two true premises and a true conclusion, and it is invalid. The quickest way to see this is to keep the form and change the content: all poets are human; Socrates is human; therefore Socrates is a poet. The premises are still true and the conclusion is now false, so the form cannot be trusted, and it never could. This is the method of counterexample, and it is the logician’s basic instrument.

反過來的情況也有。「所有詩人都是人；李白是人；所以李白是詩人」有兩個真前提、一個真結論，卻是無效的。要看出這一點，最快的辦法是保留形式、更換內容：所有詩人都是人；蘇格拉底是人；所以蘇格拉底是詩人。前提還是真的，結論卻變成假的了，可見這個形式靠不住，而且從來就靠不住。這就是反例法，邏輯學家最基本的工具。

🔊An argument that is valid and also has true premises is called sound. Soundness is what we actually want, because only a sound argument guarantees its conclusion. The division of labor is worth noticing. Whether an argument is valid can be settled from an armchair. Whether its premises are true usually cannot: that depends on biology, or history, or whatever else the premises bear on. Logic transmits truth from premises to conclusion. It does not supply the truth in the first place.

既有效、前提又都為真的論證，叫作「健全」。健全才是我們真正要的，因為只有健全的論證保證它的結論。這裡的分工值得留意。一個論證有沒有效，坐在扶手椅上就能判定。它的前提是不是真的，通常不行：那要看生物學、歷史，或前提所涉及的任何別的領域。邏輯把真從前提傳送到結論；它並不一開始就供應真。

🔊Much everyday reasoning runs on “if.” Two patterns are valid. Modus ponens: if P then Q; P; therefore Q. Modus tollens: if P then Q; not Q; therefore not P. Each has an invalid twin that looks almost the same. Affirming the consequent: if it rained, the street is wet; the street is wet; therefore it rained. (A street cleaner may have passed.) Denying the antecedent: if it rained, the street is wet; it did not rain; therefore the street is not wet. A wet street does not rule out other causes, and that is all it takes for both twins to fail.

日常推理有一大半是靠「如果」在運轉的。有兩個樣式有效。肯定前件（modus ponens）：如果 P 則 Q；P；所以 Q。否定後件（modus tollens）：如果 P 則 Q；非 Q；所以非 P。它們各有一個長得幾乎一樣的無效雙胞胎。肯定後件：如果下過雨，街道是溼的；街道是溼的；所以下過雨。（也許是洗街車剛經過。）否定前件：如果下過雨，街道是溼的；沒下過雨；所以街道不是溼的。街道溼，並不排除別的原因；單憑這一點，兩個雙胞胎就都不成立。

🔊Everything so far concerns deduction, where the premises are meant to guarantee the conclusion. Most reasoning in science and in life is of another kind. From “every swan observed so far has been white” to “all swans are white” is an inductive step: the conclusion says more than the premises do, and so it can be false even when they are true. Inductive arguments are therefore never valid, and it would be a mistake to blame them for that. They are judged by strength, and a strong one makes its conclusion probable. Whether that standard can itself be justified is a famous problem, taken up in lesson A8.

到目前為止談的都是演繹：前提要保證結論。科學與生活裡的推理，大半屬於另一種。從「目前觀察到的每一隻天鵝都是白的」到「所有天鵝都是白的」是歸納的一步：結論說的比前提多，所以即使前提為真，結論也可能為假。因此歸納論證永遠不是有效的，而為此責怪它是弄錯了對象。評判它的標準是「強度」：強的歸納論證使結論很可能為真。這個標準本身能不能被證成，是一個著名的難題，留到 A8 那一課。

🔊Logic was invented at least three times. In China the later Mohists, in the third century BCE, analyzed disputation and held that an explanation must bring out the reason for a claim: 以說出故. In India the Nyāya school set out a five-step inference that became the common property of Hindu and Buddhist philosophers: the hill has fire; because it has smoke; wherever there is smoke there is fire, as in a kitchen; this hill has smoke; therefore it has fire.

邏輯至少被發明過三次。在中國，西元前三世紀的後期墨家分析了辯論，主張解說必須把一個主張的理由提出來：「以說出故」。在印度，正理派定出了五支的推論，後來成為印度教與佛教哲學家的共同資產：此山有火；因為有煙；凡有煙處必有火，如廚房；此山有煙；所以此山有火。

🔊The Indian pattern differs from Aristotle’s in one instructive way. It insists on an example, the kitchen, so that the general rule is tied to a case both parties have seen. A Greek syllogism asks only whether the conclusion follows. An Indian inference asks in addition how the rule is known. That second question, as we shall see, is exactly where Aristotle’s critics attacked.

印度的樣式跟亞里斯多德的有一個很有啟發性的差別：它堅持要舉一個例子（廚房），把普遍規則繫在雙方都見過的個案上。希臘的三段論只問結論推不推得出來；印度的推論還多問一句：這條規則是怎麼知道的？我們接下來會看到，批評亞里斯多德的人，攻擊的正是這第二個問題。

🔊It may seem that validity is trivially easy to get, and in a sense it is: for any conclusion at all, one can invent premises that entail it. That is why a valid argument settles nothing by itself, and why serious disagreement is nearly always about premises. Yet the discipline of form is what makes such disagreement productive. Set an opponent’s reasoning out in valid form, supplying in its strongest version whatever was left unsaid, and you will see which premise carries the weight. You then know precisely what the two of you disagree about, and in turn what evidence would bear on it. Philosophers call this the principle of charity. It is good manners, and it is also the fastest way to find out who is right.

有效性似乎輕而易舉就能到手；在某個意義上確實如此：不管什麼結論，都可以編出蘊涵它的前提。正因為這樣，一個有效的論證本身什麼也沒解決，而認真的分歧幾乎總是關於前提的。可是，讓這種分歧有成果的，正是對形式的講究。把對手的推理整理成有效的形式，沒說出口的部分用最強的版本替他補上，你就會看出是哪一個前提在承重。這時你確切知道你們兩人到底在爭什麼，進而知道什麼證據跟它有關。哲學家稱之為「善意理解原則」。這是好的風度，同時也是找出誰對的最快方法。