Chapter 33.--False Inferences May Be Drawn from Valid Reasonings, and Vice Versa.
51. In this passage, however, where the argument is about the resurrection, both the law of the inference is valid, and the conclusion arrived at is true. But in the case of false conclusions, too, there is a validity of inference in some such way as the following. Let us suppose some man to have admitted: If a snail is an animal, it has a voice. This being admitted, then, when it has been proved that the snail has no voice, it follows (since when the consequent is proved false, the antecedent is also false) that the snail is not an animal. Now this conclusion is false, but it is a true and valid inference from the false admission. Thus, the truth of a statement stands on its own merits; the validity of an inference depends on the statement or the admission of the man with whom one is arguing. And thus, as I said above, a false inference may be drawn by a valid process of reasoning, in order that he whose error we wish to correct may be sorry that he has admitted the antecedent, when he sees that its logical consequences are utterly untenable. And hence it is easy to understand that as the inferences may be valid where the opinions are false, so the inferences may be unsound where the opinions are true. For example, suppose that a man propounds the statement, "If this man is just, he is good," and we admit its truth. Then he adds, "But he is not just;" and when we admit this too, he draws the conclusion, "Therefore he is not good." Now although every one of these statements may be true, still the principle of the inference is unsound. For it is not true that, as when the consequent is proved false the antecedent is also false, so when the antecedent is proved false the consequent is false. For the statement is true, "If he is an orator, he is a man." But if we add, "He is not an orator," the consequence does not follow, "He is not a man."
Chapter 34.--It is One Thing to Know the Laws of Inference, Another to Know the Truth of Opinions.
52. Therefore it is one thing to know the laws of inference, and another to know the truth of opinions. In the former case we learn what is consequent, what is inconsequent, and what is incompatible. An example of a consequent is, "If he is an orator, he is a man;" of an inconsequent, "If he is a man, he is an orator;" of an incompatible, "If he is a man, he is a quadruped." In these instances we judge of the connection. In regard to the truth of opinions, however, we must consider propositions as they stand by themselves, and not in their connection with one another; but when propositions that we are not sure about are joined by a valid inference to propositions that are true and certain, they themselves, too, necessarily become certain. Now some, when they have ascertained the validity of the inference, plume themselves as if this involved also the truth of the propositions. Many, again, who hold the true opinions have an unfounded contempt for themselves, because they are ignorant of the laws of inference; whereas the man who knows that there is a resurrection of the dead is assuredly better than the man who only knows that it follows that if there is no resurrection of the dead, then is Christ not risen.
Chapter 35 .--The Science of Definition is Not False, Though It May Be Applied to Falsities.
53. Again, the science of definition, of division, and of partition, although it is frequently applied to falsities, is not itself false, nor framed by man's device, but is evolved from the reason of things. For although poets have applied it to their fictions, and false philosophers, or even heretics--that is, false Christians--to their erroneous doctrines, that is no reason why it should be false, for example, that neither in definition, nor in division, nor in partition, is anything to be included that does not pertain to the matter in hand, nor anything to be omitted that does. This is true, even though the things to be defined or divided are not true. For even falsehood itself is defined when we say that falsehood is the declaration of a state of things which is not as we declare it to be; and this definition is true, although falsehood itself cannot be true. We can also divide it, saying that there are two kinds of falsehood, one in regard to things that cannot be true at all, the other in regard to things that are not, though it is possible they might be, true. For example, the man who says that seven and three are eleven, says what cannot be true under any circumstances; but he who says that it rained on the kalends of January, although perhaps the fact is not so, says what posssibly might have been. The definition and division, therefore, of what is false may be perfectly true, although what is false cannot, of course, itself be true.