弗雷格对逻辑的重新奠基－(牛津大学出版社，Anthony Kenny教授主编，全英文版)第四卷：现代哲学

Mill maintains that we are always, though not necessarily consciously, applying his canons in daily life and in the courts of law. Thus, to illustrate the second canon he says, ‘When a man is shot through the heart, it is by this method we know that it was the gunshot which killed him: for he was in the fullness of life immediately before, all circumstances being the same, except the wound.’ Mill’s methods of agreement and disagreement are a sophistication of Bacon’s tables of presence and absence.1 Like Bacon’s, Mill’s methods seem to assume the constancy of general laws. Mill says explicitly, ‘The propos- ition that the course of Nature is uniform, is the fundamental principle, or general axiom, of Induction.’ But where does this general axiom come from? As a thoroughgoing empiricist, Mill treats it as being itself a gener- alization from experience: it would be rash, he says, to assume that the law of causation applied on distant stars. But if this very general principle is the basis of induction, it is difficult to see how it can itself be established by induction. But then Mill was prepared to affirm that not only the funda- mental laws of physics, but those of arithmetic and logic, including the very principle of non-contradiction itself, were nothing more than very well-confirmed generalizations from experience.2 Frege’s Refoundation of Logic On these matters Frege occupied the opposite pole from Mill. While for Mill propositions of every kind were known a posteriori, for Frege arith- metic no less than logic was not only a priori but also analytic. In order to establish this, Frege had to investigate and systematize logic to a degree that neither Mill nor any of his predecessors had achieved. He organized logic in a wholly new way, and became in effect the second founder of the discipline first established by Aristotle. One way to define logic is to say that it is the discipline that sorts out good inferences from bad. In the centuries preceding Frege the most important part of logic had been the study of the validity and invalidity of a particular form of inference, namely the syllogism. Elaborate rules had been drawn up to distinguish between valid inferences such as 1 See vol. III, p. 31. 2 See Ch. 6 below. LOGIC 100 All Germans are Europeans. Some Germans are blonde. Therefore, Some Europeans are blonde. and invalid inferences such as All cows are mammals. Some mammals are quadrupeds. Therefore, All cows are quadrupeds. Though both these inferences have true conclusions, only the first is valid, that is to say, only the first is an inference of a form that will never lead from true premisses to a false conclusion. Syllogistic, in fact, covers only a small proportion of the forms of valid reasoning. In Anthony Trollope’s The Prime Minister the Duchess of Omnium is anxious to place a favourite of hers as Member of Parliament for the borough of Silverbridge, which has traditionally been in the gift of the Dukes of Omnium. He tells us that she ‘had a little syllogism in her head as to the Duke ruling the borough, the Duke’s wife ruling the Duke, and therefore the Duke’s wife ruling the borough’. The Duchess’s reasoning is perfectly valid, but it is not a syllogism, and cannot be formulated as one. This is because her reasoning depends on the fact that ‘rules’ is a transitive relation (if A rules B and B rules C, then A does indeed rule C), while syllogistic is a system designed to deal only with subject–predicate sentences, and not rich enough to cope with relational statements. A further weakness of syllogistic was that it could not cope with inferences in which words like ‘all’ or ‘some’ occurred not in the subject place but somewhere in the grammatical predicate. The rules would not determine the validity of inferences that contained premisses such as ‘All politicians tell some lies’ or ‘Nobody can speak every language’ in cases where the inference turned on the word ‘some’ in the first sentence or the word ‘every’ in the second. Frege devised a system to overcome these difficulties, which he expounded first in his Begriffsschrift. The first step was to replace the gram- matical notions of subject and predicate with new logical notions, which Frege called ‘argument’ and ‘function’. In the sentence ‘Wellington defeated Napoleon’ grammarians would say (or used to say) that ‘Wellington’ was the subject and ‘defeated Napoleon’ the predicate. Frege’s introduction of LOGIC 101 Trollope’s Lady Glencora Palliser ruled not just one but two Dukes of Omnium. Here, in Millais’ illustration to Phineas Finn, she establishes her dominion over the elder Duke by presenting him with a grandson. 102 LOGIC the notions of argument and function offers a more flexible method of analysing the sentence. This is how it works. Suppose that we take our sentence ‘Wellington defeated Napoleon’ and put into it, in place of the name ‘Napoleon’, the name ‘Nelson’. Clearly this alters the content of the sentence, and indeed it turns it from a true sentence into a false sentence. We can think of the sentence as in this way consisting of a constant component, ‘Wellington defeated . . . ’, and a replaceable element, ‘Napoleon’. Frege calls the first, fixed component a function, and the second component the argument of the function. The sentence ‘Wellington defeated Napoleon’ is, as Frege would put it, the value of the function ‘Wellington defeated . . . ’ for the argument ‘Napoleon’ and the sentence ‘Wellington defeated Nelson’ is the value of the same function for the argument ‘Nelson’. We could also analyse the sentence in a different way. ‘Wellington defeated Napoleon’ is also the value of the function ‘ . . . defeated Napoleon’ for the argument ‘Wellington’. We can go further, and say that the sentence is the value of the function ‘ . . . defeated . . . ’ for the arguments ‘Wellington’ and ‘Napoleon’ (taken in that order). In Frege’s terminology, ‘Wellington defeated . . . ’ and ‘ . . . defeated Napoleon’ are functions of a single argument; ‘ . . . defeated . . . ’ is a function of two arguments.3 It will be seen that in comparison with the subject–predicate distinction the function–argument dichotomy provides a much more flexible method of bringing out logically relevant similarities between sentences. Subject– predicate analysis is sufficient to mark the similarity between ‘Caesar conquered Gaul’ and ‘Caesar defeated Pompey’, but it is blind to the similarity between ‘Caesar conquered Gaul’ and ‘Pompey avoided Gaul’. This becomes a matter of logical importance when we deal with sentences such as those occurring in syllogisms that contain not proper names like ‘Caesar’ and ‘Gaul’, but quantified expressions such as ‘all Romans’ or ‘some province’. Having introduced these notions of function and argument, Frege’s next step is to introduce a new notation to express the kind of generality expressed by a word like ‘all’ no matter where it occurs in a sentence. If ‘Socrates is 3 As I have explained them above, following Begriffsschrift, functions and arguments and their values are all bits of language: names and sentences, with or without gaps. In his later writings Frege applied the notions more often not to linguistic items, but to the items that language is used to express and talk about. I will discuss this in the chapter on metaphysics (Ch. 7). LOGIC 103 mortal’ is a true sentence, we can say that the function ‘ . . . is mortal’ holds true for the argument ‘Socrates’. To express generality we need a symbol to indicate that a certain function holds true no matter what its argument is. Adapting the notation that Frege introduced, logicians write (x)(x is mortal) to signify that no matter what name is attached as an argument to the function ‘ . . . is mortal’, the function holds true. The notation can be read as ‘For all x, x is mortal’ and it is equivalent to the statement that everything whatever is mortal. This notation for generality can be applied in all the different ways in which sentences can be analysed into function and argument. Thus ‘(x)(God is greater than x)’ is equivalent to ‘God is greater than everything’. It can be combined with a sign for negation (‘�’) to produce notations equivalent to sentences containing ‘no’ and ‘none’. Thus ‘(x)� (x is immortal)’ ¼ ‘For all x, it is not the case that x is immortal’ ¼ ‘Nothing is immortal’. To render a sentence containing expressions like ‘some’ Frege exploited the equivalence, long accepted by logicians, between (for example) ‘Some Romans were cowards’ and ‘Not all Romans were not cowards’. Thus ‘Some things are mortal’ ¼ ‘It is not the case that nothing is mortal’ ¼ ‘� (x)�(x is mortal)’. For convenience his followers used, for ‘some’, a sign ‘(Ex)’ as equivalent to ‘� (x)�’. Frege’s notation, and its abbreviation, can be used to make statements about the existence of things of different kinds. ‘(Ex)(x is a horse)’, for instance, is tantamount to ‘There are horses’ (provided, as Frege notes, that this sentence is understood as covering also the case where there is only one horse). Frege believed that objects of all kinds were nameable—numbers, for instance, were named by numerals—and the argument places in his logical notation can be filled with the name of anything whatever. Consequently ‘(x)(x is mortal)’ means not just that everyone is mortal, but that every- thing whatever is mortal. So understood, it is a false proposition, because, for instance, the number ten is not mortal. It is rare, in fact, for us to want to make statements of such unrestricted generality. It is much more common for us to want to say that everything of a certain kind has a certain property, or that everything that has a certain given property also has a certain other property. ‘All men are mortal’ or ‘What goes up must come down’ are examples of typical universal LOGIC 104 sentences of ordinary language. In order to express such sentences in Frege’s system one must graft his predicate calculus (the theory of quan- tifiers such as ‘some’ and ‘all’) on to a propositional calculus (the theory of connectives between sentences, such as ‘if ’ and ‘and’). In Frege’s system of propositional logic the most important element is a sign for conditionality, roughly corresponding to ‘if ’ in ordinary language. The Stoic logician Philo, in ancient times, had defined ‘If p then q’ by saying that it was a proposition that was false in the case in which p was true and q false, and true in the three other possible cases.4 Frege defined his sign for conditionality (which we may render ‘!’) in a similar manner. He warned that it did not altogether correspond to ‘if . . . then’ in ordinary language. If we take ‘p ! q’ as equivalent to ‘If p then q’ then propositions such as ‘If the sun is shining, 3� 7 ¼ 21’ and ‘If perpetual motion is possible, then pigs can fly’ turn out true—simply because the consequent of the first proposition is true, and the antecedent of the second proposition is false. ‘If ’ behaves differently in ordinary language; the use of it that is closest to ‘!’ is in sentences such as ‘If those curtains match that sofa, then I’m a Dutchman’. Frege’s sign can be looked on as a stripped-down version of the word ‘if ’, designed to capture just that aspect of its meaning that is necessary for the formulation of rigorous proofs containing it. In Frege’s terminology, ‘ . . .! . . . ’ is a function that takes sentences as its arguments: its values, too, are sentences. Whether the sentences that are its values (sentences of the form ‘p! q’) are true or false will depend only on whether the sentences that are its arguments (‘p’ and ‘q’) are true or false. We may call functions of this kind ‘truth-functions’. The conditional is not the only truth-function in Frege’s system. So too is negation, represented by the sign ‘�’, since a negated sentence is true just in case the sentence negated is false, and vice versa. With the aid of these two symbols Frege built up a complete system of propositional logic, deriving all the truths of that logic from a limited set of primitive truths or axioms, such as ‘(q ! p)! ( �p ! �q)’ and ‘��p! p’. Connectives other than ‘if ’, such as ‘and’ and ‘or’, are defined in terms of conditionality and negation. Thus, ‘�q ! p’ rules out the case in which p is false and �q is true: it means that p and q are not both false, 4 See vol. I, p. 138. LOGIC 105 and therefore is equivalent to ‘p or q’ (in modern symbols, ‘p V q’). ‘p and q’ (‘p & q’), on the other hand, is rendered by Frege as ‘� ðq! �pÞ’. As Frege realized, a different system would be possible in which conjunction was primitive, and conditionality was defined in terms of conjunction and negation. But in logic, he maintained, deduction is more important than conjunction, and that is why ‘if ’ and not ‘and’ is taken as primitive. Earlier logicians had drawn up a number of rules of inference, rules for passing from one proposition to another. One of the best known was called modus ponens: ‘From ‘‘p’’ and ‘‘If p then q’’ infer ‘‘q’’ ’. In his system Frege claims to prove all the laws of logic using this as a single rule of inference. The other rules are either axioms of his system or theorems proved from them. Thus the rule traditionally called contraposition, which allows the inference from If John is snoring, John is asleep to If John is not asleep, John is not snoring, is justified by the first of the axioms quoted above. When we put together Frege’s propositional calculus and his predicate calculus we can symbolize the universal sentences of ordinary language, making use of both the sign of generality and the sign of conditionality. The expression (x)(Fx ! Gx) can be read For all x, if Fx then Gx, which means that whatever x may be, if ‘Fx’ is true then ‘Gx’ is true. If we substitute ‘is a man’ for ‘F ’ and ‘is mortal’ for ‘G ’ then we obtain ‘For all x, if x is a man, x is mortal’, which is what Frege offers as the translation of ‘All men are mortal’. The contradictory of this, ‘Some men are not mortal’, comes out as ‘�(x)(x is a man ! x is mortal)’ and its contrary, ‘No man is mortal’, comes out as ‘(x)(x is a man!� x is mortal)’. By the use of these translations, Frege is able to prove as part of his system theorems corresponding to the entire corpus of Aristotelian syllogistic. LOGIC 106 Administrator 线条 Administrator 线条 Frege’s logical calculus is not just more systematic than Aristotle’s; it is also more comprehensive. His symbolism is able, for instance, to mark the difference between Every boy loves some girl ¼ (x)(x is a boy! Ey(y is a girl & x loves y) ) and the apparently similar (but much less plausible) passive version of the sentence Some girl is loved by every boy ¼ (Ey(y is a girl & (x)(x is a boy ! x loves y) ). Aristotelian logicians in earlier ages had sought in vain to find a simple and conspicuous way of bringing out such differences of meaning in ambiguous sentences of ordinary language. A final subtlety of Frege’s system must be mentioned. The sentence ‘Socrates is mortal’, as we have seen, can be analysed as having ‘Socrates’ for argument, and ‘ . . . is mortal’ as function. But the function ‘ . . . is mortal’ can itself be regarded as an argument of a different function, a function operating at a higher level. This is what happens when we complete the function ‘ . . . is mortal’ not with a deter- minate argument, but with a quantifier, as in ‘(x)(x is mortal)’. The quantifier ‘(x)(x . . . )’ can then be regarded as a second-level function of the first-level function ‘ . . . is mortal’. The initial function, Frege always emphasizes, is incomplete; but it may be completed in two ways, either by having an argument inserted in its argument place, or by itself becoming the argument of a second-level function. This is what happens when the ellipsis in ‘ . . . is mortal’ is filled with a quantifier such as ‘Everything’. Induction and Abduction in Peirce A number of Frege’s innovations in logic occurred, quite independently, to C. S. Peirce; but Peirce was never able to incorporate his results into a rigorous system, much less to publish them in a definitive form. Peirce’s importance in the history of logic derives rather from his investigations into the structure of scientific inquiry. Deductive logic assists us in organizing our knowledge; but the kind of reasoning that extends our knowledge (‘ampliative inference’ as Peirce calls it) is of three kinds: induction, hypothesis, and analogy. All of these inferences, Peirce claimed, LOGIC 107 depend essentially on sampling. Any account, therefore, of non-deductive inference must be related to the mathematical theory of probability (EWP 177). Scientists frame hypotheses, make predictions on the bases of these hypotheses, and then make observations with a view to confirming or refuting their hypotheses. These three stages of inquiry are called by Peirce abduction, deduction, and induction. In the abductive phase the inquirer selects a theory for consideration. In the deductive phase he formulates a method to test it. In the inductive phase he evaluates the results of the test. How does a scientist decide which hypotheses are worth inductive testing? Indef- initely many different theories might explain the phenomena he wishes to investigate. If he is not to waste his time, his energy, and his research funding, the scientist needs some guidance about which theories to explore. This guidance is given by the rules of the logic of abduction. The theory must, if true, be genuinely explanatory; it must be empirically testable; it should be simple and natural and cohere with existing knowledge, though not necessarily with our subjective opinions about antecedent likelihood. (P 7.220–1) Rules of abduction, however, do not by themselves explain the success of scientists in their choice of hypotheses. We have to believe that in their investigation of nature they are assisted by nature herself. Science presupposes that we have a capacity for ‘guessing’ right. We shall do better to abandon the whole attempt to learn the truth . . . unless we can trust to the human mind’s having such a power of guessing right that before very many Modern symbolic logic no longer uses the actual symbol system of its founder Frege, which was difficult to print. The illustration shows the pattern, in his notation, for deriving results such as ‘‘If this ostrich is a bird and cannot fly, it follows that some birds cannot fly’’. LOGIC 108 hypotheses shall have been tried, intelligent guessing may be expected to lead us to the one which will support all tests. (P 6.530) This trust has to be presupposed at the outset, even though it may rest on no evidence. But in fact the history of science shows such trust to be well founded: ‘it has seldom been necessary to try more than two or three hypotheses made by clear genius before the right one was found’ (P 7.220) Once the theory has been chosen, abduction is succeeded by deduction. Consequences are derived from the hypothesis, experimental predictions that is, which will come out true if the hypothesis is correct. In deduction, Peirce maintained, the mind is under the dominion of habit: a general idea will suggest a particular case. It is by verifying or falsifying the predictions of the particular instantiations that the scientist will confirm, or as the case may be refute, the hypothesis under test. It is induction that is the all-important element in the testing, and induction is essentially a matter of sampling. Suppose a ship arrives in Liverpool laden with wheat in bulk. Suppose that by some machinery the whole cargo be stirred up with great thoroughness. Suppose that twenty-seven thimblefuls be taken equally from the forward, midships, and aft parts, from the starboard, center and larboard parts, and from the top, half depth and lower parts of her hold, and