“Ich liebe dich 3” the swains in mountain valleys of Austria inscribe on their presents to those to whom they plight their troth. The pun is a rare one in German. Only in remote valleys does the word for ‘three’ rhyme with joy; and the word for ‘true’ is usually ‘wahr’ not ‘treu’ ‘Wahr’ is more propositional, less evaluative than our ‘true’. So too in Latin and the romance languages ‘verum’, ‘Vrai’ ‘vera’ ‘verdadero’ carry a lighter load of feeling. Only the Hebrews resembled us in giving their word for ‘true’ strong moral overtones, and using the same or some cognate word to praise him who had used no deceitfulness to his neighbour as to recommend some statement as being meet to be believed.

When I began to study philosophy sense-data were in the fashion; everybody had some. Nowadays talking about sense-data, like distinguishing between “shall” and “will”, is apt to be regarded as an indication that one has stopped moving with the times. Before abandoning this old habit, I want to consider whether there may not after all be something in a doctrine adopted by so many leading philosophers in pre-war England.

What is the connection between a representation, such as a painting, statue or engraving, and its subject? For example, what makes a painting a painting of McX? The problem is not how to paint McX, which belongs to art experts. So the answer is not, for example, ‘The painter starts at the top with an egg-shape for the head …” The question is rather: what makes the results of such efforts a painting of McX? What conditions must a painting satisfy to warrant such a description? What is the force or meaning of the word ‘of’ in phrases like ‘painting of McX’?

“It Seems natural to suppose”, wrote Mill, “that rules of action must take their whole character … from the end to which they are subservient”. Many moralists have agreed. If we could establish the Summum Bonum, the foundation of morality, the rational basis of moral thinking, this would constitute a criterion, a rule, by means of which men could actually make good practical judgments. This view is radically mistaken. I first try to show this by means of an a priori argument, and then urge the replacement of the “rational model” of practical thinking embodied in this view by an “empirical model”, which should take account of man's actual capacities and limitations. This has important consequences for morality, some of which I investigate.

The first solid bit of argumentation you get in Plato's Phaedo goes something like this: Whatever comes to be, comes to be from its opposite (cf. 70e, sqq.). If at a certain time t a given thing a begins to be F, before that time t it must have been non-F. Wherever a pair of predicates, F and G, are genuine contradictories; where, that is, they stand to each other in the same relation as F stands in to non-F; it is necessarily true that if a began to be F at t before then a was G. The only trouble comes from the difficulty of finding substitutes for F and G that people will allow to be genuine contradictories. It the butter began to be soft at four o'clock, we may suggest, before four it was hard. A tiresome opponent will retort that there is a state in which butter cannot properly be called either hard or soft. We try again: If Ann began to be asleep at eight, before eight she was awake; if my shirt became dirty on Tuesday, before Tuesday it was clean; if Bonzo died last week, before last week he was alive. And when the advocate of the borderline state reminds us of the various twilight areas of consciousness or cleanliness, we are reduced to legislation: “asleep” shall henceforth apply to every mental state short of complete wakefulness, “alive” to every condition of the body before the onset of putrefaction, “clean” to every shirt incapable of producing a certain measurable discoloration in the water in which it is washed. “Let F and G be contradictories”, we still guardedly maintain: “then if a comes to be F at time t, before time t a was G”.

J. R. Lucas argues in “Minds, Machines, and Gödel”, that his potential output of truths of arithmetic cannot be duplicated by any Turing machine, and a fortiori cannot be duplicated by any machine. Given any Turing machine that generates a sequence of truths of arithmetic, Lucas can produce as true some sentence of arithmetic that the machine will never generate. Therefore Lucas is no machine.

In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is not true that Goedel's theorem proves this. At most, Goedel's theorem proves that not all minds can be explained as machines. Since this is so, Goedel's theorem cannot be expected to throw much light on why minds are different from machines. Lucas overestimates the importance of Goedel's theorem for the topic of mechanism, I believe, because he presumes falsely that being unable to follow any but mechanical procedures in mathematics makes something a machine.

The Bishop of Cloyne, George Berkeley, was the sort of philosopher who, although most genial himself, was quite apt to embroil opponents and critics of his time and of our own in long-lasting and sometimes unresolved controversies. In attacking the “infidel mathematicians”, the “minute philosophers” among the scientists, Berkeley initiated a controversy on behalf of religion by taking to task the theory of fluxions held by Sir Isaac Newton, his friends, and followers which, beginning with Berkeley's Analyst and replies to it by Jurin and Walton, was continued on over one hundred years by subsequent writers. Florian Cajori in his History of Mathematics, commenting on the affair has written:

‘We must not neglect to express our appreciation of the fact that Berkeley withdrew from the controversy after he had said all that he had to say on his subject. Some of the debates that came later were almost interminable, because the participants continued writing even after they had nothing more to say.’

It is odd that something that Wittgenstein says is clear should have been so puzzling to English-speaking philosophers. 5.542 begins:— ‘Es ist aber klar, dass “A glaubt, dass p”, “A denkt p”, “A sagt p” von der Form, “p ‘sagt p” sind.’ I would like to suggest that one reason for the difficulties that have been felt with this lies in a misleading translation, particularly of, “p ‘sagt p”. For this both English translations have “p” says p’. But since German has a single present tense where English has two, ‘sagt’ may be rendered either ‘says’, or ‘is saying’, and I want to argue that ‘“p” is saying p’ is preferable.