¹ See on this point Ernst Cassirer, Substance and Function, Chicago, 1923, Ch. I.

² See also Robert S. Hartman, Value Theory as a Formal System, Kant-Studien, L, pp. 290 ff. (1958–1959).

³ For details see Robert S. Hartman, “The Logic of Description and Valuation,” Review of Metaphysics, XIV, 2 (December 1960), pp. 191–230.

⁴ See Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science, New York, 1957.

⁵ Actually “Aha” is of profound scientific significance. See on the “Aha experience” in science, for example, Norman L. Munn, Psychology, Boston, 1946, pp. 109, 187; Jacques Hadamard, The Psychology of Invention in the Mathematical Field, Princeton, 1945; M. Polanyi, Personal Knowledge, London, 1958.

⁶ “Natural Philosophy, as a speculative science, I imagine, we have none; and perhaps I may think I have reason to say, we never shall be able to make a science of it. The works of nature are contrived by a wisdom and operated by ways far surpassing our faculties to discover, or capacities to conceive for us ever to be able to reduce them into a science.” (Some Thoughts Concerning Education).

⁷ See in particular Philebus, which in turn was the model for the ethics of Aristotle and hence of that of all the Middle Ages.

⁸ The Era of Violence, 1898–1945, The New Cambridge Modern History, Vol. XII, Cambridge, 1960.

⁹ J. Ferrater Mora “Wittgenstein, A Symbol of Troubled Times,” Philosophy and Phenomenological Research, XIV, 89–96, (1953).

¹⁰ Bertrand Russell, “My Present View of the World,” Encounter, Jan. 1959, p. 9.

¹¹ La estructura del valor, pp. 307 ff; “Value Theory as a Formal System,” op. cit., pp. 312 ff.

¹² See Robert S. Hartman, “The Logic of Value,” The Review of Metaphysics, XIV (March 1961), pp. 408–423.

¹³ La estructura del valor, pp. 307 ff; “Value Theory as a Formal System,” op. cit., pp. 312 ff.

¹⁴ G. E. Moore himself commits the fallacy of method in calling Ethics what he should have called Axiology. For, in Principia Ethica, he deals with goodness in general, rather than with the particular kind of goodness called moral goodness, which alone is the subject of Ethics.

¹⁵ This means that philosophical “systems” are not systems in the sense here defined, but pseudo-systems. That is to say, they are not axiomatic constructions using synthetic concepts and being cumulatively applied to their subject matter generation after generation. Rather, they are individual creations by individual thinkers, consisting of the analytic spinning out of some fundamental category. (See La estructura del valor, pp. 52 ff).

¹⁶ As for example, that of the unity of the definition, Metaphysics, H, vi.

¹⁷ Cassini in his sermon on the text: “Ye men of Galilee, why stand ye gazing up into heaven ?” thundered against Galileo that “geometry is of the devil” and “mathematicians should be banished as the authors of all heresies.” To the heresies of using mathematics and sense observation, and the degradation of “movement” which apparently degraded the moral and divine world itself, Galileo added the ultimate heresy of sponsoring the Copernican system which dethroned man from his moral and divine position in the universe. Considering the mortal blows Galileo dealt the medieval world it is a wonder he fared as well as he did.

¹⁸ Some of this literature advocates the procedure of explaining logic by value disciplines rather than value disciplines by logic. Logic then appears as “generalized” economics, sociology, psychology, jurisprudence, etc. Methodologically, this corresponds to the Aristotelian as against the Galilean procedure. The corresponding proposal in natural philosophy would be advocating that mathematics and natural science be abolished and mathematics be understood by the categories of natural philosophy, rather than natural philosophy by the axiomatic relations of mathematics. It would mean, in other words, going back to alchemy and astrology. Strangely enough, this parallel is not only not seen, but such proposals are made in the names of Kepler, Galileo, Newton, and Lavoisier. (For example by Stephen Toulmin, The Uses of Arguments, Cambridge, 1958, esp. pp. 257f.).

¹⁹ See Robert S. Hartman, “The Logic of Description and Valuation,” Review of Metaphysics, loc. cit., p. 205; La estructura del valor, pp. 257 ff.

²⁰ As we are actually doing. The Aristotelian procedure of applying the category of value to the system of logic is, of course, again that of the logical positivists. They regard extensional logic as the highest philosophical value, but do not analyze value.

²¹ Especially in The Practice of Philosophy, New York, 1933, pp. 203 ff.

²² Prediction, in particular, is not so much a matter of time as a matter of applying a formal frame of reference. I can predict what note Heifetz will play next when I know the score of the concerto.

²³ La estructura del valor, pp. 11 ff. and pass.

²⁴ Ibid.

²⁵ See, for applications to some one hundred value situations Robert S. Hartman, “Value Theory as a Formal System,” op. cit., pp. 301 ff.

²⁶ See also Review of Metaphysics, XIV, 2 (December 1960), “The Logic of Description and Valuation.”

²⁷ Other examples in the review of the fallacy are the following: (1) The non-natural character of “good” is based on the logical nature of “good” as a second-order quality and not on an example, as cited by the reviewer. This objection against a logical argument by means of objecting against an example is an instance of the fallacy of method. (2) The reviewer confuses the book's definition of the value term “good” with that of the value relation “good for.” While “x is good” means that x has the properties of its conceptual intension, “x is good for y” means that “x and y are in different classes but that their intensions overlap” (p. 238). Again instead of discussing the logical definitions the reviewer deals with examples. (3) There is given an exact way of distinguishing between exposition and definition. Exposition is said to contain a denumerably infinite number of predicates whereas definition contains a finite such number (pp. 130, 283). Again this logical distinction is not discussed. What are discussed are examples. (4) The relation between the position of emotivists and subjectivists, on the one hand, and that of the book, on the other, is not left up in the air (“not a single argument is offered …”). It is shown that these schools either are content with the chaos of value phenomena rather than attempting its systematic ordering, or that their attempts at ordering are futile since they use imprecise, namely analytic, rather than precise, namely synthetic concepts.

²⁸ Theodor Lessing, Studien zur Wertaxiomatik, Leipzig, 1914, pp. 104 f.

²⁹ Considering not only “good” but the other value terms as well, we may say that the difference between x is a C factually and x is a C valuationally is represented by the difference between “ϕx” and “ϕ!x”, the latter being the matrix for the second-order functions (ϕ)ϕx, (Ǝϕ)ϕx, etc., which define one aspect of the value terms.

³⁰ That there is a relation between the theory of types and value theory has been mentioned by recent writers in ethics and aesthetics. Cf. Rosamond Kent Sprague, “Negation and Evil,” Philosophy and Phenomenological Research, XI, p. 566 (1951) and Morris Weitz, Philosophy of the Arts, Cambridge, Mass., 1950, p. 142. There are writers in axiology who come close to the solution without seeing the logical connection (e.g. W. D. Ross, The Right and the Good, Oxford, 1930, pp. 121 f.) and writers in logic who come close to it without seeing the axiological connection (e.g. Bertrand Russell, Principia Mathematica, p. 56; Inquiry into Meaning and Truth, New York, 1940, pp. 250 ff.; Introduction to Mathematical Philosophy, London, 1938, p. 189). Although the predicates which Russell discusses in connection with the axiom of reducibility are all value predicates—Napoleon's greatness or viciousness, the typicalness of a Frenchman or an Englishman—he makes no use of this in his axiological writings.

³¹ W. J. Rees, “Continuous States,” Proc. Arist. Soc., LVIII, 223–244 (1957–1958).

³² Op. cit., p. 226.

³³ Op. cit., p. 235.

³⁴ Op. cit., p. 241. Also cf. Ryle's “second order” inclinations, motives, etc. e.g. Gilbert Ryle, The Concept of Mind, London, 1949, p. 112.

³⁵ See Maria Ossowska, “Qu'est ce qu'un jugement de valeur?,” Proc. X. Int. Cong. Phil., Amsterdam, 1949, pp. 443 f.

³⁶ Time, September 1, 1958, “Headline of the Week.”

³⁷ Rees, op. cit., p. 233.

³⁸ The higher-order function would be one disclosed by axiology, as in the definition of “good,” or one appearing openly in language; in which latter case any higher-order function might be regarded as a valuation. To do so would, for example, solve the value questions connected with adverbs. In He drove slowly. He ate with his hands “slowly” and “with his hands” obviously are valuations, qualifications of “he drove” and “he ate.” A value theory, of course, must account for this value character of adverbial expressions. The expansion of our theory in regarding all higher-order functions as valuations, would take care of this. Moreover, there seems to be no example of higher-order functions that is not charged with value content. Cf. the suggestive example of Franz Crahay and its instructive explanation:

“Y., a mediocre poet, to be sure, but for a poet, a very skilfull politician, and, among those most skillfull, the least obnoxious.”

To type t ₀ (individuals) belongs: “Y.”

To type t ₁ (predicates of individuals or classes) belong: “poet,” “politician.”

To type t ₂ (predicates of predicates or classes of classes) belong, classified in orders:

Of order O ₀: “mediocre,” which qualifies “poet” and does not explicitly refer to the totality of poet-individuals.

Of order O ₁: “skillfull,” which qualifies “politicians” and explicitly refers to the totality of politician-individuals.

Of order O ₂ “the least obnoxious,” which explicitly refers to the totality of “most skillfull politicians.”

(Franz Crahay, Le formalisme logico-mathématique et le problème du non-sens, Paris, 1957, p. 48). Yet, we are not prepared to identify all non-predicative functions, or even all “reduced” predicative functions, with valuation. To do so may constitute the next step in formal axiology.

³⁹ If the genus of the definiens is said to be of the same type as the definiendum, then the differentia is of a higher type than the definiendum. If x is human, to be human, means to be rationally animalic, and to be animalic means to possess a certain set of properties, then x is human, x is animalic, and to be human is to be rational are predicative functions; but x is rational is a “reduced” predicative function. Cf. D. Garcia Baca, Introduction a la logica moderna, Barcelona, 1936, pp. 59 ff.; Henry Lanz, In Quest of Morals, Stanford University, 1941, pp. 57 ff.