Inquiry Into Inquiry

For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates.  Now, however, there is nothing to prevent us from considering the predicates and sentences themselves as individuals which may serve as arguments of predicates.

Consider, for example, a logical expression of the form   This may be interpreted as a predicate whose first argument place is occupied by a sentence and whose second argument place is occupied by a monadic predicate

A false sentence is related to every by the relation   a true sentence only to those for which holds.

Further examples are given by the properties of reflexivity, symmetry, and transitivity of dyadic predicates.  To these correspond three predicates:  and whose argument is a dyadic predicate.  These three properties are expressed in symbols as follows:

All three properties are possessed by the predicate   ( is identical with ).  The predicate on the other hand, possesses only the property of transitivity.  Thus the formulas and are true sentences, whereas and are false.

Such predicates of predicates will be called predicates of second level.  (p. 135).

We have, first, predicates of individuals, and these are classified into predicates of different categories, or types, according to the number of their argument places.  Such predicates are called predicates of first level.

By a predicate of second level, we understand one whose argument places are occupied by names of individuals or by predicates of first level, where a predicate of first level must occur at least once as an argument.  The categories, or types, of predicates second level are differentiated according to the number and kind of their argument places.  (p. 152).

Finally, let us recall our real subject and, so far as the infinite is concerned, draw the balance of all our reflections.  The final result then is:  nowhere is the infinite realized;  it is neither present in nature nor admissible as a foundation in our rational thinking — a remarkable harmony between being and thought.  We gain a conviction that runs counter to the earlier endeavors of Frege and Dedekind, the conviction that, if scientific knowledge is to be possible, certain intuitive conceptions [Vorstellungen] and insights are indispensable;  logic alone does not suffice.  The right to operate with the infinite can be secured only by means of the finite.

The role that remains to the infinite is, rather, merely that of an idea — if, in accordance with Kant’s words, we understand by an idea a concept of reason that transcends all experience and through which the concrete is completed so as to form a totality — an idea, moreover, in which we may have unhesitating confidence within the framework furnished by the theory that I have sketched and advocated here.  (p. 392).

I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.

That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of.  We thus think of the thought‑sign itself, making it the object of another thought‑sign.

Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions.  Does this series proceed endlessly?  I think not.  What then are the characters of its different members?

My thoughts on this subject are not yet harvested.  I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:  Actuality, Possibility, Destiny (or Freedom from Destiny).

On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.  (CP 4.549).

§1.  This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it.  (CP 1.545).

§2.  This theory gives rise to a conception of gradation among those conceptions which are universal.  For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied;  and so on.  (CP 1.546).

Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different.  For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named.  For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different.  For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding.  Thus a man and an ox are called animals.  The name is the same in both cases;  so also the statement of essence.  For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.  (Aristotle, Categories, 1.1^a1–12).

Translator’s Note.  “Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture.  We have no ambiguous noun.  However, we use the word ‘living’ of portraits to mean ‘true to life’.”