THE THREE SPHERES OF INCLUSIVE TRUTH
 
To say that something does have what it does have,
or that it does not have what it does not have,
and to say that something can have what it can have,
or that it can not have what it can not have,
and to say that something should have what it should have,
or that it should not have what it should not have,
and to say that it is not the case
that something should have or should not have something
when it is not the case
that it should have or should not have it,
is true.

When asserting that a thing has a certain part or predicate, this does imply that the thing in question can have this part or predicate but not that it is probable, neither probable nor improbable or improbable that it has it. And it does not imply that it should have it. When asserting that it can have such a part or predicate, this does not imply that it has it or that it should have it. And when asserting that it should have the predicate concerned, this does not imply that it has it either, nor that it can have it. (Ought implies can may be a useful ethical axiom, it is not an ontological proposition.) Thus in our discussion on the question of the parts and predicates which things have, we have until now restricted ourselves to one of three spheres only: the sphere of what is, of having, as distinct from what is possible or probable, or can be had, and as distinct from what should be or ought to be had.

The sphere of having (and hence of being and doing) we shall call "the factual sphere". This sphere encompasses facts, events, situations or the world as it is, but not the mere existence of things independent of time and of their having other things or being had by other things. The three spheres only distinguish themselves with respect to relations, such as the relation of having-as-an-element, and such relations, or the absence thereof, presuppose the existence of the things related or not related. The existence of things per se is therefore not a fact as distinct from what is possible or should be. It is the existence of nonbasic things with particular parts or predicates which is a fact --or not a fact-- because such existence depends on the having of a certain kind of element. In the strict, basic sense one may speak of "a fact" when there is a relation between two things or types of thing. In a less strict, less basic sense a 'fact' may also be privative, when there is no relation between two particular things or types of thing. It may also be a complex of basic and privative facts, not unlike a 'situation'. In an even looser sense, a 'fact' may also be a true proposition in which the relation in question is mentioned. In any case it should be clear why we cannot see or feel facts in a literal sense: even if they were identical to the relations themselves, these relations would only be things in the second domain, not in the first domain to which we belong ourselves. They are not identical to those relations, because they exist 'always' in a nontemporal sense, whereas facts are temporal and only 'exist' when there is a particular relationship or type of relationship between particular things or types of thing.

The sphere of the possibility of having (being, doing) something we shall call "the modal sphere". Regardless of the question whether a state of affairs does occur or does not occur, one can ask whether it is possible or impossible, necessary or contingent, probable, neither probable nor improbable or improbable. Only when it is impossible may one conclude that it cannot and does not occur, and only when it is necessary may one conclude that it must and does occur. When it is possible that a thing has a certain part or predicate, it is improbable (but not impossible), neither improbable nor probable or probable (perhaps necessary) that it has it. When it is contingent, it is not necessary, that is, again, improbable (perhaps impossible), neither improbable nor probable or probable (but not necessary). Expressions like possible and probable are not second-order predicate expressions, because they do not denote the predicate of a first-order predicate. Their different ontological status, which is similar to that of facts, lies in the way they relate things to other things (parts, predicates or external things).

Like future and later, expressions such as possible and probable do not refer to primary predicates, while forming part of a connected series nevertheless. The first such series is, then, the series of necessary, possible (but not necessary) and impossible; the second one that of probable, neither probable nor improbable and improbable. When these two series are combined into one, the result looks very much like a catena, and one would be tempted to give such a series the name of an explicit triad: probability catena. So long as this full name is used, confusion is hardly possible, but there are two reasons why the series in question is not a catena. Firstly, its component parts are not primary predicates; and secondly, the number of component parts is not necessarily three or more. For example, if a relation or situation is either necessary or impossible, without intermediate degrees of probability, then the quantity concerned is binary and could not be catenary. (A factual quantity which is 1 when a relationship exists between two particular things, and 0 when it does not exist, is not catenary either.)

The auxiliary series of necessity, impossibility and all the elements in between is a modal auxiliary. The terms must, may, might and can used in these contexts are also called "modal auxiliaries" by grammarians. The values of these modal auxiliaries (that is, auxiliary elements) vary between -½ (or 0: impossibility) and +½ (or 1: necessity).

Adjectives typical of the modal sphere are able and capable when employed in the same or a similar sense referring to an ability or fitting quality. Such an 'ability' or 'fitting quality' is not an attribute on our ontology but is something a thing may be said to have if it can have a certain relationship with another thing, for example, if it can repair that other thing. The ability does not imply that it is probable or likely that the relationship will actually exist between the things in question. All it implies is that it is not extremely improbable in the strictest sense, that is, impossible. The concept of modality as expressed by terms such as can, (cap)able and probable needs much more analyzing but suffice it here to stress the difference between what is purely factual and what is modal, at least in some way.

Just as we can distinguish an ontological auxiliary pertaining to modal conditions in a modal sphere, so we can distinguish an ontological auxiliary pertaining to normative conditions in a normative sphere. Also these conditions are in the first instance as little factual as their modal counterparts, and also these conditions admit of degree: something can agree with a normative standard to a greater or lesser extent. Instead of speaking of "probable" and "improbable" we speak of "superior" and "inferior" in the case of such a normative auxiliary (or in similar terms like good, better, (more) proper and bad, worse, less proper or improper). Also the values of this auxiliary series vary between -½ or -1 (most inferior) and +½ or +1 (most superior or supreme). Inferior and superior are used, then, in a normative and absolute sense. They should not be confused with the same terms used in a nonnormative and/or comparative sense.

A term like ought, and terms like should and must, when used to describe normative conditions, are called "modal auxiliaries" too in traditional grammar. This universal employment of modal merely contributes to the confusion of the different ontological spheres in everyday language. On our terminology they should be termed "normative auxiliaries" (and words like will when used to form tenses, "temporal auxiliaries"). The correspondence, or lack thereof, between auxiliary terms like should and must in ordinary language on the one hand, and modal and normative conditions on the other, is illustrated in the following scheme (in which A is a primary thing, and p a primary predicate):

• A can/may have p = it is possible that A has p = (the having of) p is possible with respect to A
• A probably has/should have p = it is probable that A has p = (the having of) p is probable with respect to A
• A must have p = it is necessary that A has p = (the having of) p is necessary with respect to A
• A ought to/should/must have p = it is (normatively) superior/good that A has p = (the having of) p is (normatively) superior/good with respect to A