Noneism was created by Richard Routley. Its point of departure is the rejection of what Routley calls “The Ontological Assumption”. This assumption consists in the explicit or, more frequently, implicit belief that denoting always refers to existing objects. If the object, or objects, on which a proposition is about, do not exist, then these objects can only be one: the null entity. It is incredible that Frege believed that denoting descriptions without a real (empirical, theoretical, or ideal) referent denoted only the null set. And it is also difficult to believe that Russell sustained the thesis that non-existing objects cannot have properties and that propositions about these objects are false.
This means that we can have a very clear apprehension of imaginary objects, and quite clear intellection of abstract objects that are not real. This is possible because to determine an object we only need to describe it through its distinctive traits. This description is possible because an object is always chacterized through some definite notes. The amount of traits necessary to identify an object greatly varies. In some cases we need only a few, for instance, the golden mountain, or the blue bird; in other cases we need more, for instance, the goddess Venus or the centaur Chiron. In other instances the traits can be very numerous, even infinite. For instance the chiliedron, and the decimal number 0,0000…009, in which 9 comes after the first million zeros, have many traits. And the ordinal omega or any Hilbert space have infinite traits (although these traits can be reckoned through finite definitions). These examples show, in a convincing manner, that the Ontological Assumption is untenable. We must reject it and replace it with what Routley dubbs the Characterization Postulate. The Characterization Postulate says that, to be an object means to be characterized by determined traits. The set of the characterizing traits of an object can be called its “characteristic”. When the characteristic of an object is set up, the object is perfectly recognizable.
Once this postulate is adopted, its consequences are far reaching. Since we can characterize objects through any traits whatsoever, an object can not only be inexistent, it can even be absurd or inconsistent. For instance, the “squond” (the circle that is square and round). And we can make perfectly valid logical inferences from the premiss: x is the sqound:
(1) if x is the squond, then x is square
(2) if x is the squond, then x is round
So, the theory of objects has the widest realm of application. It is clear that the Ontological Assumption imposes unacceptable limits to logic. As a matter of fact, the existential quantifier of classical logic could not have been conceived without the Ontological Assumption. The expression “(∃x)Fx” means that there exists at least an object that has the property F (or, in extensional language, that there exists an x that is a member of the extension of F). For this reason, “∃x” is unappliable to non existing objects. Of course, in classical logic we can deny the existence of an Object, but we cannot say anything about Objects that have never existed and shall never exist (we are strictly speaking about classical logic). We cannot quantify individual variables of a first order predicate that do not refer to a real, actual, past or future entity. For instance, we cannot say “(∃x) (x is the eye of Polyphemus)”. This would be false, of course, because Polyphemus does not exist. But if the Ontological Assumption is set aside, it is true, within a mythological frame, that Polyphemus has a single eye and many other properties. And now we can understand why noneism leads to logical material-dependence.
As we have anticipated, there must be some limitations concerning the selection of the contradictory properties; otherwise the whole theory becomes inconsistent and is trivialized. To avoid trivialization neutral (noneist) logic distinguishes between two sorts of negation: the classical propositional negation: “8 is not P”, and the narrower negation: “8 is non-P”. In this way, and by applying some other technicalities (for instance, in case an universe is inconsistent, some kind of paraconsistent logic must be used) trivialization is avoided. With the former provisions, the Characterization Postulate can be applied to create inconsistent universes in which classical logic is not valid. For instance, a world in which there is a mysterious personage, that within determined but very subtle circumstances, is and is not at the same time in two different places. In this case the logic to be applied is, obviously, some kind of paraconsistent logic (the type to be selected depends on the characteristic of the personage). And in another universe there could be a jewel which has two false properties: it is false that it is transparent and it is false that it is opaque. In this kind of world we must use, clearly, some kind of paracomplete logic. To develop naive set theory (in Halmos sense), we must use some type of paraconsistent logic to cope with the paradoxes, that are produced through a natural way of mathematical reasoning; this logic can be of several orders, just like the classical. In other cases, we can use some kind of relevant and, a fortiori, paraconsistent logic; and so on, ad infinitum.
But if logic is content-dependent, and this dependence is a consequence of the Ontological Assumption’s rejection, what about ontology? Because the universes determined through the application of the Characterization Postulate may have no being (in fact, most of them do not), we cannot say that the objects that populate such universes are entities, because entities exist in the empirical world, or in the real world that underpins the phenomena, or (in a somewhat different way), in an ideal Platonic world. Instead of speaking about ontology, we should speak about objectology. In essence objectology is the discipline founded by Meinong (Theory of Objects), but enriched and made more precise by Routley and other noneist logicians. Its main division would be Ontology (the study of real physical and Platonic objects) and Medenology (the study of objects that have no existence).