Lejewski - Ontology, What Next?

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    antinomy in the ambiguity of the notion of class as used in ordinary discourse. On distinguish-

    ing between the notion of distributive class and that of collective class, he argued that some

    of the presuppositions from which the antinomy was derived, were not true of collective

    classes although they did not appear to be obviously false. This explained to him why the

    argument making up the antinomy carried with it an element of cogency although, in the

    end, it led to a contradiction.3 In order to give his concept of collective class a sound found- ation he related it by means of definitions to the notion of proper part, which to his mind

    was less controversial than the notion of class. It is this notion that Lesniewski used as a

     primitive notion to construct an axiomatized system of a theory which, subsequently, he

    called  Mereology.4 It was a very general theory of partwhole relations, and it provided a framework within which a number of statements about collective classes could be established.

    Unlike arithmetic,  Mereology  was not derivable from what Lesniewski regarded as logic,

     but it presupposed no other theory besides logic. It is significant that when Lesniewski

    decided to give the form of an axiomatized system to the logic presupposed by his  Mereology,

    he called that logic Ontology.5 And ontology it was in the sense of being a most general theory of what there is.

    It was constructed by Lesniewki in 1920, ten years after the publication of the first volume

    of Whitehead and Russell’s Principia. It is not surprising that Lesniewki’s Ontology  displays

    certain refinements not present in Principia.  The most notable concern the grammar of

    its language, definitions, and, generally, the directives, that is to say, the rules specifying

    the conditions under which a new thesis is allowed to be added to the system. However, all

    the improvements Lesniewski incorporated into Ontology  can be adapted to Principia,

    and the difference between the two systems of logic can be reduced to the difference between the types of logical language used, respectively, by the authors of Principia and by Lesniewski.

    The meaningful expressions of the language of Principia  can be made to form a hierarchy

    founded on propositions and singular referential names, whereas the hierarchy of meaning-

    ful expressions of Lesniewski’s Ontology  is based on propositions and general names, that is to say, common nouns. Some of them designate one object each, some designate more

    objects than one, while some others designate no objects at all. As a result of this sort of pre-

    supposition the quantifiers binding name or noun variables in the language of Lesniewski’s

    Ontology  have no existential import, and, in this respect, differ from the quantifiers which

     bind name variables in the language of Principia.6 However, the differences between these two

    types of logical language are irrelevant as regards their power of expression. By and large, what-

    ever can be said in the language of Principia  can be equally well expressed in the language

    of Ontology  and vice versa.  The translations from the one language into the other will be

    truth preserving, but they may not necessarily preserve the identity of grammar.

    As theories of what there is, neither the logic of Principia  nor Lesniewski’s Ontology are complete. The general description of reality they offer is open to expansion. They offer

    us, as it were, only an unfinished first chapter of such a description with further chapters

    to follow.

     Now, an axiomatized system can be expanded in various ways. We can expand its vocabulary

     by means of definitions, which may give rise to new and important theses. Alternatively, we

    can expand it by strengthening its axiomatic foundations in terms of the available vocabulary.

    Finally, we can expand a system by introducing new vocabulary that cannot be introduced

     by means of definitions but involves subjoining to the system new axioms. It is this kind of

    expansion that I have in mind when I ask in the title of my paper what comes next after

    ontology, I mean, after Lesniewski’s Ontology.  To put it in metaphorical terms, the question

    is this: what is the second chapter in the description of reality? And the answer is already at hand. It is  Mereology.  In fact, in the case of Lesniewski, the second chapter in the description

    of reality had been written before the first. It cannot be denied that the theses of  Mereology  

    describe, in very general terms, everything that there is. And the only theory that is pre-

    supposed by  Mereology, is logic, in Lesniewski’s sense, that is to say, his Ontology.  It is

    to be noted that the expansion of Ontology  achieved by subjoining to it mereological axioms

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    ices enrich the original vocabulary of Ontology  but it does not affect the original grammar

    t c  the ontological language.

    The logic of Principia  can also be expanded into a system of mereology. This has

    niready been done by Tarski7 and by Goodman in collaboration with Leonard.8

    Like the first chapter the second chapter in the description of reality is not finished.

    Tirski’s system of mereology has been expanded, by Tarski himself, into a system of atomistic

    rereology and, more recently, a similar expansion has been discussed in the literature withreference to Lesniewski’s  Mereology  as well as an expansion that turns Lesniewski’s original

    system into a system of atomless mereology.9 In accordance with atomistic mereology every

    reject has an atom as its proper or improper part, an atom being an object that has no

     proper parts. The characteristic thesis of atomless mereology asserts that every object has

    i proper part.

    For an object to have proper parts it is sufficient and necessary that it should be extended

    n time or in space. If this is the case then we have a clue as to what the third chapter in

    me description of reality should be. It should be, so it seems, a general theory of time or

    i general theory of space. General chronology,10 which appears to be a suitable name for the

    former, is likely to be simpler, from the formal point of view, than general stereology, by

    *hich I understand the latter, notwithstanding the fact that timeproperties characterizing rejects and timerelations holding between objects are perhaps less intuitive than the corres-

     ponding spaceproperties and spacerelations. It is for this reason that , in my view, general

    ±ronology deserves to be given priority.

    The preliminary, philosophical enquiry into the nature o f time has been going* on for

    omturies, and will no doubt continue, but it is also true to say that the stage has been reached

    for some researchers to pursue the study of time in either of the two directions analogous

    id   those distinguished by Russell in the case of mathematics. Indeed the progress in the

    direction towards precision in measuring the temporal extent of objects of minute duration,

    ind in the direction of greater accuracy in placing various cosmic events on a time scale that

    mends for thousands of light years into the past and for a shorter, perhaps, but not less repressive distance into the future has been amazing, and the credit for it must go to

     physicists and to astronomers.

    The study of time in the direction towards greater abstractness and logical simplicity seems

     jo have been pursued along two different routes. One of them has been very busy in recent

     jears. It has become a favourite choice of many talented researchers inspired by the ideas

    ind enthusiasm of Arthur Prior. The results of his and his followers’ researches have reached

    me form of axiomatized systems of what has become known as tense logic.11 The other route

    las been explored by relatively few, and, at present, appears to be deserted. More about

    i in a moment.

    The characteristic feature of tense logic is that in its framework certain notions relating

     zo  time in ordinary discourse are taken to be embedded in propositions forming functors

    for propositional arguments. For instance, in the expressions of the form ‘it has been the

    case that  p ’ or ‘it will be the case that  p \   where ‘p ’ stands for a proposition, the expressions

    it has been the case that’ and ‘it will be the case th at’ are taken to be functors belonging

     jo  the same semantic category (z.e., the same part of speech) to which the functor of negation,

    it is not the case that’, belongs. I entirely agree with Prior that systems of tense logic developed

    on these lines have much in common with modal logic, and like the latter can be regarded

    is manyvalued systems of propositional logic,12 but I find it difficult to see with what sort

    of contribution manyvalued systems of propositional logic can be credited as regards

    ontology. Faced with the choice between twovalued logic and a manyvalued one, I opt

    for the former because compared with the world in which I live, systems of the latter strike me as very

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