Formal logic is the logic of validity in a chosen domain of discourse -- 'the logic of the chosen domain'. The choice of domain itself, and the translation from ordinary language or informal argument to formal symbolism is, strictly speaking, not a subject of formal logic. These are questions of interpretation, of fields of study like ontology and linguistic pragmatics. Yet, from a more general point of view, the 'logic of domain choice', which precedes the logic of the chosen domain, is not less important. This reflection on the choice of formal domain itself should give us more insight into the presuppositions made explicitly or implicitly, and into the relevance or irrelevance of the distinctions drawn thereby. The question of what is relevant or not we have to leave until later; suffice it to say at this place that someone's reasoning may be as valid as it ever can be within the formal logic of a chosen domain, but that this does not finally settle an issue if the scope of the domain itself is selected on the basis of an irrelevant factor. Objects or other things would then be taken in or left out which ought to be excluded or included for a proper argument. (Consider, for example, a utilitarian who proposes that a hedonic balance of good or pleasure over evil or pain should be maximized and who exclusively chooses human beings as the objects whose happiness is to be taken account of.)

Some 'discoveries' of those working with formal logical systems depend entirely on the presuppositions made and the foreknowledge at hand before selecting the domain. Without such a priori knowledge, also if merely conventional or linguistic, it would not even have been possible to choose a domain of the kind in question. As an example, let us have a look at a formal analysis of the relations heavier than and as heavy as. The domain is said to consist of material objects only. The conversation is therefore about concrete primary things.

Heavier than and as heavy as denote two-place relations and, moreover, are comparative notions. Such concepts enable us to order individuals on the basis of the relation concerned. Let Hxy be the relation heavier than and Exy the relation as heavy as. Hab means, then, a is heavier than b and Eab, a is as heavy as b. Now, to 'conclude' that b is as heavy as a if Eab is true, or that b is less heavy or lighter than a if Hab is true, already requires a certain foreknowledge. Logically speaking, it is first necessary to define E as an equivalence relation, that is, a relation which is symmetrical, reflexive and transitive: 'symmetrical' in that Exy implies Eyx and vice versa (if a is as heavy as b, then b is as heavy as a); 'reflexive' in that for every object x Exx is true (every x is as heavy as itself); and 'transitive' in that Exz is true if Exy and Eyz are true (if a is as heavy as b, and b as heavy as c, then a is as heavy as c).

The logical properties of the relation H are that it is transitive as well, that it is irreflexive with respect to E and connected with respect to E: 'irreflexive' with respect to E in that Hxy is not true if Exy is true (if a is as heavy as b, then a is not heavier than b); and 'connected' with respect to E in that always Hxy or Hyx is true if Exy is not true (a is heavier than b, or b is heavier than a, if a is not as heavy as b ). If we take the relation R, heavier than or as heavy as, then there is a strong form of connectedness in that for every x and for every y of the domain concerned, Rxy or Ryx is true. (In a weak sense a relation R is connected over the class C of things if, and only if, whenever x and y belong to C, and are distinct, either Rxy or Ryx is true.)

With these formal requirements for E and H it is possible to arrange all individuals to which they apply in a certain order -- a quasi-linear order in which there is exactly one place for every individual, but sometimes with several individuals (which are equally heavy) at one place. But what is the class of individuals to which these formal requirements apply? And what is the foundation of these formal requirements themselves? Have they been miraculously supplied to spiritually nourish the absolutely blank minds of logicians and scientists before they started to think of qualitative, comparative and quantitative notions at all? Of course not: before anyone had ever heard of notions like strong or weak connectedness in their logical sense, expressions like being heavier than, being lighter than, being as heavy as and being as light as were already intimately connected in common parlance. All the comparative notions of the same name are linguistically even unthinkable without the corresponding 'qualitative' (or 'classificatory') notions of heavy and light (even when heaviness and lightness are derelativized relations rather than genuinely nonrelative attributes or qualities). At least from the perspective of language, speaking of "heavy" and "light" precedes all speaking of "heavier than", "lighter than" and "as heavy as". Both the absolute and the relative conceptions apply to the class of all things that are heavy or light or something in between (namely medium heavy or light). Two distinct things of this class are either equally heavy (or light) or not, and if so, then both ways -- otherwise the word equal would simply not be appropriate. If they are not equally heavy, the one is heavier than the other, and the one which is not heavier is lighter than the other as long as the two things have some weight. Formal systems do not establish these trivial truths; they start from them.

Now, we are going to choose a domain restricted to material things. What is 'material', however? There are two possibilities to be taken into consideration here: (1) material is defined in terms of weight; or (2) it is defined in other physical terms. In the first instance a material thing is a thing which has a weight. This, in turn, may be defined in an absolute (qualitative) or in a relative (comparative ) way. On the absolute account things which have a weight are light, heavy or something in between, and on the relative account they are things which are lighter than, as heavy as or heavier than other things. It is immediately obvious, when the relative approach is selected, that the connectedness of being heavier than or as heavy as (or lighter than) was presupposed or foreknown in the definition of materialness. The knowledge that the relation obtained by putting together heavier than and as heavy as in one disjunctive predicate (expression) would be connected over the class of material things and not over any class comprising immaterial things was a prerequisite to the very division between material and immaterial things itself.

Weight is not a relative notion. Just as length must be read as longness rather than long-er-ness, so weight must be read as heaviness rather than heavi-er-ness. But also on the absolute interpretation one must have foreknowledge of the connectedness as somewhere in between cannot literally be read as neither light nor heavy, since immaterial things, too, are neither light nor heavy. Neither light nor heavy is to refer to the borderline, or rather the fuzzy zone, between the 'light' and the 'heavy', that is, to the 'medium light' or the 'medium heavy'. Yet, strictly speaking, this may imply the connectedness of the qualities or attributes but not the connectedness of a relation in the logical sense. Somehow it must also be given a priori that everything between light and heavy is heavier than light and lighter than heavy. Granted that things are as heavy as themselves or as other things belonging to the same set of the three qualitative sets distinguished, we have the same conditions as on the relative interpretation (assuming, again, that we allow ourselves to construct the disjunctive relation which is connected by now).

Instead of in terms of weight, material may be defined as having a speed, as being at rest or moving (in a positive or negative direction) or in other spatiotemporal or physical terms. The crucial assumption is now that everything that is material in this sense (that is at rest or moves, for instance) is an object which is heavier than, as heavy as or lighter than other material things. It is implicitly taken for granted, then, that there could be no material object which is heavier than another one without there being any other material object being lighter than at least one other material object. Hence, also in this case the connectedness of the relation heavier than or as heavy as (or lighter than) over the class of all, and only, material objects is presupposed or part of our foreknowledge. Any 'discovery' of this formal connectedness in such a domain of material objects begs the question. (This is not to play down the issue of whether and why the class of all things that are heavier, as heavy as or lighter than one another is identical to the class of things that are, for example, at rest or in motion.)

There is another, perhaps more serious, objection to overrating a formal, logical property such as connectedness and that is that it may be a necessary condition but that it is not a sufficient condition to express the typical 'connection' between notions like heavier than and as heavy as. Imagine a domain or a world in which all objects that are arranged in a quasi-linear order from 'light' to 'heavy' are arranged in a parallel order from 'fast' to 'slow': every object that is lighter happens to be faster (and necessarily faster if also dealing with modal conditions) and every object that is heavier happens to be slower (and necessarily slower). Our actual world is not like this, but one may think of other attributes or relations thus correlated. (The example is similar to that of all cordate beings that are renate, and vice versa, with the impossibility of identifying predicates on the basis of their extension alone, whether they are proper or improper.) On our assumption not only heavier than or as heavy as is connected over the domain in question but also heavier than or as slow as and faster than or as light as. Altho people are free to fabricate any fancy, disjunctive, or other, predicate expression they wish in formal logic, this certainly has no bearing on reality. If the (logically) informal language we communicate in is to mean anything, the concepts heavier than and lighter than are related to the concept as heavy as in a way they are not related to any other concept (whether it be as slow as or something else). The formal, logical requirements for H and E do not suffice: there must also be a special relationship between these relations which connects them in a way in which they are not or cannot be connected to any other relation. But the moment we recognize such a relationship (like that between heavier than and as heavy as) we do not need to emphasize the other requirements for our purposes anymore.

©MVVM, 41-57 ASWW

Model of Neutral-Inclusivity
Book of Instruments
Catenas of Attributes and Relations
Beyond Formal Connectedness