CHAPTER 18. PERMUTATION.
We need to determine the limits of applicability of classificatory forms by looking more closely at the matter of everyday reasoning. We also need to observe common sense practises, and find out if any forms of argument, other than those dealt with previously, are instinctively used by us.
The ultimate goal of Logic is, of course, to bring out into the open for scrutiny all the details of everyday reasoning. We want to encompass into the sphere of Logic, any forms or processes capable or worthy of formal treatment.
As previously pointed out, the form ‘S is P’, as commonly understood in Logic, serves the function of classification. Such use of the copula ‘to be’ is rather abstract and specialized. In this sense, ‘S is P’ tells us that S is an individual, or some or all members of a class, which also count(s) as among the units of some other class. S and P have to some extent the roles of species and genus.
Most often, in practise, ‘S is P’ means that P is an attribute of S; we are describing an object (most typically, an entity) in terms of its qualities. We mean that P somehow is in S, something which S has, part of the being of S, one of the many phenomena which all together add up to the thing we call S, and which are distinguishable within it.
This sense of ‘is’ as attribution is strictly-speaking quite distinct from the use of ‘is’ for classifying; there is an ambiguity, the same word is used for two different relations. The common ground of the two is the information that the ‘universals’ S and P, in some degree intersect. Two domains of reality are compared for overlap, in their instances, in space and time, in causal contexts.
Thus, ‘S is P’, may mean ‘S has P-ness’ or ‘the unit(s) of S is/are unit(s) of P’. The attributive sense may be ‘permuted’ to the classificatory, by saying ‘S is P-ness having’. This puts the specific copula ‘has’ into the predicate, in a proposition with more numerical intent.
But the possessive relation remains; it has value and meaning quite apart from such permutation; otherwise, we would have no need for the concept. Classification cannot occur without prior attribution. Before one can put a unit under a class, the class-defining quality has to be attributed to the unit. Permutation merely conceals or bypasses the attribution, it does not erase it.
The classificatory ‘Ss are Ps’ contains attributive propositions within its terms, since it means ‘things which are S are things which are P’. The ‘are’ in the two terms are attributive, while the ‘are’ of their relation to each other is classificatory.
Attribution concentrates on the substantive (i.e. in terms of universals), qualitative, so-called ‘connotative’ aspect, of this relation of coincidence. Classification focuses on the enumerative, quantitative, so-called ‘denotative’ aspect. Both aspects exist out there (at whatever level the phenomenon might be); we mentally isolate the one from the other somewhat, to stress each in turn.
Although we regard these as two aspects of the same term — we say that in one case, it is taken ‘in its denotation’, and in the other case, it is taken ‘in its connotation’ — it is more accurate to say that these are two kinds of term, which have a close relationship, both objective and verbal. For example, dogs may be viewed as things which have dog-ness (or in better English, caninity), and dog-ness may be viewed as that which dogs have in common and distinctively.
In truth, dog-ness cannot be called the meaning of dogs, as some have suggested; nor is the reverse correct. Each of these terms subsumes slightly different referents: the former refers to every corporeal dog, and the latter (though we regard it as effectively singular) refers to every occurrence of the qualities which make up dog-ness. These two are indeed equal in number, and have no existence apart from each other, but the intention is not identical.
Other propositions are permutable, besides ‘S is (or has) P’. Some propositions are geometrical: they locate the subject in space, by reference to the location of the predicate; thus, ‘S is at or in, next-to, above or below, near or far-from P’. Some involve placement in time, using relatives like ‘earlier’, ‘later’, ‘at the same time’. One can say that ‘S is at P’ implies ‘S is an at-P thing’, but one cannot say that they are equivalent and identical.
Some propositions describe actions, implying change and/or causality. To do, is to change or move in a certain way, or to cause something else to be or to change somehow. The verb may be simple or continuous; ‘S does or is doing P’. Here again, permutation is possible, to ‘S is a P-doer or one of the P-doing things’. We permute when it is useful, but the original form does not lose its utility and disappear.
The logical properties of the classificatory forms are only generic properties, applicable to all permutables. We may well expect that each of the original, more specific, forms, has its own special logical properties. It is part of the long-term task of logical science to gradually intercept all such forms, and confront them for analysis of their peculiar properties. To establish the implications, oppositions, and arguments, which are peculiar to each form.
While modality can be permuted, e.g. ‘S can be P’ to ‘S is one of the things which can be P’, this easily leads to error. The reason is that, normally, when we say of a unit that it belongs to predicate P, we mean this to apply to times and circumstances when it is actually P. This is not equivalent to reference to the class of things only potentially P. As a result, as we have seen, if this process is used inattentively, one may draw wrong conclusions in syllogism.
Similarly with transitive copulae, like becoming. They are too fundamental, too much part of the structure of things, to be permuted with any more than superficial interest. To process ‘S becomes P’ to ‘S is one of the P-becoming things’ merely puts the need to investigate the logic of becoming one step removed, since it still there, but now hidden in a more detailed predicate. Likewise, with regard to causality. To find the specific properties of such a copula, we have to keep it intact, as a relation in its own right.
(Note that any goals pursued by such permutations can, as we shall later see, be fulfilled by using ‘extensional conditional’ propositions.)
Copulae represent relations. Each relation has its own nature. ‘Is’ in the sense of class-thinking, is a broad relation, with a number of rules; ‘is’ (in the attributive), ‘becomes’, ’causes’, are all perhaps narrower relational concepts, but still worthy of special attention, in search of the rules applicable to them specifically.
Relations like the verbs ‘sings’ or ‘digs’ also no doubt have their own characteristics, and are legitimate topics of inquiry; but their limited scope, assigns them to a secondary position in logical science, while they may well be of primary importance to other sciences. We might make a distinction between formal logic and material logic, on this basis.
Let us look briefly at propositions with a verb of possession or action. Their general form is ‘X / does-Y / Z’. There is a logical subject (X), a verb (here written ‘does-Y’), and some or no appendages (the ‘Z’ part).
‘Does’ is here meant to include agency and passion, doing and being done to. But not every action implies a patient, note. The agency, by the subject, of the act (verb/relation), may range from an act of free-will to a fixed absolute. An act may be static or dynamic, it may signify a posture, a movement, a forcing. In this context, even ‘has’ is an act.
Note that quantity and modality are applicable to any such proposition, as usual. The simple and continuous present tenses are not interchangeable, of course. The modality involved is often left implicit, and should always be clarified. For instance, ‘animals sleep’ implies ‘animals are sleeping some of the time’ without implying ‘all the time’ or ‘at this time’.
The appendages concern parameters such as: who, what, where, when, why, whence, how, what for, how much. They serve to further specify the relation, and delimit it. Prepositions like ‘by’, ‘at’, ‘to’, ‘for’, are definable in this context.
We may mention any combination of the following: the patient of the action, or agent of the passion (e.g. electrons repel each other); the locale of the incident in time and/or space (e.g. he went west at sunrise); the conditions or causes (e.g. water boils at 100 deg.C, at s.t.p.); the effects or consequences (e.g. it drove him to work harder); the means or ends (e.g. the water was boiled for tea).
Also, generally, any measurement, qualitative or quantitative, of the above, may be mentioned (e.g. she sang beautifully and softly). The ‘measure’ may concern the relationship itself (e.g. he gave liberally), or an appendage of it (e.g. he gave lots of money). Attaching expressions of number, magnitude or degree to a statement (e.g. everyone has some virtue), should not be confused with the attempt of some logicians to ‘quantify’ the predicates of classificatory propositions (as in ‘all X are some Y’).
Often, the proposition can be restructured from categorical form to subjunctive form, and so the logic of subjunction comes into play (e.g. ‘She sings when happy’ is more subjunctive than categorical).
Causality is often concealed in statements which do not mention it. For instance, ‘her beauty attracted him’ implies that the agent caused the patient to act or move in a certain way. Causality of course is of various degrees, ranging from mere influence on a voluntary act or probabilistic tendency, to compulsion or mechanical force.
Thus, we see that many material statements can be analyzed for more formal components, and thereby be subjected to certain rules of formal logic. They may nonetheless yet contain further logical properties, not found in the formal components.
We have looked at permutation which encloses a nonclassificatory copula and its appendages into the predicate of a classificatory proposition. But we also need to consider permutation of the subject.
Many propositions have a ‘universal’ as their subject. This may be a quality as such (e.g. turquoise is a bright color), or any act as such (e.g. ‘love is a nice feeling’, ‘running is good for you’). ‘As such’ subjects like these are being focused on, not so much in their capacity as classes embracing the various appearances of the ‘universal’ in the world, but as if the ‘universal’ is an individual thing (a whole, whose various manifestations in the world are its parts).
A class concept regards the individual manifestations of a universal, or of a complex intersection of universals, as separate entities, which happen to display that simple or complex distinct character in common. An ‘as such’ concept, instead, views the particular manifestations as merely the segments of a single, continuous thread, and refers to that more or less uniform whole as a distinct entity.
Before a proposition can be processed according to the logic of classificatories, its subject may need permuting. (Thus, for our examples above, we would say ‘turquoise things are bright-colored’, ‘people in love feel nice’, or ‘runners are healthier’).
Concerning verbs, the process of commutation should be mentioned.
Relations are generally directional, so that when something is true in one direction, then something else is true in the other direction. The ‘total’ relation between the terms includes both directions, though only one of the components might be named, leaving its correlative implicit. The correlative copulae may be identical (e.g. A=B and B=A) or quite different (e.g. owns and belongs-to, or shot and was-shot).
We may call ‘commutation’ the inference from one direction of a relation to the other. The correlative of ‘is’ is ‘is’, as conversion shows. The proposition ‘X causes Y’ is commutable to ‘Y is caused by X’. ‘He bought IBM shares’ implies and is implied by ‘IBM shares were sold to him’. A lot of our reasoning consists in rewording statements like that, changing the perspective to improve our understanding.
Commutation may apply not only to the main copula of a proposition, but to copulae implicit in its terms. For example: ‘the bodies attract with a force of 10 dynes: the force, with which the bodies attract, is 10 dynes’.