FUTURE LOGIC

CHAPTER 65. DEVELOPMENTS IN TROPOLOGY.

The study of modality on a philosophical plane may be called Tropology (from the Greek for 'figure', tropos); it is a broad field, with Ontological and Epistemological ramifications, and a direct relevance to fields like Aetiology, the study of causality (and thence Ethics), as well as to the likes of Physics, Biology and Psychology. Modal logic is a branch and accessory of Tropology, clarifying the formal aspects and processes of modality.

We have seen in this treatise, that modalities are attributes of relations (or of any things, in relation to existence). Modalities are distinguished first with reference to their 'types', and within each type with reference to their 'categories' (which are similar from type to type). We dealt in detail with four major types of modality (which may also be more briefly called 'modes'), which are defined with reference to their 'fulcrums' — namely, the de-re modalities, natural, temporal and extensional, and de-dicta or logical modality. Within each type, we distinguished the categories of necessity, actuality, possibility, and their negative versions (all of which have special names assigned to them in the various modes).

The combinations of category with type, yield specific modalities within each type; but additionally, there are modalities of compound categories and of mixed types, so that the list of specific modalities is quite long. Other types of modality exist, like the volitional (a subset of natural/temporal modality), and the teleological (from which the ethical is derived, at least in part); but we have not studied them closely in the present work (though I have myself studied them, and can assert that they fit into the general scheme here presented).

The fulcrums (or fulcra) of the four modes which here concern us are: natural circumstances, times, instances of a universal, and logical contexts; the generic fulcrum is labeled 'cases'. The general definition of modalities is that they are 'attributes of things or relations which exist in a number of cases', the number specified determines the 'category' and the 'type' of cases involved is specified by the fulcrum. For example, potentiality, or natural possibility) is defined as 'the modality of what is actual in some circumstances'. All other modalities are defined by similar statements, mutadis mutandis.

The fulcra are the focal points, the themes, the cruxes, the frames-of-reference, which distinguish one mode from another. The common relation is referred to by means of the tiny word 'in' — in some circumstances, at all times (in all segments of time), in most instances (in most manifestations of the universal), or in a few contexts, for examples. The fulcrum frequently has 'boundaries', which delimit the applicability of the relation; for example, the times or circumstances involved may be those in which the subject is actual or potential, to the exclusion of those in which it is not.

The relation of 'in-ness' plays an important role, though it is notional and intuitive. Its quantitative aspect is explicated spatially, by analogy to a dot within a larger area (or a point in a circle, say). Its qualitative aspect is the insight that things are in many cases 'affected' by their surrounds, that things somehow 'interact' in their environments. Thus, a wordless reference to causality is involved; but 'causality' at a very vague and intuitive stage. Later, this notion of causality, which is used to build up modality from a notion to a scientific concept, is in turn built up into a scientific concept, by the concept of modality, as we saw.

After such preliminaries, we proceeded with an analysis of the interdependencies and interactions of the various modalities, with reference to the oppositions, eductions, and syllogisms between different modalities. At a subsequent stage, different copulas were also considered. (See part II.)

All this concerns 'categorical' relations, but the concept of modality gives rise to parallel 'conditional' relations, which in turn further clarify what we mean by modality. Here, too, we found, we must distinguish between types of conditioning — the natural, the temporal, the extensional, as well as the logical (and not only the latter).

For each of the four modes, there exists a variety of conditionings, distinguished with reference to the 'connection' and the 'basis' intended, as well as to issues of polarity. There is also a distinction between implicative conditioning, and various manners of disjunctive conditioning. The connections and bases were defined with reference to modal (and polar) concepts, and the logic of their compounds was analyzed in detail. The concept of basis is a newly discovered one, which refers to the possibilities underlying a connective actuality. We thus gradually arrived at a greater understanding of causality. Each type of modality and category of conditioning gives rise to a distinct concept of causality. (See parts III, IV.)

Lastly, the issue of how modalities are (or are to be) known in practise, in specific cases arose (which should not be confused with the more philosophical issue, just mentioned, of how the concepts of modality as such were constructed). There is of course deduction of modal propositions from previously identified modal propositions; but ultimately, some modal propositions have to be induced somehow. We discovered precisely how modal induction works, in strictly formal terms, through the novel theories of factorial analysis, factor selection and formula revision (see part VI).

In this way, we developed a pretty thorough theory of modality, which set terms and methodological standards for Tropology. As far as I can see, this treatment is original on many crucial counts. Philosophers and logicians have of course over time done much work in this field. But the present situation seems to be as follows:

a. the four modes are by now more or less known, and it is known that they have analogous categories, which are somehow determined by quantitative issues — but no one so far has arrived at clear definitions and devised a classificatory understanding of these phenomena;

b. some work has been done since antiquity on modal syllogism, but errors were made, through failure to take into consideration the phenomena of change; also, subsidiary matters like productive argument seem to have been altogether ignored;

c. although logical conditioning has been analyzed in detail, especially in the modern era, the de-re forms of conditioning, and of course their respective and interactive logics, remain essentially unknown to this day;

d. the whole issue of modal induction has never till now been raised, and therefore of course factorial methods are totally unheard of.

This is the situation as I found it. A complete history of modality theory, is beyond the scope of this work. As we have seen, the topic dates at least from ancient Greece, and crops up thereafter again and again. Here, my purpose is rather to trace, in a very random-sample and fragmentary manner, but in somewhat more detail than thus far attempted, more recent developments, with a view to a fair evaluation of where my colleagues stand today. Also, comparing and contrasting my thesis, serves not only to defend it, but to further define it.

My methodology is far from exhaustive: it consists in gleaning information from a number of works, found in the library of the University of British Columbia and the Vancouver central public library, having to do with modal logic. True scholarship would demand a more thorough approach, with a special concentration on the Big Names in the field: the work of many years. My spot-checks are not sure to be representative. Such a method may paint an inaccurate patchwork: it estimates the shape of the whole from points on a graph.

2. Roots.

As earlier stated, I consider Aristotle's understanding of modality to be broad and profound. The evidence is to be found both in his logical works, like De Interpretatione and the Prior and Posterior Analytics, and in his discussions of change, coming into being and passing away, potentiality, actuality, and natural necessity, and causality, in works like the Metaphysics, the Physics, and De Generatione et Corruptione.

Quotations of all his direct or indirect references to issues of interest to modality theory, would no doubt fill a volume. In any case, what is relevant to note here is that Aristotle was aware of both the ontological and epistemological variants and dimensions of modality.

Rescher's Temporal Modalities in Arabic Logic describes the treatments of natural and temporal modalities found in Arabic texts, such as Avicenna's (Ibn Sina's) Kitab al Isharat wa-'l-tanbihat and Averroes' In I De Caelo, which 'are unquestionably of Greek provenience'. It is well known that the historic value of Islamic Middle-Eastern logic, lay in its bridging the gap of centuries between the worlds of Antiquity and Christian Europe.

The latter, in its firm will to overthrow paganism, had as it were 'thrown out the baby with the bath water', and indiscriminately rejected some of the more positive achievements of the old world. Arabic logic, judging from the said source, concentrated on some of the ontological aspects of modality, which the Medieval scholars dubbed de re. Over time, after that, the emphasis (at least in formal theory) shifted to more epistemological faces of modality, the de-dicta aspects.

Let us to begin with scrutinize Arab contributions more closely, in the context of our own theories. First, I would like to say that Arabic modalities are misnamed when they are called 'temporal', for it is clear that they are more precisely mixtures of natural as well as temporal modality.

The Arabs had a respectable concept of the various categories of modality (such as necessity, actuality, possibility, impossibility, inactuality, possibility-not, contingency), as well as of their interrelations (a square of opposition was presented by Averroes); but confusions arose when dealing with the types of modality. Although they were clearly conscious of the complexities involved, they were not entirely successful in separating the various issues from each other.

a. Avicenna distinguishes "absolute" (meaning, unmodalized) propositions from those which are modalized. However, strictly speaking, this should be viewed as a grammatical rather logical distinction. In common discourse, we admittedly do not always explicitly qualify our statements modally, but from the logical point of view, every proposition has some at least implicit modality; if the modality is not apparent, then the lowest possible modality may be assumed, just as an unquantified statement is considered as particular rather than as a distinct kind of quantity.

Similarly, for so-called "categorical necessity" and "general possibility", which refer to these categories without explicit qualification as regards type. Likewise, also, for 'impossibility in the primary and general sense' and for "special possibility" (by which is meant, contingency). These were erroneously considered as categories of a distinct type ("modes"); but they are simply generic concepts, of unspecified type.

b. Avicenna does not make clear initial distinctions between natural and temporal modality, nor between different modalities of subsumption, nor between the categorical and (de-re) conditional manifestations of modality, nor between different modalities of actualization. Instead, the significances of various compounds of these elements are discussed, as the following definitions make evident:

Thus, "absolute necessity" refers to what is 'in essence capable of' being predicated of a subject, throughout its duration as such (even if only in most instances of it or most of the time or in most circumstances, even if statistically there may be a few rare exceptions). In contrast, "general conditional necessity" refers to predications applicable to a subject constantly, while it is in a certain state or it is surrounded by certain conditions; whereas, "special conditional necessity" refers to similarly conditional, but temporary events, which may in turn be "temporal" ('as with an eclipse') or "spread" ('as with respiration').

Again, "general absolute possibility" refers to events which are 'not perpetual', yet 'necessary some of the time'. Alternatively, "general possibility" refers to events which are 'actual at some times but not others' without being 'necessary at all — neither at a given time' ("non-perpetual existential") 'nor under certain circumstances' ("non-necessary existential"). Underlying this distinction is a concern with the inevitability of actualization, or its absence, obviously.

We can also see the mixture of considerations in the subdivisions proposed by Avicenna for actuality: it may be, 'as long as the subject really exists', "absolute perpetual" (always there, 'but without necessity') or "general conventional" (there, 'always under certain definite circumstances') or "special conventional" (there 'at certain times,… though not perpetually'). Note the appeal to a modality of subsumption. I will not belabor the topic further: the point is made.

However, one more thing is worthy of note. It is evident (in Rescher's Table X) that Arabic logicians, if not all past logicians, regarded first-figure syllogism with a merely possible major premise, whether the minor premise is necessary or itself possible, as yielding a valid possible conclusion. Thus, according to them, the modes 1/pnp and 1/ppp (however possibility be interpreted) are valid. This is of course, as I have shown, a historic error (see ch. 15-17).

Lest the impression have been given that discussions of that period centered exclusively on de-re modality, I should briefly mention as an example the doctrine of the Mu'tazilite school of Arab philosophers, who according to A.Y. Heschel 'rejected the idea of causality and taught: What seems like a law to us is merely a "habit of nature"….' Thus they 'followed the principle that no heed to is be taken of reality, since it also rests on a habit whose opposite is equally conceivable'.

To these ideas, the Jewish philosopher Maimonides replied: 'Reality is not contingent on opinions, opinions are contingent on reality"' (117). Clearly, what was at issue in these discussions was the precise relation between de-dicta modality (the conceivable) and de-re modality (the real, the natural).

3. Shifts in Emphasis.

We find in Aristotle an interest in both the de-re and the de-dicta senses of modality. In Arabic logic, as we just saw, the emphasis was more on the former. But thereafter, as we shall now see, European thinkers put more emphasis on the latter sense. This is already evident in Ockham's discussion of modal propositions, in the early 14th century. Of course, some logicians, like J.S. Mill in the mid-19th century, in the context of his study of causality, continued to focus on modality in a more objective sense.

The tone was perhaps set by the great, 18th century German philosopher Immanuel Kant who, at the turn of the 19th century, defined modality in a more subjective sense, as determining 'the relation of [an] entire judgment to the faculty of cognition'. He distinguished between problematic, assertoric and apodeictic judgments; defining these, respectively, as 'accompanied with the consciousness of the mere possibility,… actuality,… [and] necessity of judging' (115). Incidentally, Kant's definitions seem circular to me, unless possibility, actuality and necessity are defined elsewhere, or considered obvious notions.

Other influential distinctions were suggested by Kant, among them that between analytical and synthetic judgements. In the former, the predicate is 'contained [though covertly] in the conception' of the subject; in the latter, the predicate 'lies completely out of the conception' of the subject (Joseph, 207). Again, these definitions are open to technical criticism, since the terms used in them are very ambiguous, but that need not concern us here. Kant was apparently trying to distinguish between the self-evident (which he considered purely 'a priori') and the empirical.

But Kant's understanding of self-evidence was very naive. For him, a proposition like 'cats are animals' would be analytical, because 'the definition' of cats includes that they are animals. But this is an error: such a statement is synthetic.

When we perceive an object, we distinguish various attributes in it; perceiving many objects, we find that they have some attribute(s) in common, and others not in common; lastly, we assign a different name to each distinct uniformity. Thus, a statement like '(all or some) X are (or are not) Y' signifies:

(All or some of) the things which had the resemblance(s) we labeled "X",

also have (or lack) the distinct resemblance(s) we labeled "Y"

— which may or not also be found in things not having the resemblance(s) we labeled "X". That 'cats are animals' is therefore an empirical finding. That 'cats' is a subclass of 'animals', simply means that all cats are animals, but not all animals are cats (see ch. 44). The selection of the animal attribute of cats, as essential, as the defining genus although it is one of many overclasses, may be due to its being obviously very 'striking' (the automobility of animals is impressively different from the growth of plants, for instance), or to some logical intuition based on wider considerations (adductive processes).

In the absence of any deciding factor, the decision may indeed, as a last resort, be very conventional. Some definitions are admittedly mere word equations, like 'Bachelors are unmarried men', but even then we always draw on some underlying experience (in this case, the fact that some men are not bound to a woman by public vows). Kant evidently focused on a very minor contingency at the tail of the conceptual process. The same can be said concerning many later philosophers (and certain modern logicians), who took the more extreme position that definition is linguistic and conventional. Reasoning is impossible without some sort of empirical data behind it.

The situation at the turn of the 20th century may be illustrated in the writings of H.W.B. Joseph, an Oxford logician. His virtue lay in unfettered discussion of issues, gently bringing many examples to bear, without the compulsion to quickly and rigidly systematize; I personally learned logic from his work. With regard to modality, Joseph evidently echoes Kant in focusing on the distinction between assertoric, problematic, and apodeictic judgments.

Incidentally, he points out that the word 'tropos' (Greek for 'modality') first occurs in the Commentary of Ammonius, where it is taken as 'signifying how the predicate belongs to the subject'. This might be interpreted as widely applying to any adverb; but logic is more concerned with adverbs which 'determine the connexion' (according to Michael Psellus), or more precisely which 'attach to the copula, and not to the subject or predicate' (according to Buridanus).

Discussing the modal qualifications, Joseph pronounces them 'clearly logical', but adds that it is not 'the act of judging' nor 'the matter judged' as such which they concern; rather, they somehow 'mark the distinction between knowledge and opinion… and differences in certainty'. Judgements not modally qualified, he calls 'pure'; these as Bain suggested express mere 'primitive credulity', they are 'assertions… made without reflection; we do not ask whether they are consistent with others'.

Apodeictic judgment is such that its 'ground… is seen to lie within the nature of' the terms involved: it is 'self-evident', but that does not mean that 'it is evident without need for understanding'. Mathematical statements are not commonly explicitly expressed as necessary, but they all (those proven) fall in this category. Empiricists 'rightly insisting that there is no knowledge without experience, wrongly suppose that we cannot by thinking discover the nature of anything that we have not perceived'. Others, including G.W. Hegel, F.H. Bradley, H.H. Joachim, counter that 'only in apprehending everything could we know anything as it is'.

Problematic judgment signifies 'our belief of certain facts which are not sufficient ground for the judgement… though we believe that along with other facts they would be'. It is 'provoked by knowledge', yet it so qualified 'because of ignorance' (Bosanquet is cited in connection with these insights). Lastly, assertoric judgment is distinguished from pure, in 'being not a bare unreflective assertion, but expressing besides our mental attitude towards a suggested doubt'.

Joseph goes on to say 'these distinctions of modality do not then express differences in the necessity with which the elements connected in reality are connected'. He also introduces the concept of probability, as being related. He concludes: 'what gave modality to a judgment was the presence of the thought of grounds for what is alleged… the grounds [being] given in other judgements', so that 'a modal judgement expresses reflection upon the question of the truth of what is judged or suggested'.

We see here that the initially almost psychological definitions by Kant, have subtly shifted over into explanations which have more to do with logic as such. Modalities (of the type under consideration) are neither entirely subjective, nor entirely objective, but somehow relate to causal and inductive reasoning (in the largest sense). We thus return full circle to the 'old distinction between ratio essendi and ratio cognoscendi, a reason for being of a fact, and a reason for acknowledging its being'.

In my own theory of modality, I attempt to contain all these trends. There are various types of de-re modality, and there is de-dicta modality, and the interrelations of the two groups are probed by formal methods. De-dicta modality is founded in epistemic notions akin to those of Kant, but the concepts of logical modality as such are more complex constructs. They depend on an interplay of rational insights, empirical data, and a holistic approach.

A couple of final comments. First, I want to say that, though I respect Kant highly, as an imaginative and stimulating philosopher, I view him as a not always very powerful formal logician or practitioner. Secondly, with regard to Joseph, although his main treatment of modality (188-207) ostensibly revolves around the de-dicta concepts, he should not be construed to have at all discarded de-re concepts.

That is evident for instance in his distinction between disjunctions which 'express the state of our knowledge' from disjunctions 'in the facts' (giving as examples 'Plato was born either in 429 or 427 B.C'. and 'Number is either odd or even'). He was of course in this case referring to the difference between logical and extensional disjunction. I do not recall whether he made a similar distinction with regard to implication. But he also points out that 'X may be Y' in some cases signifies a particular proposition 'Some X are Y', or 'that under certain conditions, not specified, though perhaps known, X is Y' — here again, the interpretation is factual (extensional, natural) rather than pertaining to knowledge.

It is also interesting to note Joseph's comment that hypothetical propositions are often used even 'when we do not see the consequent to be necessarily involved with the antecedent'. He thus anticipated what I have called 'lower case' hypotheticals, which only describe a possible consequence of an antecedent. He also mentions the use of possible disjunctions, as in 'a G may be either S1, S2, or S3'. It is a pity that other logicians did not follow up on these observations.

4. Setting the Stage.

Now to the current century. (The reader is referred once and for all to Part III of this book, for a fuller discussion of the topics raised here.)

An interesting development was that of 'many-valued' logics, like the one proposed by Lukasiewicz in 1917. The 'values' under scrutiny were truth and falsehood — again, purely de-dicta concepts — and the suggestion was that intermediate values, any number of them, were conceivable. An example given was 'I shall be in Warsaw in a year's time', which 'is neither true nor false'; this of course resembles Aristotle's 'There will be a sea-battle tomorrow'. (Bochenski, 405.)

As I have said before, I have no argument in principle with such an idea: it is applicable, to the different degrees of credibility or of logical probability, or even to 'partial truths' (propositional compounds some of whose elements are true and some false), as well as specifically to future events conditioned by voluntary factors. My only objection has been that, ultimately, such logics have to be reducible to the two-valued kind: they do not displace the latter kind, because it is used to judge their proposals.

That is, we still have to decide, with regard to any proposition of many-valued logic, whether it is true or false. Modal nuances are not primaries, but merely quantifications of certain primaries. In any case, many-valued logic was interesting, as an effort to formally recognize the philosophical truth that reality and illusion are extremes between which lie uncertain cases.

'Up to 1918 all mathematical logicians — unlike the Megarians, Stoics and Scholastics — used only one notion of implication, the Philonian or material'. At that time, C.I. Lewis 'introduced a new notion of implication and with it a modal logic'. (Bochenski, 403). This refers to 'strict' implication, which was definitely a welcome improvement in my view. Lewis clearly brought out the interrelatedness of modality and logical relations.

Note that Frege, not long before, considered modal qualifications to have 'no place in pure logic'; but H. MacColl 'had included some suggestions for modal logic' in his work (Kneale, 548-549).

However, Lewis' modality was only of a logical kind, since its focal points were the alethic concepts of truth and falsehood; with few exceptions, this seems to be the main concern of contemporary logic. More importantly, the relation between these concepts and more stringent concepts such as 'impossibility' was left undefined, with the latter taken as an irreducible primary. This failure to define the logical modalities, whether impossibility or any other category is taken as the starting point, plagued modern logic with manifold problems.

Attempts to effectively define logical modality were woefully weak. Consider for instance that of Rudolph Carnap in his intricate 1947 symbolic study of logical modalities, with reference to semantics. His system centered on necessity; and the 'explicata' he gives for it, which he finds 'clear and exact' unlike 'the vague concepts… used in common language and in traditional logic', is as follows:

it applies to a proposition [whose truth] is based on purely logical reasons and is not dependent upon the contingency of facts; in other words, if the assumption [of its negation] would lead to a logical contradiction, independent of facts (174).

Look at the description: it is technically no better, indeed much worse, than most common or traditional 'understandings'. Most of the words used (like 'applies', 'based', 'logical', etc.) refer to complex modal and logical concepts, which themselves need to be defined; indeed, many of these concepts require for their definitions prior definition of modality, so they can hardly be used to define a modality. Not only is the philosophical background vague, but the author fails to make an obvious self-test for a petitio principii.

The impression of rigor given by subsequent symbolic manipulations, however admittedly 'conventional', is entirely illusory, since no formal rigor was exercised with regard to the starting points. Apart from such criticisms, let us anyway note that Carnap was interested, at least primarily, in logical modality, rather than any de-re concept of modality.

Instead of seeking definitions, which would conceptually explain the accumulated intuitions of logical science, in the way of a theory to cover the facts — Lewis and those after him used certain known logical phenomena as axioms from which the others were to be derived, following the model of the Principia Mathematica. Consider, for instance, the axioms of Lewis' modal system S1.

One 'axiom' was that a proposition implies the negation of its own negation, or that 'P' implies 'not nonP'. For me this is not an axiom. The law of contradiction, as I have argued at length, cannot strictly-speaking be construed as a general first premise from which others are deduced; rather, it must be viewed as an inductive summary of countless specific logical insights. A fact and its negation never occur together in our experience; or if they seem to, that event itself is experienced as somehow faulty and needing some sort of correction. The word 'not' merely labels such phenomena; it does not invent or create them.

Likewise, some of the 'axioms' relate to the logical relation of mere conjunction, for instance that 'P and Q' implies 'Q and P' or even just 'P' and just 'Q'. For me these are not axioms. We commonly 'experience' situations where two or more propositions all seem true in a context. The word 'and' is used to refer our attention to such situations. It is evident from the experiences we intend by it, that the 'togetherness' is a nondirectional relation and does not exclude separate existence (had we experienced something else, we would have said so).

Such underlying experiences can be, and often are, wordless; the words (or symbols) cannot therefore be regarded as conventional determinants. The relation of implication cannot in any case be used to define 'not' and 'and', because it is itself much more complex than them, and anyway (as it turns out) they are required to define it. All that these so-called axioms do is report what we commonly and invariably all intuit: they do not serve to justify these intuitions, which are primary givens. If they are neither 'conventions' nor definitions nor first premises, then it is misleading to call them axioms.

All the more reason, propositions like 'if P implies Q and Q implies R, then P implies R' (the primary mood of hypothetical syllogism) and 'if P implies Q, and P is true, then Q is true' (the primary mood of logical apodosis), cannot be characterized as axioms. They cannot be placed, as Lewis and others have done, at the fountainhead of logical science. They are only at all meaningful provided we first come to an understanding of logical modality, which would allow us to define implication in such a way that, indeed, these properties emerge.

Similarly, that 'possibly{P and Q} implies possibly P and possibly Q', is a very derivative propositions, which depends for its recognition on a preceding understanding of logical modality. Once we know that possibly means 'in some contexts', it is easily seen that if 'there are contexts where both P and Q seem true', then 'there are contexts where P seems true and contexts where Q seems true, and some of those contexts at least are the same'. Likewise, that 'there exists cases where P neither implies Q nor implies nonQ' is a common observation: some propositions seem unrelated to each other.

In each case, we have an appeal to the idea of subsumption; or, if you wish, to the topological principle that the whole is made up of and includes the part. But even here, the relation involved is repeatedly intuited as 'logically forceful'; our statement of it is a mere verbalization of information, and not the source of our conviction.

Additionally, we must distinguish the intuitive notion of implication, reflecting our experiences of one thing seeming to lead to another, from the more conceptual construct of implication, defined with reference to modality. For this reason, we can make statements like the above, reporting common logical experiences, even before we have proposed a theory as to the definition of implication (with reference to modality). Common intuitive knowledge (what we call 'common sense') is used to test and tailor the eventual definitions.

All this to say: such 'axiomatic systems' grossly oversimplify the conceptual complexities involved in logical concepts. They fail to pay attention to what might be called the genealogy of the ideas involved. One cannot avoid having to define the logical modalities, and they are not definable arbitrarily. There is a step by step process, layer upon layer:

a. first, we have specific intuitions and experiences of the kind we label 'logical', about various things;

b. notions are formed about these logical phenomena: they are pointed to, distinguished from each other, and variously named;

c. regularities in behavior are faithfully observed and duly recorded: these will serve as the database for subsequent theorizing;

d. concepts can now be formed, which are capable of embracing the said regularities: such construction itself involves reliance on ad hoc logical insights;

e. only finally, do we have complex logical principles, to play around with symbolically, and order into axioms and theorems: and even these actions depend on the intuitive experience of their logical 'rightness' or lack of it.

The issues are further complicated by the fact that the progressive developments of different ideas impinge on each other at different stages of the proceedings. The logical intuitions and notions concerning them all come into play at all stages of each concept's development; and additionally the concepts are tiered relative to each other. We have therefore to zigzag from one idea to the other; there is no way to hierarchize whole sequences.

Thus, at the lowest level, we have: appearances, their apparent credibilities, their apparent impacts on each other, their apparent contextuality. Next, the concepts of truth and falsehood are defined, by comparing the seeming credibilities of seemingly opposite appearances in a given context. Next, modal concepts, like necessity or possibility, are defined with reference to numbers of contexts in which truth or falsehood are found. Next, logical relations, like implication, are defined, by modalizing conjunctions and their negations. At the highest level, probabilities can be analyzed, with reference to all the preceding.

Let us now consider the kind of proposition modern logicians considered as optional axioms or as theorems worth deriving; specifically, we shall consider the doctrine of 'orders of modality'.

Many different 'modal systems' were proposed by Lewis and others, according to which established logical principles were taken as primary: if p implies q and q implies p, then either of p and q can be taken as more primary. That assumption ignored conceptual considerations, as already made clear. Also, certain logical relations are ab-initio of undetermined value: therefore, different systems could be constructed, by arbitrarily assuming an additional proposition or its negation as one of the axioms. Thus, what is an axiom in one system might be denied in another, or it might be derived in the way of a theorem.

A great fascination arose with one kind of question especially: that of 'superimposition of modalities'. Starting with O. Becker (according to Feys), logicians like Lewis, Carnap, Godel, Von Wright, McKinsey, Parry, debated it with the utmost seriousness.

They called logical categories like necessity and possibility 'first order' modalities; their reiterations were called 'second (or higher) order' modalities. This refers to verbal constructs like 'necessity of necessity' or 'possibility of necessity', and similarly in other combinations. The questions they asked were: Does necessity imply necessity of necessity, or what? Does possibility imply necessity of possibility, or perhaps possibility of necessity, or maybe possibility of possibility? (Why not necessity of necessity, for that matter?) These were called 'reduction principles'.

Carnap, for instance, claimed to demonstrate, on purely semantical grounds, that necessity implies necessity of necessity (174-175). In view of the controversies, the Kneales, in 1962, wondered 'if it is not possible [to resolve such queries] how shall we ever be able to settle the question? What sort of evidence should we seek and where?' (556).

But, I say, once a quantitative definition of the logical modalities has been constructed, these questions appear utterly puerile. If the reiterated categories in question are of uniform modal type, that is, all de-dicta, logical concepts — the answers easily proceed from purely quantitative considerations. 'All of all the contexts' is equivalent to 'all the contexts' (necessity). 'All of some', 'some of all' and 'some of some' of the contexts, all signify 'some of the contexts' (possibility), although their precise extensions may well vary.

Admittedly we commonly do repeat modal qualifications. But often, the intent is only to emphasize, a mere linguistic flourish: I am sure that I am sure, I am unsure that I am sure, and so forth. Sometimes, perhaps, we intend a sequence: I am still sure, I am no longer sure, and so on. Such statements tell us whether a verification has or has not taken place, and whether further research is or is not called for.

Knowledge and opinion, as we have seen, vary over time, in the transition from context to context; thus, assumed (that is, contextually imposed) modalities do change, and logical science may have an interest in analyzing such changes in precise detail. But in its essence, logical modality is a static phenomenon: we are not so much interested in the history of our present modal qualifications, but rather in specifying how the present context is determining them.

Thus, the logical modality of a logical modality is not in practise meaningful: the weakest component determines the whole. In any case, such issues cannot be construed as having so much importance in modal logic that they may play an axiomatic role, even optionally.

The only significant way such nestings of categories of modality occur in practise, is when the categories are of mixed modal type. Concepts like 'the logical necessity of a natural necessity' are quite legitimate. In this example, we are asking whether the proposition concerned is not only naturally necessarily but logically so; I believe, in this case, the reply to be that logical necessity implies natural necessity, though a natural necessity may well not be logically necessary.

Any mix of logical, natural, temporal, and extensional modalities can similarly be considered. My analysis of compound, fractional and integral modal propositions is intended as an exhaustion of all the combinations of natural, temporal and extensional modalities with each other, for categorical propositions. Logical modality is not included, because the other modalities are viewed as the ultimate objects of study; they are the goal, logical modality is only a means (ch. 51, 52).

But in practise, we should not rush to judgment, for the intent is often more complex than it seems. Thus, taken simply, 'X must always be Y' is redundant, since 'must' implies 'always'. But the intent may rather be that, each time 'X is Y' actualizes, it does so inevitably (rather than spontaneously or voluntarily). Likewise, 'X can sometimes be Y' may be simply viewed as an abbreviation of the compound 'X can and sometimes is Y', or in more clever ways. Such statements may also be intended to refer to acquisitions or losses of powers. (See ch. 34.3, 51 for introductory comments on these topics).

The issue is never verbal or grammatical or symbolic. Words do not affect the issue: what matters to logic is what we intend by them. It makes no difference whether we say 'It is necessary that X is Y' or 'X is necessarily Y' or 'That X is Y is of necessity true', contrary to what the Kneales suggest (553). It makes no difference whether 'must' refers to 'logically must' or 'naturally must' or 'always' or 'all', so long as we agree on a terminology: we all have access to the underlying concepts, anyway.

In any case, note, logicians cannot analyze such mixed-type stacks of modalities in a generic way, because for all we know to start with, different combinations may have different explanations. Some general rules might emerge as a final conclusion; but they should not be predicted offhand. The issue is complicated by the interrelations of modal types, which are not entirely continuous (see for instance ch. 38.2).

Logical modality differs radically from the de-re types, in that it concerns a different domain, that of 'contexts', instead of that of 'circumstances' or 'times', or again that of 'instances'. Yet logical necessity implies the natural and temporal, and natural and temporal possibility implies the logical, in categorical forms (ignoring issues of modality of subsumption). However, in conditioning, these continuities are inhibited, because logical forms have been designed as mere connectives, whereas de-re forms must have a proper basis.

Natural modality likewise surrounds the temporal, but their categorical continuity is broken in conditional frameworks, because of their different bases. Again, extensional modality stands somewhat apart from the natural and temporal, since it concerns groups rather than individuals. (See ch. 25, 34, 38, 39).

5. Contemporary Currents.

We find reference to the 'resemblance to quantity' of logical modality, in a 1962 book by A.N. Prior. Again, the focus is on that specific type of modality — 'assertoric', 'apodeictic' and 'problematic' are the words used for its categories (185). But in any case this shows that the analogy of logical to extensional modality, which was obvious since antiquity (with reference to the oppositions of modal categories and to modal syllogism), was acknowledged in modern times.

However, this analogy is useless without a significant explanation: the quantitative aspect in the logical modalities can only be brought out by defining them. The given datum of similar logical behavior between modality and quantity, should have been seen as a clue to some essential similarity: it was a missed opportunity for constructing a fitting definition of the modalities.

Robert Feys suggests that, already in the 19th century, 'it would have been rather natural that the calculus of propositions be conceived as… a modal one' in analogy to the 'calculus of classes'. Because, 'propositions were conceived as applicable to (verified in) various "cases", "circumstances"", "moments of time", "states of affairs"', so that 'an implication was a proposition asserting that all cases in which p is verified are also cases in which q is verified' (3).

This statement shows that, at least at the time it was written, in 1965, logicians did indeed come very close to a precise, quantitative definition of modality and conditioning, at least in a generic sense (if we take the latter use of the expression "cases" at its broadest). But, on second thoughts, the statement seems more intended to define the form-content relation, rather than the modal underpinning of implication.

In any event, as far as I know, modern logicians did not explicitly work out distinct modal and conditional logics for each of the types of modality implied by the words they used. They should have taken, as I did, "cases" (in a narrow sense) as the focus for an extensional logic, "circumstances" as that for a natural one, "moments of time" for a temporal one, and "states of affairs" (in the sense of knowledge contexts) for a logical one. For, when one does so, it becomes clear that these various logics have distinct (though parallel) properties, which justify their separate developments. My impression is that they lumped all types together, and considered the logical sense to be generic.

This impression is not entirely offset by, for instance, Rescher's listing of many types of modality, including the 'alethic' and 'likelihood' (logical), the causal (natural), the temporal, the deontic and evaluative (ethical). For, though it is clear that he is aware that there are varieties, he also lumps such intentional relations as 'believing' and 'hoping' into the list, showing that he is not aware of the characteristic pattern which defines a relation as modal. One may well argue that such attitudes are determined by modal judgments, but they are not themselves modal (White, 168).

Questions posed by Feys concerning modality reveal the state of knowledge of current logicians; he asks if it can be used 'for the description of the physical world', or 'perhaps… in the analysis of causality'. I infer that they had not yet formally treated the relations between logical and de-re modality, and had not yet developed the logic of de-re conditionings.

It is also interesting to note that modern logicians seem still disturbed by the existence of paradoxical propositions (like p implies notp, or notp implies p). Thus we find the Kneales complaining about the lack of success of logicians 'in excluding these so-called paradoxes without also excluding at the same time arguments which everyone regards as valid' (549).

Again, had they had good definitions in mind, they would have seen that there is no antinomy in such statements provided they occur singly, not in pairs. On the contrary, precisely the existence of paradoxical forms allows us to define the concepts of logical necessity and impossibility, as self-evidence (a proposition implied by its negation) and self-contradiction (a proposition implying its negation), respectively.

Let us now look at how a recent (1976) Dictionary of Philosophy discusses modalities. It focuses on 'alethic' modalities; these include the necessity or possibility 'of something being true'; the 'factual' is defined as neither necessary nor impossible nor merely possible. But such a definition is admitted to be circular: 'it is hard to define modal terms without begging the question'.

A statement without explicit modal qualification is called 'assertoric'; only if the word 'necessary' or 'possible' appears in it may the statement be called 'apodictic' or 'problematic'. The author, A.R. Lacey, admits that 'Kant uses "apodictic", etc. slightly differently to indicate how judgments are thought, not expressed'.

But this difference is far from 'slight', it is a still more massive confusion of the issues. Whether or not certain words are used, the logical status of the proposition is not affected; moderns do not seem to understand that. We could say that the words used indicate whether the maker of a statement is aware or not (or wants to stress or ignore) its logical modality. But, as far as logical science is concerned, within the context of knowledge of that person, the logical modality of the proposition is determinate, whether known and stated or not.

Admitting that 'modal logic… is not always limited to the alethic modalities', the article goes on: 'a difficult and controversial distinction, of medieval origin, is that between de re and de dicta modality', applying the former 'to the possession of an attribute by a subject' and the latter 'to a proposition'. Some 'view that de re modality is intelligible, and that there are cases of it, even if ultimately they must be analyzed in terms of de dicto modality', while others deny this view in some way.

Further on: 'the nature of physical necessity and possibility has been disputed for centuries, especially since Hume. Are they independent of logical necessity and possibility, or ultimately reducible to them, or merely illusory?' Also: 'can there be possibilities which remain possibilities throughout all time but are never actualized? Aristotle and Thomas Hobbes, among others, said no'.

It is suggested that the logically or physically necessary may be 'what happens… in all conceivable worlds or all worlds compatible with certain laws', though admittedly the words '"conceivable" and "compatible" have modal endings'. (Incidentally, Bradley and Swartz seem to attribute this suggestion to Wittgenstein (7), but as I recall the phrase 'all possible worlds' dates from Leibniz; in any case, such a phrase obviously cannot be used as a definition of possibility.)

I think that the results presented in my book adequately answer all these questions. The fact that they (and others like them) are still asked, in so recent an article, tells me that similar results have not been obtained by others. The objectivity and scientific knowability of natural modality is demonstrated by my formal theory of modal induction.

Natural necessity may be deduced from logical necessity (or of course from other natural necessities), or induced by generalization (according to strict rules) from actuality or potentiality, or even logical possibility (by adduction); actuality is observable, as well as inferable deductively or inductively; potentiality may be deduced from actuality or necessity, or be discovered indirectly by syllogistic means (from other potentialities), or even induced from logical possibility (as a last resort, by adduction). And so forth: these issues are easily resolved, very formally, once the categories and types of modality are clearly defined.

It might be contended that I am being too picky, in evaluating modern understanding of modality. Are my definitions of the modalities so far different from the current ones? For instance, Paul Snyder of Temple University, in 1971, writes: 'Alethic modality is concerned with what must be the case in every possible state of affairs (necessity) or in some possible state of affairs (possibility)'. Is that so different from my own proposals in ch. 21?

The point is well-taken. The quantitative aspect of logical modality is by now, in Snyder, clearly brought out. The choice of words 'must be the case that' and 'possible state of affairs', may be excused, as not a circularity but a parenthetical emphasis. 'State of affairs' is equivalent in intent to what I call 'context'. It is sufficient to remark that 'while contexts exist (mentally or perceivably), they are actual'; there is no need to say that they are 'possible'. I am open, but still at least insist on my own wording.

The necessary is that given in all contexts of knowledge, the possible is that given in some. We do not need to specify that some of the contexts are merely 'possible', because that is understood from the awareness that contexts change across time and from person to person, that is, that there 'are' (in the widest, timeless sense) actual contexts other than here and now. And of course, it is important to stress that the environments concerned are (in the widest senses) empirical and logical data, that is, items of 'knowledge'.

Indeed, sticking with Snyder, we can regard the idea that there are many 'systems' of (modal) logic with more generosity. He says, provocatively, 'There is not precisely one correct system of logic. There are many'. But later he makes clear that by that he means alethic modal logic, temporal logic, deontic logic, and so forth. Some rules hold 'generically for all the senses of "necessary" and "possible"', while 'some features… will not be shared' for instance by logical and physical modality.

Again, I cannot but congratulate and agree. I also subdivide modal logic into various 'types' (the extensional, the natural and temporal, the logical, the ethical), each of which has to be treated as a separate field, because of their distinct properties, though eventual parallelisms do emerge. Note again that I do not regard tense as such, nor knowing and believing, nor intending, willing or preferring, or the like, as 'modalities' in a strict sense, but as considerations which may underlie modal concepts (which are distinguished by quantification of certain phenomena).

One example of distinct properties of modal types is paradox. We have seen that logical conditioning need not be based on logical possibility; here, a de-dicta connection without basis, that is, based only on problemacy (a mere mental consideration of the theses) is quite thinkable; and paradox may therefore arise. In contrast, de-re conditioning must be based on the corresponding de-re possibility; a de-re connection without a corresponding basis does not exist (or more precisely, is too formal), and problemacy does not suffice; for this reason, there is no equivalent of paradox in de-re logic. You could say that we design our forms of conditioning, in such a way as to avoid such embarrassments on a de-re level, and keep them on a logical level. But in any case, the logics of de-dicta and de-re conditionings end up looking rather different.

In my view, in any case, it is misleading to call these fields 'systems', because that would suggest relativistically that there are many Truths. That each type of modality has distinct properties implies nothing of the kind, anymore than saying that 'cats and dogs behave in disparate ways' would do so. However, Snyder's statement does in fact proceed from the modern approach we have already noted, that according to the 'axioms' we more or less arbitrarily adopt, radically different complete systems of logic emerge, which may or not find practical application. For this reason, he adds that there are 'literally hundreds of distinct systems of formal logic'. With such attitudes, I cannot agree.

In my view, generic logic sets the common 'axioms'; the subsidiary 'axioms' serve for purposes of specification. There are no systems which qualify as logical, outside the general framework of laws like non-contradiction; special fields of logic merely add additional laws of their own. The perverse delight of relativism is not serious, and should be avoided; the logician is dedicated to strengthening common sense, not to try and debunk it (since that is in fact impossible, as repeatedly shown).

I also cannot accept the modern view, which is a direct consequence of such extreme axiomatism, and more deeply of conventionalist interpretations of language — that, once the axioms are declared, the theorems follow relentlessly, to quote Snyder: as 'a straight-forward mechanical procedure… that could just as well be done by a computer (and, as a matter of fact, has been done by an IBM 7074)' (1-12).

The work of logicians can never be divorced from philosophical considerations. Logic is inextricably interwoven with epistemology and ontology; the three evolve in tandem, stimulating inquiries in each other, mutually informed and informing. The directions taken by logical science result from a mass of insights into the world and our knowledge of it.

The totality of primary propositions required to develop a mechanical model of logic, is far greater than a few limited 'axioms': it is an innumerable number of experiences and intuitions related in very complex ways. The attempt to ignore that subtext is a sad falsification.

At every stage in the development of any theory of logic, one is called upon to consider countless, interrelated philosophical issues. The success of the theorist depends mainly on his ability to resolve such issues with vision, with the broadest regard for available data. There are some points in our progress, from which a series of developments follow more or less mechanically. But even then, the logician must be present, to determine what is relevant to human experience; to test, modify or reject. All a computer, which has no consciousness, only symbols of data (whether fed by humans or robotically acquired), can do, is duplicate the said mechanical segments of logic's growth.

The validation of logic is a function of a large number of insights. The reordering of the propositions formed from these insights, in accordance with the model of axioms and derived theorems, is perhaps an interesting and worthy research, but it is an auxiliary and ex-post-facto development. To suggest, as modern logicians do, that a dozen or a score of 'axioms' suffice to construct a logical system from scratch, is ridiculous.

Apart from that, we may severely doubt that the 'axioms' they propose are all indeed primary propositions. Most are certainly not conceptually primitive, but in practise and in theory the end-products of very significant preliminary, philosophical positionings, whether conscious or effective. This does not mean that logic is a derivative science, but only that its mental separation from philosophy is an artifice; the two are part of and depend on each other.

The more geometrico concept is itself an outcrop of logic and cannot be regarded as validating it; rather, logic confirms it for us, and encourages us in its use thenceforth. Logic develops from innumerable individual experiences (in the largest sense), including perceptual and conceptual insights, and intuitions of logical correlation, that we call consistency, conflict, implication or alternation. Intellectual validity is merely a subset of notional validity.

The geometrical model emerges from the theory of adduction, not from purely deductive motives. The perceptual and conceptual insights may be taken for what they are, or eventually grouped into forms; the logical intuitions may be taken ad hoc, or generalized into logical science. The 'common sense' art of logic, is the parent, not a poor cousin of the science. As the patterns of our inductions emerge, we come to see the value and importance of deductive ordering of the information into 'axioms and theorems', as a final step and a test. But the validity of the whole stems from the roots.

The relation of logical science to logical practise is not merely one of consistent one-way implication — in the manner, say, that physical theories