1.Seeds of Growth.

2.Categories of Modality.

3.Types of Modality.

4.Extensional Modality.

5.Temporal Modality.

6.Tense and Duration.

7.Natural Modality.

8.Other Types.

1.Seeds of Growth.

Aristotelean Logic, we have seen, deals with categorical propositions of the form ‘S is P’. The copula ‘is’ is often conceived as having an absolute or timeless quality; it is viewed as the essential relationship between things in a scientific body of knowledge. Although this knowledge may involve particular statements, their role is merely that of either stepping stones towards eventual general statements or tools for denying general statements. Science’s goal is mainly to discover universals. For this reason, time, change, and causality were not given formal attention in the traditional approach.

But if Logic as a science is to be universal in scope, it must go into deeper detail, and analyze the full range of existing phenomena reflected in language and everyday thought processes. This is painstaking, perhaps never-ending, work. Logic is not to be confused with grammar; it is not primarily concerned with the structure of sentences, which may vary from one language to another, and indeed sometimes seem illogical. But Logic can observe commonplace statements to identify possible areas of interest for treatment in its peculiar way. In any case, its ultimate goal is to say some general things about reality, and about how we may properly think about it.

In this perspective, then, classical Logic is but a beginning, a specialized investigation which needs to be pushed further gradually. In this chapter, we will indicate some of the possible areas of expansion for our discipline.

The concept of modality is extremely interesting, because its detailed development has a powerful systematizing effect on logical science. From the seeds of thought provided by a few insights, like postulates, a large chunk of knowledge can be organized into a formal whole. A relentless progression of problem solutions and predictions is put in motion, providing us with exciting tools for the growth of knowledge.

A theory is ultimately judged not only by its consistency and truth, but also by its fruitfulness. The distinctions and classifications, the understandings and guidance, which modality generates, show its importance to advanced logic, and thereby to the broader concerns of philosophy.

Modalities are certain qualifications of relations, expressing the frequency of events, within some framework. In the deepest sense, modality is concerned with the differing and varyinglevels of being; hence its central place in both ontology and epistemology. The study of modality could be called ‘Tropology’: it is a broad field.

The term modality may be used in the sense of a ‘category of modality’, or in the sense of a ‘type of modality’. Category refers to the frequency aspect, type defines the framework. When referring to the modality of a relation, we may mean either of these senses, or their intersection. For we find, within each type of modality, the same categories of modality, only with a somewhat different meaning.

The types and categories define the multiplicity of ways in which anything may said to ‘exist’.

2.Categories of Modality.

We call categories of modality the concepts of possibility or necessity, impossibility or unnecessity, contingency or incontingency, probability or improbability and their degrees — as well as presence or absence.

These terms will all be more fully defined further on. Meanwhile, let us note that they are interrelated in various ways. The following tree illustrates some aspects of their interrelationships.

Presence signifies the occurrence of an ostensible individual phenomenon, a unit clearly defined in time and place; and absence is the negation of this. Presence is a class standing under possibility and above necessity; absence, between unnecessity and impossibility. Presence or absence occur either because of incontingency, or through the realization of contingency.

Diagram 11.1Tree of Modalities.

Possibility may be viewed as a generic concept which embraces either contingency or necessity. Likewise, contingency and impossibility may be viewed as mutually exclusive species of unnecessity. Contingency signifies possibility and unnecessity taken together. Incontingency a genus for necessity or impossibility. The various degrees of probability are subcategories of possibility or unnecessity.

In practise, these concepts are expressed in sentences by words like ‘in some cases’, ‘sometimes’, ‘can’, ‘may’, ‘might’, ‘possibly’, ‘potentially’, ‘permissibly’, ‘perhaps’, and all their related terms. The differences between these modal expressions are not merely verbal.

Indeed, in normal discourse, we tend to interchange terminology indiscriminately. For instances, in some cases we say ‘always’ to mean ‘all’; in some cases, ‘can always’ means ‘all can’. This is not our concern as logicians: we identify the connotations closest to what we are trying to discuss, and henceforth adopt restrictions which serve our purposes.

3.Types of Modality.

I have identifiedfivemain types of modality, five senses in which the various categories of modality may be understood. Within each type, all the categories occur, but with other meanings than in the other types. The categories have similar interrelationships and properties within each type. These uniformities allow us to abstract them, but ultimately each type needs to be considered separately. The interactions between types must also be analyzed.

Quantity, or extensional modality, is the primary type of modality, and is the one which was thoroughly dealt with by Aristotle. Two more, temporal modality and natural modality, will presently be analyzed in detail; they interact intimately with quantity. The last two types, logical modality and ethical modality, are each sui generis, and require independent treatment.

It will be soon be evident that the temporal and natural modalities have characteristics in common with quantity. They represent different ways the subject and predicate might be related. They can be combined in certain ways with quantity, to form complex propositions. They are mutually related, in fact form a continuum, although they cannot be compounded together as they can be with quantity. They are subject to rules resembling those found for quantity, because they derive from the same geometric fundamentals.

Each type of modality has its own character. Quantity refers to the proportion of a whole class that is subject to a certain relation to a predicate. Temporal modality refers to the proportion of its whole existence in time that any individual subject happens to have a certain relation to a predicate. Natural modality expresses the degree of causal conditionality concerning such relation.

Extensional modality recognizes the variations which can be found to exist between instances of similar phenomena, be they static or dynamic. Temporal modality proceeds from the occurrence of change in individual things during their existence. Natural modality stems from the belief that ‘laws’ guide events. Our world is diverse in all these senses. There is thus an ontological basis for such distinctions.

Furthermore, Logic must investigate the differences and similarities in behavior of such phenomena, and the results of their interplay. Here, then, is a possible area of new activity for Logic, clarifying the meanings of forms involving modality, and analyzing their oppositions, the eductions possible from them, and the syllogistic arguments involving them. This topic will be dealt with in considerable detail in this treatise.

The two types of modality we are introducing here are effectively qualifications of terms similar to distribution, although strictly speaking they apply to the relationships of terms. Such propositions are complex variations of the standard forms researched by Aristotle, involving an additional factor, modality, which can be subjected to whole-and-part, inclusion-exclusion type analyses, as was done with quantity.

4.Extensional Modality.

Consider the ways in which we use expressions of possibility or necessity. As stated previously, we are in everyday discourse not consistent in our use of terms like ‘sometimes’, ‘can’, ‘may’, ‘might’, ‘must’, and so on. Ultimately these are semantic issues, not important to us, though they need pointing out. Logic simply establishes conventions for terminology, and focuses on the material issues.

Now, it happens, for instance, that when we say ‘S may be P’ we mean ‘some S are P’. This use of a modal-looking qualification to express quantity is not accidental. When considering a specimen of S, we may want to note that the fact that other S have been found P suggests that this one also could fall in that group for all we know.

But is there an ontological basis for considering quantity a type of modality, or are we just dealing with a mode of thinking, a useful artifice? Quantity is essentially a qualification of universals, which we suppose to have some kind of reality, although we cannot yet understand their nature adequately. When we say that some S are P, we are not merely intending to express a quantitative fact, but to affirm the compatibility between ‘S-ness’ and ‘P-ness’.

This is traditionally known as the distinction between viewing a concept in its extension (the units it applies to) and its intension (its meaning). A universal may be viewed as a ‘substance’ (or stuff) which is scattered in the world. When two universals, S and P, coincide in some entities, we learn more than simply the fact of contiguity; we learn that the natures of the two universals do not intrinsically prevent such occurrences, and this is for us significant information.

Similar argument is possible for the other quantities. They tell us of the compatibility or incompatibility, necessity or contingency, high or low probability of coincidence between universals. The numerical aspect of quantity is incidental, though Logic develops by concentrating on it because of its manageability.

Social statistics, for example, are mostly based on this approach. The information we obtain concerning a social group is applied to each individual in the group, with the corresponding degree of probability. The mere fact that most individuals in a sample behave in a certain way, should not imply that there is at any possibility that individuals who did not behave in that way at all could have. And yet we do feel justified in so reasoning, because we believe that reality functions through the forces inherent in universals.

As stated before, although many skeptical philosophers have denied validity to such modes of thinking, my position is pragmatic optimism. This is the position of science: that even if an appearance is not fully understood, it is received with an open mind, provided or so long as no inconsistency arises from the belief.

Humans inevitably conceive the world in terms of universals; therefore it appears that they exist. That they are difficult to fully grasp does not mean they are untrue. Only if they were logically contradictory to evidence, would doubt be reasonable. But no credible cause for doubt has arisen. Indeed, most importantly, to deny universals through some speech, is using universals to deny them: that position is the inconsistent one of the two, and therefore absolutely false.

So when we say that ‘some S are P and some are not P’, we still believe that there was a ‘possibility’ even for S which are not P, to have been P (or vice versa), although they did not happen to concretize in this way. We think this, because the universals S and P (or nonP) have displayed compatibility in some cases of their existence.

Thus, we mentally distribute not only generals, but even particulars to each of all the individuals involved, via the universals, while remaining aware that the factual concretization of the universals in that contingent way is final. In this sense, quantity can legitimately be viewed as a type of modality.

It must be stressed, however, that extensional modality differs radically from temporal and natural modalities, in that it can be combined with either of them, whereas they cannot be superimposed with each other. That is because they are really part of the same modal continuum of individual capabilities, whereas quantity remains essentially a factor concerning groups of phenomena.

Temporal and natural modality may be called ‘intrinsic’ modalities, because they concern the properties of concrete individuals; extensional modality is comparatively ‘extrinsic’, in that it focuses on abstract universals.

5.Temporal Modality.

While it is true that often the copula ‘is’ is intended in a timeless sense, we sometimes use the word with a more restrictive connotation involving temporal limits.

The temporal equivalent of what is a singular instance in extension, is a momentary occurrence; this is the unit under consideration here. When we say ‘S is P’ we may mean either that S is always P, or that S is now P, or even that S is sometimes P. This ambiguity must be taken into consideration by Logic explicitly. A possible modification of standard propositions is therefore through the factor of temporal frequency.

We can say of an individual S that it is now or not-at-this-time P, or sometimes or always, or sometimes-not or never P, or usually or rarely P. We recognize that a thing can vary in attributes during time, and often use such forms to express such experiences. Such propositions can in turn be quantified, so that complex combinations emerge.

According to the traditional approach, we are supposed to deal with these forms simply by attaching the frequency qualification to the predicate, to obtain a new predicate. This process is called permutation; we encountered it previously, in the context of changing propositions into the “is” form, and obversion is a sample of it, too. Tradition has assumed that once permuted, such propositions can be processed in the normal way, through Aristotelean syllogism.

But this first impression was wrong; the device is misleading where modality is concerned, for three reasons. Firstly, it fails to account for a large number of practical inference, whose validity can only be established through analysis of the propositions in their original forms. With such propositions in their permuted forms, syllogisms would contain a middle term which is not identical in the two premises (for example, ‘S is sometimes-M, M are always-P’), or a minor or major term not identical in premises and conclusion (for example, the conclusion ‘S is sometimes-P’ from the said premises).

Secondly, and even more importantly, permutation can result in erroneous inference. For, in fact, as analysis shows, we cannot always transmit a frequency unchanged from premises to conclusion (for example, as in ‘S is M, M are always-P, therefore S is always-P’), and sometimes not at all (for example as in ‘S is M, M are sometimes-P, therefore S is sometimes P’).

Thirdly, in some cases, we can deduce from a given frequency, not capable of being itself simply transmitted, another, lower frequency; if we merely relied on permutation, the conclusion would not be formally valid. (For example, in ‘S is M, M are always-P, therefore S is P’). So we have no choice but to demand special treatment; the issues are more complex than we are led to believe by the permutation theory.

All this will become clearer by and by. It will be seen, as the analysis of modal forms proceeds in full detail, that, although our method of analysis is similar to Aristotle’s, we cannot mechanically reduce temporal modality arguments to traditional forms. Such situations must be investigated systematically, and special principles must be formulated to guide our reasoning in relation to them. The results obtained are often unexpected and instructive, and justify our research effort.

Temporal modality is especially useful, when reporting the behavior patterns of organisms; this is especially true for animals, who have powers of volition, and even more so for humans, who we consider as having free will. For, with regard to certain actions or states of such subjects, we cannot say that they ‘must’ or ‘cannot’ do or have them, in the sense of natural determinism, but only that they always or occasionally or never do so.

Thus, for instance, we can study the psychology of people, and predict their reactions to some extent, without having to postulate a more rigid degree of necessity than mere constancy, and before being able to explain volition or free will.

6.Tense and Duration.

We have indicated that the unit considered by temporal modality is a moment of existence. But ‘now’ is not the only individual moment we can refer to. The individual moment involved may be located anywhere in time, past, present, or future; and that location may be expressed precisely, by date and time o’clock, or roughly. This issue is known to grammar as tense, and we may adopt the same name for it in logic.

Also, the individual moments we speak of vary in size. The segment of time involved may be a fleeting moment, or an extended period of time; it may be expressed vaguely, or precisely, as a year, week, hour, or microsecond. This is an issue of duration.

These different units in the continuum of time, defined by the tense and duration of existence, of the subject and predicate relation under scrutiny, are the instances of the ‘class’ under consideration in the context of temporal modality, in analogy to the cases of a universal in the context of quantity.

We can in principle thus develop an infinite list of possible tense/duration characterizations for propositions, according to where in the time continuum the event is projected, and for how long. Thus ‘things S and P’ could mean: things now S and P, or which were or had earlier been S and P, or which will be or are later going to be S and P; and the time locations and periods tacitly intended could be specified explicitly.

Here again, following the permutation idea, we would suppose it possible to merge the tense into a term so related, to form a new term capable of timeless treatment; for example, ‘S was P’ would become “S is a ‘was-P'”. This presupposes that, provided no equivocation was involved, a proposition so altered could then enter into a syllogism without causing problems.

However, in fact, this artifice does not work; it conceals the validity of certain arguments which it assumes false, and it causes us to assume certain arguments correct, which closer inspection reveals false. So a specific analysis is required. These claims will be seen evident as formal treatment proceeds.

Tense is not in itself a distinct type of modality qualification; but an integral part of the doctrine of frequency. It simply defines the possible variety of locations in time, besides the elementary ‘now’; without awareness of them, we might make logical mistakes.

However, apart from these general guidelines, the topic of tense will not be developed in full detail in this paper. It is enough for our present purposes to make the reader aware that our use of the expression ‘now’ is intended to include past and future nows, and nows of any size. So long as the now involved in any argument is one and the same, the rules we will establish for such arguments will work. The possible interactions of different nows will not be covered, however.

7.Natural Modality.

The most significant type of modality is what I call natural modality. This refers to propositions such as ‘S can be P’, ‘S cannot be P’, ‘S can not-be P’, and ‘S must be P’, with the sense of real, out-there potential or necessity. These relations were effectively recognized by Aristotle in his philosophical discussions, but were not systematically dealt with in the framework of his logic works.

Note in passing that often, when people write ‘S can not be P’, they mean ‘cannot be’ rather than ‘can not-be’; in the former case, the ‘not’ negates ‘can be’ (it means ‘not-can be’ in spite of its position in the phrase), whereas in the latter, the ‘not’ only negates the ‘be’.

Such modality differs radically from temporal modality. We do not here merely recognize that something may be sometimes one thing and sometimes another, or always or never so and so. We tend to go a step further, and regard that there is a character intrinsic to the object which makes it able to behave in this way or that, or incapable of doing so or forced to do so. Thus, temporal and natural modalities represent distinct outlooks, which cannot be freely interchanged.

We can infer from S being sometimes P, the implication that it can be P, arguing that otherwise it would never be P; likewise that S is sometimes not P, implies that it can not-be P, or else it would always be P. But when we say that S can be P or nonP, we mean something deeper than merely an observed conjunction. We often claim, through indirect discovery, to know that S can be P (or nonP), even though this potentiality is never actualized. Whereas, with S is sometimes P (or nonP), we are making a statement that requires the relation of S and P to be actualized at least once.

Similarly, we may induce, in the way of a generalization from experience, from S always being P (or never being P), that it must (or cannot) be P. But when we say that S must (or cannot) be P, we intend a more profound relationship than mere constant recurrence (or nonoccurrence, as the case may be). We claim knowledge of the inner nature of the object (whence my choice of the term ‘natural modality’, by the way); we claim to be explaining why the observed constancy took place. We may thereafter discover indirectly that S can be and can not-be P; we would then conclude that, although S is always (or never) P, this is not a case constancy due to necessity, but just the way a contingency was actualized.

The indications here given should be enough to clarify ostensibly what phenomenon we are trying to refer to. Before discussing the concept of natural modality further, on a more philosophical plane, a pragmatic definition, sufficient for the needs of logical science, will be proposed.

An event is said to be potential if it occursin some circumstances; it is said to be naturally necessary if it occursin all circumstances. Unnecessity is, then, nonoccurrence under some circumstances, and impossibility occurrence under no circumstances.

This concept of circumstance refers us, then, not to time as did temporal modality, but to the assumption that, scattered in the environment of an event, are certain causative factors, be they known or unknown, specified or unspecified.

‘S can be P’ thus means ‘When certain causes occur, S is P’, ‘S can not-be P’ means ‘Under certain conditions, S is nonP’, ‘S must be P’ means ‘In all situations, S remains P’, ‘S cannot be P’ means ‘Whatever the surrounding circumstances, S remains nonP’.

That definition justifies our calling this phenomenon a type of modality, because, like the previous types of modality (temporal and extensional), it is reducible to an issue of enumeration: we use the same ideas of whole and part, inclusion and exclusion, all/this/some, frequency.

In the case of extensional modality, we are dealing with instances of a universal; in that of temporal modality, with moments of an existence; in natural modality, with causal conditions. All these implicit concepts are admittedly inscrutable in their essences, but their applications are numerical and so capable of systematic treatment by logical science.

We can argue, as we did for temporal modality, that natural modality is not permutable. I will not repeat the arguments here, especially since this truth becomes so obvious once we start dealing with formal issues.

8.Other Types.

Two other main types of modality, the logical and the ethical, need to be indicated to complete our introductory synopsis of the topic. As previously stated, these types are each sui generis, and worthy of thorough treatment on their own. Logical modality will be dealt with later in this work, but ethical modality is left to some future volume.

What distinguishes these types from those previously considered, is their object of attention. Extensional, temporal and natural modalities tell us something concerning the subject and predicate related themselves. Logical and ethical modalities, in contrast, either report about the state of our knowledge, or make recommendations for action, in connection to those objects.

a.Logical Modality.This expresses the compatibility or otherwise of a proposed assumption with the general framework of our knowledge to date. Logical modality makes use of terms such as ‘might’ (or perhaps) and ‘surely'(or certainly), for possibility and necessity. Remember that we defined truth and falsehood as contextual, so this definition fits in consistently.

To the extent that such an evaluation is scientific, based on rigorous process, thorough, common public knowledge, and so on, it is objective information. To the extent that thought is deficient in its methodology, such modality is subjective.

Whereas the extensional, temporal and natural types of modality may be called ‘materialistic’, in that they refer directly to the world out there, which is mainly material or in any case substantial, logical modality may be called ‘formalistic’, because it operates on a more abstract plane.

b.Ethical Modality.Ethical statements tacitly refer to some value to be safeguarded or pursued, and consider the compatibility or otherwise of some proposed event with that given standard. We use terms such ‘may’ (for permissibles) and ‘should’ (for imperatives), to indicate ethical possibility or necessity.

Ethical modality is of course relative to standards of value. The complex issue of how to establish absolute standards, or whether we are able to, will not be discussed here. Suffices to say that, within a given framework, an ethical statement can in principle be judged true or false like any other.

Subjectivity comes into play here, not only in the matter of selecting basic values, but also to the extent that, in this field more than any other, factual knowledge is often very private.

Logic must, of course, eventually analyze such modality types in detail. But for our present purposes, let us note only that, in either case, the resemblance to the other types of modality is the aspect of conditionality. They are defined through the conditions for their realization.

Their distinction is that they do not concern the object in itself (i.e. the S-P relationship as such) like the others, but involve an additional relation to man the knower of that object, or man the eventual agent of such object. The latter relation is thus a new object, which includes the former, but is not identical with it. Such modalities, then, are not essentially subjective, though they can degenerate into subjectivity, but rather concern another object.

The reader should beware of the various ways the words ‘modality’ or ‘modal’ will be used in this volume. In its broadest sense, ‘modality’ applies to any type and category of modality, which details should be specified, and every proposition is ‘modal’.

In practise, we sometimes use the word ‘modality’ to refer specifically to the natural, temporal or extensional types of modality, to the exclusion of the logical. Sometimes, the sense is restricted to only natural and temporal modality, as distinct from quantity. Likewise, we may in some cases call a proposition ‘modal’, to signify that it is other than actual or singular or factual.

The context should always make the intent clear.