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by Jos Kunst (1978)
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Modal logic [1] is most commonly interpreted as explicating the (socalled alethic) notions of the necessary and the possible. These concepts are easily seen to be interdefinable: "it is necessary that A" can be defined as "it is not possible that notA"; or, in symbols, using "¬" for "not", "□" for "it is necessary that", and "◊" for "it is possible that":
and, conversely, "it is possible that A" can be defined as "it is not necessary that notA":
In the model theory currently associated with them, these operators are read, respectively,
and
From this sort of talk stems the expression "possible worlds semantics"; however, the idea underlying it can be used to explicate a great variety of notions, which give then rise to calculi often called "intensional logics" . To quote a few examples, the notions "obligatory" and "permitted" of deontic logic, "known" and "credible" of epistemic logic, and "believed" and "plausible" of doxastic logic, if written, respectively, as "□" and "◊", exhibit precisely the interdefinability mentioned in (1) and (2); accordingly, to interpret, e.g., the deontic operators, we would have to read "deontically perfect worlds" instead of "possible worlds", etc. (cf in particular Hintikka 1969 and Snyder 1971).
To go back to the notions of necessity and possibility, it is clear that they exist in different variants. In the foregoing chapter we have been talking about representing, among other things, perception through suitably qualified modality statements. Now one important way of making precise such "qualifications" is through relations between possible worlds, of which the best studied is the socalled relation R or "accessibility" ("alternativity") relation: world w_{1} has access to world w_{2} (w_{2} is accessible to (an alternative for) w_{1}) if, roughly, w_{2} is "possible with respect to" w_{1}. By way of a simple example: if one wants to represent possibility in the sense of, e.g., "thinkability", one might argue that, on the one hand, an eighteenthcentury world without radio or television is quite "thinkable" for the inhabitants of the XXth century world, but that, on the other hand, our world might very well not be "thinkable" to any inhabitant of theirs. In that case the accessibility relation would not be symmetrical. (The interested reader is referred to the "parlour games" described in Hughes and Cresswell 1968 Ch 4.)
Thus, where specific interpretations are intended, one can, by imposing suitable conditions on the relation R, capture important intuitions one has about the field of one's inquiry, incorporate them in the calculus, and test them out that way.
A close structural similarity with modal logics can be found in the logic of tenses as developed since the late fifties and during the sixties by its pioneer, Arthur Prior. In tense logics, the "possible worlds" mentioned above are generally interpreted as representing moments arranged linearly on the timeline(s). Consequently, the Rrelation, here interpreted as the "after" ("before")ness relation, will often be thought of as transitive (if a moment w_{2} is situated after (before) moment w_{1}, and moment w_{3} after (before) w_{2}, then w_{3} is situated after (before) w_{1}). Different ideas concerning time (branching time, "relativistic" time, (non)ending/(non)beginning time, converging time, circular time, dense/discrete time, etc. etc.) may be captured by suitable conditions on the Rrelation (cf Prior 1967, Rescher & Urquhart 1971).
A rather much debated conceptual point of contact between tense logic and alethic modal logic is to be found in the socalled 'Diodorean' modalities. They have been formulated by the Stoic logician Diodorus Cronus, and are defined as follows:
At first reading, it is probably (6) that will appear most implausible to some readers (obviously, given the interdefinability, the same implausibility affects the righttoleft part of (5)) but if, on the other hand, we understand time as branching towards the future, as in Fig 2, where timelines represent possible courses of events,
Fig. 2
most of the implausibility dissolves [2]. In one of the eight possible worldlines now open to us, A.
We have brought up the subject of Diodorean modalities because an extremely simple transformation of them gives us modalities at least very akin to our concept of a "musical law" as defined in section 0.2.4 above. The transformation meant is purely interpretational, and consists of the turning through 180° of the graphic model given in Fig 2, so that past and future exchange places. We will call them, in that interpretation, the Diodorean* modalities (with asterisk). Accordingly, as we could now propose to use them, applied to music,
Again, as with (6), the admission of more than one possible past in the role of the given music's past may do much to dissolve any awkwardness about (8). Thus, in the following case,
Fig. 3
where on all points in the past of w1 we specify that ¬A, at w_{1} we have □¬A (¬◊A). Now if I come to think of any plausible possibility of A, then, on the proposed interpretation, all I have done just amounts to discovering a possible perspective in which to "place" the music, something like a "possible past", in which, somewhere, A. Consequently, in w_{2}, ◊A. (Cf 1.2.3 for a systematic weakening of all musical laws and interdictions.)
We are now describing our logic chiefly in the modeltheoretic way, and this has not been how, historically speaking, it has come into being. The axiomatic approach antedates in some cases even by half a century the, fairly recent, modeltheoretic one – cf 0.2.3. But most of the axioms manipulated by mathematicians like C.I. Lewis have been proven to be, in a very simple and elegant way, mathematically related to specific modeltheoretic conditions, most of them involving the relation R, the "accessibility" relation (1.0.2; cf notably Fitch 1973). We will now take a closer look at some plausible conditions we might want to impose on our models for music listening, viz.
and
in that order.
To impose the condition of reflexivity on the relation R amounts to stipulating that each world bear the relation R to itself, be accessible to itself. On our intended interpretation, where the' worlds' in question represent suitable 'moments' arranged along timelines which represent possible pasts this will mean that whenever at some such moment or 'point in musical spacetime' a musical behaviour A arises, it will interfere, at that moment or point of musical spacetime, with the formation and/or disruption of 'musical laws'. Alternatively, whenever at some point in musical spacetime a law is asserted, there cannot be, at that point no more than at any other point accessible from it, any behaviour that goes against the law asserted.
Because of the fact that, as argued at some length in chapter 0, we take the music to be the listener's activity, what we loosely call "musical behaviour A" is really the listener's hearing, conceptualizing it, and "musical spacetime" is constructed by the listener. Therefore we feel justified in accepting reflexivity. The things the listener hears and conceptualizes will immediately interfere with any "musical laws" he uses for understanding them.
We will also assume transitivity. We mentioned its tenselogical interpretation in 1.1.0; narrowed down to our Diodorean* interpretation of musical laws, it will mean that if at any given point w_{i} of musical spacetime I have access to some points that constitute its past, then, at all points in w_{i}'s future, not only w_{i}, but also its past remains accessible to me. Consequently, if at any given moment we have p, then at all moments in the future of that moment we will have (at least) ◊p.
This might be taken to imply nothing more and nothing less than that memory works. But it is perhaps a little more specific than that. One could read it as claiming that if our models actually represent essential aspects of the way we orient ourselves in music, what is stored in memory is not just a set of formulas ("truths", in pedestrian parlance), but also the way these formulas ("truths") have been arrived at: something like their "genetic history" . To us, and until further notice, this seems entirely plausible.
Nevertheless, some readers will object that memory does not always work, and, less trivially, that the workings of memory are (co)determined by the musical context at hand. We acknowledge that fact; for the moment, we will be using as musical material only fragments, and one "onephase" little piano piece. Later, when tackling longer and more complicated musics, we will not give up transitivity, but use operators of different strengths □_{m}, □ _{n}, (◊_{m}, ◊_{n}), with natural numbers as subscripts, as described in Goble 1970 and Rennie 1970, which will correspond to the different memoryspans a listener may need to activate.
The last condition on our models, viz. the presence in them of socalled "nonnormal" worlds (henceforth n.n. worlds) will be made plausible in two (obviously related) ways, according to whether or not the n.n. worlds in question reflect behaviour of some present music.
Let us first state what exactly an n n. world is. It is a world in which all modalized statements have their truthvalues settled a priori, i.e. independently of the expressions occurring within the scope of the modal operators, in the following way. All statements prefixed by □ count as false, and, consequently, by the interdefinability of □ and ◊, all statements beginning by ◊ count as true. No world is, or, in view of the foregoing, need be accessible to it – except, perhaps, the impossible (Tertullian's) world (cf Woods 1974, p. 97). But it may itself be accessible to "normal" worlds. (In that case, the normal worlds in question will no longer contain true statements prefixed by □ □: and, on the other hand, all statements prefixed by ◊◊ will automatically (i.e. axiomatically) be true in them.)
Now what sense or purpose can n.n. worlds possibly have within the context of our "musical spacetime" models? As promised, two lines of attack; here goes the first.
It is clear that among the numerous 'laws' that one may construct in understanding the behaviour of any given piece of music, there will be a certain number that have a purely ad hoc character, viz. the laws that govern those aspects of the piece that guanrantee its recognizability, its uniqueness. In other words, some aspects of any music's behaviour assure its inner coherence, and not its coherence with the cultural background emhodied by previously heard musics. If this were not so, the piece would have no uniqueness at all. In case it just provides a unique blend of borrowed elements, it is the blend we are talking about. (We do not really mean to assume anything like ontological differences between "elements" and their "blending". But let us not quibble.)
Consequently, it will always be the case that a piece exhibits some lawlike behaviour, in our Diodorean* sense, by reference not to any previous music, but to its own early stages. This can be pictured in a diagram; but first we lay down some rules for designing and reading diagrams. These rules will hold for all diagrams representing models for music listening in the remainder of this book.
Now for our diagram.
Fig. 4
The letter p is supposed to represent, for the moment, a uniquenessguaranteeing musical behaviour, as meant above; the music (as always, a listener's music!), call it M, is supposed to have started just before w_{0}; w_{0}, therefore, represents M's "early stages", as meant above ("early" may be defined with respect to p). Now if at w_{1} we have the law □p, this will have to be, according to rules D and E, by reference to w_{0} and w_{1}; w_{0} (cf rule F) is a n.n. world, and, consequently, we can have there nothing modally stronger than p, specifically no law □p. It is best to think of M as a music heard for the first time [3]. Then w_{0} may be thought of as reflecting, through its being n.n., the arbitrariness of a unique behaviour's first occurrence (cf the "anything can happen"feeling bound up with our first perceiving it); or, alternatively, the arbitrariness of our first conceptualizing of new musical behaviour (cf the feeling of "I don't know if I have this right; in a few moments perhaps I'll know"). In fact, the absolute arbitrariness of all things true in n.n. worlds is taken by us, in our interpretation of them, to be one of the main features determining their usefulness in our models. They reflect the newness of new musical behaviour, by showing on the one hand its initial lawlessness with respect to musical laws, and by permitting on the other hand suitable new musical laws to be worked out later, and in retrospect.
Now for our second line of attack.
No music is all new. Therefore we will now interpret Fig 4 as reflecting a musical behaviour, call it M', which is supposed to have started at a point in time situated between w_{1} and w_{2}. (Obviously, M and M' could be different aspects of one and the same music.) In this case the letter p will reflect a behaviour known to the listener from some foregoing music(s), represented by the couple w_{1}w_{0}. This case will perhaps be felt by many to be a much more common one, because it is, in practical situations, so much easier to tackle verbally: names and titles of composers and pieces we have in stock, and therefore our verbal machinery picks out M' aspects, giving them considerable extra weight in our consciousness.
Now our contention is that also in the M' case n.n. worlds can be given a very plausible, and indeed needed, interpretation, viz., as reflecting the conventional character inherent in the framework of musical thinking. Indeed, we may, in the Diodorean* perspective, find some music's behaviour lawlike by reference to some earlier music; this earlier music may again be lawlike in the same way, and so on; but this process cannot go on indefinitely (cf 0.6.5). In fact we have the intuition that the behaviour p when it occured for the first time may well have been understandable, lawlike, with respect to natural law, some psychological or sociosemiotic theory, or whatnot, but that those understandings, even in case they are not lost by now, simply do not function in a conventiongoverned perspective. Convention is a phenomenon that may be explained by suitable theories (e.g. the theory of games, as in Lewis 1969), but within the conventiongoverned semiotic systems conventional laws obtain just because once and for all, in some beginning, some decision, some agreement has been arrived at which, within the system, can only count as arbitrary, but which on the other hand has become the basis of conventional laws.
One possible objection must be taken care of immediately. In the very first sentence of 1.2.3 I have claimed that my two lines of attack were going to be "obviously related", and so they are, through our idea of 'initial arbitrariness', but we also see that the first centers upon some individual listener, and the second upon something like the whole historical listening community. The reader may require of me to make sense of the M' case also within the perspective of some individual listener. Our answer, for the moment, is a short one.
I simply cannot define any clear boundary between musical learningprocesses that belong to my personal past and the continuous genesis of the conventional system as a whole, for the "community of listener ", as I can, and do, reconstruct it in my mind. Therefore, and until further notice, let us assume a continuum, and refrain from separating our general idea of "some individual listener" from our concept of "the (historical) community of listeners". This will leave us also with a whole gamut of "conventions", from old ones to recent ones, from widescoped ones to purely adhoc ones. I think this could prove a particularly fruitful idea, because our theory of music listening would, if we accepted it, be also a specialized microtheory of how subjects handle conventions. It is interesting to note, in this context, that both w_{1} and w_{2} (w_{2} has, in virtue of the transitivity mentioned in rule E, direct access to w_{0}) have ¬□□p as a logical law, i.e. whatever truthvalues p may take in those or any other worlds, ¬□□p is always true. Given the Diodorean* interpretation, if p is any musical fact, □p will be any musical law, and ¬□□p will say: "of any musical law it is true that it did not always obtain". This squares beautifully with our intuitions about conventions, for it says that all lawlike situations are preceded by some situation(s) characterized by arbitrary decisions: p was true in them, but not because of any law □p, which did not obtain.
Fig 4, as it stands, is somewhat redundant: w_{1} and w_{2} have exactly the same formulas. Therefore we introduce the following additional rule G:
In this way, in Fig 4, in the M case we would delete w_{2}, in the M' case w_{1}.
We have now three important conditions on our models, which, by the correspondence mentioned in 1.2.0, add up to the axiomatics of the two systems known in the literature as S7 and S8. These systems have been considered up till now, along with related ones (S6, S9), as "philosophical oddities", with no clear practical importance. Nevertheless Kripke states that they have "an elegant model theory" (Kripke 1965 p. 206). The reader will not be surprised if we argue for bettering their status somewhat: they seem useful not only in musical contexts, but, we argue, in all contexts based on conventiondetermined lawlikeness, and hence generally in practical semiotic research (cf the widely different uses we put our Bivalence Function to, below (2.4)). For the orientation of those of our readers who are acquainted with the essentials of the Lewis systems of modal logic we give here a table summarizing their relations with some more widely known systems of the Lewis family.
assume reflexivity 
assume reflexivity transitivity 
assume reflexivity transitivity symmetry 

no decision as to n.n. worlds  S2  S3  S3.5 
n.n. worlds not admitted  System T  S4  S5 
n.n. worlds assumed  S6  S7 and S8  S9 
(Names of systems as in Hughes & Cresswell 1968)
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