аще бы кто могУщий объяснил мне на пальцах, в чем логический (а не компутерно-сайенсный) смысл статьи про рекурсию как модальность, я был бы весьма благодарен и отметил бы в тексте (который когда-нибудь нескоро превратится в публикацию, а до тех пор будет прочитан разными достойными людьми)...
знак "меньше или равно" превратился в фунт стерлингов. тоже символический, к сожалению.
1 Quantifier more than
The quantifier more than belongs to the ‹1, 1, 1› type because it presupposes a relation (of comparison) between two sets A and C of the same universe M within a given modal state, that is, in respect to the set of the modal states B. In other words, we have to add to the already familiar to us ‹1, 1› type quantifier Q(A, B) the third component C, which results in quantifier Q(A, B, C).
The signs normally used for designation of more than are > (strong preference) and ³ (weak preference = conjunction of strong preference and equality in value). Q(A, B, C) means that A > C (A is strongly preferred to C) in the modal state B. Below I omit “in the modal state B” as meant by default.
A and C are considered as incompatible preferences. The comparison of preferences is considered as a binary relation, that is, there are only two possibilities, either A or C. Thus, the sets A and C are complement sets to each other, non-A = C and vice versa.
Let us consider the negative operations on this quantifier.
External negation: ¬Q = not more than, A £ C (C is weakly preferred to A).
Internal negation: Q¬ = more than non-C = more than A, that is, reversion of “A is strongly preferred to C” to C > A (“C is strongly preferred to A”).
Dual: Qd = ¬Q¬ = no more than A, that is, C £ A (A is weakly preferred to C).
Strong and weak preferences are duals of each others. Reversion of preference is the operation of inner negation.
The quantifier more than, in its four avatars (that is, the whole square of this quantifier) should be added to the table of Doležel. It is not place here to go into the purely narratological aspects of this addition to the set of quantifiers (cf. my “Theory of Narrative,” forthcoming). It is important here only to mention that the difference between all and not all could be easily comprised as the difference between more and less. Thus, “not completely necessary,” “not completely obligatory,” “not completely indifferent,” and “not completely/exclusively believed” are interpreted as “necessary less than absolutely,” “obligatory less than absolutely,” “indifferent less than completely,” and “less than completely/exclusively believed.” These phrases are sharply distinct in their inner logical structure (by using the quantifiers of different types). Nevertheless, to the human perception, this logical difference is not “phonological.”
Our “phonological” quantifiers are the triplets of the quantifiers of ‹1, 1› type (square of ‹1, 1› minus internal negation) and the whole square of those of ‹1, 1, 1› type.
2 Quantifying in narrative modalities
The list of the narrative modalities is still open, as it was in Doležel’s studies. However, it is not too short even now when it contains four items. Some generalizations would be at place.
The events in the narrative are either changes of modal states or changes of preferences within the same modal state. In the first case, these changes are governed by the quantifiers of ‹1, 1› type (forming a triplet structure = square minus internal negation), in the second case by the quantifiers of ‹1, 1, 1› type (forming the full square of quantifier).
Thus, the formula of any motive (an elementary part of the plot) looks as
where Q is either triplet of ‹1, 1› type or square of ‹1, 1, 1› type.
Probably, some quantifiers of other types play some part in the game, too. However, in any case, our system Q is the basic one.