I. M. (1998) Talking with Alex: logic and speech in parrots. In K. Brown, In particular: (i) on the diagonals, the duality relation corresponds to the Aristotelian relation of contradiction, (ii) on the vertical edges, the duality relation corresponds to the Aristotelian relation of subalternation, and (iii) on the horizontal edges, the duality relation corresponds to the Aristotelian relations of contrariety and subcontariety. Since the input and output of the functions , , and are of the same type (namely: operators ), they can be applied repeatedly. of Plurals, All, and the nonuniformity of collective predication predication. For example, since , we can write , and say that is the (unique) dual of . In general, an -ary modal operator is called a triangle (), and its dual a nabla (): The equivalences (15–16) again clearly illustrate the interaction between internal and external negation. Löbner has also used this analysis to account for asymmetries in lexicalization patterns: already and still are less marked than not yet, which in turn is less marked than no longer (also see Section 5). This was a time when several researchers were analyzing data of attempts to teach apes language (see, for example, Gardner and Gardner, 1969; Premack, 1970, 1971; Terrace, 1979). ~ 2014), internal/external negation plays a less central role. dish"), Prevarication: speakers can - intentionally - make utterances This duality could be a simplification but crucial for making meaning. Since each of these 3 negations may or may not be applied, gives rise to operators. Natural Language Semantics, 2:71–82. → observed Figure 9: (a) Aristotelian hexagon (for a modal system that is at least as strong as , (b) duality ‘hexagon’, and (c–d) its two components. which means that is the trivial Boolean algebra in which (in logical terms: is the Lindenbaum-Tarski algebra of a logical system that is inconsistent). Logica Universalis, 10:233–292. Note that if , then also , and thus collapses into a group that is isomorphic to ; see the left and middle Cayley tables below and also recall Figure 5(b). van Benthem, J. Discreetness of language is somewhat connected to duality. Factors 4. It has the highest productivity. This shows that is dual to itself, which illustrates the fact that in GQT, proper names are self-dual (Gamut 1991, p. 238)). In Zaefferer, D., editor, Semantic Universals and Universal Semantics, volume 12 of Groningen-Amsterdam Studies in Semantics, pages 17–36. In a similar vein, since for all operators it holds that , we can write . 9 units The primary difference is known as duality of patterning, or structure. This shows that in GQT, too, the dual of is . For example, Figure 6 shows a duality cube for the composed operator ; analogously, Westerståhl (2012) draws a duality cube for possessives with multiple quantifiers. Concessive relations as the dual of causal relations. The most prototypical duality in logic, namely that between the propositional connectives of conjunction and disjunction, only plays a minor role, if any, in the linguistic realm. Its Cayley table looks as follows: The fact that duality behavior can be described by means of V4 was already noted by authors such as Piaget (1949), Gottschalk (1953), Löbner (1990), van Benthem (1991) and Peters and Westerståhl (2006). Already a member? Belgium, Hans Smessaert 88 (Jan or Sept?) For example, since, it is easy to see that is true iff and that is true iff . Part I. Cambridge University Press, Cambridge. Similarly, the modal auxiliary may in the lefthand side of (25–26) interacts differently with the negative particle not depending on the type of modality involved: in its epistemic use, it gets the internal negation reading (25), whereas in its deontic use, it gets the external negation reading (26). ; Konig 1991, p. 201); also see Section 4. For example, natural language conjunction very often conveys additional aspects of causality ( and and therefore ) or sequentiality ( and and afterwards ), whereas disjunction is notoriously ambiguous between an inclusive interpretation ( or or , and perhaps both) and an exclusive interpretation ( or or , but not both). Again, they show the interaction between an internal negation (which occurs inside the scope of the quantifier) and an external negation (which occurs outside the scope of the quantifier). Lyons, J. If , then , and thus, the duality square in Figure 5(a) degenerates into the binary horizontal duality diagram in Figure 5(b).

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