More on this in lecture 4. Some vocal distinctive features used by gelada monkeys. A more linguistically motivated example, viz. Recall that in the case of single operators, duality can be seen as the combination of the external and internal negations ( ). Essai de logistique operatoire. Accordingly it can be argued that the view regarding productivity from a dichotomous perspective somehow neglecting diachronic linguistics should be abandoned. false or meaningless. The Aristotelian relations. Duality phenomena of this kind are highly important. singing in birds, b) The border wars: A neo-Gricean perspective. For any relation , one can show that. Pepperberg, 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. Techniques to Improve Productivity 9. but no evidence (yet) of structural complexity. Saying duality in European Languages. As an example, consider the binary operator of conjunction: In comparison to the ordinary duality behavior of a binary operator, we thus have rather than independent negations, and generalized Post duality behavior is governed by the group rather than (Libert 2012). Dissimilarities. The argument above is based on the fact that is ‘closed under duality’, in the sense that applying any of the -, -, – or -functions to its elements only yields operators that already belong to . Structures of oppositions for public announcement logic. Aristotle’s Categories and De Interpretatione. These linguistic phenomena are also explained by Löbner (1990, 2011), but his phase quantification account is based on the duality relations, rather than the Aristotelian relations. Already a member? But variations in songs appear not to be Note that the diagonal plane that spans the front left and back right vertical edges of this cube corresponds to the ‘ordinary’ duality square for conjunction (see Figures 2(c) and 3(a)). and thus we find that is true iff , iff . How does human language differ from animal languages? The first three squares in Figure 3 correspond to the first three examples of duality in logic that were discussed in Section 1: (a) the propositional connectives of conjunction and disjunction, (b) the universal and existential quantifiers, and (c) the modal operators of necessity and possibility. This article focuses exclusively on duality phenomena involving the interaction between an ‘external’ and an ‘internal’ negation of some kind, which arise primarily in logic and linguistics. Acta Analytica, 28:71–93. Lorenz Demey As to the technical prerequisites for this article, Sections 1 and 2 should be accessible to everyone with a basic understanding of philosophical logic. the observation that natural languages have primitive lexical items for the quantifiers all, some and none, but not for not all (the latter’s lexicalization as a single word—for example: *nall— does not occur in natural language). This shows that (32) and (33) are not equivalent, and that (32) and (34) are not contradictory either. Measurement 6. There are two types of such transitions: the truth value of can change from false into true, or alternatively, from true into false. For example, if the operator is the dual of the operator , then for all , the element can be said to be the dual of the element . Other linguistic expressions. Displacement: Allows language users to communicate about events or things that are absent from the current environment (what you did last week, your friend who is not with you, what you will do on vacation, etc. In this way, we obtain a large number of functional identities that descibe the behavior of the duality and internal/external negation functions: These identities can be summarized by stating that the functions , , and jointly form a group that is isomorphic to the Klein four group (German: Kleinsche Vierergruppe). Visually speaking, duality squares can be presented in a number of different ways, depending on which aspects the author wishes to emphasize. The main reason is the ambiguity of natural language and and or, which is often explained pragmatically in terms of conversational implicatures (Horn 2004). observed Similarly, corresponds to , which represents external negations and internal negation (keeping in mind that internal negation applies to all arguments). /u, a, i/), combine to create primary level (word) units, e.g. 2001), and thus a fortiori a Boolean algebra). IT or computer language (digital language) is also based on two modes (two digits: there is current vs. there is no current). 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. This shows that some but not all is its own internal negation. Since this holds independently of the concrete operator , we can write , which means that applying and then (to some operator) yields the same result as applying (to that same operator). What does it mean to be a professional teacher? Temporal reasoning with aspectual adverbs. Another well-known case of duality concerns the universal and existential quantifiers in classical first-order logic (). For example, if we consider , we find that. I stopped (2) smoking (1): (stop: give up). Figure 2(c) shows an alternative layout, in which the -relations occupy the diagonals, thereby graphically reflecting the fact that is the combination of (which constitutes the vertical edges) and (which constitutes the horizontal edges) (Löbner 1990, p. In this case, one can also show that , i.e. Gelada monkeys have a very rich inventory /p, b, t, d, k, g/) or vowels (e.g. Importance of Productivity 3. ch_sid = "Chitika Premium"; The duality relations are defined as follows: and are, Special cases. A well-known example from linguistics concerns the duality between the aspectual particles already and still in natural language: already outside means the same as not still inside, and hence, not already outside means the same as still inside (where inside is taken to be synonymous with not outside). America, http://dx.doi.org/10.1126/science.298.5598.1569. Beyond external and internal negation. (2012). This is probably as far as one would wish to go in defining the concept of duality. MIT Press, Cambridge, MA. Generalized quantifiers and natural language. These can naturally be partitioned into three families, based on their number of ‘basic’ operations (i.e. Routledge, London. In Beziau, J.-Y. Linguistics, 2nd edition. ch_color_border = "FFFFFF"; Duality is quite an old-fashioned concept – by which I mean that it was popularized in the late 1970s. Some scholars define it by six properties: productivity, arbitrariness, duality, discreetness, displacement, and cultural transmission. In Zaefferer, D., editor, Semantic Universals and Universal Semantics, volume 12 of Groningen-Amsterdam Studies in Semantics, pages 17–36. Furthermore, both Aristotelian and duality diagrams have been used by linguists to explain certain lexicalization patterns in natural languages. Modal operators. structure (phrases and notes). Smessaert, H. and Demey, L. (2014). Journal of Symbolic Logic, 18:193–196. larger, meaningful units; 2) arrangements of meaningful units into more This means exactly that (, ), and hence the duality relation holding between and holds independently of the specific axioms of the logical system . Barwise, J. and Cooper, R. (1981). and Jacquette, D., editors, Around and Beyond the Square of Opposition, pages 313–339. For any element , it holds that, and thus , i.e. The present or future by contrast is an open system. Springer, Basel. ‘s external negation and dual coincide with each other. (2008) The Articulate Mammal (5 rev. Linguists since Hockett in the 1960s have described this hallmark property of language as duality of patterning." (1996). Rhythm and melody in gelada vocal exchanges. Logic, Language, and Meaning. Degenerate duality patterns. Since each of these 3 negations may or may not be applied, gives rise to operators. studies; Recall the standard presentation of the duality square (with horizontal – and vertical -edges) in Figure 2(b), which is repeated here as Figure 5(a). True of, Traditional transmission: the communication system is learned. Several such lists were constructed at the time, each differing in the number of properties that the proponents claimed were minimally sufficient for answering the question, ‘What is language?’. Hypercubes of duality. Who are the experts?Our certified Educators are real professors, teachers, and scholars who use their academic expertise to tackle your toughest questions.