Multivalued Logics 387 grammatical mode, or tense and aspect. There is also Crang M (1998). Cultural geography. London: Routledge. an increased interest among cultural geographers in Dunne M D (2003). Democracy in contemporary Egyptian discourse analysis and social theory. Hence, these political discourse. Amsterdam/Philadelphia: John Benja- multimodal theorists now see political life as a com- mins Publishing Company. plex process by which meaning is produced across Kress G & van Leeuwen T (1998). Reading images: The grammar of visual design. London: Routledge. many modes; and therefore, the analysis of the lan- Kress G & van Leeuwen T (2001). Multimodality. London: guage of politics is beginning to incorporate methods, Edward Arnold. and theoretical tools from disciplines that only a few Lemke J L (2002). ‘Travels in hypermodality.’ Visual years ago were rarely referenced by linguists and Communication 1(3), 299–325. discourse analysts. McNeill D (1992). Hand and mind: What gestures reveal about thought. Chicago, IL: University of Chicago Press. Bibliography Norris S (2004). Analyzing multimodal interaction: A methodological framework. London: Routledge. Blommaert J (2005). Discourse: A critical introduction. Reisigl M & Wodak R (2001). Discourse and discrimi- New York: Cambridge University Press. nation: rhetorics of racism and antisemitism. London: Chouliaraki L (2004). ‘The moral spectator: Distant Routledge. suffering in the September 11th live footage.’ In LeVine Scollon R & Scollon S W (2003). Discourses in place: P & Scollon R (eds.) Discourse and technology: Multi- language in the material world. London: Routledge. modal discourse analysis. Georgetown University Round Wodak R (2001). ‘The discourse-historical approach.’ In Table on Languages and Linguistics Washington, DC: Wodak R & Meyer M (eds.) Methods of critical discourse Georgetown University Press. analysis. London: Sage. Crang M & Thrift N (2000). Thinking space. London: Routledge.

Multivalued Logics P A M Seuren, Max Planck Institute for ‘true’ (T) and ‘false’ (F), without any transitional Psycholinguistics, Nijmegen, The Netherlands values, and without any values outside T and F. Some ß 2006 Elsevier Ltd. All rights reserved. logicians have queried the validity of PET with regard to natural language. Strawson’s attempt at infringing PET (Strawson, Logic is the formal discipline establishing calculi of 1950, 1952) is most widely known among linguists entailments. A sentence, or set of sentences, A entails and semanticists. His ‘gapped bivalent logic’ keeps a sentence B just in case in all cases where A is true, the values T and F but introduces the possibility of a B must also be true on account of the meanings of ‘truth-value gap’ for sentences without any truth va- A and B (on analytical grounds). Since meanings are lue. Strawson thus violated subprinciple (a) of PET, in varied and manifold, it is, in general, not possible that he allowed for propositions (sentences) without a formally to compute entailments: most entailments truth value. He never provided a formal account of are recognized intuitively. Aristotle discovered, how- this logic, but one surmises that it is identical to ever, that in some cases entailments can be computed standard propositional calculus, except that a logical- with the help of specific words, the logical constants, ly complex sentence lacks a truth value when at least whose semantic properties are precise enough to allow one of the component propositions (sentences) does for formal computation. The logical constants usually so: lack of truth value is ‘infectious.’ Lack of truth recognized are all and some for quantification theory value, in Strawson’s analysis, is the fate incurred by (predicate logic), and not, and, or, and if for both pre- sentences that suffer from presupposition failure. The dicate and propositional logic. When an entailment is main empirical problem with this analysis (besides computable by virtue of one or more of the logical problems of a metalogical nature) is that such sen- constants, it is a logical entailment. tences are generally judged to be false, while their Standard modern logic is firmly based on the negations are judged to be true (see Presupposition). Aristotelian Principle of Bivalence, also known as Other logicians have developed nonbivalent logics the Principle of the Excluded Third (PET), which that infringe subprinciple (b) of PET. The linguistic says that (a) all propositions always have a truth justification for such logics is the consideration that value, and (b) there are exactly two truth values, if the predicates ‘true’ and ‘false’ in natural language 388 Multivalued Logics allow for degrees, or if there are different kinds of as ‘A _ B.’ One sees that the negation ‘’ turns T into truth and/or falsity, then it should be possible to devise F and vice versa, and leaves I unaffected; conjunction logical calculi reflecting the different degrees or kinds selects the lowest of the values of the component of truth and falsity. It has turned out that this is not sentences A and B (T > I > F); disjunction selects the only possible but also highly enlightening. highest of the values of A and B. This accounts for Apart from PPC3, to be discussed in a moment, the the fanlike arrangement of the values T, I, and F in main developments in formal multivalued logics oc- the tables for conjunction and disjunction. Kleene’s curred between the 1920s and the 1950s, as an out- calculus has no operator turning I into T. Such an flow of the formalization of logic that had just operator can be added to the system by letting ‘?A’ occurred, with C. S. Peirce as an early precursor produce T for I, and F for both T and F. The standard (Rescher, 1969: ch. 1). The multivalued logics devel- bivalent negation operator ‘:’ also finds a place in this oped in that period are all restricted to propositional system, in that it turns T into F and all other values calculus – that is, to truth tables for the logical con- into T (it simply says that A is not true). One notes that stants (the so-called connectives) not, and, or,and : is the disjunction (union) of  and ?: A _ ?A :A. if. Extensions to predicate logic, where the logic of The algorithm for generating Kleene logics with the quantifiers all and some is defined, were not an unlimited number of values is simple. For an n- attempted. Moreover, all multivalued logics (apart valued Kleene logic (n  2), let the highest value be from Strawson’s gapped bivalent logic) developed T, and the lowest F. All intermediate values Ii during the period mentioned work with a value (i  n–2) are optional and are ranked between T and ‘intermediate’ or ‘undefined.’ They were devised to F. Then  turns T into F and vice versa, and leaves provide logical accounts for future contingency all Ii unaffected; ^ selects the lowest of the values (J. Łukasiewicz), modalities (C. S. Peirce), intuitionism of A and B; _ selects the highest of the values of (A. Heyting), logical paradoxes (D. A. Bochvar), or A and B. ?i turns the intermediate value Ii into T undecidable mathematical statements (S. C. Kleene). and all other values into F. An extension of the value There is a great deal of overlap among the logical I to an infinite number of intermediate values gen- machineries proposed by these authors. erates the multivalued ‘fuzzy’ logic of Zadeh (1975). The more interesting multivalued logics are exten- Standard bivalent logic is merely the minimal variant sions of standard bivalent logic, in the sense that there of this Kleene family of logics, with n ¼ 2 and without is, for each type of extension Lm, an algorithm that any intermediate values. generates the precise form of Lm for any desired num- Although Kleene developed his logic with a mathe- ber of truth values in such a way that the theorems of matical purpose in mind (to account for undecidable standard logic, with the operators :, ^, and _, are mathematical statements), it is of specific linguistic preserved. That is, they are closed under {:, ^, _}. interest in that it appears to account for cases of One may repeat what Peirce wrote in a letter to vagueness, in the sense of neither clearly true nor William James in 1909 (Rescher, 1969: 5): ‘‘The rec- clearly false, as when one can say of a dimly lit room ognition [of multivalued logics] does not involve either that it is dark or light. any denial of existing logic, but it involves a great A different trivalent propositional calculus, like- addition to it.’’ wise extendable to any number of truth values, was The best known, and most fruitful, system is Kleene’s developed in Seuren (1985) for the specific purpose of trivalent logic (Kleene, 1938, 1952), with the values capturing the logical properties of presupposition. ‘true’ (T), ‘false’ (F), and ‘intermediate’ (I). This logic This calculus, trivalent presuppositional proposition- thus violates subprinciple (b) of PETas defined earlier. al calculus, or PPC3, distinguishes between two kinds Kleene’s truth tables are as shown in Figure 1, with ‘’ of falsity. For logically atomic sentences, radical falsi- for Kleene’s trivalent negation, ‘^’ for conjunction, ty (F2) ensues when there is presupposition failure; and ‘_’ for disjunction (the implication is definable minimal falsity (F1) ensues when all presuppositions

Figure 1 Truth tables of Kleene’s trivalent propositional calculus. Multivalued Logics 389

Figure 2 Truth tables of PPC3. are true but an update condition of the main predicate negation 1, which then equals standard :. Again, : is not satisfied (see Presupposition; Lexical Condi- is the disjunction (union) of 1 to n–1: 1A _2A _ tions). Thus, a sentence like Thepresentqueenof ..._n–1A :A. Englandisbald is valued F1, because no presupposi- The two families – the Kleene family and the PPCn tion is violated, but the person in question happens family – can be combined into one large family with not to be bald. By contrast, a sentence like Thepres- any number of intermediate values between any two entkingofFranceisbald is valued F2 because the values of the PPCn system. Which selection of this existential presupposition that there is such a person wealth of possible logical options suits the facts of is not fulfilled. natural language best is an empirical question. The The first proposal to this effect was made in theory of language benefits substantially from this Dummett (1973: 421), also on grounds of presup- widening of the range of available options. Before positional phenomena, though more from a philo- multivalent logics were discovered, there was only sophical than from an observational point of view: one option for the logic of the propositional opera- Dummett does not provide an actual trivalent logical tors, the standard bivalent system, and there was, system incorporating the two kinds of falsity. He also therefore, no empirical question because there was considered the possibility of two kinds of truth – a no choice. The extension of logical space in various suggestion that should be taken seriously but is not directions has given rise to the empirical question of elaborated here. The logical system that corresponds what the correct logic of language amounts to. to Dummett’s suggestion is PPC3. Yet it must be Research into the empirical question just formu- emphasized that PPC3 does not define presupposition lated is relatively recent. What has been found so far in natural language. Presupposition is defined by its suggests that a system like PPC3 with an infinite set purely linguistic and semantic characteristics (see Pre- of Kleenean intermediate values between T and F1 supposition). PPC3 is merely meant as a description of appears to stand a reasonable chance of providing a the epiphenomenal logical properties of presupposi- suitable frame for an adequate analysis and descrip- tion. tion of the logical properties of the propositional In PPC3 there are two negations, a minimalnega- connectives, including negation, in human language. tion‘’ turning T into F1, F1 into T, and leaving F2 unchanged, and a radicalnegation‘’’ turning both DiscourseSemantics;LexicalConditions;Pre- Tand F1 into F1 and F2 into T. The standard bivalent See also: supposition;PropositionalandPredicateLogic:Linguistic negation : can be added for good measure: it turns Aspects;Vagueness:PhilosophicalAspects. T into F1 and all other values into T. Moreover, the conjunction ^ selects the lowest (T > F1 > F2), and the disjunction _ selects the highest of the component Bibliography values. This gives the tables of Figure 2, again with the typical fanlike arrangement for conjunction and Borkowski L (1970). Jan Łukasiewicz: selected works. disjunction observed in Figure 1. Amsterdam: North-Holland. The algorithm for extension to n truth values with Dummett M (1973). Frege. Philosophy of language. London: one value T and n–1 values Fi (i n–1) is again Duckworth. Kleene S C (1938). ‘On notation for ordinal numbers.’ simple. Define a specific negation ‘i’ for each kind Journal of Symbolic Logic 3, 150–155. of falsity. i turns Fi into T, all higher values (T > F1, n–1 1 Kleene S C (1952). Introduction to metamathematics. ...> F ) into F and leaves all lower values unaf- Amsterdam: North-Holland. fected. ^ selects the lowest, and _ selects the highest Łukasiewicz J (1934) ‘Z historii logiki zda´n.’ Przegla d B of the component values. Again, standard bivalent filozoficny 37, 417–437. [German translation: ‘Zur logic is merely the minimal variant of this family of Geschichte der Aussagenlogik.’ Erkenntnis 5 (1935), logics, with the values T and F1 and the one specific 111–131.] [English translation: On the history of the 390 Multivalued Logics

logic of propositions. In McCall (1967: 66–87); reprinted Seuren P A M, Capretta V & Geuvers H (2001). ‘The logic in: Borkowski (1970: 197–217).] and mathematics of occasion sentences.’ Linguistics & McCall S (1967). Polish logic 1920–1939. Oxford: Oxford Philosophy 24(5), 531–595. University Press. Strawson P F (1950). ‘On referring.’ Mind 59, 320–344. Rescher N (1969). Many-valued logic. New York: Strawson P F (1952). Introduction to logical theory. London: McGraw-Hill. Methuen. Seuren P A M (1985). Discourse semantics. Oxford: Zadeh L A (1975). ‘Fuzzy logic and approximate Blackwell. reasoning.’ Synthese 30, 407–428.

Munda Languages G D S Anderson, University of Oregon, Eugene, OR, (Gata , Didey), Kharia, and Juang. In terms of further USA subgrouping, it is clear that Sora and Gorum form a ß 2006 Elsevier Ltd. All rights reserved. branch of their own, as do the closely related Gutob and Remo. Gta , which is properly speaking two separate languages: the poorly known Hill Geta ‘Munda’ is a group of languages belonging to the and Plains/Riverside Geta , has been traditionally Austroasiatic language family spoken in eastern linked at a slightly higher taxonomic level with and central , primarily in the states of Orissa, Gutob-Remo (so-called Gutob-Remo-Gta ), and Jharkand, , and and in adjacent Kharia and Juang have been linked together in a areas of , , and Andhra branch as well. These latter two classifications are Pradesh. Some Munda speakers are found in expatri- tenuous and remain to be adequately demonstrated ate or diaspora communities throughout India, Nepal, (Anderson, 2001). South Munda languages range in and western as well. In earlier literature, total number of speakers from 300 000 or more Sora Munda is often referred to as Kol or Kolarian. speakers to between 150 000 and 200 000 Kharia The Munda language family recognizes a major speakers to Gutob with approximately 30 000– split between a North Munda and a South Munda 50 000 total speakers and Juang with around 15 000 subgroup. Within the North Munda subgroup, there speakers. The remaining South Munda languages is a binary opposition between Korku and a large have around 2000–4000 speakers each. group of Kherwarian languages, which is perhaps Typologically speaking, all Munda languages are more properly described as a dialect continuum. moderately to extensively agglutinating, show SOV Kherwarian includes both the largest of the Munda basic clause structure, and possess preglottalized or languages, Santali, with nearly 7 million speakers, as unreleased ‘checked’ consonants. This latter feature well as some of the smallest, such as Birhor, with is quintessentially Munda, and readily distinguishes under 1000. Other noteworthy North Munda lan- Munda languages from the surrounding languages of guages include Mundari and Ho, each with approxi- the subcontinent. mately 1 million speakers, and smaller languages such The shift to SOV word order from SVO to VSO as Agariya, Asuri, Bhumij (Mundari), Karmali San- may be attributed to influence from Indo–Aryan or tali, Koraku, Korwa, Mahali, and Turi. Publications (Anderson, 2003), which varies may be found in the larger of the Kherwarian lan- from moderate to strong in individual Munda lan- guages (Mundari, Ho, Santali), including a range of guages. Generally speaking, the southernmost South Santali publications in a native orthography (the Ol’ Munda languages show the greatest degree of struc- Cemet script). Short wave radio broadcasts are also tural influence from local tribal Dravidian and lexical available in Santali. The newly founded ‘tribal’ state influence from the local tribal Indo–Aryan (e.g., of has a Munda-speaking majority and is Desia/Kotia Oriya). In addition, Kharia has been lobbying to have some form of Kherwarian declared heavily influenced by Kherwarian North Munda another state language of India. languages such as Mundari. The South Munda subgroup is older and more Kherwarian North Munda languages are character- internally diversified than North Munda. At least ized by a high degree of morphological complexity, the following languages belonging to this subgroup: standing out even within the morphologically rich lan- Sora (Savara), Juray, Gorum (Parengi, Parenga), guages of . Among the most noteworthy Gutob (Gadaba, Bodo), Remo (Bonda, Bondo), Gta characteristic features of Kherwarian verb structure is