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Updates and Boolean Universals Updates and Boolean Universals Fausto Carcassi1,* and Giorgio Sbardolini1,* 1Institute for Logic, Language and Computation; Universiteit van Amsterdam *Co-first authors. Abstract Languages across the world have simple lexical terms that convey Boolean opera- tors, such as conjunction and disjunction. Of the many logically possible operators of this kind, only three are in fact attested: and, or, nor. Moreover, there are patterns in the way they co-occur in lexical inventories, e.g. nor is not observed by itself. In this paper, we develop a proposal to explain the occurrence of lexical inventories in natural language based on previous work in update semantics. We present a bilat- eral extension of standard update semantics, which allows to calculate how linguistic information can be accepted in a context but also rejected. Boolean operators are en- coded as particular forms of update, on the basis of which we define a cognitively plausible measure of conceptual simplicity. We show that the attested operators are conceptually simpler than the remaining Booleans. Moreover, we show that the pat- terns in co-occurrence of Boolean operators can be explained by balancing conceptual simplicity and simplicity of using the language. Our perspective reveals a way that formal and cognitive constraints on dynamic aspects of the use of sentences may have contributed to shaping language evolution. 1 Introduction English and expresses Boolean conjunction: the compound ‘p and q’ is true just in case p and q are both true. Assuming two truth values, True and False, there are 16 Boolean truth functions of two arguments, like conjunction. However, only few of them are expressed in natural language by lexical primitives. For instance, English has morphologically simple lexical entries and, or, and nor, expressing ^, _, and nor, respectively. Other Booleans can only be expressed compositionally. In English one can only express negated conjunction p nand q by the compound ‘not (both) p and q’. The lexicon in other languages patterns somewhat differently: in Iraqw there is no nor (Mous, 2004), there is no _ in Wari’ (Mauri, 2008; Everett and Kern, 1997), and no ^ in Maricopa and Warlpiri (Gil, 1991; Bowler, 2015). On the other hand, no language has been observed to have a lexical primitive that expresses nand (Horn, 1972). 1 We present and discuss a novel explanation of these observations. We start from a version of update semantics in which the Boolean connectives can be encoded. Update semantics is a well-established approach to the study of semantic interpretation, with applications to phenomena such as anaphora resultion and default reasoning (Veltman, 1996; Heim, 1983). We now put it to new use in the study of language evolution, introduc- ing what can be in effect regarded as a Dynamic Language of Thought. Specifically, we aim at accounting for the attested lexical inventories of Boolean operators. For example, in languages in which nor occurs such as English and Italian, it never does so without conjunction and disjunction also being present. In line with previous work, on our account language finds the best compromise be- tween opposing pressures. The first pressure tends to minimize conceptual complexity. Boolean operators may be defined in the update system we introduce, according to the structure of complex updates. The conceptual complexity of a Boolean operator is as- signed on the basis of the complexity of the update procedure by which the operator is encoded in the update system. Our first conclusion is that, with respect to the logic of updates, the naturally occurring Boolean functions tend to minimize the conceptual com- plexity of information processing. The second pressure we discuss tends to minimize usage complexity, which can be thought of as the effort involved on average in expressing an observation. We measure usage complexity as the average length of a sentence expressing a combination of truth values in a given lexicon. We show that the naturally attested lexical inventories (English, Warlpiri, and so on) are different ways of optimizing the trade-off between conceptual and usage complexity. The paper is structured as follows. We start by discussing the typological data in Sec- tion 1. Then, we introduce our Dynamic Language of Thought in Section 2. In Section 3, we use our Dynamic LOT to calculate a measure of conceptual complexity for the indi- vidual Boolean concepts. In Section 4, we apply this measure, together with a measure of usage complexity, to inventories of Boolean concepts, and show how they explain the lexicalization patterns. In Section 5, we review previous accounts of this phenomenon and compare it with ours. 1.1 Overview of the Evidence Cross-linguistic evidence points to a few universal generalizations regarding the presence of lexically simple expressions that convey Boolean connectives. We assume a common understanding of the notion of ‘lexical simplicity’ as monomorphemicity, following Horn (1972, 1989) and Keenan and Stavi (1986). In this sense, English nor is lexically simple.1 In 1There is, however, a debate on whether nor and similar operators in English and other Germanic lan- guages hide complex syntactic structure. See Sauerland (2000) and Zeijlstra (2011) for arguments, and Geurts (1996) and De Swart (2000) for counterarguments. 2 some languages, such as Iraqw (Cushitic), nor is absent (Mous, 2004). In others, includ- ing several European languages, there are specialized lexical items for nor, along with conjunction and disjunction. Many languages lexicalize conjunction and disjunction differently (Payne, 1985). In some languages, however, a single coordinating expression can take conjunctive or dis- junctive meanings, depending on context. Examples are ASL (Davidson, 2013) and Japanese (Ohori, 2004; Sauerland et al., 2015). Bowler (2015) argues that the coordinator manu of Warlpiri (Pama-Nyungan) is a disjunction that can be pragmatically strengthened to a conjunction. Otherwise, there is no lexical conjunction in Warlpiri. A relatively common strategy to express coordination is simple juxtaposition, as in ‘Would you like coffee, tea?’. In Maricopa (Yuman; Gil, 1991), juxtaposition expresses conjunction, and an optional evidential element leads to disjunctive interpretations. In Maricopa there is a lexical disjunction, and no conjunction. The strategy of indicating uncertainty among alternatives to express disjunction (ex- pressing ‘p or q’ by something like ‘Perhaps p, perhaps q’) is observed quite frequently. Ex- amples include Kuskokwin (Athabaskan; Kibrik, 2004), Wari’ (Chapacuran; Mauri, 2008; Everett and Kern, 1997), Aranda (Pama-Nyungan; Wilkins, 1989). These languages lack a lexical disjunction. Several generalizations are apparent. For instance, nor is not attested unless _ and ^ are, but both ^ and _ can be present without the other. Moreover, there is no language that includes nand, confirming Horn (1972). Indeed, all remaining logically possible Boolean operators are missing. 1.2 The Language, Informally We present now an informal summary of the Dynamic Language of Thought discussed below. The aim is to define a language that can encode a specific class of concepts, namely all the possible Boolean operators of two arguments or, in other words, all possible func- tions from two truth values to one truth value. We assume three conceptual primitives for the Dynamic LOT. The first conceptual primitive is assertion. When a proposition p is asserted, the agent gives high plausibility to p being true. The second conceptual primitive is rejection. When a proposition p is rejected, the agent gives low plausibility to p being true. The third and last conceptual primitive is restriction, where the agent updates their model of the world so that only the propositions with high plausibility are true in the model. All Boolean concepts can be defined using these three simple conceptual primitives. For instance, conjunction p ^ q can be defined as the following process: (1) accept p, (2) restrict the world model, (3) accept q, (4) restrict the world model again. The end result of this process is a model of the world where both p and q are true. In order to define dis- junction, high plausibility is attributed to p and to q before the world model is restricted. 3 Other operators may require a more complex procedure. For instance, in the proce- dure for p nand q, the agent has to keep in mind the original world model through vari- ous restrictions to use it later in the process: (1) start with world model c, (2) accept p, (3) restrict the world model, (4) accept q, (5) restrict the world model again, (6) create a world model that is like the starting model c, but which excludes a world like the one defined in step (5). In the world model resulting from this procedure, p and q are not both true, which is the meaning of p nand q. In Section 2, we formalize this intuitive explanation as a bilateral update system. 2 A Language of Thought for Boolean Connectives According to a popular account, the information accepted by all participants in a conver- sation is stored in a context (Stalnaker, 1978). A declarative sentence carries information that modifies the context in which the sentence is uttered. This dynamic perspective on semantics underlies a rich paradigm of linguistic analysis that has been employed for the study of a variety of phenomena, including presupposition projection, modal subordi- nation, default reasoning, conditional and hypothetical reasoning (Veltman, 1996; Heim, 1983; Gillies, 2004). We expand on standard update semantics by making a basic conceptual distinction. The informational content of a declarative sentence p can be “added” to the context c if p is asserted, or “removed” from it, if p is denied (Incurvati and Sbardolini, xb). Thus we distinguish the positive c[+ p] and the negative update c[− p] generated by p, using + and − as force indicators for assertion and denial. The resulting update system is called bilateral. In this section we explain the logic of a bilateral update system, and how it can be used to encode the basic truth functions.
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