Counting the languages we could speak

The Principles and Parameters (P&P) approach to the study of the human language faculty, stemming from Chomsky\u2019s investigations in the 1980s, is based on the idea that syntactic knowledge is determined by the interaction of at least two sets of abstract entities: a) Principles, i.e., unvarying linguistic universals; b) Parameters, i.e., open choices between binary values innately predefined by Universal Grammar (UG), that must be closed (set) by language learners on the basis of environmental evidence. Parameters are supposed to be finite in number and to form a list of universal options that define the whole space of variation of biologically acquirable human grammars; thus, the apparently enormous amount of language variation seems reducible to the possible combinations of parameter values. This reduction represents one of the major contributions of P&P theories to the biolinguistic framework, in order to explain cultural variation and to provide a reliable model of language acquisition. Furthermore, recent developments suggest that parametric analyses have the potential to renovate the evolutionary and historical study of cognitive domains.Discovering the whole list of UG parameters is a challenging task which is far from completion. A feasible strategy to systematically analyse parameter variation is to adopt heuristic models which reduce the magnitude of grammatical diversity, though preserving its complexity. The analysis of the (ideally) whole list of parameters concerning a single, independent and internally coherent module of the grammar (the nominal domain) proposed by Longobardi and coauthors is a first contribution in this direction that hints at interesting theoretical and historical results. In particular, partial interactions between parameters and their impact on the downsizing of grammatical variation have been explored; a partial interaction occurs when the setting of a certain (set of) parameter(s) induces the setting of a specific value of another parameter. Focusing on a list of 50 parameters, Longobardi and coauthors show that the phenomenon is quite pervasive: 38 out of 50 parameters undergo the effect of at least one other parameter; 28 out of 50 parameters induce an effect on at least one other parameter. As a consequence, the total number of possible languages comes to be considerably reduced. Assessing the amount of this reduction is the goal of this work.Principles and Parameters models have been implemented since the Eighties in order to solve the contrast between the invariance of the universal language faculty and the actual diversification of languages. According to these theories, the whole space of possible grammars would be defined by means of a complete list of universal parameters and the whole cluster of possible combinations among their value: thus, assuming that parameters are binar, and that they form a universal list of indipendent n members, the whole class of possible Grammars would amount to 2 elevated n members. Yet, parameters are not independent: the values of certain parameters can be set only if other specific parameters are set on a certain specific value; this is the well-known phenomenon of implications among parameters. Working on a cluster of parameters within the nominal domain, Guardiano and Longobardi (since 2005) have shown that the phenomenon of implications is more pervasive than expected, and that implications contribute, in fact, to a sensible downsizing of the possible space of grammatical variation. In this paper we suggest one of the possible methods for calculating the average amount of such a downsizing