22.1 Preliminaries The terms 'variation' and 'optionality' in phonology describe a situation where one phonological input has more than one output. 1 Consider the following examples from American English and South-Eastern Finnish. In both cases, variation results from the optional application of one or more phonological processes. (1) (a) wEst saId (b) lentä-i wEst saId wEs saId lénti lénsi lént léns 'west side' 'fly-PAST' In (1a), a coronal stop is variably deleted at the edge of a complex coda (t/d-deletion) (Labov 1997). In (1b), we have two variable processes working together: /t/ becomes [s] before [i] (Assibilation) and /i/ is deleted in an unstressed syllable (Apocope). This yields four logically possible outcomes, all attested (Laalo 1988). Before embarking upon the discussion, it will be useful to draw some pre-theoretical distinctions. First, variation may occur within an individual (the same individual uses different forms at different times) or across individuals (different individuals use different forms). Second, variation may be free or contextually conditioned. Contextual conditions are usually divided into internal and external factors. Internal factors may be phonological, such as stress, syllable structure, foot structure, or segment quality, or they may be morphosyntactic, such as part of speech, morpheme type, or morphosyntactic domain. External factors include age, gender, style, register, identity, ethnicity, social class, and target audience. Third, contextual conditioning may be categorical or quantitative. In the first case, the occurrence of a variant is completely predictable from the context − i.e. we have a 'rule'. In the second case, the occurrence of a variant is not completely predictable, but there is a systematic quantitative pattern − i.e. we have a 'tendency'. Variation and optionality are pervasive in the phonologies of natural languages and for this reason optional rules have always been part of the generative phonologist's descriptive toolbox (Chomsky and Halle 1968). It is much less clear how to go beyond pure optionality. One view holds that quantitative regularities have no place in the theory 1 I thank Andries Coetzee, Paul de Lacy, and Kate Ketner for helpful comments. All errors are mine. Arto Anttila 2 of linguistic competence, but belong to the theory of performance (see e.g. Newmeyer 2003) and in practice quantitative aspects of phonological variation have been studied mainly by sociolinguists and phoneticians. There are two observations that are in a deep conflict with this view. First, categorical and quantitative regularities are often conditioned by the same grammatical factors. If the phonological grammar simply delivers the phonologically possible forms, it follows that any quantitative regularities must be explained by external factors, but in reality such regularities often refer to the grammar. Second, phonological variation may involve morphological and lexical conditioning and phonologically conditioned allomorph selection. This means that variation and quantitative regularities are potentially present at every level of phonology and cannot be reduced to 'low-level' phonology or phonetics. We conclude that a satisfactory theory of phonology must be able to provide an explicit theoretical interpretation for variable and quantitative regularities and show how such regularities relate to the more familiar invariant and categorical ones. The main goal of this chapter is to give a brief overview of three current approaches to phonological variation. All these approaches assume some version of Optimality Theory (Prince and Smolensky 1993/2004, henceforth P&S 1993). 2 For other brief overviews that complement the picture given here, see especially Sankoff (1988) and Pierrehumbert (2003). The examples have been kept simple in the interest of conceptual clarity. For more detailed analyses, including several examples of quantitative modelling of phonological variation in large naturalistic corpora, the reader is referred to the work listed in the bibliography.