Chapter 5 examines central aspects of Wittgenstein’s view on mathematical proof, as integrated within his normativist eliminativism. It elucidates and defends the claim that “the proof creates a new concept.” His (‘interactionist’) conception of proof is spelled out, as a clarification of the idea that proofs “change the grammar of our language.” The main outcome is an explication as to why the puzzle generated by the seemingly absolute power of a mathematical proof – to produce an unbreakable link between the (concepts appearing in the) premises and the (concepts in the) conclusion of a mathematical argument – vanishes. The chapter appeals to examples from both advanced and ‘pure’ mathematics. An important concern will be expounding on Wittgenstein’s point that proofs ensure the objectivity of (higher) mathematics, since for him proofs justify changes in the network of mathematical norms. Advanced, new rules are introduced on the basis of proofs, making the management of the network a nonarbitrary process. The chapter also argues for Wittgenstein’s priority over Lakatos in advocating interactionism.

Before we can talk about how we learn, use, and lose language, we need to define what it is, and what it isn’t. Over the centuries, many people have attempted to describe language. In this chapter was ask: what is language? This is a simple enough question although the answer is much more complicated, and intriguing, than it seems at first. The question of “what is language?” isn’t something we can just sum up in a single pithy sentence. Language is a system of communication, but it’s so much more than that. Language is multi-faceted. It involves signs and symbols, smaller and bigger segments, dialects and accents, and also writing and signing. This chapter serves as a primer to introduce us to the basic underlying principles of language sceince, so we can now talk the talk.

A survey of phrase and sentence grammar, including discussion of the difference between grammaticality and acceptability, parts of speech-word classes, word order, case marking, verbal inflection, constituency, structural ambiguity, argument structure, anaphora, quantifiers, and negation.

Over the last fifty years, research into the development of second or other grammatical systems has been largely concerned with the stages through which language might evolve in the individual learner or in groups of learners. Explaining exactly how learner grammars undergo transition from one stage to the next has a much shorter history. However, there has in recent years been a rise in interest in the mechanisms that drive grammatical transition, particularly in online processing. Different theoretical approaches to the transition problem will be reviewed and exemplified here, including those in the few years that have elapsed since the first edition. Issues covered will also include (a) alternative definitions of grammar, (b) the acquisition of additional grammars after the second, (c) various other factors in the learner’s mind that help to shape the emergence of new grammatical structures. The chapter will close with an overview of the dominant questions still facing researchers embarking on investigations of grammatical transition.

This chapter provides you with some guidelines on how to write a paper that conveys your findings in an attractive, easily readable, and convincing way. We have divided the guidelines into four parts: Part 1 explains how to present your content in a way that is compelling and convincing. Part 2 deals with style issues. Part 3 deals with language and grammar issues. Finally, Part 4 deals with commonly misused words.

This chapter centers on theory, research and instructional practices related to input and input processing. It is divided into five main sections. The first addresses how input, or samples of a target language to which a learner is exposed during communication, can be broken down in terms of different modalities (spoken, written and others) and different levels of linguistic analysis (phonology, lexis, morphosyntax and others). The second section underscores the role of input and input processing in the development of form-meaning mapping during language acquisition. The third offers a critical analysis of several models and theories related to input processing. The fourth section then focuses on language instruction, with particular attention to how the acquisition of grammar and vocabulary may be facilitated by instructional approaches grounded in research on input processing. This section, which highlights the use of structured input and input-based incremental vocabulary instruction, also proposes a graded taxonomy for classifying different techniques of input enhancement. The fifth section concludes by delineating current limitations on research in this area and spotlighting ways in which future research can help us to advance our understanding with regard to theory and to increase our options when it comes to evidence-based language instruction.

This chapter explores the challenge of teaching Homer and Vergil in the Roman Empire, focusing on the pleasures of fiction in epic poetry. Using the Phaeacian books of the Odyssey (6–13) and Carthaginian books of the Aeneid (1–4) as case studies, it shows how educators reckoned with the poetic seduction that threatened to derail heroic virtue and integrity. Drawing on philosophical critiques of these canonical poets, the chapter traces evolving responses to the nexus of “Phaeacian pleasures” in their episodes. In the second half, it analyzes how four educators – Plutarch, the anonymous author of the Essay on Homer, Tiberius Claudius Donatus, and Augustine – developed distinctive approaches to epic pleasure. While Plutarch disciplines poetic deception into a propaedeutic for philosophy, the Essay embraces Homeric fiction as a new pedagogy of pleasure. Donatus treats the Aeneid as rhetorical panegyric, while Augustine transforms the affective power of Aeneas and Dido into a new Christian grammar. Together, these authors reveal the centrality of epic pleasure to Imperial education and the divergent strategies by which students learned to navigate literary enchantment

This chapter applies concepts defined and argued for throughout the book to the area of L2 teaching/pedagogy. It is shown that Second Language Learning/Teaching that is informed by linguistic findings would be beneficial to the students and would promote easier, faster, and less stressful learning experiences.

This chapter provides a detailed description of the socio-historical background of English in Fiji, Samoa and the Cook Is lands discussing the spread of English from the first contact with English explorers and traders to the impact of English after political (semi-)independence. It further includes the first real-time study on language variation and change from the Cook Islands. Three Cook Island women of three generations were recorded twice within 10 years. Comparing intra- and inter-speaker variation in terms of lexis, grammar and accent features, the study shows differences between the three individuals, which opens up a debate on the role of the individual in language change and the future of L2 Englishes in the South Pacific. The chapter confirms that empirical diachronic research on English in the South Pacific yields insights into variety formation.

New Zealand English (NZE) is one of the most well-researched varieties of English in the world. This is largely due to the existence of several high-quality spoken corpora, including the Origins of New Zealand English (ONZE) corpus. This corpus has provided the foundation for work on new dialect formation, sociolinguistic variation and sound change. However, published descriptions of the emergence of NZ English are based on data from only a sub-set of speakers from the ONZE corpus. More recent work has begun to utilise data from the full ONZE corpus, and other spoken and written corpora of NZ English. This chapter will first provide an overview of the existing received wisdom on the development of New Zealand English, before focussing on several recent studies showing how much further we have now come in our understanding of the history and development of this variety.

It is also possible to investigate the morphology of the Minoan language—that is, its grammar and methods of word-formation—by studying Linear A. For example, many words in the script occur multiple times, but with different endings (suffixes) or beginnings (prefixes), thus yielding hints about the nature of Minoan grammar. Chapter 4 systematically addresses the morphology of Minoan through a refinement of a statistical method of analysis devised by David W. Packard in the 1970s. His method was designed to help determine the spoken values of some signs whose values were not yet known back then, whereas my refinement is designed to determine which syllabic signs are most likely to be indicating the presence of Minoan prefixes and suffixes at the beginnings and ends of words. This process isolates six signs—two very common at the beginnings of words, and four very common at the ends of them—that are highly likely to be grammatical prefixes and suffixes, including one suffix in particular that is highly likely to denote a genitive (showing possession), an ablative (showing origin), or a toponymic adjective (such as the “-ish” in “British,” again showing origin).

In its history, the phonology of Irish English went through a number of stages in which features arose and subsequently declined. Many of the traits to be seen in the textual record for early modern Irish English were lost by the nineteenth century, with others being retained, such as the incomplete long vowel shift and dentalisation of stops before R. The early twentieth century saw a change in supraregional Irish English given the endonormative reorientation which set in after independence in 1922. Language contact between Irish and English has been a consistent theme in Ireland’s history and has led to a prolonged language shift, which culminated in the accelerated switch in the mid nineteenth century with the vast majority of the population being native speakers of English by the onset of the twentieth century. The language shift also resulted in many instances of grammatical transfer from Irish to English, a small number of which remained emblematic of Irish English and have survived to this day.

This chapter focuses on the history of English speaking in Wales and on the character of Welsh English and its varieties. After a short examination of the terms ‘English in Wales’ and ‘Welsh English’, the chapter proceeds with an outline historical account of anglicisation in Wales considered in relation to topography and geology, and charting the geographical spread of English speaking and the growth of speaker numbers. This is followed by a brief survey of academic research on dialects of Welsh English from the earliest work at the end of the seventeenth century to the present day, and from shorter works on specific localities and features to national surveys. The chapter ends with a descriptive synopsis containing overviews of the phonology, grammar and lexis of Welsh Englis with a concise discussion of a selection of features from each level. Also included are previously unpublished linguistic maps from the archives of the Survey of Anglo-Welsh Dialects, prepared by the Survey’s director, David Parry.

This chapter analyses Wittgenstein’s notion of ‘form of life’ as his final model of context. It argues that the deliberate vagueness of this concept represents Wittgenstein’s attempt to avoid the metaphysical commitments involved in previous formal models while still providing tools for understanding context. This strategic formlessness contrasts with how the concept was later interpreted in anthropology.

This chapter analyses Wittgenstein’s transitional period and his shift from logical to linguistic models of context. Centred on his work in early 1930s and on his ‘Remarks on Frazer’s Golden Bough’, it shows how Wittgenstein moved from seeing context as singular logic to viewing it as multiple ‘logical spaces’ or ‘grammars’. This shift prefigures later anthropological moves away from formal systems while retaining some commitment to structure through language as model.

This chapter surveys the scholarship on linguistics in education and analyzes the status of linguistics in language education master’s programs. Using the National Center for Education Statistics’ Classification of Instructional Programs (CIP) system to define the data field, we searched the Integrated Postsecondary Education Data System for institutions conferring degrees between 2011-2020 for CIP code 13.14, the designation for master’s programs in Teaching English as a Second or Foreign Language. From this list, we analyzed available curricula for both MA/MS TESOL degree and MA/MS in elementary, secondary, or adult education language teacher training. Considering programs across 255 institutions, we analyzed linguistics courses as required, elective, or not present. Results reveal that linguistics is integrated at variable rates. Focusing on four core subdisciplines, TESOL fares better: Introductory linguistics was required 69% of the time, sociolinguistics 41.5%, syntax/grammar 53%, and second language acquisition (SLA) 70%. A more dismal picture appears for the other language education group: Introductory linguistics was required 6% of the time, sociolinguistics 6%, syntax/grammar 4%, and SLA 12%. This study demonstrates that language teachers require more robust linguistics education to better equip them for serving language learners’ needs.