Meaning versus Grammar : an inquiry into the computation of meaning and the incompleteness of grammar

9 789087 282134

LUP

ME

ANIN

G VERSUS

GRAMMAR

Cremers

, Hi

jzelendoorn

& Rec

kman

Meaning versus Grammarinvestigates the complicated relationship between grammar, computation, and meaning in natural languages. It details conditions under which meaning-driven processing of natural language is feasible, discusses an operational and accessible implementation of the grammatical cycle for Dutch, and offers analyses of a number of further conjectures about constituency and entailment in natural language.

Crit Cremersis an associate professor in formal semantics and computational linguistics at Leiden University. Maarten Hijzelendoornis a senior

software engineer at the Faculty of Humanities of Leiden University. Hilke Reckmanis a senior researcher in natural language processing.

“This book is unique in my experience in giving this amount of detail to setting out the Dutch case with respect to separately a parser and generator, and the motivation for this as a property of the framework itself is wholly novel. But what makes this book such a winner is its style: I know of no other book at this level of formal explicitness that is such a joy to read.”

ruth kempson, professor of linguistics, king’s college london

www.lup.nl

l e i d e n u n i v e r s i t y p r e s s

MEANING

GRAMMAR

An Inquiry into

the Computation of Meaning

and

the Incompleteness of Grammar

Crit Cremers

Maarten Hijzelendoorn

Hilke Reckman

”‹–”‡‡”•

ƒƒ”–‡‹Œœ‡Ž‡†‘‘”

‹Ž‡‡…ƒ

An Inquiry into the Computation of Meaning

and the Incompleteness of Grammar

MEANING

Lay-out: Jurgen Leemans

Cover illustration: Matthias Stom, Samson and Delilah, 1630s, Palazzo Barberini, Galleria Nazionale d’Arte Antica, Rome, Italy

ĎĘćē ͻ͹ͺͻͲͺ͹ʹͺʹͳʹ͹ e-ĎĘćē ͻ͹ͺͻͶͲͲ͸Ͳͳͺ͵͵ȋ‡Ǧ’†ˆȌ e-ĎĘćē ͻ͹ͺͻͶͲͲ͸ͲͳͺͶͲȋ‡Ǧ’—„Ȍ ēĚė 616

̹”‹–”‡‡”•Ȁƒƒ”–‡‹Œœ‡Ž‡†‘‘”Ȁ‹Ž‡‡…ƒȀ‡‹†‡‹˜‡”•‹–›”‡••ǡʹͲͳͶ

ŽŽ”‹‰Š–•”‡•‡”˜‡†Ǥ‹–Š‘—–Ž‹‹–‹‰–Š‡”‹‰Š–•—†‡”…‘’›”‹‰Š–”‡•‡”˜‡†ƒ„‘˜‡ǡ‘’ƒ”–‘ˆ–Š‹• book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any ˆ‘”‘”„›ƒ›‡ƒ•ȋ‡Ž‡…–”‘‹…ǡ‡…Šƒ‹…ƒŽǡ’Š‘–‘…‘’›‹‰ǡ”‡…‘”†‹‰‘”‘–Š‡”™‹•‡Ȍ™‹–Š‘—– –Š‡™”‹––‡’‡”‹••‹‘‘ˆ„‘–Š–Š‡…‘’›”‹‰Š–‘™‡”ƒ†–Š‡ƒ—–Š‘”‘ˆ–Š‡„‘‘Ǥ

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

9

0. INTRODUCTION

11

0.1 A Language Machine 11

0.2 Language and computability 16

0.3 The book 21

1. SYNTAX:

the game of recursion and discontinuity

25

1.1 The need for syntax 25

1.2 Forms of Dutch 30

1.3 The task for syntax 34

1.4 The logic and the ƒŽ‰‡„”ƒ‘ˆŽ‹•–•ǡϔŽƒ‰•ǡ–›’‡•ƒ†‘†‡• ͺͷ

ͳǤͶǤͳ ƒ–‡‰‘”‹‡•ƒ†…‘’Ž‡š•›„‘Ž• Ͷͳ

ͳǤͶǤʹ ƒ•‹…• ͶͶ

ͳǤͶǤ͵ ‡”‰‡ Ͷ͹

ͳǤͶǤͶ ‘†ƒŽ‹–‹‡• Ͷͻ

ͳǤͶǤͷ ”‰—‡–Ž‹•– ͷͲ

ͳǤͶǤ͸ ‡†—…–‹‘•…Š‡‡• ͷʹ

ͳǤͶǤ͹ Š‡ƒŽ‰‡„”ƒ‘ˆ•–”‹‰• ͷ͵

ͳǤͶǤͺ ‹•Šƒ”‘‹……‘’‘•‹–‹‘ƒ†…ƒ–‡‰‘”‹ƒŽŽ‘‰‹…ǣ

‰”ƒƒ”ƒ†”‡ƒ•‘‹‰ ͷ͸

1.5 The calculi 60

ͳǤͷǤͳ ‡”‰‡ƒ†‘˜‡ ͸ʹ

ͳǤͷǤ͵ Š‡ˆ—†ƒ‡–ƒŽƒ•›‡–”›‘ˆ‡”‰‡ ͸ͺ ͳǤͷǤͶ ‘ƒ‹’—Žƒ–‹‘‘ˆ•–”—…–—”‡ ͹Ͳ

1.6 The case for Dutch 71

ͳǤ͸Ǥͳ ‡‡”ƒŽ ‘”ƒ– ͹ͳ

ͳǤ͸Ǥʹ …‘…‹•‡•›–ƒš‘ˆ—–…Š ͹Ͷ

ͳǤ͸Ǥ͵ ‡ƒŽ‹‰™‹–Šƒ†Œ—…–• ͻͲ

1.7 The grammar of discontinuity and coordination 94

ͳǤ͹Ǥͳ Š‡•‘—”…‡‘ˆ†‹•…‘–‹—‹–›ƒ†…‘‡…–‡†‡•• ͻͶ

ͳǤ͹Ǥʹ ‹•…‘–‹—‹–›‹ ͻͷ

ͳǤ͹Ǥ͵ ƒ––‡”•‘ˆ—–…Š ͳͲʹ

ͳǤ͹ǤͶ ‘‘”†‹ƒ–‹‘ƒ•†‹•…‘–‹—‹–› ͳͲ͸

1.8 Parsing the syntax 114

ͳǤͺǤͳ Š‡•›–ƒš‘ˆ ͳͳͷ

ͳǤͺǤʹ ‘’ƒ”‹‰–Š‡•›–ƒš‡•‘ˆƒ† ͳͳ͸ ͳǤͺǤ͵ ‘’ƒ”‹‰–Š‡‰‡‡”ƒ–‹˜‡…ƒ’ƒ…‹–›‘ˆƒ† ͳʹͲ ͳǤͺǤͶ ƒ†–Š‡Š‘•›Š‹‡”ƒ”…Š› ͳʹͶ

ͳǤͺǤͷ ƒ”•‹‰• ͳʹͷ

ͳǤͺǤ͸ ƒ”•‹‰ ͳʹͻ

ͳǤͺǤ͹ š–‡†‹‰ƒ†”‡•–”‹…–‹‰ƒ’ƒ”•‡”ˆ‘” ͳ͵Ͳ

ͳǤͺǤͺ ƒ”•‹‰…‘‘”†‹ƒ–‹‘ ͳ͵͸

1.9 Generating by syntax: agendas and linearization 138

ͳǤͻǤͳ ™‘Ǧ†‹”‡…–‹‘ƒŽ‰”ƒƒ” ͳ͵ͺ

ͳǤͻǤʹ ƒ–‡‰‘”‹‡•ƒ•ƒ‰‡†ƒ• ͳͶͷ

2. SEMANTICS:

the game of scope and intensionality

153

2.1 The ways of meaning 153

2.2 The forms of meaning 157

͸Ǥ͹ …‘’‡ƒ†’‡…‹ϔ‹…ƒ–‹‘ ͷͼ͸

ʹǤ͵Ǥͳ —ƒ–‹ϐ‹‡”• ͳ͸ʹ

ʹǤ͵Ǥʹ Š‡•–”—…–—”‡•‘ˆ“—ƒ–‹ϐ‹…ƒ–‹‘ ͳ͸͹ ʹǤ͵Ǥ͵ ‘’‘•‹–‹‘ƒŽ‹–›ƒ†—†‡”•’‡…‹ϐ‹…ƒ–‹‘ ͳ͹Ͳ

2.4 Intensionality and semantic dependency 175

ʹǤͶǤͳ –‡•‹‘ƒŽ‹–› ͳ͹ͷ

ʹǤͶǤʹ ‘Ž‡‹œƒ–‹‘‘ˆ†‡’‡†‡–‡˜‡–• ͳͺͲ

͸Ǥͻ ˜‡–•ƒ†•–ƒ–‡•ǣ”‡‹ϔ‹…ƒ–‹‘‘ˆ’”‡†‹…ƒ–‹‘ ͷ;͸

ʹǤͷǤͳ ‡ˆ‡”‡…‡–‘‡˜‡–• ͳͺʹ

2.6 Exploiting Logical Form for Parsing 196

ʹǤ͸Ǥͳ ‘‰‹…ƒŽ ‘”ǡ”ƒƒ”ƒ†‘’—–ƒ–‹‘ ͳͻ͸

ʹǤ͸Ǥʹ ‘‰‹…ƒŽˆ‘”‹ĊđĎđĆč ͳͻ͹

ʹǤ͸Ǥ͵ –‘”‡†‘‰‹…ƒŽ ‘” ͳͻͺ

ʹǤ͸ǤͶ ’’Ž‹‡†‘‰‹…ƒŽ ‘” ʹͲͺ

ʹǤ͸Ǥͷ Žƒ–‘‰‹…ƒŽ ‘” ʹͳͲ

2.7 Generating from Logic 217

ʹǤ͹Ǥͳ ‡‡”ƒ–‹‘ƒ•–”ƒ•Žƒ–‹‘ ʹͳ͹ ʹǤ͹Ǥʹ ”‘Ž‘‰‹…ƒŽˆ‘”–‘Ž‡š‹…ƒŽŽ‹‡—’ ʹͳͻ ʹǤ͹Ǥ͵ ”‘Ž‡š‹…ƒŽŽ‹‡Ǧ—’–‘•‡–‡…‡ǣ‹–‡”–™‹‹‰ƒ‰‡†ƒ• ʹʹ͵ ʹǤ͹ǤͶ ‡•–‹‰Ž‘‰‹…ƒŽˆ‘”„›‡–ƒ‹Ž‡– ʹʹ͸

3. LEXICON:

the language’s encyclopaedia and database

229

3.1 Storing knowledge of language 229

͵ǤͳǤͳ ‡š‹…ƒŽ‹•ǣƒ–‘•ƒ†‘Ž‡…—Ž‡• ʹʹͻ

͵ǤͳǤʹ ĊđĎđĆ胐† ʹ͵Ͷ

͵ǤͳǤ͵ ‘”†•ƒ†™‘”Ž†•ǣƒŽ‡š‹…‘‹•‘–ƒ†‹…–‹‘ƒ”› ʹ͵͹ ͵ǤͳǤͶ ‡š‹…ƒŽ…Š‡‹•–”›ǣ…‘‘‹‰–Š‡‰”ƒ’Š• ʹͶͲ

3.2 Modes of lexical knowledge 242

͵ǤʹǤͳ Š‘‘Ž‘‰‹…ƒŽˆ‘”ǣ–Š‡‘‡Ǧ†‹‡•‹‘ƒŽ‰”ƒƒ” ʹͶʹ ͵ǤʹǤʹ ‘”’Š‘Ž‘‰›ǣ–Š‡…‘„‹ƒ–‘”‹…‰—‹†‡ ʹͶͶ ͵ǤʹǤ͵ ›–ƒšǣ–Š‡—‹ϐ‹…ƒ–‹‘ƒ‰‡†ƒ ʹͶͷ ͵ǤʹǤͶ ‡ƒ–‹…•ǣ—Ž–‹ƒ–‡‘™Ž‡†‰‡‘ˆŽƒ‰—ƒ‰‡ ʹͶͻ ͵ǤʹǤͷ ˆ‘”ƒ–‹‘–”—…–—”‡ǣ–‘–Š‡Ž‹‹–•‘ˆ†‡…‹†ƒ„‹Ž‹–› ʹͷͺ

͹Ǥ͹ ‹ϔ‹…ƒ–‹‘ǣ’‘™‡”‹‰grammar conservatively 259

͵Ǥ͵Ǥͳ ”‘…‡†—”‡•ƒ†•’‡…‹ϐ‹…ƒ–‹‘• ʹͷͻ ͵Ǥ͵Ǥʹ ”‘„Ž‡•™‹–Š”‡Ǧ‡–”ƒ…‡ ʹ͸Ͷ

3.4 The making of the lexicon 265

͵ǤͶǤͳ ”‰ƒ‹œ‹‰Ž‡š‹…ƒŽ‘™Ž‡†‰‡ǣ

‘Ž‡ƒ•Ȃ‡…‘‘›˜•ϐŽ‡š‹„‹Ž‹–› ʹ͸ͷ ͵ǤͶǤʹ ‡‡”ƒŽƒ†•’‡…‹ƒŽǣ‡˜‡”›–Š‹‰ƒ•ƒ‰”ƒ’Š ʹ͹Ͳ ͵ǤͶǤ͵ Š‡”—Ž‡•–Šƒ–ƒ‡–Š‡Žƒ‰—ƒ‰‡ ʹ͹ʹ ͵ǤͶǤͶ Š‡…‘•–”—…–‹˜‡Ž‡š‹…‘ ʹ͹ͻ

͵ǤͶǤͷ Š‡Ž‡š‹…‘‹•Ž‘…ƒŽ ʹͺ͵

3.5 Disclosing the lexicon: object-orientation and speed for semantic

generation 284

͵ǤͷǤͳ Š‡‡–‡”’”‹•‡ ʹͺͶ

͵ǤͷǤ͵ ‡–Š‘†• ʹͻʹ ͵ǤͷǤͶ ”‡ƒ–‹‰ƒ†ƒ……‡••‹‰–Š‡‘„Œ‡…–Ž‡š‹…‘ ʹͻͶ

͵ǤͷǤͷ ‡”•’‡…–‹˜‡• ʹͻ͹

3.6 The lexicon while parsing 298

4. GRAMMAR:

the reward of incompleteness

301

4.1 The three duals of grammar 301

4.2 The conservativity of syntax 302

ͶǤʹǤͳ Š‡›‹‡Ž†‘ˆ•›–ƒš ͵Ͳʹ

ͶǤʹǤʹ Š‡”‡•–”‹…–‡†‡••‘ˆ—‹ϐ‹…ƒ–‹‘ ͵Ͳ͵ ͶǤʹǤ͵ Š‡‰‡‡”ƒŽ‹–›‘ˆ—‹ϐ‹…ƒ–‹‘ ͵Ͳͷ

4.3 The destructivity of semantics 308

ͶǤ͵Ǥͳ ‹˜‘”…‹‰…‘•–‹–—‡–•ƒ†‡–ƒ‹Ž‡–• ͵Ͳͺ ͶǤ͵Ǥʹ –ƒ‹Ž‡–•ƒ•’”‘†—…–•‘ˆ…‘’‘•‹–‹‘ƒ††‡†—…–‹‘ ͵Ͳͻ

4.4 The denial of structure 314

4.5 The mismatch of structure and meaning 316

ͶǤͷǤͳ Š‡‹•ƒ–…Š‹‡š…‡’–‹‘’Š”ƒ•‡• ͵ͳͺ ͶǤͷǤʹ Š‡‹•ƒ–…Š‹…‘’ƒ”ƒ–‹˜‡• ͵ʹͲ ͶǤͷǤ͵ Š‡‹•ƒ–…Š‹‡ŽŽ‹’•‹• ͵ʹʹ ͶǤͷǤͶ …‘Œ‡…–—”‡‘–Š‡‹–‡”ˆƒ…‡ ͵ʹ͵

4.6 The lexicon as an oracle: the case of „‡ŠƒŽ˜‡ 324

4.7 The incompleteness of grammar 327

4.8 The fruit of incompleteness 329

REFERENCES

333

ACKNOWLEDGEMENTS

Š‹• „‘‘ ”‡’‘”–• ‘ ƒŽ‘•– –™‡–› ›‡ƒ”• ‘ˆ •–”—‰‰Ž‹‰ ™‹–Š ‰”ƒƒ”Ǧ‹Ǧ ƒ…–‹‘ǤŠ‡Žƒ‰—ƒ‰‡ƒ…Š‹‡‹–†‡•…”‹„‡•Šƒ•„‡‡…‘–”‹„—–‡†–‘„›ƒ› •–—†‡–•‹–Š‡‡‹†‡‹˜‡”•‹–›Ž‹‰—‹•–‹…’”‘‰”ƒ‡••‹…‡ͳͻͻͷǤ••‡–‹ƒŽ …‘’‘‡–•™‡”‡…‘Ǧ†‡•‹‰‡†„›Š”‹•–‹ƒƒ˜ƒ†‡‘‡•–‹Œ‡ƒ†ƒ–Š‹‡— ƒ‡‡Ǥ••‡–‹ƒŽ…”‹–‹…‹•™ƒ•†‡Ž‹˜‡”‡†„›ƒ”‹—•‘‘”‡„ƒŽǡ”‹…Š‘‘”Ǧ Ž‡‡”ǡ‡–‡”‹ǡƒ”–‹‘…‘‘’ǡ‘„ƒ”•‡‹Œ‡˜‹…ǡ‹Ž˜ƒ†‡‡”‰ǡ ‘…Š‡ ‘‡†‡”ǡ ƒ› ƒ†•†ƒƒŽǡ ‘„‹ ƒ‰‡†‹Œǡ ‹‡–‡” ƒ‡”•ǡ ƒ† ‘Š ‡”Š‡‹Œ†‡Ǥ‡™ϐ‹‡Ž†•ƒ”‘—†–Š‡ƒ…Š‹‡Šƒ˜‡„‡‡‡š’Ž‘”‡†„›ƒ”‹Œ˜ƒ Ǯ– ‡‡”ǡ ƒ”Œƒ —†‡‰ƒǡ ƒƒ”–Œ‡ …Š—Ž’‡ǡ ƒ˜‹† Šƒ‘—”‹ǡ ‹ƒ ‘éǡ ƒ ƒ†‡›‰ƒ‡”†ǡƒ†‘”‹•‘œŽ‘˜Ǥ

ƒ› ‘”‡ •–—†‡–• ‹•’‹”‡† —• „› †‹•…—••‹‰ ’‹–ˆƒŽŽ• ƒ† ’‡”•’‡…–‹˜‡• ‘ˆ ‡ƒ‹‰ˆ—Ž‰”ƒƒ”ǡƒ†„›™”‹–‹‰’ƒ’‡”•ƒ†–Š‡•‡•‘ƒ•’‡…–•‘ˆ‹–Ǥ‹‡ ƒ› …‘ŽŽ‡ƒ‰—‡• ƒ– ‡‹†‡ ‹˜‡”•‹–›ǡ –Š‡› ƒ†‡ –Š‡ •™‡ƒ–‹‰ ™‘”–Š™Š‹Ž‡ǡ •Š‘™‹‰‘”‡–ŠƒŒ—•–’‘Ž‹–‡‹–‡”‡•–‹‘—”‡ˆˆ‘”–•–‘‡‡’‰”ƒƒ”’”‡…‹•‡ǡ ‡š’Ž‹…‹–ǡ†‡‡’ƒ†‘’‡”ƒ–‹‘ƒŽǤ”‡•—‹‰‘––‘Šƒ˜‡•–‡’’‡†ƒ•‹†‡ˆ”‘”‡ƒŽ Ž‹‰—‹•–‹…•‘”‡–Šƒ‡…‡••ƒ”›ǡ™‡‘™‡–Š‹•‘”‹‡–ƒ–‹‘–‘–Š‡Ǥ

›ƒ†Žƒ”‰‡ǡ–Š‡”‡•‡ƒ”…Š„‡Š‹†–Š‹•‘‘‰”ƒ’Š™ƒ•ˆƒ…‹Ž‹–ƒ–‡†„›–Š‡‡‹Ǧ †‡‹˜‡”•‹–›‡–”‡ˆ‘”‹‰—‹•–‹…•Ǥ‹Ž‡‡…ƒǯ•”‡•‡ƒ”…Š™ƒ•ˆ—†‡† „›–Š‡—–…Š…‹‡…‡”‰ƒ‹œƒ–‹‘ƒ•’ƒ”–‘ˆ–Š‡ƒ””ƒ–‘”’”‘Œ‡…–ǤŠ‡ ϐ‹”•–ƒ—–Š‘”Šƒ•„‡‡Š‘•–‡†„›–Š‡‡–Š‡”Žƒ†••–‹–—–‡‘ˆ†˜ƒ…‡†–—†Ǧ ‹‡•–‘ƒŽ‘•–Ǧ…‘’Ž‡–‡–Š‡„‘‘Ǥ‡‰”ƒ–‡ˆ—ŽŽ›ƒ…‘™Ž‡†‰‡–Š‡•—’’‘”–‘ˆ –Š‡•‡‹•–‹–—–‹‘•Ǥ

‹ƒŽŽ›ǡ™‡ƒ”‡‹†‡„–‡†–‘ƒ•‹ƒ†‡ƒƒˆ‘”™ƒ–…Š‹‰‘˜‡”–Š‡„‘‘ǯ•ŽƒǦ ‰—ƒ‰‡Ǥ

”‹–”‡‡”•

0. INTRODUCTION

0.1 A LANGUAGE MACHINE

‹†•–‡ˆˆ‘”–•–‘Š”‹•–‹ƒ‹œ‡—•Ž‹•ǡ–‘†‡•…”‹„‡ǮǤǤǤ’”‘ˆƒ‡Ž‘˜‡‹•…‡‡• ‘ˆ •—…Š ”‡’—Ž•‹˜‡ ”‡ƒŽ‹• –Šƒ– –Š‡› ™‘—Ž† •Š‘… ‡˜‡ ƒ ƒ†‹”‡” ‘ˆ ‡”› ‹ŽŽ‡”ǯ•ϐ‹…–‹‘ǯȋƒ”†‡”ͳͻͷͺǣ͹Ȍǡ–‘†‡˜‡Ž‘’˜‘–‹‰•›•–‡•ȋ™Š‹…Šƒ”‡•–‹ŽŽ‹ —•‡Ȍǡƒ†–‘•“—ƒ”‡–Š‡…‹”…Ž‡ȂŠ‹•…‘•–”—…–‹‘ƒ’’”‘ƒ…Š‡†ƒ…‹”…Ž‡ǯ••—”ˆƒ…‡ —’–‘ͲǤͲͶΨȂƒ×Ž—ŽŽ‹˜‡–‡†logic machinesǤͳ͵–Š_{Ǧ…‡–—”›ƒ–ƒŽ‘‹ƒǡ}

Š‡†‡•‹‰‡†ƒ†‡š’Ž‘‹–‡†–Š‡•›•–‡ƒ–‹…ƒ†‡…Šƒ‹…ƒŽ’”‘†—…–‹‘‘ˆ’”‘’Ǧ ‘•‹–‹‘•Ǥ ‘”‡šƒ’Ž‡ǡŠ‡™”‘–‡’Š”ƒ•ƒŽ…‘…‡’–•‘–Š‡‡†‰‡•‘ˆ…‘…‡–”‹… …‹”…Ž‡•Ǥ‘‡…–‹‰–Š‡…‘…‡’–•‹‘‡•›•–‡‘ˆ…‹”…Ž‡•ȋ‘”†‹••Ȍǡ„›‘˜‹‰ –Š‡…‹”…Ž‡•™‹–Š”‡•’‡…––‘‡ƒ…Š‘–Š‡”ǡ›‹‡Ž†•’”‘’‘•‹–‹‘•™‹–Šƒ†ƒ„‘—––Š‡•‡ …‘…‡’–•ǡ ƒ……‘”†‹‰ –‘ –Š‡ •‡ƒ–‹… ˆ”ƒ‡ ‘ˆ –Š‡ ’ƒ”–‹…—Žƒ” •›•–‡Ǥ ‡”‡ ‹• Ž—ŽŽǯ•‰‡‡”ƒ–‘”ˆ‘”–”—‹••‘–Š‡•‘—Žȋˆ”‘ǣArs demonstrativaǡ…Ǥͳʹͺ͵ȌȂ ‘‡ƒ‘‰Š—†”‡†•‘ˆ•›•–‡•‘ƒŽŽϐ‹‡Ž†•‘ˆ’Š‹Ž‘•‘’Š‹…ƒŽ–Š‡‘Ž‘‰›Ǥ

–’”‘†—…‡•‹‡ƒ…Š•–ƒ–‡ˆ‘—”…‘Œ—…–‹‘•‘ˆ–Š”‡‡Ǯ•ƒŽŽ…Žƒ—•‡•ǯǡ™Š‡”‡‡ƒ…Š …‘Œ—…–‹‘†‡•…”‹„‡•ƒ•–ƒ–‡‘ˆ–Š‡•‘—Žǡƒ……‘”†‹‰–‘ƒ”†‡”ȋͳͻͷͺǣͳ͵Ȍ—’ –‘ƒ–‘–ƒŽ‘ˆͳ͵͸…‘„‹ƒ–‹‘•ǤŽ—ŽŽ…‘•‹†‡”‡†–Š‡•‡‰‡‡”ƒ–‘”•ƒ•Š‹•Ars Magnaǡ Š‹• ƒŒ‘” –”‹…Ǥ † –Š‘—‰Š Ž—ŽŽ …‘–”‹„—–‡† Ž‹––Ž‡ –‘ –Š‡ ƒ–‡”‹ƒŽ ‘™Ž‡†‰‡‘ˆŠ‹•†ƒ›•ǡ‡‹„‹œ”‡…‘‰‹œ‡†Š‹ƒ•ƒ’‹‘‡‡”‹…‘„‹ƒ–‘”› Ž‘‰‹…ǣlet us calculate ‹•–‡ƒ† ‘ˆ ‡†Ž‡••Ž› †‹•’—–‹‰Ǥ Ž—ŽŽǯ• †‹•• ƒ”‡ ‰”ƒǦ ƒ”•ǡ‰‡‡”ƒ–‹˜‡‰”ƒƒ”•‘ˆ‡ƒ‹‰ˆ—Ž•‡–‡…‡•Ǥˆ‘”–—ƒ–‡Ž›ǡŽ—ŽŽ…‘Ǧ •‹†‡”‡†Š‹•‰”ƒƒ”•–‘‰‡‡”ƒ–‡ȋƒŽŽƒ†‘Ž›Ȍ–”—‡’”‘’‘•‹–‹‘•Ȃƒ“—‹–‡ …‘‘‘˜‡”‡•–‹ƒ–‹‘‘ˆ–Š‡’‘™‡”‘ˆŽ‹‰—‹•–‹…•Ǥ‡–ǡ–Š‡‹†‡ƒ–Šƒ–’”‘’Ǧ ‘•‹–‹‘•ƒ”‡‡ƒ‹‰ˆ—Žbecause‘”to the extent that–Š‡›ƒ”‡‰‡‡”ƒ–‡†„› ƒ •›•–‡ ƒ† –Šƒ– ‡ƒ‹‰ˆ—Ž‡•• ‹• ‘– ƒ ƒ……‹†‡–ǡ ™‡ ƒ› ‰”ƒ– –‘ –Š‡ Ǯ‘„•…—”ƒ–‹•–ǯƒ‘Ž—ŽŽǤ

–‡ŽŽ•™Š›Ǥ•™‡”‹‰–Š‡“—‡•–‹‘™Š›…‘’—–‡”Ǧ‰‡‡”ƒ–‡†’‘‡–”›ǮǤǤǤ”‡ƒ†• Ž‹‡•‘‡‘†‡‘”‘–Š‡”‘ˆ…‘–‡’‘”ƒ”›‡”‹…ƒƒ˜ƒ–Ǧ‰ƒ”†‡™”‹–‹‰ǤǤǤǯ Š‡ •—‰‰‡•–• –Šƒ– •‘‡ ‘ˆ –Š‡•‡ ƒ˜ƒ–Ǧ‰ƒ”†‡ –‡š–• Ǯ ǤǤǤ ƒ› –Š‡•‡Ž˜‡• Šƒ˜‡ „‡‡‰‡‡”ƒ–‡†Dz‡…Šƒ‹…ƒŽŽ›dzǮǤ‡’”‘…‡‡†•–‘†‹•–‹‰—‹•Šˆ‘—”‡–Š‘†•‘ˆ ‡…Šƒ‹…ƒŽ…‘’‘•‹–‹‘ǣ‹’‘•‹‰…‘•–”ƒ‹–•ǡƒ’’Ž›‹‰’”‘…‡†—”‡•ǡ‹’‘•Ǧ ‹‰•‡Ž‡…–‹‘ƒ†—•‹‰…‘„‹ƒ–‹‘Ȃ‹†‡‡†–Š‡‹‰”‡†‹‡–•‘ˆ‘†‡”…‘Ǧ •–”—…–‹‘ ‰”ƒƒ” ˆ‘” ƒ–—”ƒŽ Žƒ‰—ƒ‰‡Ǥ ‡ǯ• „—‹Ž–Ǧ‹ –‡š– ‰‡‡”ƒ–‘” ƒ› „‡Šƒ˜‡‡…Šƒ‹…ƒŽŽ›‘™ƒ†–Š‡Ǥ

Š‡ „ƒ•‹… Š›’‘–Š‡•‹• –Šƒ– ™‡ ™‹ŽŽ ‹˜‡•–‹‰ƒ–‡ ‹• –Šƒ– Žƒ‰—ƒ‰‡ ‡–‡”–ƒ‹• ƒ ‘Ǧ–”‹˜‹ƒŽŽ‡˜‡Ž‘ˆ•‡ƒ–‹……‘’—–ƒ„‹Ž‹–›ǡƒ†–Šƒ–‡ƒ‹‰‹•–Š‡–”ƒ…–ƒ„Ž‡ ’”‘†—…–‘ˆƒȋƒŽ‘•–Ȍ‡…Šƒ‹…ƒŽƒ”…Š‹–‡…–—”‡Ȃ‡ƒ‹‰ǡ‘––”—–Šǡ–Šƒ–‹•ǡ ‘”ƒ†‡“—ƒ…›ǡ‹ˆ‘”ƒ–‹‘ǡ‘”ƒŽ‹–›‘”ƒ›‘–Š‡”‘Ǧ–”—–ŠǦˆ—…–‹‘ƒŽ‡’‹Ǧ ’Š‡‘‡‘‘ˆŽƒ‰—ƒ‰‡Ǧ‹Ǧ—•‡Ǥ—”…Žƒ‹‹•‘––Šƒ–ƒŽŽ‡ƒ‹‰ˆ—Žƒ•’‡…–• ‘ˆŽƒ‰—ƒ‰‡ƒ”‡…‘’—–ƒ„Ž‡Ǣ‹’ƒ”–‹…—Žƒ”ǡ™‡†‘ǯ–„‡Ž‹‡˜‡–Šƒ––Š‡…‘–‡šǦ –—ƒŽ ƒ•’‡…–• ‘ˆ ‡ƒ‹‰ ƒ”‡ ‹ –Š‡ •…‘’‡ ‘ˆ †‡–‡”‹‹•–‹…ƒŽ‰‘”‹–Š•Ǥ ”ƒ‰Ǧ ƒ–‹…•‘ˆ”‹…‡ƒ„”‡‡†ƒ›„‡—†‡”•–‘‘†ƒ•ƒŽ‡˜‡Ž‘ˆ‡ƒ‹‰ˆ—ŽƒƒŽ›•‹• „›‡˜‹•‘ȋʹͲͲͲȌƒ†‘–Š‡”•ǣ…‘˜‡”•ƒ–‹‘ƒŽ‹’Ž‹…ƒ–—”‡•e.g.ƒ••—‡‹–‡ŽǦ Ž‹‰‡…‡ǡƒ™ƒ”‡‡••ƒ†ƒ–Š‡‘”›‘ˆ‹†™Š‹…Š™‡…‘•‹†‡”–‘”‡ƒ…Š„‡›‘† –Š‡‹‹ƒŽƒ••‡••‡–ˆ‘”Š—ƒŽƒ‰—ƒ‰‡Ž‡ƒ”‹‰ǡ—•ƒ‰‡ƒ†—†‡”•–ƒ†Ǧ ‹‰Ǥ‡–ƒŽƒ„‘—––Š‘•‡ƒ•’‡…–•‘ˆ‡ƒ‹‰–Šƒ–ƒ”‡‘–ƒˆˆ‡…–‡†„›ƒŠ—ƒǯ• †‡‰”‡‡‘ˆƒ—–‹•Ǥ–Š‡‘–Š‡”Šƒ†ǡ‘•‡–‡…‡˜ƒ”‹‡•‹–•‡ƒ‹‰‘˜‡”–Š‡ ‹ϐ‹‹–‡Ž›ƒ›…‘–‡š–•‹™Š‹…Š‹–…ƒ„‡—•‡†ǤŠ‘•‡…‘’‘‡–•‘ˆ‡ƒǦ ‹‰–Šƒ–†‘‘–˜ƒ”›™‹–Š…‘–‡š–•„—–”ƒ–Š‡”•–ƒ›…‘•–ƒ–ƒ”‡–Š‡‘‡•™‡ ƒ”‡ƒˆ–‡”Ǥ ‘”‡šƒ’Ž‡ǡ–Š‡”‡ƒ”‡‘…‘–‡š–•‹™Š‹…Šƒ•‡–‡…‡ƒ†‹–•‡‰ƒǦ –‹‘‡ƒ–Š‡•ƒ‡ǤŠ‡‹”’”ƒ‰ƒ–‹…‹’ƒ…–ƒ›„‡–Š‡•ƒ‡ǡŠ‘™‡˜‡”ǡˆ‘” ‡šƒ’Ž‡‹ƒ•‹–—ƒ–‹‘™Š‡”‡‘‡Œ—•–•ƒ›••‘‡–Š‹‰‹””‡Ž‡˜ƒ––‘„”‡ƒƒ ’ƒ‹ˆ—Ž•‹Ž‡…‡ǤŠ‡”‡ˆ‘”‡ǡ–Š‡†‹ˆˆ‡”‡…‡„‡–™‡‡ƒ•‡–‡…‡ƒ†‹–•‡‰ƒ–‹‘ ‹•ƒ†‹‡•‹‘‘ˆ‡ƒ‹‰–Šƒ–™‡™ƒ––‘‡š’Ž‘”‡„›…‘’—–ƒ–‹‘ǤŠ‹•„‘–Ǧ –‘Ž‹‡‘ˆ‡ƒ‹‰™‡…‘•‹†‡”–‘„‡ǮǤǤǤ–Š‡DzŠ‘Ž›‰”ƒ‹Ždz‘–‘Ž›‘ˆŽ‹‰—‹•–‹…•ǡ „—–ƒŽ•‘‘ˆ’Š‹Ž‘•‘’Š›ǡ’•›…Š‘Ž‘‰›ƒ†‡—”‘•…‹‡…‡Ȃ‘––‘‡–‹‘‘”‡ †‹•–ƒ–†‘ƒ‹••—…Šƒ•…—Ž–—”ƒŽƒ†Ž‹–‡”ƒ”›–Š‡‘”›ǯȋƒ…‡†‘ˆˆʹͲͲʹǣʹ͸ͺȌǤ

–Šƒ–•‡•‡ǡ–Š‹•„‘‘‹•ƒ„‘—––Š‡‰”ƒƒ–‹…ƒŽ…‘†‹–‹‘•ˆ‘”…‘’—–‹‰–Š‡ ’”‘’‘•‹–‹‘ƒŽ‘Ž‡…—Ž‡•‘ˆ‡ƒ‹‰‹‹–•„”‘ƒ†‡•–•‡•‡Ǥ

‡–‡…‡• ƒ”‡ ‡ƒ‹‰ˆ—Ž „‡…ƒ—•‡ ‘•– •–”‹‰• Ȃ •–ƒ–‹•–‹…ƒŽŽ›ǣ ƒŽ‘•– ƒŽŽ •–”‹‰•Ȃ‘ˆ™‘”†•ƒ”‡‘–™‡ŽŽˆ‘”‡†Ǥ‘–‡ŽŽ–Š‡ˆ”ƒ…–‹‘‘ˆ™‡ŽŽǦˆ‘”‡†ƒ† ‡ƒ‹‰ˆ—Ž•–”‹‰•ˆ”‘–Š‡˜‹”–—ƒŽ‘˜‡”ϐŽ‘™‘ˆ˜‡”„ƒŽ‘•‡•‡‹•–Š‡–ƒ•‘ˆ ‰”ƒƒ”ǡƒ†‘ˆƒ‰”ƒƒ”ƒ…Š‹‡Ǥ‡ƒ”‡‘––ƒŽ‹‰ˆ—”‹‘—•Ž›•Ž‡‡’‹‰ ‰”‡‡‹†‡ƒ•Š‡”‡Ǥ‘‡•‡–‡…‡•ƒ›•‘—†™‡‹”†because–Š‡›ƒ”‡‡ƒǦ ‹‰ˆ—ŽǤŠ‡‰”ƒƒ”…ƒ‘–„—–ƒ’’”‘˜‡‘ˆ–Š‡ǡˆ‘”‡šƒ…–Ž›–Šƒ–”‡ƒ•‘ǤŠ‡ ‰”ƒƒ”Ǧ‹Ǧƒ…–‹‘—•–‹†‡–‹ˆ›–Šƒ–—––‡”Ž›•ƒŽŽǦ–Š‘—‰ŠǦ‹ϐ‹‹–‡•—„•‡–‘ˆ •‡”‹ƒŽ™‘”†•–Šƒ–Šƒ’’‡•–‘…‘–ƒ‹–Š‡‡ƒ‹‰ˆ—Ž’”‘’‘•‹–‹‘•ǡ™Š‡–Š‡” –Š‡›ƒ”‡‘”–—”‘—––‘„‡™‡‹”†ǡˆƒŽ•‡ǡ‹•—Ž–‹‰‘”‹ƒ†‡“—ƒ–‡Ǥ‰”ƒƒ” ƒ…Š‹‡—•–‘–•–‹…™‹–Š‘‘Ž‡ƒŒ—†‰‡‡–•Ǥ–ƒ””‹˜‡•ƒ–ƒ†‡…‹•‹‘„› …‘’—–‹‰•–”—…–—”‡ǡƒ†–Š‹••–”—…–—”‡…‘˜‡›•–Š‡•‡ƒ–‹…•ǤŽŽ™‡Šƒ˜‡ –‘•Š‘™ǡ–Š‡ǡ‹•–Šƒ––Š‹••‡ƒ–‹…•‹•‘––”‹˜‹ƒŽǤŠƒ–ƒ‘—–•–‘–Š‡˜‹‡™ –Šƒ––Š‡”‡‹•‡ƒ‹‰–‘ƒ’”‘’‘•‹–‹‘ǡƒ†‘–Œ—•––‘™‘”†•Ǥ ‘”–Š‡’”‡•‡– ’—”’‘•‡ǡ™‡™‹ŽŽ‡˜‡ƒ••—‡–Šƒ––Š‡‡ƒ‹‰‘ˆ™‘”†•‹•not–Š‡‘Ǧ–”‹˜‹ƒŽ Ž‡˜‡Ž‘ˆ‡ƒ‹‰™‡ƒ”‡ƒˆ–‡”ǤŠ‡Š›’‘–Š‡–‹…ƒŽŽ‡˜‡Ž‘ˆƒƒŽ›•‹•‹•–Š‡‘‡ƒ– ™Š‹…Š•‡ƒ–‹……‘•‡“—‡…‡•…ƒ„‡‡•–ƒ„Ž‹•Š‡†ǤŠƒ–Ž‡˜‡Ž‹•’”‘’‘•‹–‹‘ƒŽǡ „›ƒ–—”‡Ǥ–‹•ǡŒ—•–Ž‹‡•›ŽŽ‘‰‹••ǡ‘–†‡’‡†‡–‘–Š‡’ƒ”–‹…—Žƒ”‡ƒǦ ‹‰‘ˆ™‘”†•ǡ„—–‹–ƒ••—‡•–Š‡‡ƒ‹‰ˆ—Ž‡••‘ˆ™Šƒ–‡˜‡”…‘•–‹–—‡––Šƒ– ’Žƒ›•ƒ”‘Ž‡Ǥ

‡ƒ”ƒ„Ž›ǡ–Š‡‡ƒ‹‰‘ˆƒ•‡–‡…‡‹•‡‹–Š‡”–”‹˜‹ƒŽ‘”˜ƒ‰—‡ǤŽ–Š‘—‰Š –Š‹•‡ƒ‹‰‹•‡‹–Š‡”Ž‹•–‡†‘”Ž‡ƒ”‡†ǡ‹–‹•ƒŽ‘•–„‡›‘††‡„ƒ–‡™Š‡ …‘’ƒ”‡†–‘–Š‡‡ƒ‹‰•‘ˆ™‘”†•‘”–‘–Š‡’”ƒ‰ƒ–‹…•‘ˆƒ–‡š–Ǥ‡‡‡† ‘– ƒ‰”‡‡ ‘ ƒ› ’ƒ”–‹…—Žƒ” ™‘”† ‡ƒ‹‰ ‹ ‘”†‡” –‘ ƒ‰”‡‡ ϐŽƒ™Ž‡••Ž› ‘ –Š‡‡ƒ‹‰‘ˆ•‡–‡…‡•Ǥ—–ƒ‰”‡‡‡–ǡ‹–Š‹•…ƒ•‡ǡ’”‡•—’’‘•‡••Šƒ”‡† ‘™Ž‡†‰‡ ƒ† •Šƒ”‡† ‹–‡”‡•–•Ǥ ‘•‡“—‡–Ž›ǡ ‘—” ƒ‰”‡‡‡– ‘ •‡–‡…‡ ‡ƒ‹‰ …ƒ‘– •–‡ ˆ”‘ ‹•’‡…–‹‘ ‘ˆ Ǯ–Š‡ ™‘”Ž†ǯǣ –Šƒ– ‹• ‡šƒ…–Ž› ™Š‡”‡ ‘—”˜‹‡™•ƒ†‹–‡”‡•–†‹˜‡”‰‡Ǥ‡…ƒ‘Ž›Ȃ•‹Ž‡–Ž›Ȃƒ‰”‡‡‘ƒ†‘ƒ‹ ™‡‡˜‡”†‡„ƒ–‡ǣ–Š‡•–”—…–—”‡‘ˆ–Š‡Žƒ‰—ƒ‰‡ǤŠ‡”‡ˆ‘”‡ǡ–Š‡‘Ǧ–”‹˜‹ƒŽ„—– —“—‡•–‹‘‡†Ž‡˜‡Ž‘ˆ•‡ƒ–‹……‘’—–ƒ„‹Ž‹–›…ƒ‘–„‡‡•–ƒ„Ž‹•Š‡†‡š…‡’– „›Žƒ‰—ƒ‰‡•–”—…–—”‡Ǥ”‘’‘•‹–‹‘ƒŽ‡ƒ‹‰‹•‡‹–Š‡”ƒƒ……‹†‡–‘”ƒ ‹†‹˜‹†—ƒŽ†‡…‹•‹‘ǡ„—–ƒ‹–”‹•‹…ƒ––”‹„—–‡‘ˆ…‘’—–‡†•–”—…–—”‡Ǥ

‘–…‡”–ƒ‹ƒ•’‡…–•‘ˆŽƒ‰—ƒ‰‡ƒ”‡—”‹‰Ǧ…‘’—–ƒ„Ž‡ǤŠ‡‘Ž›‹’‘”–ƒ…‡ ‘ˆ–Š‹•‰ƒ‡Ž‹‡•‹–Š‡“—‡•–‹‘whichƒ•’‡…–•‘ˆŽƒ‰—ƒ‰‡…ƒ„‡•Š‘™–‘„‡ —”‹‰Ǧ…‘’—–ƒ„Ž‡Ǥ‡”‡ǡ™‡’”‘’‘•‡–Šƒ–’”‘’‘•‹–‹‘ƒŽ‡ƒ‹‰ƒ†Ž‘‰‹Ǧ …ƒŽ ‡–ƒ‹Ž‡– ƒ”‡ ‹ –Š‹• ”ƒ‰‡Ǥ ‘ –Š‡ ‡š–‡– –Šƒ– ™‡ …ƒ ‡•–ƒ„Ž‹•Š –Š‹•ǡ ‹– •Š‘™•–Šƒ–ƒ‹–‡”‡•–‹‰’ƒ”–‘ˆ–Š‡Š—ƒŽƒ‰—ƒ‰‡…ƒ’ƒ…‹–›‹•”‡…—”•‹˜‡ ‡—‡”ƒ„Ž‡‘”‡˜‡‘”‡”‡•–”‹…–‡†ǤŠ‡’ƒ”–‘ˆ–Š‡Š—ƒŽƒ‰—ƒ‰‡…ƒ’ƒ…Ǧ ‹–› ™‡ ‡˜‹•ƒ‰‡ Š‡”‡ǡ ‹• ‹–‡”‡•–‹‰ „‡…ƒ—•‡ ‹– …‘–ƒ‹• •‡ƒ–‹… …‘„‹ƒǦ –‘”‹…•Ǥ ‹˜‡ –Š‡ …‘’—–ƒ„‹Ž‹–› ‘ˆ •›–ƒš ƒ† ƒ …‘’‘•‹–‹‘ƒŽ‹–› ‹†‡‘Ž‘‰›ǡ ‘‡…ƒƒ”‰—‡–Šƒ––Š‡…‘’—–ƒ„‹Ž‹–›‘ˆ•‡ƒ–‹…•…‘‡•ˆ‘”ˆ”‡‡ǤŠƒ–‹•‘– –”—‡ǤŠ‡”‡ƒ•‘Š‡”‡‹•–”ƒ•’ƒ”‡–ǣ•›–ƒ…–‹…•–”—…–—”‡†‘‡•‘–‹’‘•‡ƒ —’’‡”Ž‹‹––‘–Š‡—„‡”‘ˆ’”‘’‘•‹–‹‘•‹–‹†—…‡•ǤŠ‡’‘••‹„Ž‡’”‘’‘•‹Ǧ –‹‘•‡ƒ•‹Ž›‘—–—„‡”–Š‡‰‹˜‡•›–ƒ…–‹…•–”—…–—”‡•–Š‡›ƒ›„‡”‡Žƒ–‡†–‘Ǥ ‹‰—‹•–‹…•ǡ –Š‡ǡ ‹• „‘—† –‘ †‡‘•–”ƒ–‡ –Šƒ– …‘’‘•‹–‹‘ƒŽ •‡ƒ–‹…• ‹• …‘’—–ƒ„Ž‡ƒŽŽ–Š‡™ƒ›—’ƒ††‘™Ǥ‘Š‘’‡ǡ–‘•ƒ›‘”–‘…Žƒ‹‹–‹•†‘‡•‘– •—ˆϐ‹…‡Ǥ‹‰—‹•–•—•–†‘™Šƒ–‡‹„‹œƒ’’”‡…‹ƒ–‡†Ž—ŽŽˆ‘”ǣ…ƒŽ…—Žƒ–‡‹•–‡ƒ† ‘ˆ†‡„ƒ–‹‰–Š‡’‘••‹„‹Ž‹–‹‡•‘ˆ…ƒŽ…—Žƒ–‹‘Ǥ

‘•‡“—‡–Ž›ǡ–Š‹•„‘‘‹•‘Ž‹‰—‹•–‹…‡‰‹‡‡”‹‰ǡ”ƒ–Š‡”–Šƒ‘Ž‹‰—‹•–‹… –Š‡‘”›Ǥ—––Š‡ǡŽ‹‰—‹•–‹…–Š‡‘”›‹••‘‹…‘’Ž‡–‡‘”—†‹”‡…–‡†–Šƒ–‡‰‹Ǧ ‡‡”‹‰Šƒ•–‘ϐ‹ŽŽ–Š‡‰ƒ’•Ǥ‘–Š‡”™ƒ›–‘Ž‘‘ƒ––Š‡’”‘„Ž‡‹˜‘Ž˜‡†‹•–‘ •ƒ›–Šƒ––Š‡‘”‹‡•ǡ‘†‡Ž•ƒ†’”‘„Ž‡•‘Ž˜‹‰‹Ž‹‰—‹•–‹…•ƒ”‡”ƒ–Š‡”—„ƒŽǦ ƒ…‡†ǤŠ‡–Š‡‘”‹‡•ƒ›„‡about–Š‡†ƒ–ƒǡƒ†•‘‡–‹‡•–Š‡†ƒ–ƒare–Š‡ –Š‡‘”›ǡ ƒ• ‹ ƒ‹˜‡ …‘•–”—…–‹‘‹•Ǥ Š‡”‡ ‹• –‘‘ Ž‹––Ž‡ †‹•–ƒ…‡Ǣ –Š‡”‡ ‹• •‘ Ž‹––Ž‡†‹•–ƒ…‡–Šƒ––Š‡–Š‡‘”›…ƒŠƒ”†Ž›‰—‹†‡–Š‡‡‰‹‡‡”‹‰ǤŠ‡’‘‹–‹• ƒ†‡„›Š‹‡„‡”ȋͳͻͺͺȌƒ†Šƒ•Šƒ”†Ž›Ž‘•–”‡Ž‡˜ƒ…‡ǤŠ‹•„‘‘ǡ–Š‡”‡ˆ‘”‡ǡ ’”‡•‡–•Ž‹‰—‹•–‹…‡‰‹‡‡”‹‰‘ƒŽŽŽ‡˜‡Ž•‘ˆ•‡–‡…‡ƒƒŽ›•‹•ǡ‹…Ž—†‹‰–Š‡ –Š‡‘”›‘ˆ‰”ƒƒ”Ǥ†‘‹‰•‘ǡ‹–ƒ••—‡•™Šƒ–™‡…ƒŽŽthe language machine hypothesis:

Ȉ –Šƒ–Žƒ‰—ƒ‰‡‹•’ƒ”–Ž›…‘’—–ƒ„Ž‡ǡ

Ȉ –Šƒ– Š—ƒ Žƒ‰—ƒ‰‡ …ƒ’ƒ…‹–‹‡• ƒ”‡ –‘ „‡ ‘†‡ŽŽ‡† „› ‡…Šƒ‹•• ”ƒ–Š‡”–Šƒ„›”‡’”‡•‡–ƒ–‹‘•ǡ

Ȉ –Šƒ– Š—ƒ ‰”ƒƒ” ‹• ƒ ’”‘…‡•• Ȃ ƒ ’ƒ”•‹‰ ‘” ‰‡‡”ƒ–‹‰ ’”‘…‡•• Ȃ‹–Š‡•’‹”‹–‘ˆƒ”…—•ȋͳͻͺͲȌǡƒ†–Šƒ–‹–‹•ˆƒ•–ǡˆƒ—Ž–›ǡ…‘””‡…–ƒ„Ž‡ ƒ†‰‘ƒŽ‘”‹‡–‡†ǡ”ƒ–Š‡”–Šƒ„‡‹‰‰—‹†‡†„›’”‹…‹’Ž‡•‘ˆ‡ˆϐ‹…‹‡…›ǡ ‡Ž‡‰ƒ…‡ƒ†‡…‘‘›ǡƒ†

0.2 LANGUAGE AND COMPUTABILITY

‘–‡˜‡”›•…Š‘Žƒ”™‹ŽŽ„‡…‘˜‹…‡†–Šƒ––Š‡…‘Œ—…–‹‘Š‡ƒ†‹‰–Š‹•…Šƒ’Ǧ –‡”ƒ‡••‡•‡Ǥ˜‡‹…‘’—–ƒ–‹‘ƒŽŽ‹‰—‹•–‹……‹”…Ž‡•ǡ–Š‡…‘’—–ƒ„‹Ž‹–› ‘ˆƒ–—”ƒŽŽƒ‰—ƒ‰‡‹•†‹•’—–‡†ǡ‹ƒ–Ž‡ƒ•––™‘•‡•‡•Ǥ‘‡”‡•’‡…–ǡ•‘‡ …‘’—–ƒ–‹‘ƒŽŽ‹‰—‹•–•„‡Ž‹‡˜‡–Šƒ–”‡Ž‡˜ƒ–’ƒ”–•‘ˆŽƒ‰—ƒ‰‡ƒ”‡‡ˆˆ‡…–‹˜‡Ž› „‡›‘†–Š‡–›’‡‘ˆˆ‘”ƒŽ‹œƒ–‹‘–Šƒ–‹•”‡“—‹”‡†ˆ‘”—”‹‰Ǧ…‘’—–ƒ„‹Ž‹–›Ǥ ƒ‘–Š‡” –Š‘—‰Š ‘– —”‡Žƒ–‡† ”‡•’‡…–ǡ •‘‡ …‘’—–ƒ–‹‘ƒŽ Ž‹‰—‹•–• †‘—„– ™Š‡–Š‡”•—…Šˆ‘”ƒŽ‹œƒ–‹‘…ƒ„‡†‘‡‡ˆϐ‹…‹‡–Ž›ǤŠ‹••‡…‘†–›’‡‘ˆ‘„Œ‡…Ǧ –‹‘‹•–ƒ‡ˆ‘”‰”ƒ–‡†Š‡”‡Ǥƒ…‡”–ƒ‹•‡•‡ǡ‡˜‡–Š‡ƒ—–Š‘”•‘ˆ–Š‹•„‘‘ ƒ”‡…‘˜‹…‡†–Šƒ–ƒ™Š‘Ž‡…Žƒ••‘ˆŽƒ‰—ƒ‰‡–ƒ••…ƒ„‡’‡”ˆ‘”‡†‘”‡ ‡ˆϐ‹…‹‡–Ž›„›•–ƒ–‹•–‹…ƒŽ‘†‡Ž•–Šƒ„›”‡…—”•‹˜‡ƒƒŽ›•‹•Ǥ‡†‘‘–„‡Ž‹‡˜‡ǡ Š‘™‡˜‡”ǡ–Šƒ–‹–‡”‡•–‹‰•‡ƒ–‹…–ƒ••ƒ”‡‹–Šƒ–…Žƒ••Ȃ‘–™‹–Š•–ƒ†‹‰ –Š‡—‡”‘—•‡ˆˆ‘”–•–‘’‡”ˆ‘”‹ˆ‡”‡–‹ƒŽ–ƒ••„›•ŠƒŽŽ‘™’”‘…‡••‹‰ǡ‰ƒ–ŠǦ ‡”‹‰‹–Š‡ƒ†‘–Š‡”…ŠƒŽŽ‡‰‡•Ǣˆ‘”ƒ…”‹–‹“—‡•‡‡‡…ƒȋʹͲͲͻȌǤ ‡Œ—•––”›–‘•Š‘™–Šƒ–‡ˆˆ‡…–‹˜‡…‘’—–ƒ–‹‘‘ˆ†›ƒ‹…•‡ƒ–‹…•‘–Š‡ „ƒ•‹•‘ˆƒŽ‡š‹…ƒŽ‹œ‡†…ƒ–‡‰‘”‹ƒŽ‰”ƒƒ”‹•™‹–Š‹”‡ƒ…Šǡˆ‘”ƒ‘Ǧ–”‹˜‹ƒŽˆ”ƒ‰Ǧ ‡–‘ˆ—–…Šǡ‘–Š‡’ƒ”•‹‰–”ƒ…ƒ•™‡ŽŽƒ•ˆ‘”‰‡‡”ƒ–‹‘Ǥ

‘”–Š‡„ƒ•‹……‘•–”—ƒŽŠ‡”‡ǡ–Š‡‘—–Ž‹‡‘ˆ™Š‹…Š™‡‘™‡–‘ƒ˜ƒ‡‡—™‡ ȋ’Ǥ…ǤȌǡ™‡ƒ”‡‹–‡”‡•–‡†‹–Š‡”‡Žƒ–‹‘Means „‡–™‡‡•‡–‡…‡•‘ˆ—–…Š ȋȌƒ†’”‘’‘•‹–‹‘•ˆ”‘ƒˆ‘”ƒŽŽƒ‰—ƒ‰‡ǤMeans(ɔǡɗ)Š‘Ž†•„‡–™‡‡ ɔאƒ†ɗא‹ˆˆɗ‹•ƒ‡ƒ‹‰‘ˆɔǤ —”–Š‡”‘”‡ǡ™‡™‹ŽŽ•ƒ›–Šƒ–‹• •‡ƒ–‹…ƒŽŽ›…‘’—–ƒ„Ž‡‹ˆˆ–Š‡”‡‹•ƒ’ƒ”–‹ƒŽ”‡…—”•‹˜‡ˆ—…–‹‘f•—…Š–Šƒ– ˆ‘” ƒŽŽ ’ ‹ ǡ ‹ˆ ˆȋ’Ȍ α ɗ –Š‡Means(’ǡ ɗ),ƒ† ‹• ‡—‡”ƒ„Ž‡Ǥ Ž‡ƒ”Ž›ǡ –Š‹•ˆ—…–‹‘f‹•ƒ’ƒ”•‡”ǡ•‡†‹‰’–‘ɗǤ—’’‘•‡™‡Šƒ˜‡–Šƒ–’ƒ”•‡”ǤŠ‡ǡ ‹••‡ƒ–‹…ƒŽŽ›…‘’—–ƒ„Ž‡‹ˆˆ‹•‡—‡”ƒ„Ž‡ǤˆŠƒ•ƒϐ‹‹–‡Ž‡š‹…‘Ȃ ™Š‹…Š™‡ƒ••—‡„›†‡ϐ‹‹–‹‘Ȃ–Š‡‹–••‡–‡…‡•ƒ”‡‡—‡”ƒ„Ž‡„›‹†—…Ǧ –‹‘‘–Š‡‹”Ž‡‰–ŠǤ ‘”‡˜‡”›“‹–Š‹•‡—‡”ƒ–‹‘ǡ…Š‡…™Š‡–Š‡”ˆȋ’Ȍα“Ǥ Š‡”‡—•–„‡ƒ–Ž‡ƒ•–‘‡ǡ„›–Š‡†‡ϐ‹‹–‹‘‘ˆfǢ–Š‡ϐ‹”•–‘‡‹•ˆ‘—†ƒˆ–‡”ƒ ϐ‹‹–‡—„‡”‘ˆ•–‡’•ǤŠ—•ǡ‹••‡ƒ–‹…ƒŽŽ›…‘’—–ƒ„Ž‡‹ˆMeans(ɔǡɗ)‹• †‡…‹†ƒ„Ž‡Ȃ‡“—‹˜ƒŽ‡…›‹•‘–ƒ–•–ƒ‡Š‡”‡Ǥ

ˆ‘”—Žƒ•‹Ǥ‘™…‘•‹†‡”–Š‡”‡Žƒ–‹‘NLMeans(ɔǡɗ)„‡–™‡‡•‡–‡…‡• ‹ Ǥ Š‡ ”‡Žƒ–‹‘ Š‘Ž†• ‹ˆ ɔmeans ɗǡ –Šƒ– ‹•ǡ ɗ …ƒ ”‡’Žƒ…‡ ɔsalve veri-tate.NLMeans‡š’”‡••‡•ƒ‰”‡‡‡–„‡–™‡‡Š—ƒ„‡‹‰•‘–Š‡•‡ƒ–‹… ‹†‡Ǧ’‘–‡…›‘ˆɔƒ†ɗǤ‡–ƒ‡‹––‘„‡†‡…‹†ƒ„Ž‡ǣ‹ˆŠ—ƒ•…ƒ‘–ƒ‰”‡‡ ‘•‡ƒ–‹…‡“—‹˜ƒŽ‡…‡ǡ™Š‘…ƒǫ‡š–ǡ•—’’‘•‡™‡Šƒ˜‡ƒ”‡Ž‹ƒ„Ž‡‰‡‡”ƒǦ –‘”gƒ’’‹‰‡ƒ‹‰•‘–‘•‡–‡…‡•Ǥ‡–g„‡•—…Š–Šƒ–ˆ‘”ƒŽŽ’”‘’‘•‹–‹‘• “ǡMeans(ɔǡ“)‹ˆˆ‰ȋ“ǡɖȌƒ†NLMeans(ɔǡɖ)ˆ‘”ɔǡɖאǤ–Š‡’”‡•‡…‡‘ˆ •—…Šƒ‰‡‡”ƒ–‹‰ˆ—…–‹‘g,Means‹•†‡…‹†ƒ„Ž‡‹ˆ NLMeans‹•ǤǤ

–Š‹•„‘‘™‡–”›–‘’ƒ”–‹ƒŽŽ›†‡ϐ‹‡–Š‡ˆ—…–‹‘•fƒ†g‘ˆ–Š‡ˆ‘”‡‰‘‹‰ •‡–…Š ‘ˆ ’”‘‘ˆǤ Š‡f ‹• ƒ ’ƒ”•‡” ƒ••‹‰‹‰ ˆ‘”ƒŽ ‡ƒ‹‰• –‘ —–…Š •‡Ǧ –‡…‡•ƒ†gƒ‰‡‡”ƒ–‘”ƒ••‹‰‹‰—–…Š•‡–‡…‡•–‘ˆ‘”ƒŽ‡ƒ‹‰•Ǥ‘ –Š‡‡š–‡––Šƒ––Š‡›–—”‘—––‘„‡•‘—†ƒ†”‡Ž‹ƒ„Ž‡ǡ–Š‡ˆ”ƒ‰‡–‘ˆ—–…Š …‘˜‡”‡†„›–Š‡“—ƒŽ‹ϐ‹‡•ƒ•ƒ•‡ƒ–‹…ƒŽŽ›…‘’—–ƒ„Ž‡Žƒ‰—ƒ‰‡„›–Š‡”‡ƒǦ •‘‹‰ƒ„‘˜‡Ǥ

parserf: NL ฽ Ե(L)

generator g: L ฽ Ե(NL)

NL

Ե(NL)

L

Ե(L) grammar & lexicon

Figure 2 The parser and the generator as functions

Š—•™‡…Žƒ‹–Šƒ–”‡˜‡”•‹„Ž‡‰”ƒƒ”•‘Ž›‡š‹•–‘—–•‹†‡–Š‡”‡ƒŽ‘ˆˆ—ŽŽ› •’‡…‹ϐ‹‡†‡ƒ‹‰ǤŠ‡ˆ—…–‹‘•fƒ†g…ƒ‘–„‡‹˜‡”–‡†ǡƒ•–Š‡›–ƒ”‰‡– †‘ƒ‹•‘ˆƒ†‹ˆˆ‡”‡–‘”†‡”Ǥ

ƒŽ‹‰ƒ„‘—–‡ƒ‹‰ǡ™‡…ƒŠƒ”†Ž›‡•…ƒ’‡–Š‡“—‡•–‹‘™Šƒ–‹–‹•–Šƒ–ŽƒǦ ‰—ƒ‰‡‹•ƒ„‘—–Ǥ—’’‘•‡–Šƒ––Š‡‹–‡”’”‡–ƒ–‹‘‘ˆƒŽƒ‰—ƒ‰‡‡š’”‡••‹‘™‡”‡–‘ †‡’‡†‘ƒ†˜ƒ”›™‹–Š–Š‡•–”—…–—”‡‘ˆ‹–•‹–‡†‡†‘†‡ŽǤ–‹•ƒ‰‘‘†‰—‡•• –‘ƒ••—‡–Šƒ––Š‡”‡ƒ”‡‘”‡–Šƒ…‘—–ƒ„Ž›ƒ›’‘••‹„Ž‡‘†‡Ž•ˆ‘”ƒŽƒǦ ‰—ƒ‰‡‡š’”‡••‹‘ǤŠ‹•‹–‡”’”‡–ƒ–‹‘ƒŽ†‡’‡†‡…›‘ˆƒŽƒ‰—ƒ‰‡‡š’”‡••‹‘ ‘‹–•‘†‡Ž™‘—Ž†ǡ–Š‡ǡ”—‹–‘‹…‘’—–ƒ„‹Ž‹–›‘ˆ‡ƒ‹‰Ǥ‘•–…‡”–ƒ‹Ž›ǡ Š‘™‡˜‡”ǡ–Š‡’‘™‡”–‘”‡ˆ‡”†‘‡•‘–”‡•‹†‡‹Žƒ‰—ƒ‰‡ǯ•…ƒ’ƒ…‹–›–‘”‡ˆ‡”–‘Ȃ ‘”‹–‘‘”‘–‘Ȃƒ’ƒ”–‹…—Žƒ”™‘”Ž†Ǥ–‹•”ƒ–Š‡”–Š‡‘–Š‡”™ƒ›”‘—†ǣŽƒ‰—ƒ‰‡ …ƒ”‡ˆ‡”„‡…ƒ—•‡‹–†‘‡•‘–’”‡•—’’‘•‡ƒ›’ƒ”–‹…—Žƒ”‘”‰ƒ‹œƒ–‹‘‘ˆ™Šƒ– ‹–‹•ƒ„‘—–Ǥ—–…Š…ƒ„‡—†‡”•–‘‘†‡˜‡‹ˆ–Š‡™‘”Ž†–—”•‘—––‘„‡…”‡ƒ–‡† ‹•‹š†ƒ›•ƒ†…‘•–‹–—–‡•ƒ’”‘–‘–›’‹…ƒŽ„Žƒ…Š‘Ž‡ǡ‹Šƒ„‹–‡†„›•“—ƒ”‡…‹”Ǧ …Ž‡† †‡‘•Ǥ • ƒ ƒ––‡” ‘ˆ ˆƒ…–ǡ ™‡ –ƒŽ ‹ —–…Š ƒ„‘—– ‡˜‡”› ’‘••‹„Ž‡ ƒ† ‹’‘••‹„Ž‡™‘”Ž†Ǥ ‘”Žƒ‰—ƒ‰‡–‘”‡ˆ‡”ǡ–Š‡‹‹ƒŽ‘†‡Ž‹•ƒ‘Ǧ‡’–›•‡–Ǥ Šƒ–•‡–’”‘˜‹†‡•†‡‘–ƒ–‹‘•‘ˆ–›’‡eȂ„—––Š‡›ƒ›„‡†‹•…ƒ”†‡†‹ˆƒ˜‘—” ‘ˆ‰‡‡”ƒŽ‹œ‡†“—ƒ–‹ϐ‹‡”•ǡƒ……‘”†‹‰–‘‘–ƒ‰—‡ȋͳͻ͹ʹȌǤŠ‡„ƒ•‹…•‡–•‰‹˜‡ ”‹•‡ǡ™‹–Š‘—–ƒ›ƒ††‹–‹‘ƒŽ‘–‘Ž‘‰›ǡ–‘•—„•‡–•ǡ’”‘˜‹†‹‰†‡‘–ƒ–‹‘•‘ˆ–›’‡ <e,t>ǡ–Š‡’”‡†‹…ƒ–‡•Ǥ‡•–‡’ˆ—”–Š‡”ǡ™‡Šƒ˜‡•‡–•‘ˆ•‡–•ǡ‘ˆ–›’‡<<e,t>,t>ǡ–Š‡ “—ƒ–‹ϐ‹‡”•ǤŠƒ–‹•ƒŽŽ™‡‡‡†Ǥ‡ˆ‡”‡…‡–‘†‡‘–ƒ–‹‘•‘ˆ–›’‡δ<<e,t>,t>, tn_>

–‹‘•ǡ–‹‡•ǡ„‡Ž‹‡ˆ•ǡǤǤǤǤǤŠ‹•‹•™Šƒ–ƒ–—”ƒŽŽƒ‰—ƒ‰‡••Šƒ”‡™‹–Š’”‘‰”ƒ‹‰ Žƒ‰—ƒ‰‡•ǣ–Š‡‘–‘Ž‘‰›‹•‘–ϐ‹š‡†ǡ„—––Š‡•›–ƒš‹•ǤŠ‡†‹ˆˆ‡”‡…‡„‡–™‡‡ ƒ–—”ƒŽ Žƒ‰—ƒ‰‡• ƒ† ’”‘‰”ƒ‹‰ Žƒ‰—ƒ‰‡• ‹• –Šƒ– –Š‡ Žƒ––‡”ǡ „—– ‘– –Š‡ ˆ‘”‡”ǡ …ƒ •™‹–…Š ‘–‘Ž‘‰‹‡• ‹ ‘‡ •‹‰Ž‡ ’”‘’‘•‹–‹‘ǡ ™‹–Š‘—– †‡…Žƒ”ƒ–‹‘Ǥ Š‹• ‹• ™Šƒ– ƒ‡• ƒ–—”ƒŽ Žƒ‰—ƒ‰‡ —‹˜‡”•ƒŽ ƒ† …‘—‹…ƒ–‹˜‡ǡ ƒ† –Š‡ –Š”‡‡Ǧ–›’‡™‹†‘™‹•ƒ‡š…‡ŽŽ‡–…ƒ†‹†ƒ–‡ˆ‘”the—‹˜‡”•ƒŽˆ‡ƒ–—”‡‘ˆƒ–—”ƒŽ Žƒ‰—ƒ‰‡Ǥ ‘”–Š‹•™‹†‘™ƒ†‹–•†‡”‹˜ƒ–‹˜‡•Ž‹‡ˆ—…–‹‘•ˆ”‘‘‡–›’‡‹–‘ ƒ‘–Š‡”ǡ–Š‡Žƒ„†ƒ…ƒŽ…—Ž—•’”‘˜‹†‡•‘”‡–Šƒ‡‘—‰Š’‘••‹„Ž‡”‡…‡‹’–•–‘ …ƒ–…Š‡˜‡”›’‘••‹„Ž‡†‡‘–ƒ–‹‘Ž‹˜‹‰–Š‡”‡Ǥ‡‹–Š‡”‡š–‡•‹‘ƒŽŽ›‘”‹–‡Ǧ •‹‘ƒŽŽ›ǡ™‹ŽŽ™‡‡ƒ•‹Ž›”—‘—–‘ˆ‰ƒ•ˆ‘”‘—”•‡ƒ–‹…ƒ…Š‹‡”›Ǥ

Š‡ „ƒ•‹… …‘…‡’– „‡Š‹† —”‹‰ǯ• ƒ† Š—”…Šǯ• ‘–‹‘• ‘ˆ …‘’—–ƒ„‹Ž‹–› ‹• ”‡…—”•‹‘ǤŠ‡…‘’—–ƒ„Ž‡ˆ—…–‹‘•ƒ”‡–Š‡’ƒ”–‹ƒŽŽ›”‡…—”•‹˜‡‘‡•ǣ–Š‘•‡–Š‡ …‘’—–ƒ–‹‘‘ˆ™Š‹…Š…ƒ„‡’‡”ˆ‘”‡†„›ƒ•‡“—‡…‡‘”…‘’‘•‹–‹‘‘ˆˆ—…Ǧ –‹‘•™‹–ŠǮ•ƒŽŽ‡”ƒ”‰—‡–•ǯ‘”‘ˆŽ‡••ƒ”‹–›ǡ—’–‘–Š‘•‡ˆ—…–‹‘•–Šƒ–ƒ”‡ …ƒŽŽ‡†’”‹‹–‹˜‡”‡…—”•‹˜‡Ǥ‡…—”•‹‘ǡ‘”‡–Šƒƒ›–Š‹‰‡Ž•‡ǡ‹•™Šƒ–Ž‹•–Š‡ …‘’—–ƒ–‹‘‘ˆ‡ƒ‹‰‹ƒ–—”ƒŽŽƒ‰—ƒ‰‡–‘—”‹‰…‘’—–ƒ„‹Ž‹–›Ǥˆ‘”–—Ǧ ƒ–‡Ž›ǡ”‡…—”•‹‘‘ˆ•‡ƒ–‹……‘•–”—ƒŽŠƒ•„‡‡‹š‡†—’™‹–Š…‘’‘•‹–‹‘Ǧ ƒŽ‹–›‹Ž‹‰—‹•–‹…•…‡‡•ǣ–Š‡‡ƒ‹‰‘ˆƒ•–”—…–—”‡†‡’‡†•‘–Š‡‡ƒ‹‰ ‘ˆ‹–•’ƒ”–•Ǥ‘‡…ƒŽŽ‹–Frege’s principleȋ•‡‡e.g.‡‹ƒ†”ƒ–œ‡”ͳͻͻͺȌ„—– ƒ••‡ȋͳͻͺ͸Ȍ†‡‹‡•–Š‹•ƒ––”‹„—–‹‘ǤCompositionality‹•ƒ—ˆ‘”–—ƒ–‡–‡” „‡…ƒ—•‡‹–Šƒ•„‡‡—•‡†–‘†‡–”ƒ…–ˆ”‘–Š‡’”‘…‡••ǣ–Š‡‡ƒ‹‰‘ˆ–Š‡™Š‘Ž‡ ‹•ƒˆ—…–‹‘‘ˆ–Š‡‡ƒ‹‰‘ˆ–Š‡’ƒ”–•Ǥ–‡”’”‡–ƒ–‹‘ǡ–Š‘—‰Šǡ‹•ƒ’”‘…‡••ǡ ‘–ƒ’”‘’‡”–›ǡƒ†‹–‹•–Š‡”‡…—”•‹‘‘ˆ–Š‡…‘’‘•‹–‹‘’”‘…‡••ǡ”ƒ–Š‡”–Šƒ –Š‡‡ƒ‹‰ˆ—Ž‡••‘ˆ–Š‡’”‘’‡”’ƒ”–•‘ˆƒ…‘•–‹–—‡–ǡ–Šƒ–—•–‰—ƒ”ƒ–‡‡ –Š‡•‡ƒ–‹…˜ƒŽ‹†‹–›‘ˆ–Š‡‘—–…‘‡Ǥ–Š‡‘–Š‡”Šƒ†ǡƒ–Š‡ƒ–‹…•‹•–Š‡ƒ”– ‘ˆ†‡•…”‹„‹‰’”‘…‡••‡•‹ƒ•–ƒ–‹˜‡™ƒ›Ǥ—–•–‹ŽŽǡ™Šƒ–™‡ƒ”‡Ž‘‘‹‰ˆ‘”‹•ƒ procedure–‘ƒ’’Ž›‡ƒ‹‰•–‘‡ƒ…Š‘–Š‡”ǡ”‡…—”•‹˜‡Ž›Ǥ‡ƒ–‹…”‡…—”•‹‘ƒ† …‘’‘•‹–‹‘ƒŽ‹–›ƒ”‡‘–‡“—‹˜ƒŽ‡–…‘…‡’–•ǡ‹–Š‹•”‡•’‡…–ǤŽƒ›‡”‡†–”‡ƒ–Ǧ ‡–‘ˆƒ‘”ƒ…Ž‡ǯ•’”‘…Žƒƒ–‹‘‘ˆ‡ƒ‹‰ƒ›„‡”‡…—”•‹˜‡ǡ„—–‘–…‘’‘Ǧ •‹–‹‘ƒŽǡƒ†–Š‡•‹’Ž‡‘‡Ǧ•–‡’…‘…ƒ–‡ƒ–‹‘‘ˆ‡ƒ‹‰•ƒ›„‡…‘’‘•‹Ǧ –‹‘ƒŽ„—–‘–”‡…—”•‹˜‡ǤŠ‹•‘„•‡”˜ƒ–‹‘‹•…”—…‹ƒŽǤŠ‡’”‘’‘•‹–‹‘ƒ••‹‰‡† ƒ•ƒ‡ƒ‹‰–‘ƒ•‡–‡…‡ƒ›‹–•‡Žˆ„‡…‘’‘•‡†‘ˆ’”‘’‘•‹–‹‘•ǡ„—–ƒ–—”ƒŽ Žƒ‰—ƒ‰‡‹•‘–•—…Š–Šƒ–•‹’Ž‡…‘…ƒ–‡ƒ–‹‘•—ˆϐ‹…‡•–‘ƒ••—”‡–Šƒ–ƒ••‹‰Ǧ ‡–ǤŠƒ–‹•ǣ‹ˆabc‹•ƒ™‡ŽŽǦˆ‘”‡†•‡–‡…‡‡ƒ‹‰ɔƒ†–Š‡‡ƒ‹‰‘ˆaǡ bƒ†cƒ”‡–Š‡…‘Œ—…–‹‘•‘ˆ’”‘’‘•‹–‹‘•p_a1& ... p_ai ... &... p_anǡp_b1& ... p_bi ... &... p_bmƒ†p_c1& ... p_ci ... &... p_ck”‡•’‡…–‹˜‡Ž›ǡ–Š‡™‡‘™ˆ‘”•—”‡–Šƒ–ɔءp_a1& ... p_ai ... &... p_an & p_b1& ... p_bi ... &... p_bmƬp_c1... p_ci ... &... p_ckǤ ‘”‡‰‡‡”ƒŽŽ›ǣ

Š‡„ƒ•‹…‹†‡ƒ„‡Š‹†–Š‹•Ž‡ƒ‹•–Šƒ––Š‡‡ƒ‹‰•‘ˆ–Š‡’ƒ”–•ƒ”‡•—’Ǧ ’‘•‡†–‘„‡‹–‡”–™‹‡†‹ƒ™ƒ›–Šƒ–…ƒ‘–„‡‘†‡ŽŽ‡†„›•‹’Ž‡…‘…ƒ–‡Ǧ ƒ–‹‘Ǥ’ƒ”–‹…—Žƒ”ǡ™‡ƒ••—‡–Šƒ––Š‡’ƒ”–•Šƒ˜‡˜ƒ”‹ƒ„Ž‡ǦŽ‹‡’Žƒ…‡Š‘Ž†‡”• –Šƒ–Šƒ˜‡–‘„‡•›…Š”‘‹œ‡†ǤŠ‹•‹•™Š‡”‡‘–ƒ‰—‡ǯ•”‘’‡””‡ƒ–‡–‘ˆ —ƒ–‹ϐ‹…ƒ–‹‘‹”†‹ƒ”›‰Ž‹•Š…‘‡•‹ȋ‘–ƒ‰—‡ͳͻ͹ʹȌǤ‡”‡‡ƒ‹‰• ƒ”‡…‘’‘•‡†„›–›’‡†Žƒ„†ƒ…‘˜‡”•‹‘ǡ‹’ƒ”–‹…—Žƒ”„›ˆ—…–‹‘…‘’‘Ǧ •‹–‹‘ǤŠ‡ˆ—…–‹‘•ƒ”‡•—…Š–Šƒ––Š‡‘—–’—–‘ˆ‘‡‹•‹’—––‘–Š‡‘–Š‡”Ǥ•ƒ …‘•‡“—‡…‡ǡ˜ƒ”‹ƒ„Ž‡•ƒ”‡Ǯ’‹’‡Ž‹‡†ǯƒ†–Š‡”‡•—Ž–‹‰’”‘’‘•‹–‹‘•„‡…‘‡ …‘‡…–‡†ǣ ‡˜‡”› ˜ƒ”‹ƒ„Ž‡ –Šƒ– ™ƒ• ƒ„•–”ƒ…–‡† ‘˜‡” ‹• …‘˜‡”–‡† Ǯƒ™ƒ›ǯ ȋ‘” „‘—† „› ‘’‡”ƒ–‘”• ‘—–•‹†‡ ‹–• ‘”‹‰‹Ȍ ‹ –Š‡ …‘’‘•‹–‹‘ ’”‘…‡••Ǥ ‘†‡ŽǦ Ž‹‰–Š‹•”‡…—”•‹˜‡’”‘…‡••‘ˆ…‘’‘•‹–‹‘‹•–Š‡…ŠƒŽŽ‡‰‡ˆ‘”…‘’—–ƒ–‹‘ƒŽ •‡ƒ–‹…•ǤŠ‡”‡…—”•‹‘—•–„‡•–‡‡”‡†‹•—…Šƒ™ƒ›–Šƒ––Š‡’Žƒ…‡ǦŠ‘Ž†Ǧ ‹‰˜ƒ”‹ƒ„Ž‡•ƒ”‡ƒ……‘—–‡†ˆ‘”’”‘’‡”Ž›Ǥ‘Œ—…–‹‘ƒŽ‘‡™‘ǯ–†‘Ǥ —…Ǧ –‹‘…‘’‘•‹–‹‘‹‰Š–Ǥ–‹ŽŽǡ™‡™‹ŽŽƒ”‰—‡‹…Šƒ’–‡”ʹ–Šƒ––Š‡…‘’‘•‡† ‡ƒ‹‰…ƒ„‡†‡•‹‰‡†ƒ•ƒ…‘Œ—…–‹‘‘ˆ•ƒŽŽ…Žƒ—•‡•ǣ—‘”†‡”‡†•‡ŽˆǦ …‘–ƒ‹‡†•‡ƒ–‹…‘„Œ‡…–•‹–Š‡‹”‘™”‹‰Š–Ǥ—––Š‹•…‘Œ—…–‹‘‹•‡‹–Š‡” ‹‡†‹ƒ–‡‘”ƒ–”‹˜‹ƒŽ…‘•‡“—‡…‡‘ˆ•–”—…–—”‡Ǥ

‘ƒ‡ƒŽ‘‰Š‹•–‘”›•Š‘”–ǣˆ”‘‘—”’‡”•’‡…–‹˜‡ǡ”‹•–‘–Ž‡ǡ ”‡‰‡ǡŠ—”…Šǡ Š‘•›ǡ‘–ƒ‰—‡ǡƒ†ƒ‡–Š‡‡“—ƒŽŽ›…‘–”‹„—–‡†–‘–Š‡‹•‹‰Š––Šƒ– Žƒ‰—ƒ‰‡…ƒ„‡…ƒ’–—”‡†„›ˆ‘”ƒŽ‡ƒ•Ǥ”‹•–‘–Ž‡”‡˜‡ƒŽ‡†–Š‡Žƒ™•‘ˆ•›ŽǦ Ž‘‰‹• ƒ† –Š‡ ƒŽ‰‡„”ƒ ‘ˆ ”‡ƒ•‘‹‰ ™‹–Š ™‘”†•Ǥ ”‡‰‡ †‡–‡…–‡† –Š‡ ’‘Žƒ”Ǧ ‹–›‘ˆ”‡ˆ‡”‡…‡ƒ†‡ƒ‹‰ǤŠ—”…Šƒ†‡ˆ—…–‹‘•‘’‡”ƒ–‹‘ƒŽƒŽŽ–Š‡™ƒ› †‘™ǤŠ‘•›”‡Ǧ•–›Ž‡†‰”ƒƒ”ǡˆ‘”Ž‹‰—‹•–‹…•ƒ†’•›…Š‘Ž‘‰›Ǥ‘–ƒ‰—‡ ‰ƒ˜‡‡ƒ‹‰„ƒ…–‘‰”ƒƒ”Ǥƒ‡–Š‡–‹‡†Žƒ„†ƒ•ǡ–›’‡•ƒ††‡”‹Ǧ ˜ƒ–‹‘•–‘‰‡–Š‡”Ǥ†–Š‡ǡ–Š‡”‡™‡”‡‰‡‡”ƒŽ‹œ‡†“—ƒ–‹ϐ‹‡”•ȋƒ”™‹•‡ƒ† ‘‘’‡”ͳͻͺͳǡ™ƒ”–•ͳͻͺͳȌƒ†–Š‡›™‡”‡–Š‡ϐ‹”•–•‡ƒ–‹…‘„Œ‡…–•–Š‡ƒŽ‰‡Ǧ „”ƒ‹… •–”—…–—”‡ ‘ˆ ™Š‹…Š –—”‡† ‘—– –‘ „‡ Ž‹‰—‹•–‹…ƒŽŽ› ”‡Ž‡˜ƒ–ǣ †‡ϐ‹ƒ„Ž‡ǡ –‡•–ƒ„Ž‡ǡ ”‡ˆ—–ƒ„Ž‡ …”‡ƒ–—”‡• ‹ ƒ ƒŽ‰‡„”ƒ‹… Žƒ†•…ƒ’‡ǡ ‹–‡”ˆ‡”‹‰ ™‹–Š –Š‡ †‹•–”‹„—–‹‘‘ˆ’Š”ƒ•‡•ƒ†…‘‡…–‡†–‘˜‡”›‰‡‡”ƒŽǡ‹ˆ‘–—‹˜‡”•ƒŽǡ’”‘’Ǧ ‡”–‹‡•‘ˆƒ–—”ƒŽŽƒ‰—ƒ‰‡ȋ™ƒ”–•ͳͻͺ͵ȌǤŠ‡‹”„‡‹‰–Š‡”‡ƒ†–Š‡‹”ƒ–—”‡ ƒ†‡…Ž‡ƒ”–Šƒ–Žƒ‰—ƒ‰‡Ž‹˜‡••‘‡™Š‡”‡‹–Š‡„”ƒ‹ǡƒ†–Šƒ–‹–ƒ‡•‹–•‡Žˆ ‘™ǤŠ‡”‡‹•‘”‡ƒ•‘–‘—†‡”‡•–‹ƒ–‡‘”‘„•…—”‡‘”‡‰Ž‡…–‹–ǤŠ‡”‡ ‹•ƒ•›•–‡–Šƒ–ˆƒ…‹Ž‹–ƒ–‡•…‘˜‡›ƒ…‡‘ˆ–Š‘—‰Š–•ǡϐ‹†‹‰•ǡ‡‘–‹‘•Ǥ–ƒ› Šƒ˜‡ƒ”‹•‡ˆ”‘–Š‡Ž—•–ˆ‘”…‘—‹…ƒ–‹‘ǡ‘”ˆ”‘–Š‡Ž—•–ˆ‘”‡š’”‡••‹‰ ‘‡ǯ•–Š‘—‰Š–‘”ˆ”‘„‘–ŠȂ…‘—‹…ƒ–‹‘‹•–Š‡‹–‡”ƒ…–‹‘„‡–™‡‡–Š‡ ˜‡”„ƒŽ‹œ‡† •–”ƒ™ ‡ ‘ˆ ‘—” ‹†•Ǥ ‹‡ ‘•– –Š‹‰• ‹ ‡˜‘Ž—–‹‘ǡ ‹– ‹• ‘– †‡•‹‰‡†–‘†‘™Šƒ–‹––—”•‘—––‘†‘ǡ„—–‹–‹•Š‡ƒ˜‹Ž›ƒˆˆ‡…–‡†„›‹–•ˆ—…–‹‘Ǥ Šƒ– •›•–‡ ‹• …‘’—–ƒ„Ž‡ǡ ’ƒ”–Ž› „› •›„‘Ž‹… ‡ƒ• ƒ……‘”†‹‰ –‘ ƒ”…—• ȋʹͲͲ͵Ȍǡƒ†‹–ƒ›ƒ•™‡ŽŽŠƒ˜‡ƒ—–…Šˆƒ…‡ǤQuod sit demonstrandum.

0.3 THE BOOK

• ™‡ ƒ‹–ƒ‹ ƒ ”ƒ–Š‡” …‘•‡”˜ƒ–‹˜‡ ˜‹‡™ „‘–Š ‘ …‘’—–ƒ–‹‘ƒŽ Ž‹‰—‹•Ǧ –‹…•Ȃ‹–‹•Ž‹‰—‹•–‹…•Ȃƒ†‘•‡ƒ–‹…•Ȃ‡ƒ‹‰‡š‹•–•ƒ†…ƒ„‡…‘Ǧ ’—–‡†Ȃ–Š‡„‘‘Šƒ•ƒ”ƒ–Š‡”…‘•‡”˜ƒ–‹˜‡•‡–Ǧ—’Ǥ–…‘•‹•–•‘ˆ–Š”‡‡…Šƒ’Ǧ –‡”•†‡˜‘–‡†–‘–Š‡ƒ‹Ž‹‰—‹•–‹…‡‰‹‡•‘ˆ–Š‡Žƒ‰—ƒ‰‡ƒ…Š‹‡ǣ•›–ƒšǡ •‡ƒ–‹…•ƒ†–Š‡Ž‡š‹…‘Ǥ†‹–…‘…Ž—†‡•™‹–Šƒˆ‘—”–Š…Šƒ’–‡”‹™Š‹…Š–Š‡ ”‡Žƒ–‹‘•Š‹’„‡–™‡‡–Š‡–Š”‡‡‡‰‹‡•‹•”‡˜‹‡™‡†Ǥ

’Ž‡š•›„‘Ž•ƒ––Š‡•ƒ‡–‹‡Ǥ —”–Š‡”‘”‡ǡ–Š‡…Šƒ’–‡”•…”—–‹‹œ‡•Š‘™–Š‡ ‹•‡š’Ž‘‹–‡†‹’ƒ”•‹‰ƒ†‰‡‡”ƒ–‹‘„›–Š‡ĊđĎđĆ蕛•–‡Ǥ

Š‡ …Šƒ’–‡” ‘ •‡ƒ–‹…• ’”‡•‡–• –Š”‡‡ †‹ˆˆ‡”‡– „—– ”‡Žƒ–‡† ‹•–ƒ…‡• ‘ˆ Ž‘‰‹…ƒŽˆ‘”ǡ†—„„‡†–‘”‡†ǡ’’Ž‹‡†ƒ† Žƒ– ǤŠ‡›†‹ˆˆ‡”‹•’‡…‹ϐ‹…‹–›ƒ† –Š‡ ™ƒ› –Š‡› ƒ”‡ †‡”‹˜‡†Ǥ Š‡ ‡‡† ˆ‘” –Š‡ –Š”‡‡ˆ‘Ž† ‹• ‡•–ƒ„Ž‹•Š‡† „› ”‡ˆǦ ‡”‡…‡–‘–Š‡˜ƒ”‹‡–›‘ˆ–ƒ••™‹–Š™Š‹…Šƒˆ‘”ƒŽ‹–‡”’”‡–ƒ–‹‘‘ˆƒ–—”ƒŽ Žƒ‰—ƒ‰‡ Šƒ• –‘ …‘’Ž›Ǥ  ’ƒ”–‹…—Žƒ”ǡ –Š‡ ‹–‡”ˆ‡”‡…‡ ‘ˆ …‘’‘•‹–‹‘ƒŽ‹–› ™‹–Š”‡ƒ†‹‰•‡Ž‡…–‹‘ǡ•…‘’ƒŽ’Š‡‘‡ƒǡ‹–‡•‹‘ƒŽ‹–›ƒ†•‡ƒ–‹…‹ˆ‡”Ǧ ‡…‡‹•Š‹‰ŠŽ‹‰Š–‡†ǤŠ‡…Šƒ’–‡”†‡•…”‹„‡•Š‘™’ƒ”•‹‰‹–Š‡ĊđĎđĆ蕛•–‡ Ž‡ƒ†•–‘ƒˆ—ŽŽ›•’‡…‹ϐ‹‡†•‡ƒ–‹…”‡’”‡•‡–ƒ–‹‘ƒ†Š‘™–Š‹•ˆ—ŽŽ›•’‡…‹ϐ‹‡† •‡ƒ–‹…”‡’”‡•‡–ƒ–‹‘…ƒ†”‹˜‡‰‡‡”ƒ–‹‘Ǥ

Š‡–Š‹”†…Šƒ’–‡”†‡ƒŽ•™‹–Š–Š‡†ƒ–ƒˆ‘”‡ƒ‹‰Ǧ†”‹˜‡’ƒ”•‹‰ƒ†‰‡‡”ƒǦ –‹‘ǡ–Š‡Ž‡š‹…‘ǤŠ‡ĊđĎđĆ蕛•–‡‹•Ž‡š‹…ƒŽ‹•–‹…ǡ‹ƒƒŽ‘•–‡š–”‡‡™ƒ›ǣ ‹–Š‘Ž†•‡–”‹‡•ȂŽ‡ƒ•ǡ‹‡ˆˆ‡…–Ȃˆ‘”‡ƒ…Š—•‡‘ˆ‡ƒ…Š’Š”ƒ•‡ǤŠ‡‡š’Ž‘Ǧ •‹‘‘ˆ’Š”ƒ•‡•ˆ‘ŽŽ‘™‹‰ˆ”‘–Š‹••–”ƒ–‡‰›‹•…‘—–‡”‡†™‹–Š•ƒ”–‹†‹…‡• ƒ†—––‡”Ž›ˆƒ•–”‡–”‹‡˜ƒŽǤŠ‡Ž‡š‹…‘…‘•‹•–•‘ˆ—‹“—‡ˆ—ŽŽ›•’‡…‹ϐ‹‡†…‘Ǧ ’Ž‡š•›„‘Ž•ǡ‹–Š‡ˆ‘”‘ˆ‰”ƒ’Š•‘’‡–‘—‹ϐ‹…ƒ–‹‘ǤŠ‡…Šƒ’–‡”†‡•…”‹„‡• –Š‡‘ˆˆǦŽ‹‡…‘•–”—…–‹‘ƒ†–Š‡‘ǦŽ‹‡”‡–”‹‡˜ƒŽ‘ˆ–Š‡†ƒ–ƒ„ƒ•‡ǤŠ‡™ƒ›™‡ ‡š’Ž‘‹––Š‡Ž‡š‹…‘‡ˆϐ‹…‹‡–Ž›ˆ‘”’ƒ”•‹‰ƒ†ˆ‘”‰‡‡”ƒ–‹‘‹•”‡˜‡ƒŽ‡†Ǥ•ƒ ƒ––‡”‘ˆˆƒ…–ǡ–Š‡Ž‡š‹…‘†‡•…”‹„‡†Š‡”‡‹•‘”‡‘ˆƒconstructiconǣ‹–Š‘Ž†• –Š‡’Š”ƒ•ƒŽ…‘’‘‡–•‘ˆ—–…Š‹ƒ‘ǦŠ‹‡”ƒ”…Š‹…ƒŽǡƒ–”‹šǦŽ‹‡†ƒ–ƒ„ƒ•‡ǡ –Š‡ˆ”‘–Ǧ‘ˆϐ‹…‡ƒ……‡••–‘™Š‹…Š‹•—…Š‘”‡”‡Ž‡˜ƒ–ˆ‘”…‘’—–ƒ„‹Ž‹–›–Šƒ ‹–•„ƒ…Ǧ‘ˆϐ‹…‡‰‡‡•‹•ƒ†Š‹††‡•–”—…–—”‡Ǥ–Š‡•‡…Šƒ’–‡”•ǡ‡••‡–‹ƒŽ†ƒ–ƒ •–”—…–—”‡•ƒ”‡–ƒ‡ˆ”‘–Š‡ĊđĎđĆ葒‡”ƒ–‹‘‘†‡ǤŠ‡ƒ‡Ǧ—’‘ˆ–Š‡ †ƒ–ƒ•–”—…–—”‡•ƒ›•Ž‹‰Š–Ž›†‹ˆˆ‡”†‡’‡†‹‰‘–Š‡’”‘…‡••‘†—Ž‡Šƒ†Ž‹‰ –Š‡Ǥ ‘–™‹–Š•–ƒ†‹‰ –Š‡•‡ †‹ˆˆ‡”‡…‡•ǡ –Š‡› ƒŽ™ƒ›• ”‡’”‡•‡– –Š‡ ˜‡”› •ƒ‡–›’‡‘ˆ—†‡”Ž›‹‰‰”ƒ’ŠǤ†‹•…—••‹‰–Š‡†ƒ–ƒ•–”—…–—”‡•™‡†‘‘– †‹ˆˆ‡”‡–‹ƒ–‡„‡–™‡‡…ƒ–‡‰‘”‹‡•ƒ†–›’‡•‹ƒˆ—†ƒ‡–ƒŽ™ƒ›Ǥ

‘•—ƒ”‹œ‡ǡ–Š‡„‘‘‹•‡ƒ––‘„‡ƒ†‡ˆ‡…‡‘ˆƒ–Ž‡ƒ•––Š‡ˆ‘ŽŽ‘™‹‰…Žƒ‹•ǣ Ȉ –Š‡’”‘’‘•‹–‹‘ƒŽ…‘–‡–‘ˆƒ—–…Š•‡–‡…‡…ƒ„‡…‘’—–‡†Ǣ Ȉ ˆ‘”‡˜‡”›’”‘’‘•‹–‹‘ǡƒ—–…Š•‡–‡…‡…ƒ„‡…‘’—–‡†–Šƒ–‡–ƒ‹Ž•

–Šƒ–’”‘’‘•‹–‹‘Ǣ

Ȉ –Š‡’”‘’‘•‹–‹‘ƒ••‘…‹ƒ–‡†™‹–Šƒ—–…Š•‡–‡…‡…ƒ„‡”‡’”‡•‡–‡† ƒ•ƒϐŽƒ–…‘Œ—…–‹‘‘ˆ•ƒŽŽ…Žƒ—•‡•ǡ‹†‡š‹‰–Š‡’”‘’‘•‹–‹‘ƒŽ…‘Ǧ –‡–ǡ™‹–Š•‘Ž‡‹œ‡†‡˜‡–•ƒ†•–ƒ–‡•ƒ•ƒ„ƒ…„‘‡Ǣ

Ȉ ™Š‡…‘’—–‹‰‡ƒ‹‰ǡ—†‡”•’‡…‹ϐ‹‡†ƒ†•’‡…‹ϐ‹‡†Ž‡˜‡Ž•‘ˆ”‡’Ǧ ”‡•‡–ƒ–‹‘ƒ”‡’”‘†—…‡†„›‡••‡–‹ƒŽŽ›†‹ˆˆ‡”‡–ƒŽ‰‘”‹–Š•Ǣ

Ȉ ‰‹˜‡–Š‹••‡’ƒ”ƒ–‹‘ǡ‘Š‘‘‘”’Š‹•„‡–™‡‡•›–ƒ…–‹…ƒ†•‡ƒǦ –‹…’”‘…‡†—”‡•‡‡†•–‘„‡‡•–ƒ„Ž‹•Š‡†ˆ‘”…‘’—–ƒ–‹‘ƒŽ”‡ƒ•‘•Ǣ Ȉ –Š‡’ƒ”•‡”…‘’—–‹‰’”‘’‘•‹–‹‘•ƒ†–Š‡‰‡‡”ƒ–‘”’”‘†—…‹‰•‡Ǧ

–‡…‡•ƒ”‡‘–‡ƒ…Š‘–Š‡”ǯ•‹˜‡”•‡•„—–‡š’Ž‘‹––Š‡•ƒ‡Ž‡š‹…ƒŽƒ† ‰”ƒƒ–‹…ƒŽ”‡•‘—”…‡•Ǣ

Ȉ —†‡”•’‡…‹ϐ‹‡†Ž‡˜‡Ž•‘ˆ‡ƒ‹‰…ƒ„‡…‘’—–‡†„›—‹ϐ‹…ƒ–‹‘Ǣ Ȉ –Š‡Ž‡š‹…‘‘ˆ—–…Š‹•’Š”ƒ•ƒŽǢ

Ȉ –Š‡Ž‡š‹…‘‘ˆ—–…Š…ƒ„‡‘”‰ƒ‹œ‡†ƒ†ƒ……‡••‡†ƒ•ƒ‘ǦŠ‹‡”ƒ”…Š‹Ǧ …ƒŽˆƒ‹Ž›‘ˆ…‘’Ž‡š•‹‰•Ǣ

1. SYNTAX:

the game of recursion and

discontinuity

1.1 THE NEED FOR SYNTAX

—–…Š‹•ƒƒ–—”ƒŽŽƒ‰—ƒ‰‡Ȃ–‘ƒ…‡”–ƒ‹‡š–‡–Ǥ••—…Šǡ‹–‹•–‘”„›ϐ‹‹–‡Ǧ ‡••ƒ†–™‘‹ϐ‹‹–‹‡•Ǥ ‹”•–ǡ–Š‡”‡‹•–Š‡ϐ‹‹–‡•‡–‘ˆƒŽŽŽ‡š‹…ƒŽƒ–‘•ǡ–Š‡ Ž‡š‹…‘Ǥ ‘” –Š‡ ’”‡•‡– ’—”’‘•‡ǡ ‹– ‹• ‹ƒ–‡”‹ƒŽ Š‘™ ™‡ †‡ϐ‹‡ –Š‡ ‡Ž‡Ǧ ‡–•‘ˆ–Š‹••‡–ǡƒ•™‘”†•ǡ‘”’Š‡‡•‘”’Š”ƒ•‡•ǤŠ‡”‡Šƒ•–‘„‡ƒϐ‹‹–‡ •‡––Šƒ–‹•‘–ƒƒŽ‰‡„”ƒǣ‹–•‡„‡”•ƒ”‡‘–†‡†—…‡†„—–”ƒ–Š‡”‡Ž‡…–‡†‘” ƒ••‹‰‡†‘”’”‘’‘•‡†‘”‡ˆ‘”…‡†ǡ„—–„›‘‡ƒ•ƒ”‡–Š‡›–Š‡‘”‡•‘ˆƒ •›•–‡Ǥ‹–Š‘—––Š‹••‡–„‡‹‰ϐ‹‹–‡ǡƒŽƒ‰—ƒ‰‡™‘—Ž†„‡‡‹–Š‡”Ž‡ƒ”ƒ„Ž‡ ‘”†‡…‹†ƒ„Ž‡Ǥ‡…‘†ǡ™‡Šƒ˜‡–Š‡Ž‡š‹…‘ǯ•Kleene closureǣ–Š‡‹ϐ‹‹–‡•‡– ‘ˆ ƒŽŽ ϐ‹‹–‡ •–”‹‰• ‘˜‡” –Šƒ– Ž‡š‹…‘Ǥ ‹…‡ –Š‡”‡ ‹• ‘ ”‡ƒ•‘ –‘ ƒ••—‡ ƒ Ž‘‰‡•–•–”‹‰ǡ–Š‡”‡ƒ”‡‹ϐ‹‹–‡Ž›ƒ›‘ˆ–Š‡ǡ„—–‡ƒ…Š‹•‘ˆϐ‹‹–‡Ž‡‰–ŠǤ Š‹”†ǡ–Š‡”‡‹•–Š‡‹ϐ‹‹–‡•‡–‘ˆϐ‹‹–‡•–”‹‰•‘˜‡”–Š‡Ž‡š‹…‘–Šƒ–™‡–ƒ‡ –‘ „‡ ‡š’”‡••‹‘• ‘ˆ –Š‡ Žƒ‰—ƒ‰‡ Ȃ –Š‡ Žƒ‰—ƒ‰‡ ‹ ‡š–‡•‹‘Ǥ ‰ƒ‹ǡ ‹– ‹• ‹ƒ–‡”‹ƒŽŠ‘™™‡†‡ϐ‹‡–Š‹••‡–Ȃ„›™‡ŽŽǦˆ‘”‡†‡••ǡ„›‹–‡”’”‡–ƒ„‹Ž‹–›ǡ „›’”ƒ‰ƒ–‹……”‹–‡”‹ƒ‘”ƒ›‘–Š‡”•‡•‹„Ž‡‘”Ǥ

Š‡ „ƒ•‹… ˆƒ…– ƒ„‘—– Žƒ‰—ƒ‰‡Ǧƒ•ǦƒǦ•›•–‡ ‹• –Šƒ– –Š‡ –™‘ ‹ϐ‹‹–‡ •‡–• †‘ ‘– …‘‹…‹†‡ǣ ƒ Žƒ‰—ƒ‰‡ ‹• ƒproper •—„•‡– ‘ˆ –Š‡ Ž‡‡‡ …Ž‘•—”‡ ‘˜‡” ‹–• Ž‡š‹…‘Ǥ ‘ ‰”ƒƒ”‹ƒ ‡˜‡” ’”‘’‘•‡† ƒ› Žƒ‰—ƒ‰‡ –‘ …‘‹…‹†‡ ™‹–Š ‹–• Ž‡‡‡ …Ž‘•—”‡Ǥ ‘ ƒ—–Š‘” ‘ˆ ˆ‘”ƒŽ Žƒ‰—ƒ‰‡• ‡˜‡” ’”‘’‘•‡† ƒanything goes Žƒ‰—ƒ‰‡Ǥ —…Š ƒ Žƒ‰—ƒ‰‡ ™‘—Ž† „‡ „‘—† –‘ …ŠƒŽŽ‡‰‡ ‡ƒ‹‰ˆ—ŽǦ ‡••ǣ ‡ƒ‹‰ ”‡•—Ž–• ˆ”‘ †‹•–‹…–‹‘ ƒ† †‹•–‹…–‹˜‹–›ǡ „—– ‹ –Š‡ Ž‡‡‡ …Ž‘•—”‡ƒŽŽ•–”‹‰•ƒ”‡„‘”‡“—ƒŽǤŽ›Ž‡‰–Š‘”–Š‡‘……—””‡…‡‘ˆ‹†‹˜‹†—ƒŽ ƒ–‘•…‘—Ž†…‘—–ƒ•…‘–”‹„—–‹‘•–‘‡ƒ‹‰Ǥ‹…‡‡˜‡”›ƒ–‘‘……—”•‹ ƒ›‘”†‡”™‹–Š‘–Š‡”ƒ–‘•ǡ‹–‹•Šƒ”†–‘•‡‡™Šƒ–ƒ‹†‹˜‹†—ƒŽƒ–‘‹ƒ ”ƒ†‘•–”‹‰…‘—Ž†…‘–”‹„—–‡–‘‹–••’‡…‹ϐ‹…‡ƒ‹‰Ǥˆ…‘—”•‡ǡ–Š‡Ž‡‰–Š ‘ˆƒ•–”‹‰…‘—Ž†…‘—–ƒ•‡ƒ‹‰ˆ—ŽǤ‹…‡–Š‡Ž‡‰–Š‹•ƒϐ‹š‡†‡š–‡•‹‘ƒŽ ’”‘’‡”–›‘ˆƒ•–”‹‰ǡŽ‡‰–Š…ƒ‘Ž›„‡‡ƒ‹‰ˆ—Ž‹–Š‡™ƒ›‹–‡‰‡”•ƒ”‡ ‡ƒ‹‰ˆ—ŽǤ—„‡”•ǡŠ‘™‡˜‡”ǡƒ”‡‡ƒ‹‰ˆ—Ž‹ƒ™ƒ›Žƒ‰—ƒ‰‡•…ƒ‘– ƒˆˆ‘”†–‘”‡•–”‹…––Š‡•‡Ž˜‡•–‘Ǥ—„‡”•ƒ”‡”‹‰‹††‡•‹‰ƒ–‘”•Ǥ‘‘””‘™ǡ –Š‡›™‹ŽŽ”‡ˆ‡”‹‡šƒ…–Ž›–Š‡•ƒ‡™ƒ›ƒ•–Š‡›†‹†›‡•–‡”†ƒ›ǤŠ‡›ƒ”‡one-world†‡•‹‰ƒ–‘”•ǡƒ††‘‘–’”‘˜‹†‡…‘–‡–‘—–•‹†‡–Šƒ–™‘”Ž†Ȃ—Ž‡•• ™‡—•‡—„‡”•‹‘–Š‡”Žƒ‰—ƒ‰‡•–Šƒ–ƒ”‡‘–”‹‰‹†ǤŠƒ–Ž‡ƒ˜‡•‘Ž›–Š‡ ’‘••‹„‹Ž‹–›ˆ‘”•–”‹‰•‹–Š‡Ž‡‡‡…Ž‘•—”‡–‘…ƒ””›‡ƒ‹‰”ƒ†‘Ž›‘”„› ‘”ƒ…Ž‡Ǥ–‹•Šƒ”†ǡŠ‘™‡˜‡”ǡ–‘Ž‘‘ƒ–‡ƒ‹‰ƒ•ƒƒ……‹†‡–ǡ‘–•–‡‡”‡†„› ƒ–‡”‹ƒŽ’”‘’‡”–‹‡•‘ˆ–Š‡‡š’”‡••‹‘Ǥ–™‘—Ž†ƒ‡›‘—”’”‡•‡–‡–‡”’”‹•‡ ‘ˆ”‡ƒ†‹‰–Š‹•–‡š–˜ƒ…—‘—•ǤŠ‡ˆƒ‘—•„—–‘–—…‘–”‘˜‡”•‹ƒŽ…‘…‡’–‘ˆ compositionality‘ˆ‡ƒ‹‰”‡’”‡•‡–•ƒ‡ˆˆ‘”––‘‡š’Žƒ‹™Š›–Šƒ–‡–‡”Ǧ ’”‹•‡‹•‘–…‘’Ž‡–‡Ž›˜ƒ…—‘—•ȋ‡˜‡‹ˆ–Š‡…‘…‡’–…ƒ‘–„‡ƒ––”‹„—–‡†–‘ ‘––Ž‘„ ”‡‰‡ǡƒ•ƒ••‡ȋͳͻͺ͸Ȍ…‘˜‹…‹‰Ž›ƒ”‰—‡•ȌǤ……‹†‡–ƒŽ‡ƒ‹‰ ™‘—Ž†‘–…‘˜‡›‹ˆ‘”ƒ–‹‘ǡŒ—•–•‘—†Ǥ

Ž‘•–„›†‡ϐ‹‹–‹‘ǡ–Š‡ǡŽƒ‰—ƒ‰‡•ƒ”‡Ǯ•ƒŽŽ‡”ǯ–Šƒ–Š‡‹”Ž‡‡‡…Ž‘•—”‡Ǥ ‹…‡ ™‡ Šƒ˜‡ ‘ ”‡ƒ•‘ –‘ ƒ••—‡ –Šƒ– –Š‡ ‘”†‡” ‘ˆ ‹ϐ‹‹–› „‡–™‡‡ –Š‡ •‡–•‹•†‹ˆˆ‡”‡–Ȃƒ•‹•–Š‡…ƒ•‡™‹–Še.g.–Š‡•‡–•‘ˆ”ƒ–‹‘ƒŽƒ†”‡ƒŽ—Ǧ „‡”•ȂǮ„‡‹‰•ƒŽŽ‡”ǯŠ‡”‡—•–„‡ƒ’’”‘ƒ…Š‡†•–‘…Šƒ•–‹…ƒŽŽ›ǣ–Š‡…Šƒ…‡–Šƒ– ƒ•–”‹‰ǡ”ƒ†‘Ž›…Š‘•‡ˆ”‘–Š‡Ž‡‡‡…Ž‘•—”‡ǡ‹•ƒŽ•‘‹–Š‡Žƒ‰—ƒ‰‡‹• Ž‡••–Šƒ‘‡Ǥ•ƒƒ––‡”‘ˆˆƒ…–ǡ–Š‡…Šƒ…‡‹•…Ž‘•‡–‘œ‡”‘Ǥ

…‘—–™‘”†•‘”–‘ϐ‹†•–”‹‰•Ǥ—–‹ˆ‡ƒ‹‰…‘‡•‹–‘’Žƒ›ǡ†‹•–‹…–‹‘‹• ”‡“—‹”‡†Ǥ‘ƒ—–‘ƒ–‘…ƒ”‡Ž‹ƒ„Ž›ƒ••‹‰‡ƒ‹‰•–‘•‡–‡…‡•ƒ†‘––‡ŽŽ ‹–‡”’”‡–ƒ„Ž‡ˆ”‘—‹–‡”’”‡–ƒ„Ž‡‘‡•ǡˆ‘”–Š‡•ƒ‡”‡ƒ•‘–Šƒ–ƒ…‘’—–‡” –Šƒ–ƒ††•ȗƒ†͓ǡ…ƒŠƒ”†Ž›„‡–”—•–‡†™Š‡ƒ††‹‰͵ƒ†ͶǤˆƒ›–Š‹‰‹• ‡ƒ‹‰ˆ—Žǡ‡ƒ‹‰‡˜ƒ’‘”ƒ–‡•Ǥƒ‰—ƒ‰‡Ž‹˜‡•‘•‡Ž‡…–‹˜‹–›Ǥ

‡‘™Šƒ˜‡–Š‡ˆ‘ŽŽ‘™‹‰•‹–—ƒ–‹‘ǣƒϐ‹‹–‡Ž‡š‹…‘ǡ‹–•Ž‡‡‡…Ž‘•—”‡ȗα ȓδ™_ͳǤǤǤ™_εȁ™_iאȔƒ†ƒ–—”ƒŽŽƒ‰—ƒ‰‡ؿǡ™Š‡”‡‹•†‡ϐ‹‡†„› ƒ ‘‘Ž‡ƒ ˆ—…–‹‘ ˆ_NLǡ ‹–• ‰”ƒƒ”Ǥ ‹• ‘– ‡…‡••ƒ”‹Ž› ™‡ŽŽǦ†‡ϐ‹‡†Ǥ –• …Šƒ”ƒ…–‡”‹•–‹…ˆ—…–‹‘ƒ›„‡ˆ—œœ›ǡƒ†…‡”–ƒ‹Ž›ƒˆƒ‹Ž›‘ˆŽƒ‰—ƒ‰‡•–Šƒ– ‹–‡”•‡…–• ™‹–Š …ƒ „‡ †‡ϐ‹‡† ‘”ǡ ‘”‡ ’”‡…‹•‡Ž›ǡ ƒ ˆƒ‹Ž› ‘ˆ Žƒ‰—ƒ‰‡• –Šƒ–•Šƒ”‡•‹ϐ‹‹–‡•—„•‡–•™‹–ŠǤŠ‹•ˆƒ‹Ž›…‘˜‡”•–Š‡˜ƒ”‹‡–‹‡•‘ˆ—–…Š ‘‡‹‰Š–™ƒ––‘ˆ‘…—•‘Ǥ–‹•‘”‡‹–”‹‰—‹‰™Š‡–Š‡”™‡…‘—Ž†ϐ‹†ƒ–—Ǧ ”ƒŽ Žƒ‰—ƒ‰‡• ‹ –Š‡ …‘’Ž‡‡– ‘ˆ ‹ ȗǡ ‹ –Š‡ •‡– ȗ Ǧ ǣ Žƒ‰—ƒ‰‡• –Šƒ– Ž‹˜‡ ‘ –Š‡ •ƒ‡ Ž‡š‹…‘ „—– ™‘—Ž† ‘– •Šƒ”‡ ƒ› ‡š’”‡••‹‘ ™‹–Š ‘”ǡ‘”‡’”‡…‹•‡Ž›ǡ–Šƒ–•Šƒ”‡ƒ–‘•–ϐ‹‹–‡•—„•‡–•™‹–Šǡe.g.ƒ•‡–‘ˆ‘‡ ™‘”†‡š’”‡••‹‘•Ǥ”‘„ƒ„Ž›ǡ‘•—…ŠŽƒ‰—ƒ‰‡•™‘—Ž†„‡ƒ……‡’–‡†ƒ•’‘••‹„Ž‡ ƒ–—”ƒŽŽƒ‰—ƒ‰‡•Ǥ ‘”ˆ‘”ƒŽŽƒ‰—ƒ‰‡•ǡ–Š‘—‰Šǡ…‘’Ž‡‡–ƒ”›Žƒ‰—ƒ‰‡• …ƒ„‡†‡ϐ‹‡†ƒ†‡š’Ž‘‹–‡†‹•‹’Ž‡™ƒ›•Ǥ‡‹‰Š–™ƒ––‘Šƒ˜‡ǡ–Š‡ǡ ƒ –Š‡‘”› –Šƒ– †‡–‡”‹‡• ™Š‹…Š •—„•‡–• ‘ˆ ȗ “—ƒŽ‹ˆ› ƒ• ƒ–—”ƒŽ Žƒ‰—ƒ‰‡•Ǥ Šƒ– –Š‡‘”› ‹• –Š‡theory of grammar, ‡„‘†›‹‰ –Š‡ …Žƒ••‹…ƒŽ Š‘•›ƒ …‘…‡’–‘ˆ’‘••‹„Ž‡ƒ–—”ƒŽŽƒ‰—ƒ‰‡Ǥ‡™‘ǯ––‘—…Š—’‘–Šƒ––Š‡‘”›Š‡”‡ †‹”‡…–Ž›ǡƒ•™‡ƒ”‡†‡ƒŽ‹‰ƒ‹Ž›™‹–Š–Š‡–Š‡‘”›‘ˆ—–…ŠǤ‡ƒ••—‡ǡŠ‘™Ǧ ‡˜‡”ǡ–Šƒ––Š‡–Š‡‘”›‘ˆ‰”ƒƒ”™‹ŽŽ…‘˜‡”‘—”ƒƒŽ›•‹•‘ˆ—–…ŠǤ†‡ˆ‡†‹‰ –Šƒ–ǮŽ‘…ƒŽǯ–Š‡‘”›ǡ™‡™‹ŽŽŠƒ˜‡–‘”‡ˆ‡”–‘‘”‡‰‡‡”ƒŽ’‘‹–•ǡ‘™ƒ†–Š‡ǡ ‹’ƒ”–‹…—Žƒ”™Š‡ƒ––‡”•‘ˆ”‡•–”‹…–‹˜‹–›ƒ”‡ƒ–•–ƒ‡Ǥ

”ƒƒ”—•–„‡ϐ‹‹–‡–‘„‡ƒ’’Ž‹…ƒ„Ž‡ǤŠƒ–‹•™Š›ƒ‰”ƒƒ”…ƒ‘–„‡ ‹†‡–‹ϐ‹‡† ™‹–Š –Š‡ Žƒ‰—ƒ‰‡ ‹ ‡š–‡•‹‘Ǥ ‡ ȋŠ—ƒ•ǡ …‘’—–‡”•Ȍ ‡‡† ƒ ϐ‹‹–‡ǡ ‹–‡•‹‘ƒŽ †‡•…”‹’–‹‘ Ȃ ‹ ™Šƒ–‡˜‡” ˆ‘”ǣ ’Š”ƒ•‡ •–”—…–—”‡ ”—Ž‡•ǡ ƒŽ‰‘”‹–Š•ǡ ’ƒ”•‹‰ ”‘—–‹‡•ǡ ‹–‡‰”ƒ–‡† …‹”…—‹–• Ȃ –‘ Šƒ†Ž‡ –Š‡ ‹‡˜‹–ƒ„Ž‡ ‹ϐ‹‹–›‘ˆŽƒ‰—ƒ‰‡Ǥš’Ž‘‹–‹‰ϐ‹‹–‡‡ƒ•–‘†‡ƒŽ™‹–Š‹ϐ‹‹–‡•‡–•’”‘˜‘‡• ”‡…—”•‹˜‡—•‡‘ˆ–Š‘•‡‡ƒ•ǤŠƒ–‹•ǣ‹ϐ‹‹–‡ǡ•–”—…–—”‡†Žƒ‰—ƒ‰‡•ƒ›Šƒ˜‡ •‡–‡…‡•–Šƒ–•Š‘™„”ƒ…‡–‹‰’ƒ––‡”•Ž‹‡

ȋʹȌ ǥǤȏ_šǥȏǥȏǥȐǥȐǥȐ

–Š‹•’”‘’‡”–›‘ˆ”‡Ǧ‡–”ƒ…‡™‹–Š‹–Š‡•‡Ž˜‡•Ǥ”—…‹ƒŽŽ›ǡ–Š‡’ƒ––‡”’‘•–Ǧ ’‘‡•–Š‡ˆ—ŽŽ‹–‡”’”‡–ƒ–‹‘‘ˆ–Š‡—’’‡”‘……—””‡…‡—–‹Žƒˆ–‡”–Š‡‹–‡”’”‡Ǧ –ƒ–‹‘‘ˆ–Š‡Ž‘™‡”‘……—””‡…‡Ǥ†–Šƒ–‹•™Š‡”‡•›–ƒš…‘‡•‹ǣ–Š‡•›Ǧ –ƒš•’‡…‹ϐ‹‡•™Š‹…Š’Š”ƒ•‡†‡’‡†•‘™Š‹…Š’Š”ƒ•‡Ǥ›–ƒš‹•–Š‡…ƒ””‹‡”‘ˆ ”‡…—”•‹˜‡‡„‡††‹‰ƒ††‡’‡†‡…›Ǥ–†‡ϐ‹‡•ƒϐ‹‹–‡ˆƒ‹Ž›‘ˆ…ƒ–‡‰‘”‹‡• ‘”‡“—‹˜ƒŽ‡…‡…Žƒ••‡•ƒ†ƒϐ‹‹–‡…Žƒ••‘ˆ”‡Žƒ–‹‘•„‡–™‡‡–Š‡Ǥ–ƒ••‹‰• Ž‡š‹…ƒŽƒ–‘•–‘ƒ–Ž‡ƒ•–‘‡…Žƒ••‘”…ƒ–‡‰‘”›Ǥ–‰—ƒ”ƒ–‡‡•–Šƒ–‡˜‡”›•–”‹‰ of the languageǡ‘ƒ––‡”Š‘™Ž‘‰‘”…‘’Ž‡šǡ…ƒ„‡†‹˜‹†‡†‹–‘”‡’”‡Ǧ •‡–ƒ–‹˜‡•‘ˆ–Š‘•‡…ƒ–‡‰‘”‹‡•ǤŠ‡’”‘…‡••‘ˆ–Šƒ–†‹˜‹•‹‘‹•…ƒŽŽ‡†parsingǡ ƒ†‹–‹•‘‡‘ˆ–Š‡‘”‡‹–‡”‡•–‹‰™ƒ›•–‘‡š‡…—–‡ƒ…Šƒ”ƒ…–‡”‹•–‹…ˆ—…–‹‘ ‘˜‡”–Š‡•‡–‘ˆƒŽŽ•–”‹‰•Ǥ

Š‹• ‹• –Š‡ Ȃ „› ‘™ Ȃ …Žƒ••‹…ƒŽ ƒ’’”‘ƒ…Š –‘ ‰”ƒƒ”ǡ ƒ• ’”ƒ…–‹•‡† ‹ –Šƒ– „”ƒ…Š‘ˆƒ–Š‡ƒ–‹…•–Šƒ–•–—†‹‡•ˆ‘”ƒŽŽƒ‰—ƒ‰‡•ƒ†ƒ—–‘ƒ–ƒȋ…ˆǤ‘’Ǧ …”‘ˆ–et al.ʹͲͲͳȌǤ–”‡Žƒ–‡•‰”ƒƒ”–‘…‘’—–ƒ–‹‘ǣƒ‡š’Ž‹…‹–ˆ‘”ƒŽ‰”ƒǦ ƒ”†‡–‡”‹‡•ƒ…Žƒ••‘ˆƒ—–‘ƒ–ƒ™‹–Š”‡•’‡…––‘‹–•—•‡‘ˆ”‡•‘—”…‡•ǡƒ† ‡˜‡”› …‘’—–ƒ„Ž‡ ’”‘„Ž‡ …ƒ „‡ ”‡’”‡•‡–‡† ƒ• ƒ …Šƒ”ƒ…–‡”‹•–‹… ˆ—…–‹‘ ‘˜‡”ƒ•‡–‘ˆ•‡“—‡…‡•ǤŠ‡‘–‹‘‘ˆƒ—–‘ƒ–‘‹•—•‡†Š‡”‡‹–Š‡ƒ–ŠǦ ‡ƒ–‹…ƒŽ •‡•‡ǡ ƒ• ƒ —‹– ‘ˆ †ƒ–ƒǡ ’”‘…‡••‹‰ǡ …‘–”‘Žǡ ƒ† •–‘”ƒ‰‡ǡaka –Š‡ —”‹‰ƒ…Š‹‡ǤŠ‡ϐ‹”•––”ƒ…‘ˆ–Š‡”‡Žƒ–‹‘Šƒ•„‡…‘‡…ƒ‘‹•‡†ƒ•–Š‡ Š‘•›Š‹‡”ƒ”…Š›ǡ”‡Žƒ–‹‰’”‘’‡”–‹‡•‘ˆŽƒ‰—ƒ‰‡•ǡ‰”ƒƒ”•ƒ†ƒ—–‘Ǧ ƒ–ƒǤŠ‡•‡…‘†–”ƒ…‹•†—‡–‘†‹‰‹–ƒŽ‹œƒ–‹‘ǣƒŽŽƒ—–‘ƒ–ƒ…ƒ„‡†‡ϐ‹‡†‹ ƒ†‹‰‹–ƒŽŽƒ‰—ƒ‰‡ǡ™‹–Šƒ‰”ƒƒ”Ǣ–Š‡•‡–‡…‡•‘ˆ–Šƒ–Žƒ‰—ƒ‰‡ƒ”‡‹–‡”Ǧ ’”‡–ƒ„Ž‡ƒ•ƒ—–‘ƒ–ƒǤ–Šƒ–•‡•‡ǡ–Š‡…‹”…Ž‡‹•…Ž‘•‡†ǣŽƒ‰—ƒ‰‡•Šƒ˜‡‰”ƒǦ ƒ”•ǡ‰”ƒƒ”•†‡–‡”‹‡ƒ—–‘ƒ–ƒȂ–Š‡›…ƒ„‡™”‹––‡ƒ•’”‘‰”ƒ•Ȃƒ† ƒ—–‘ƒ–ƒƒ”‡†‡ϐ‹‡†‹Žƒ‰—ƒ‰‡•Ǥ

‡‰‘”‹‡•ǡ…‘•–”—…–•ǡ‡„‡††‹‰’”‘…‡†—”‡•Ȍ‹•‡••‡–‹ƒŽǤŠƒ–‹•”‡…—”•‹‘ǣ Ž‹‹‰’Š”ƒ•‡•–‘‡ƒ…Š‘–Š‡”‹ƒƒ•›‡–”‹…ƒŽ•–›Ž‡ǡad libitumǤ‡…—”•‹‘ ‘ˆˆ‡”• ƒ ˆ”ƒ‡™‘” ˆ‘” ƒ‹‰ ‡š’”‡••‹‘• •‡ƒ–‹…ƒŽŽ› †‡’‡†‡– —’‘ ‡ƒ…Š ‘–Š‡”ǡ ƒ• ™‡ŽŽ ƒ• ˆ‘” –Š‡ƒ–‹… ”‡Žƒ–‹‘•ǡ ”‡ˆ‡”‡–‹ƒŽ‹–›ǡ ‹–‡•‹‘ƒŽ‹–›ǡ ‘‘†ǡ ƒ•’‡…–ǡ ƒ† •‘ ‘Ǥ Š‹• ˜‹‡™ Ž‡ƒ˜‡• ‘’‡ ƒ› ™ƒ›• ‘ˆ ‡„‡††‹‰ ‡š’”‡••‹‘••‡ƒ–‹…ƒŽŽ›‹–‘‡ƒ…Š‘–Š‡”ǤŠ›•‹…ƒŽȂ’”‘•‘†‹…Ȃ‡„‡††‹‰ ƒ•‹ȋʹȌ‹•Œ—•–‘‡‘ˆ–Š‡Ǥƒ’Š‘”ƒ‘”ƒ–ƒ’Š‘”ƒ‹•ƒ‘–Š‡”ǣ–Š‡•‡ƒ–‹… ‡ˆˆ‡…–‘ˆȋʹȌȂ–Š‡‹–‡”‡•–‹‰‡ˆˆ‡…–Ȃ…ƒ„‡‹‹…‡†„›ƒ•–”—…–—”‡Ž‹‡ȋ͵Ȍǡ ™Š‡”‡–Š‡‹†‹…‡•‹†‹…ƒ–‡…‘Ǧ‡˜ƒŽ—ƒ–‹‘Ǥ

ȋ͵Ȍ ǥȏXǥ‹–ŒǥȐǥȏ›ǥȏXǥȐǥȐŒ

‡”‡–‘‘ǡ–Š‡˜ƒŽ—ƒ–‹‘‘ˆ–Š‡ǮŠ‹‰Š‡•–ǯ‹•†‡’‡†‡–‘–Š‡‹–‡”’”‡–ƒ–‹‘‘ˆ –Š‡Ž‘™‡”Ǥ ‘”‡šƒ’Ž‡ǡ–Š‡ˆ‘ŽŽ‘™‹‰–™‘–‡š–•ƒ”‡‹†‡’‘–‡–ǤŠ‡›„‘–Š •—„•—‡‡ƒ‹‰ˆ—Ž‡š’”‡••‹‘•—†‡”‡ƒ…Š‘–Š‡”Ǥ

ȋͶȌ Š‡ƒ™Š‘”‡ƒ†ƒ„‘‘•ƒ‹†–Šƒ––Š‡”‹˜‡”™ƒ•–‘‘™‹Ž†

ȋͷȌ Š‡”‡™ƒ•ƒƒǤ‡”‡ƒ†ƒ„‘‘Ǥ‡•ƒ‹†–Š‹•ǤŠ‡”‹˜‡”‹•–‘‘™‹Ž†Ǥ

Š‡‹”Ž‘‰‹…ƒŽˆ‘”•—•–„‡–Š‡•ƒ‡ǡƒ•–Š‡‹”‡–ƒ‹Ž‡–•…‘˜‡”‰‡ǣ‹„‘–Š –‡š–•ǡ –Š‡ •’‡ƒ‹‰ •—„Œ‡…– ‹• †‡•…”‹„‡† ƒ• ƒ ƒ ”‡ƒ†‹‰ ƒ „‘‘ǡ ƒ† –Š‡ ™‹Ž†‡••‘ˆ–Š‡”‹˜‡”‹•‘–’”‡•‡–‡†ƒ•ƒˆƒ…–ǡŽ‹‡–Š‡•’‡ƒ‹‰‡˜‡–ǡ„—–‹• ƒ––”‹„—–‡†–‘–Šƒ–ƒǤ

ƒ’Š‘”ƒǡ”‡ƒŽ‹œ‡†„›…‘’›‹‰ǡ„‹†‹‰‘”…‘Ǧ”‡ˆ‡”‡…‡ǡ’Žƒ…‡•–Š‡ƒ–‡…‡†Ǧ ‡–—†‡”–Š‡•‡ƒ–‹…”‡‰‹‡‘ˆ–Š‡•‡–‡…‡…‘–ƒ‹‹‰–Š‡’”‘Ǧˆ‘”Ǥ• ƒ ƒ––‡” ‘ˆ ˆƒ…–ǡ ƒƒ’Š‘”ƒ ƒ† ƒ–ƒ’Š‘”ƒ †‘overtly ™Šƒ– ‹ –Š‡ •‡ƒ–‹… ”‡’”‡•‡–ƒ–‹‘ ‘ˆ ƒ …‘’Ž‡š •‡–‡…‡ ‹• …‘†—…–‡†covertly „› ”‡‹ϐ‹…ƒ–‹‘ ƒ†„‹†‹‰Ǣ–Š‡”‡ǡlogicalƒƒ’Š‘”ƒƒ”‡…ƒŽŽ‡†ˆ‘”Ǥ˜‡”–ƒƒ’Š‘”ƒƒ†ƒ–ƒǦ ’Š‘”ƒ‡ˆ‘”…‡”‡…—”•‹‘Ǥ

1.2 FORMS OF DUTCH

•ƒƒ–—”ƒŽŽƒ‰—ƒ‰‡ǡ—–…Š•Š‘™••‹‰•‘ˆ•›–ƒšǤŠ‹••‡…–‹‘’”‡•‡–•ƒ ˜‡”› …‘…‹•‡ …Šƒ”ƒ…–‡”‹œƒ–‹‘ ‘ˆ –Š‘•‡ ƒ•’‡…–• ‘ˆ —–…Š •›–ƒš –Šƒ– †‡–‡”Ǧ ‹‡†–Š‡ƒ–—”‡‘ˆĊđĎđĆčǯ•‰”ƒƒ”Ǥ–‹•‡‹–Š‡”…‘’Ž‡–‡‘”„ƒŽƒ…‡†ǡ ƒ•‹–ˆ‘…—••‡•‘–Š‘•‡’”‘’‡”–‹‡•‘ˆ—–…Š’Š”ƒ•‡Ǧ•–”—…–—”‡ˆ‘”ƒ–‹‘–Šƒ– ”‡•‹•–—…‘’Ž‹…ƒ–‡†ƒ……——Žƒ–‹˜‡…‘„‹ƒ–‘”‹…•Ǥ

—–…Š‹•ƒŽƒ‰—ƒ‰‡‹–Š‡ˆ‘ŽŽ‘™‹‰•‡•‡ǣ–Š‡‘‹ƒŽƒ†’”‡’‘•‹Ǧ –‹‘ƒŽ‘„Œ‡…–•‘ˆƒ˜‡”„’”‡…‡†‡‹–Ǥ‡”„ƒŽƒ†’”‘’‘•‹–‹‘ƒŽ…‘’Ž‡‡–•ˆ‘ŽǦ Ž‘™–Š‡˜‡”„Ǥ”‡’‘•‹–‹‘ƒŽ…‘’Ž‡‡–•…ƒ‘……—”–‘–Š‡”‹‰Š–‘ˆƒ˜‡”„ǡƒ† ˜‡”„ƒŽ…‘’Ž‡‡–•ƒ›‘……—”–‘–Š‡Ž‡ˆ–ǤŠ‡–Š‡”‡••‹•ƒ†‡”‹˜‡†‘”ƒ ˆ—†ƒ‡–ƒŽ’”‘’‡”–›‘ˆŽƒ‰—ƒ‰‡•‹•‘–”‡Ž‡˜ƒ–Š‡”‡Ǥ

‘‡ ˜‡”„• •‡Ž‡…–‹‰ ‹ϐ‹‹–‹˜ƒŽ …‘’Ž‡‡–• —•– ‘” –‡† –‘ …Ž—•–‡” ™‹–Š –Š‡˜‡”„ƒŽŠ‡ƒ†‘ˆ–Šƒ–…‘’Ž‡‡–ǤŠ‹•ƒ›Ž‡ƒ†–‘—Ž–‹Ǧ…Ž—•–‡”•‘ˆ˜‡”„•Ǥ —–…Š‹•‹ˆƒ‘—•ˆ‘”ƒ‹–ƒ‹‹‰ƒ’ƒ”–‹…—Žƒ”‘”†‡”‹–Šƒ–…Ž—•–‡”ǣ–Š‡•‡Ž‡…–Ǧ ‹‰˜‡”„…ƒ‘”—•–’”‡…‡†‡–Š‡Š‡ƒ†‘ˆ‹–•…‘’Ž‡‡–Ǥ—Ž–‹Ǧ…Ž—•–‡”•‘ˆ ˜‡”„•–Šƒ–ƒŽ•‘•‡Ž‡…–‘‹ƒŽ…‘’Ž‡‡–•ǡ–Š‹•‘”†‡”‰‹˜‡•”‹•‡–‘…”‘••‹‰ †‡’‡†‡…‹‡•„‡–™‡‡‘‹ƒŽƒ†’”‡’‘•‹–‹‘ƒŽ‘„Œ‡…–•ƒ†–Š‡‹””‡•’‡…Ǧ –‹˜‡˜‡”„•Ǥ‡”‡‹•ƒ‡šƒ’Ž‡ǡ‹•’‹”‡†„›˜‡”•ȋͳͻ͹͸ȌǤ

ȋ͸Ȍ ǥ†ƒ–ƒ†‡ƒƒ”‹‡‡‡‘‡œƒ‰Š‡Ž’‡Ž‡”‡†ƒ•‡ ǥthat Jan the man Marie a cow saw help teach dance

ǥǮ–Šƒ–ƒ•ƒ™–Š‡ƒŠ‡Ž’ƒ”‹‡–‘–‡ƒ…Šƒ…‘™ȋŠ‘™Ȍ–‘†ƒ…‡ǯ

Š‹••–ƒ–‡‘ˆƒˆˆƒ‹”•‹•”ƒ–Š‡””ƒ”‡ƒ‘‰–Š‡Žƒ‰—ƒ‰‡•‘ˆ–Š‡™‘”Ž†Ǥ–Šƒ• „‡‡ —•‡† –‘ …ŠƒŽŽ‡‰‡ –Š‡ ‹†‡ƒ –Šƒ– –Š‡ ‰”ƒƒ” ‘ˆ ƒ–—”ƒŽ Žƒ‰—ƒ‰‡• „‡ …‘–‡š–Ǧˆ”‡‡ǤŠ‡’‘‹–ˆ‘”—–…Š™ƒ•ƒ†‡„›—‹Œ„”‡‰–•ȋͳͻ͹͸ȌǤŠ‡…‘Ǧ …‡’–‘ˆ…‘–‡š–Ǧˆ”‡‡‡••ƒ‘—–•–‘–Š‡…Žƒ‹–Šƒ––Š‡‹–‡”ƒŽ•–”—…–—”‡‘ˆ …ƒ–‡‰‘”‹‡• ‹ ƒ–—”ƒŽ Žƒ‰—ƒ‰‡ ‹• ‘– ‡••‡–‹ƒŽŽ› †‡–‡”‹‡† ˆ”‘ ‘—–•‹†‡ǡ ƒ†ǡ‡“—‹˜ƒŽ‡–Ž›ǡ–Šƒ–ƒ–—”ƒŽŽƒ‰—ƒ‰‡•…ƒ„‡’ƒ”•‡†Ȃƒ†‰‡‡”ƒ–‡†Ȃ™‹–Š ƒ ‡‘”› ƒƒ‰‡‡– –Šƒ– ‹• ‘”‡ ”‡•–”‹…–‹˜‡ –Šƒ ”ƒ†‘ ƒ……‡••‹„‹Ž‹–›Ǥ Š‡†‹•…—••‹‘‘–Š‡•–ƒ–—•‘ˆ…”‘••‹‰†‡’‡†‡…‹‡•ƒ†”‡Žƒ–‡†’Š‡‘Ǧ ‡ƒŠƒ•„‡‡’ƒ”–‘ˆ–Š‡‰‡‡•‹•‘ˆ‡‡”ƒŽ‹•‡†Š”ƒ•‡–”—…–—”‡”ƒƒ”Ǣ cf. —ŽŽ— ƒ† ƒœ†ƒ” ȋͳͻͺʹȌǤ ‹…‡ …”‘••‹‰ †‡’‡†‡…‹‡• ƒ”‡ ™Šƒ– ƒ†‡ —–…Šˆƒ‘—•ƒ‘‰Ž‹‰—‹•–•ǡ–Š‹•’Š‡‘‡‘‹ϐŽ—‡…‡†‘—”…Š‘‹…‡‘ˆ–Š‡ ’ƒ”–‹…—Žƒ”…ƒ–‡‰‘”‹ƒŽ‰”ƒƒ”‡š’Ž‘‹–‡†‹ĊđĎđĆčǤ‘•‹†‡”ƒ–‹‘•”‡Ž‡˜ƒ– –‘–Š‹•…Š‘‹…‡ƒ”‡†‹•…—••‡†„›–ƒ„Ž‡”ȋʹͲͲͶȌ‹ƒ‡š–”‡‡Ž›—•‡ˆ—Ž‡š‡”…‹•‡ ‘‰”ƒƒ–‹…ƒŽ…ƒ’ƒ…‹–‹‡•„‡›‘†…‘–‡š–Ǧˆ”‡‡‡••ǤŠ‡–‘’‹…™‹ŽŽ„‡ˆ—”–Š‡” †‹•…—••‡†‹•‡…–‹‘ͳǤͺǤ

”‘••‹‰†‡’‡†‡…‹‡•ƒ†˜‡”„Ǧ…Ž—•–‡”‹‰ƒ”‡‹’‘”–ƒ–•‘—”…‡•‘ˆdisconti-nuity‹–Š‡•›–ƒš‘ˆ—–…Šǣ–Š‡’Š‡‘‡‘–Šƒ––™‘’Š”ƒ•‡•–Šƒ–‡•–ƒ„Ž‹•Š ƒ‡••‡–‹ƒŽ‰”ƒƒ–‹…ƒŽ”‡Žƒ–‹‘•Š‹’‘……—”•‡’ƒ”ƒ–‡†ˆ”‘‡ƒ…Š‘–Š‡””ƒ–Š‡” –Šƒ ƒ†Œƒ…‡– ‹ –Š‡ •‡–‡…‡Ǥ ˆ …‘—”•‡ǡ ƒ ˜‡”„ ƒ† ‹–• ‘„Œ‡…– ƒ”‡ •–”‹‹‰ ‡šƒ’Ž‡•‘ˆ•—…Šƒ’ƒ‹”–Š‡•‡Ž˜‡•Ǥ—–ƒ†˜‡”„‹ƒŽ‘”’”‘’‘•‹–‹‘ƒŽƒ†Œ—…–• …ƒƒŽ•‘‘……—”ƒ‹†•–’Š”ƒ•‡•–‘™Š‹…Š–Š‡›…ƒ‘–„‡Ž‹‡††‹”‡…–Ž›ǣ

ȋ͹Ȍ ǥ†ƒ–ƒ†‡ƒ‹••…Š‹‡ƒ”‹‡‡‡‘‡œƒ‰Š‡Ž’‡Ž‡”‡†ƒ•‡ ǥthat Jan the man maybe Marie a cow saw help teach dance ǥǮ–Šƒ–‘Šƒ›„‡•ƒ™ǥǯ

Misschien Ǯƒ›„‡ǯ ‹• Ž‹‡† –‘ –Š‡ ˜‡”„ …‘’Ž‡š ƒ† ‘†‹ϐ‹‡• •‘‡ ‡˜‡–ǡ „—–‘……—”•Š‡”‡ƒ–ƒ•‡‡‹‰Ž›”ƒ†‘’‘•‹–‹‘‹–Š‡•‡“—‡…‡‘ˆ‘‹ƒŽ ’Š”ƒ•‡•Ǥ–•’‘•‹–‹‘ƒ›Šƒ˜‡ƒ‡ˆˆ‡…–ƒ––Š‡Ž‡˜‡Ž‘ˆ‹ˆ‘”ƒ–‹‘•–”—…–—”‡ǡ –Š‘—‰Šǡ„—––Šƒ–‹•‘–™‡ŽŽ‡•–ƒ„Ž‹•Š‡†Ǥ‡”„…Ž—•–‡”‹‰’”‘˜‘‡•–Š‹•–›’‡‘ˆ †‹•…‘–‹—‹–› ‹ –Š‡Mittelfeldǣ ˜‡”„ …Ž—•–‡”‹‰ Ǯ‘„•…—”‡•ǯ …‘•–‹–—‡…›ǡ ƒ† …‘•‡“—‡–Ž›–Š‡•‡ƒ–‹…‘”•›–ƒ…–‹…–ƒ”‰‡–‘ˆƒ†Œ—…–•Šƒ•–‘„‡†‡–‡”‹‡† ƒ†Ȁ‘””‡…‘•–”—…–‡†„›‘”‡…‘’Ž‡šƒ‘‡—˜”‡•–ŠƒŒ—•–…‘…ƒ–‡ƒ–‹‘Ǥ

—•– Ž‹‡ •‘‡ ‘–Š‡” Žƒ‰—ƒ‰‡•ǡ —–…Š Šƒ• ƒ ˜‡”„Ǧ•‡…‘† ‡ˆˆ‡…–Ǥ  –Š‡ ƒ„•‡…‡‘ˆƒ…‘’Ž‡‡–‹œ‡”ǡƒϐ‹‹–‡˜‡”„‘”ƒ—š‹Ž‹ƒ”›Ȃ‹—–…Šǡϐ‹‹–‡‡•• ‹• ‡š…Ž—•‹˜‡Ž› ‘”’Š‘Ž‘‰‹…ƒŽŽ› ƒ”‡† Ȃ …ƒ ‘” —•– ‘……—’› –Šƒ– ’‘•‹–‹‘ǡ –Š‡”‡„›†‡–‡”‹‹‰–Š‡•–ƒ–—•‘ˆ–Š‡•‡–‡…‡Ǥ

ȋͺȌ ‹‡ƒ†Š‡‡ˆ–†‡†”‘‡‘‘†—”˜‡…‘””‹‰‡”‡

nobody has the drunken uncle dare correct