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Unifying Logical Form and The Linguistic Level of LF

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Unifying Logical Form and The Linguistic Level of LF

  1. 1. ECAP 6 — Sixth European Congress of Analytic Philosophy Krakow August, 26, 2008“Unifying Logical Form & The linguistic Level of LF”  Mouhamadou El Hady BA Institut Jean Nicod ENS-EHESS Paris
  2. 2. Overview of the Talk: The Generative Framework The Linguistic level of LF Logicians Logical Form Unification via generalized quantification Interest & problems Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 2
  3. 3. Introduction: Logic: an artificial & well defined language + rules of inference − Looking for sound derivations − regardless of actual implementation into the human mind Generative Linguistics: define an abstract grammar but above all an empirical science looking for the laws actually used by the speaker Claim: these two research programs converge via their use of Generalized quantification − logical laws of thought Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 3
  4. 4. The Generative Framework:Whats a Grammar? "The grammar of a language purports to be a description of the intrinsic competence of an ideal speaker-listener. If the grammar is, moreover, perfectly explicit – i.e. if it doesnt simply accept the understanding if intelligent reader but gives an explicit analysis of the activity that the speaker-listener display- we can, with some redundancy, call it a generative grammar. "Noam Chomsky (1965) : Aspects of the theory of syntax Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 4
  5. 5. The Generative Framework:Whats a Grammar? − (p) Colorless green ideas sleep furiously − (q) Walk pixel hands color john foolishly Distinction Phonology/Syntax/Semantics: distinct and somewhat independent modules of the language faculty manage these aspects of natural languages Thats why we recognize (p) as a sentence, understand it and judge it absurd when we cant even say of (q) that it really is a sentence. Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 5
  6. 6. The linguistic Level of LF “The basic elements we consider are sentences; the grammar generates mental representations of their form and meaning. I will call these representations, respectively, phonetic forms and LF (which I will read, logical form, though with a cautionary note)”Chomsky[1980], Rules & Representations, p. 143 representations of their form and meaning→ LF a semantic level? Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 6
  7. 7. The Linguistic Level of LFLF as a semantic level? “LF is thus the interface between grammar and the conceptual-intentional properties of language, just as the level of Phonetic Form (PF) is an interface between grammar and the audio-perceptual properties of utterances. LF is not to be equated with the level of semantic structure anymore than PF is to be treated as a level specifying the sound waves of any given utterance. It expresses only aspects of semantic structure that are [...] contributed by grammar.”CT James Huang [1994], “Logical Form” Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 7
  8. 8. The Linguistic Level of LF  SS: Surface Structures  PF: Phonetic Forms  LF: Logical Form We are only interested in the SS/ LF branch Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 8
  9. 9. The Linguistic Level of LF Defining Move α Minimalism: • Only one rule: Movement (Move α) with constraints • E.g.: Move NPs to the head of the sentence leaving traces at their former place “The transformational mapping to S-structure can be reduced to (possibly multiple application of) a single general rule “Move α,” where α is an arbitrary phrase category” Chomsky [1980], Rules & Representations, p. 145 Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 9
  10. 10. The Linguistic Level of LF Why should we add a covert level of LF? To understand the interpretation of Wh-phrases To uncover the reference of pronouns To explain syntactically why some ambiguities arise: Ex: Lets consider this sentence with two NPs : (d) Every philosopher loves some linguist Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 10
  11. 11. The Linguistic Level of LF (d) Every philosopher loves some linguist Application of Move α : Quantifier Raising – Move the NPs at the head of the sentence – Leave traces at their former place – Traces are kind of variables like in FOLs predicate language Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 11
  12. 12. The Linguistic Level of LF Two possibles LF formula: (d1) [Every philosopher]1[Some linguist]2 [t1 loves t2] (d2) [Some linguist]2 [Every philosopher]1[t1 loves t2] Hence the ambiguity: possibly many linguists in (d1) but same beloved linguist in (d2). Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 12
  13. 13. The Linguistic Level of LF Traces: (d1) [Every philosopher]1[Some linguist]2 [t1 loves t2] (d2) [Some linguist]2 [Every philosopher]]1[t1 loves t2] The traces t1 & t2 variables binded by the NPs Variables, Quantified Noun Phrases, Rings a bell? Thats logic, isnt it? But lets do some real logic and we will come back to see if we can unify logic andlinguistics Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 13
  14. 14. Logicians Logical FormLogic: Consensus till Montague: natural languages cant handle the task offormalizing our reasonings“Languages are not made so as to match logics ruler. Even the logical element inlanguage seems hidden behind pictures that are not always accurate.”Frege [1906] Letter to Husserl“The syntax of ordinary language, as is well known, is not quite adequate for thispurpose. It does not in all cases prevent the construction of nonsensical pseudo-propositions”Wittgenstein [1929]: Some remarks on Logical Form Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 14
  15. 15. Logicians Logical FormRussell On denoting:Breaking up Denoting Phrases: “The phrase per se has no meaning, because in any proposition in which it occurs the proposition, fully expressed, does not contain the phrase, which has been broken up” “Consider the next the proposition all men are mortal. This proposition is really hypothetical and states that if anything is a man, it is mortal.” Russell (1905) ∀ x [M(x) →D(x)] → Grammatical structure is irrelevant Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 15
  16. 16. Logicians Logical FormSolution:Translate NL utterances into an artificial language displaying their “real” logicalstructure and using correct laws of entailment Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 16
  17. 17. Logicians Logical FormFrege on quantification: Numbers and quantifiers (∀, ∃) are second order predicates expressing relations between concepts rather than properties of objects For example: (f) The shirt is red (g) Every shirt is red Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 17
  18. 18. Logicians Logical FormFrege on quantification:Universe: {shirts} (f): ( ) is red is attributed to an object (a particular shirt) (g): a bit more complexAnalyse of (g) according to Frege: two concepts: ( ) is a shirt and ( ) is red, (g) statesa relation between these concepts namely that whatever satisfy ( ) is a shirt satisfy ( )is red. So quantifiers are second order concepts expressing something about first orderconcepts Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 18
  19. 19. Logicians Logical FormRussell On denoting:Variables first :“I take the notion of the variable as fundamental...” → Quantifiers are just variable binding operators Do not really state that Quantifiers are not second order predicates but limits the explorations in this direction Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 19
  20. 20. Logicians Logical FormFrege, Russell, Quine....Consensus: Natural languages are improper for good reasoning. A logician has to use an artificial language in order to display the real structure and make correct inferences. Enters Montague! Respected philosopher & logician Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 20
  21. 21. Unification via generalized QuantifiersMontague: English as a formal Language, PTQ, UG1- English, like any other NL, is just an interpreted formal language “There is in my opinion no important theoretical difference between natural language and artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory.” Universal Grammar2- NPs are Generalized quantifiers Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 21
  22. 22. Unification via generalized QuantifiersGeneralized quantifiers:Every king of France is bald∀x [K(x) → B(x)]What about Most Kings of France are Bald?We cant analyse it in predicate logic.Even if we introduce “Q”, a quantifier, meaning “most” Qx [K(x) → B(x)] Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 22
  23. 23. Unification via generalized QuantifiersGeneralized quantifiers: Qx [K(x) → B(x)] This impossibility shows that Quantifiers are not just variable binding operators but true second order concepts letting us define and manipulate sets into the universe Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 23
  24. 24. Unification via generalized QuantifiersGeneralized quantifiers:Back to Frege: with quantifiers, we compare sets “Every King of France is bald” means that the set of kings of France is a subset of the set of Bald persons. Likewise, “Most Kings of France are bald” means that the set of bald kings of France is larger than the set of hairy Kings of FranceHow could we formalize this? Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 24
  25. 25. Unification via generalized QuantifiersGeneralized quantifiers:How could we formalize this? [Qx K(x)] [B(x) ]Reading: “For most x such that x is King of France, x is bald ”Works for every and some [∀x K(x)] [B(x) ] [∃x K(x)] [B(x) ] Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 25
  26. 26. Unification via generalized QuantifiersGeneralized quantifiers:It even works for proper names [Johnx K(x)] [B(x)]Reading: “For John x such that John is King of France, John is bald ”Nota: [Johnx K(x)] is not a name but a quantifier denoting the intersection of all the sets of which John King of France is an element Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 26
  27. 27. Unification via generalized QuantifiersGeneralized quantifiers: [∀x K(x)] [B(x) ] [∃x K(x)] [B(x) ] [Qx K(x)] [B(x) ] [Johnx K(x)] [B(x)]Note that we are back to the grammatical NP/VP structure.Moreover, in generative linguistics also QNPs are Generalized Quantifiers cf. for example Fox[2002] Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 27
  28. 28. Unification via generalized QuantifiersFox (2002): «QNPs denote second order predicates. They convey information about basic (firstorder) predicates like tall; they tell us something about the set of individuals that agiven (first order) predicate is true of. So in the sentences in (1) the relevantpredicate is tall. And, given the meaning of the specific QNPs, the sentences conveythe information that the predicate is true of at least one girl, (1)a, many girls, (1)b,every girl, (1)c, or no girl, (1)d. » Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 28
  29. 29. Unification via generalized QuantifiersQuantifier Raising just reveal the true logical structure of Natural Languagesentencesergo Linguists LF = Logicians Logical Form Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 29
  30. 30. Interest & Problems Interest: * Naturalising logic * Flesh out the LOT hypothesis * Clarify the debates over contextualism [Wordnet, conceptual graphs → Free Enrichment] * Translate our reasoning mechanisms into a well known mathematical theory: set theory Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 30
  31. 31. Interest & Problems Problems: * How to go from parallelism to identification * Even if LF is like a GQ Logic, it doesnt necessarily mean that rules of Grammar are logical rules of inference * Is using grammar identical to reasoning? Looks awfully whorfian Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 31
  32. 32. Many Thanks to you,to Pierre Stanislas Grialou & to Santiago Echeverry Hady Ba, "Unifying Logical Form & FL", ecap 6, Krakow, August 2008 32

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