Unifying Logical Form and The Linguistic Level of LF

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

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

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


The Generative Framework:

What's 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
doesn't 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


The Generative Framework:

What's 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


That's why we recognize (p) as a sentence, understand it and judge it absurd when
we can't even say of (q) that it really is a sentence.


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?


The Linguistic Level of LF

LF 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”




SS: Surface Structures


PF: Phonetic Forms


LF: Logical Form

We are only interested in the SS/ LF branch



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




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: Let's consider this sentence with two NPs :

(d) Every philosopher loves some linguist



(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 FOL's predicate language



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).



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? That's logic, isn't it?

But lets do some real logic and we will come back to see if we can unify logic and
linguistics


Logician's Logical Form
Logic: Consensus 'till Montague: natural languages can't handle the task of
formalizing our reasonings

“Languages are not made so as to match logic's ruler. Even the logical element in
language 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 this
purpose. It does not in all cases prevent the construction of nonsensical pseudo-
propositions”

Wittgenstein [1929]: Some remarks on Logical Form


Russell 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



Solution:

Translate NL utterances into an artificial language displaying their “real” logical
structure and using correct laws of entailment



Frege 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



Frege on quantification:

Universe: {shirts}
(f): ( ) is red is attributed to an object (a particular shirt)
(g): a bit more complex
Analyse of (g) according to Frege: two concepts: ( ) is a shirt and ( ) is red, (g) states
a relation between these concepts namely that whatever satisfy ( ) is a shirt satisfy ( )
is red.
So quantifiers are second order concepts expressing something about first order
concepts



Russell 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



Frege, 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


Unification via generalized Quantifiers

Montague:

English as a formal Language, PTQ, UG
1- 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 Grammar
2- NPs are Generalized quantifiers



Generalized quantifiers:

Every king of France is bald

∀x [K(x) → B(x)]

What about Most Kings of France are Bald?

We can't analyse it in predicate logic.
Even if we introduce “Q”, a quantifier, meaning “most”

Qx [K(x) → B(x)]




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



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 France

How could we formalize this?




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) ]




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




[∀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]



Fox (2002):

«QNPs denote second order predicates. They convey information about basic (first
order) predicates like 'tall'; they tell us something about the set of individuals that a
given (first order) predicate is true of. So in the sentences in (1) the relevant
predicate is tall. And, given the meaning of the specific QNPs, the sentences convey
the 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. »



Quantifier Raising just reveal the true logical structure of Natural Language
sentences

ergo

Linguist's LF = Logicians Logical Form


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


Interest & Problems


Problems:

* How to go from parallelism to identification

* Even if LF is like a GQ Logic, it doesn't necessarily mean that rules of
Grammar are logical rules of inference

* Is using grammar identical to reasoning? Looks awfully whorfian


Many Thanks to you,

to Pierre Stanislas Grialou & to Santiago Echeverry


Unifying Logical Form and The Linguistic Level of LF

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