natural logic? lauri karttunen cleo condoravdi annie zaenen palo alto research center

Natural Logic?Natural Logic?

Lauri Karttunen

Cleo Condoravdi

Annie Zaenen

Palo Alto Research Center

OverviewOverview

Part 1 Why Natural Logic

MacCartney’s NatLog

Part 2PARC Bridge

Discussion

Anna Szabolski on the semantic Anna Szabolski on the semantic enterprise (2005)enterprise (2005)

On this view, model theory has primacy over proof theory. A language may be defined or described perfectly well without providing a calculus (thus, a logic) for it, but a calculus is of distinctly limited interest without a class of models with respect to which it is known to be sound and (to some degree) complete. It seems fair to say that (a large portion of) mainstream formal semantics as practiced by linguists is exclusively model theoretic. As I understand it, the main goal is to elucidate the meanings of expressions in a compositional fashion, and to do that in a way that offers an insight into natural language metaphysics (Bach 1989) and uncovers universals of the syntax/semantics interface 4.

Anna SzabolskiAnna Szabolski

The idea that our way of doing semantics (model-theory) is both insightful and computationally (psychologically) unrealistic has failed to intrigue formal semanticists into action. Why? There are various, to my mind respectable, possibilities.

(i) Given that the field is young and still in the process of identifying the main facts it should account for, we are going for the insight as opposed to the potential of computational / psychological reality.

(ii) We don’t care about psychological reality and only study language in the abstract.

(iii) We do care about potential psychological reality but are content to separate the elucidation of meaning (model theory) from the account of inferencing (proof theory). But if the machineries of model theory and proof theory are sufficiently different, option (iii) may end up with a picture where speakers cannot know what sentences mean, so to speak, only how to draw inferences from them. Is that the correct picture?

Why model theory is not in fashion in Why model theory is not in fashion in Computational LinguisticsComputational Linguistics

Computers don’t have realistic models up to now; everything is syntax

Moss (2005)Moss (2005)

If one is seriously interested in entailment, why not study it axiomatically instead of building models? In particular, if one has a complete proof system, why not declare it to be the semantics? Indeed, why should semantics be founded on model theory rather than proof theory?”

Why full-fledged proof theory is not in Why full-fledged proof theory is not in fashion in Computational Linguisticsfashion in Computational Linguistics

Too big an enterprise to be undertaken in one go

FOL is undecidable.

Ambitious attempt: Fracas (DRS)

Need to work up our way through decidable logics: Moss’s hierarchy

Unfortunately limited to the logical connectives

Natural Logic?Natural Logic?

Long tradition: Aristotle, scholastics, Quine(?) Wittgenstein(?), Davidson, Parsons, ...

LakoffLakoff(i) We want to understand the relationship between grammar and reasoning.

(ii) We require that significant generalizations, especially linguistic ones, be stated.

(iii) On the basis of (i) and (ii), we have been led tentatively to the generative semantics hypothesis. We assume that hypothesis to see where it leads.

Given these aims, empirical linguistic considerations play a role in determining what the logical forms of sentences can be. Let us now consider certain other aims.

(iv) We want a logic in which all the concepts expressible in natural language can be expressed unambiguously, that is, in which all non- synonymous sentences (at least, all sentences with different truth conditions) have different logical forms.

(v) We want a logic which is capable of accounting for all correct inferences made in natural language and which rules out incorrect ones.

We will call any logic meeting the goals of (i)-(v) a 'natural logic'.

Basic ideaBasic idea

Some inferences can be made on the basis of linguistic form alone.John and Mary danced.

John danced. Mary danced.

The boys sang beautifully. The boys sang.

But:Often studied by philosophers interested in a limited

number of phenomena

Often ignoring the effect of lexical items.

Problem 1: impact of lexical items is Problem 1: impact of lexical items is pervasivepervasive

John and Mary carried the piano.?? John carried the piano.

The boys sang allegedly.?? The boys sang.

There are no structural inferences without lexical items playing a role.

When lexical items are taken into account, the domain of ‘natural logic’ goes beyond what has been studied under that name up to now.

Problem 2: Need for disambiguation Problem 2: Need for disambiguation we cannot work on literal stringswe cannot work on literal strings

The members of the royal family are visiting dignitaries.

visiting dignitaries can be boring.

a. Therefore, the members of the royal family can be boring.

b. Therefore, what the members of the royal family are doing can be boring.

Advantages of natural logicAdvantages of natural logic

Lexico-syntactic

Incremental

What is doableWhat is doable

‘Syntactic’ approaches geared to specific inferences

Examples: MacCartney’s approach to Natural Logic

PARC’s Bridge

Textual entailment (minimal world knowledge)

Geared to existential claims (What happened, where, when)

Existential claimsWhat happened? Who did what to whom?

Microsoft managed to buy Powerset.

Microsoft acquired Powerset.

Shackleton failed to get to the South Pole.

Shackleton did not reach the South Pole.

The destruction of the file was not illegal.

The file was destroyed.

The destruction of the file was averted.

The file was not destroyed.

MonotonicityWhat happened? Who did what to whom?

Every boy managed to buy a small toy.

Every small boy acquired a toy.

Every explorer failed to get to the South Pole.

No experienced explorer reached the South Pole.

No file was destroyed.

No sensitive file was destroyed.

The destruction of a sensitive file was averted.A file was not destroyed.

The creation of a new benefit was averted.A benefit was not created.

Recognizing Textual InferencesRecognizing Textual Inferences

MacCartney’s Natural Logic (NatLog)MacCartney’s Natural Logic (NatLog)

Point of departure: Sanchez Valencia’s elaborations of Van Benthem’s Natural Logic

Seven relevant relations:x≡y equivalence couch ≡ sofa x=yx y ⊏ forward entailment crow bird⊏ x y⊂x y ⊐ reverse entailment Asian Thai⊐ x y⊃x^y negation able^unable x y = 0 x y=U⋂ ⋀ ⋃x|y alternation cat|dog x y = 0 x y≠U⋂ ⋀ ⋃x y ‿ cover animal non-ape‿ x y ≠ 0 x y=U⋂ ⋀ ⋃x#y independence hungry#hippo

Table of joins for 7 basic entailment Table of joins for 7 basic entailment relationsrelations

≡ ⊏ ⊐ ^ | ‿ #

≡ ≡ ⊏ ⊐ ^ | ‿ #

⊏ ⊏ ⊏ ≡⊏⊐|# | | ⊏^| #‿ ⊏|#

⊐ ⊐ ≡⊏⊐‿# ⊐ ‿ ⊐^| #‿ ‿ ⊐‿#

^ ^ ‿ | ≡ ⊐ ⊏ #

| | ⊏^| #‿ | ⊏ ≡⊏⊐|# ⊏ ⊏|#

‿ ‿ ‿ ⊐^| #‿ ⊐ ⊐ ≡⊏⊐‿# ⊐‿#

# # ⊏‿# ⊐|# # ⊐|# ⊏‿# ≡⊏⊐^| #‿

Cases with more than one possibility indicate loss of information. The join of # and # is totally uninformative.

Entailment relations between expressions Entailment relations between expressions differing in atomic edits (substitution, differing in atomic edits (substitution, insertion, deletion)insertion, deletion)

Substitutions: open classes (need to be of the same type)

Synonyms: ≡ relation

Hypernyms: relation (crow bird)⊏Antonyms: | relation (hot|cold) Note: not ^ in most cases, no

excluded middle.

Other nouns: | (cat|dog)

Other adjectives: # (weak#temporary)

Verbs: ??

…

Geographic meronyms: (in Kyoto in Japan) but note: not ⊏ ⊏without the preposition Kyoto is beautiful Japan is ⊏beautiful

Substitutions: closed classes, example quantifiers:

all ≡ every

every some (non-vacuity assumed)⊏some ^ no

no | every (non-vacuity assumed)

four or more two or more⊏exactly four | exactly two

at most four at least two (overlap at 2, 3, 4)‿most # ten or more

Deletions and insertions: default: (upward-⊏monotone contexts are prevalent)e.g. red car car⊏But doesn’t hold for negation, non-intersective

adjectives, implicatives.

CompositionComposition

Bottom up

nobody can enter without a bottle of wine

nobody can enter without a bottle of liquor

(nobody (can (enter (without wine)))

lexical entailment:

wine liquor⊏without: downward monotone

without wine without liquor⊐can upward monotone, nobody downward monotone

nobody can enter without wine nobody can enter without liquor⊏

Composition: projectivity of logical Composition: projectivity of logical connectivesconnectives

connective ≡ ⊏ ⊐ ^ | ‿ #

Negation (not) ≡ ⊐ ⊏ ^ ‿ | #

Conjunction (and)/intersection

≡ ⊏ ⊐ | | # #

Disjunction (or) ≡ ⊏ ⊐ ‿ # ‿ #

Conditional antecedent

≡ ⊐ ⊏ # # # #

Conditional consequent

≡ ⊏ ⊐ | | # #

Biconditional ≡ # # # # # #



Conjunction (and)/intersection

≡ ⊏ ⊐ | | # #

happy ≡ glad kiss touch⊏ kiss and hug touch and hug⊏human ^ nonhuman living human | living nonhumanFrench | German French wine | Spanish wineMetallic nonferrous‿ metallic pipe # nonferrous pipeswimming # hungry



Disjunction (or) ≡ ⊏ ⊐ ‿ # ‿ #

happy ≡ glad happy or rich ≡ glad or richkiss touch⊏ kiss or hug touch or hug⊏human ^ nonhuman human or equine ^ nonhuman or equineFrench | German French or Spanish # German or Spanishmore that 4 less than 6‿ 3 or more than 4 3 or less than 6‿

Composition: projectivity of logical Composition: projectivity of logical connectivesconnectivesconnective ≡ ⊏ ⊐ ^ | ‿ #

Negation (not) ≡ ⊐ ⊏ ^ ‿ | #

happy ≡ glad not happy ≡ not gladkiss touch⊏ not kiss not touch⊐human ^ nonhuman not human ^ not nonhumanFrench | German not French not German‿more that 4 less than 6‿ not more than 4 | not less than 6swimming # hungry not swimming # not hungry

Compositionality: projectivity of Compositionality: projectivity of quantifiersquantifiers

quantifier 1st argument 2nd argument

≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

some ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊏ ⊐ º # ⏝ #

no ≡ ⊐ ⊏ | # | # ≡ ⊐ ⊏ | # | #

every ≡ ⊐ ⊏ | # | # ≡ ⊏ ⊐ | | # #

not every ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊐ ⊏ ⏝ ⏝ # #

at least two ≡ ⊏ ⊐ # # # # ≡ ⊏ ⊐ # # # #

most ≡ # # # # # # ≡ ⊏ ⊐ | | # #

exactly one ≡ # # # # # # ≡ # # # # # #

all but one ≡ # # # # # # ≡ # # # # # #

Compositionality: projectivity of Compositionality: projectivity of quantifiers: some, first argumentquantifiers: some, first argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

some ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊏ ⊐ ⏝ # ⏝ #

couch,sofa: some couches sag ≡ some sofas sag

finch,bird: some finches sing ⊏ some birds sing

boy,small boy: some boys sing ⊐ some small boys sing

human, non-human: Some humans sing some non-humans sing⏝boy,girl: Some boys sing # Some girls sing

animal,non-ape: Some animals breathe some non-apes breathe⏝

≣

⊏

⊐

^

|

⏝#

Compositionality: projectivity of Compositionality: projectivity of quantifiers: some, second argumentquantifiers: some, second argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

some ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊏ ⊐ ⏝ # ⏝ #

beautiful,pretty: some couches are pretty ≡ some couches are beautiful

sing beautifully,sing: some finches sings beautifully some finches sing⊏sing, sing beautifully: some finches sing some finches sing beautifully⊐human,non-human: some humans sing some non-humans sing⏝late|early: some people were early # some people were late

≣

⊏

⊐

^

|

Compositionality: projectivity of Compositionality: projectivity of quantifiers: no; first argumentquantifiers: no; first argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

no ≡ ⊐ ⊏ | # | # ≡ ⊐ ⊏ | # | #

≡⊏

⊐

^

|

⏝

couch,sofa: no couches sag ≡ no sofas sag

finch,bird: no finches sing ⊐ no birds sing

boy,small boy: no boys sing ⊏ no small boys sing

human, non-human: no humans sing | no non-humans sing

boy, girl: no boys sing # no girls sing

Animal,non-ape: no animals breathe | no non-apes breathe

Compositionality: projectivity of Compositionality: projectivity of quantifiers: every; first argumentquantifiers: every; first argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

every ≡ ⊐ ⊏ | # | # ≡ ⊏ ⊐ | | # #

≡⊏

⊐

^

|

⏝

couch,sofa: every couch sags ≡ every sofa sags

finch,bird: every finch sings ⊐ every bird sings

boy,small boy: every boy sings ⊏ every small boy sings

human, non-human: every human sings | every non-human sings

boy, girl: every boy sings # every girl sings

animal, non-ape: every animal breathes | every non-ape breathes

Compositionality: projectivity of Compositionality: projectivity of quantifiers: not every; first argumentquantifiers: not every; first argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

not every ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊐ ⊏ ⏝ ⏝ # #

≡⊏

⊐

^

|

⏝#

couch, sofa: not every couch sags ≡ not every sofa sags

finch, bird: not every finch sings not every bird sings⊏boy, small boy: not every boy sings not every small boy sings⊐human, non-human: not every human sings not every non-human sings⏝boy, girl: not every boy sings # not every girl sings

animal, non-ape: not every animal breathes not every non-ape breathes⏝

Projectivity of VerbsProjectivity of Verbs

most verbs are upward monotone and project ^, |, and ⏝ as #

humans ^ nonhumans

eats humans # eats non-humans

but there are a lot of exceptions

verbs with sentential complements require special treatment: factives, counterfactives, implicatives… (Parc verb classes)

FactivesFactives

Class Inference Pattern

Positive

Negative

++/-+ forget that

is odd that

forget that X ⇝ X, not forget that X ⇝ X

is odd that X ⇝ X, is not odd that X ⇝ X

+-/-- pretend that

pretend to

pretend that X not ⇝ X, not pretend that X not ⇝X

pretend to X not ⇝ X, not pretend to X not ⇝ X

Abraham pretended that Sarah was his sister. Sarah was not his sister⇝

Howard did not pretend that it did not happen. It happened.⇝

host polarity complement polarity+ +host polarity complement polarity- +host polarity complement polarity+ -

host polarity complement polarity- -

forget that

forget that

pretend that

pretend that

ImplicativesImplicatives

++/-- manage to

+-/-+ fail to

manage to X ⇝ X, not manage to X not ⇝ X

fail to X not ⇝ X, not fail to X ⇝ X

++ force to force X to Y ⇝ Y

+- refuse to refuse to X not X⇝

-- be able to not be able to X not X⇝

-+ hesitate to not hesitate to X X⇝

Class Inference Pattern

Two-wayimplicatives

One-wayimplicatives

Translating PARC classes into the Translating PARC classes into the MacCartney approachMacCartney approach

sign del ins example

implicatives ++/-- ≡ ≡ He managed to escape ≡ he escaped

++ ⊏ ⊐ He was forced to sell he sold⊏

-- ⊐ ⊏ He was permitted to live he did live⊐

+-/-+ ^ ^ He failed to pay ^ he paid

+- | | He refused to fight | he fought

-+ ‿ ‿ He hesitated to ask he asked‿

factives +-/+

+-/-

Neutral # # He believed he had won/ he had won

does not take the presuppositions of the implicatives into account

T. Ed didn’t forget to force Dave to leave

H. Dave left

i f(e) g(xi-1,e)projections

h(x0,xi)joins

0 Ed didn’t fail to force Dave to leave

1 Ed didn’t force Dave to leave DEL(fail) ^ Context downward monotone: ^

^

2 Ed forced Dave to leave DEL(not) ^ Context upward monotone: ^

Join of ^,^: ≡

3 Dave left DEL(force) ⊏ Context upward monotone: ⊏

Join of ≡, : ⊏ ⊏

t: We were not able to smoket: We were not able to smokeh: We smoked Cuban cigarsh: We smoked Cuban cigars

i xi ei f(ei) g(xi-1,ei) h(x0,xi)

0 We were not able to smoke

1 We did not smoke DEL(permit)

⊐ Downward monotone:⊏

⊏

2 We smoked DEL(not) ^ Upward monotone: ^ Join of ,^: ⊏|

3 We smoked Cuban cigars

INS(C.cigars)

⊐ Upward monotone: ⊐ Join of |, : ⊐|

We end up with a contradiction

Why do the factives not work?Why do the factives not work?

MacCartney’s system assumes that the implicatures switch when negation is inserted or deleted

But that is not the case with factives and counterfactives, they need a special treatment

Other limtationsOther limtations

De Morgan’s laws: Not all birds fly some birds do not fly

Buy/sell, win/lose

Doesn’t work with atomic edits as defined.

Needs larger units

Bridge vs NatLogBridge vs NatLog

NatLogSymmetrical between t and h

Bottom up

Local edits, more compositional

Surface based

Integrates monotonicity and implicatives tightly

BridgeAsymmetrical between t and h

Top down

Global rewrites possible

Normalized input

Monotonicity calculus and implicatives less tightly coupled

We need more power than NatLog allows for but it needs to be deployed in a more constrained way

than the current Bridge system demonstrates

PARC’s BRIDGE System PARC’s BRIDGE System

Anna Szabolski 2005Anna Szabolski 2005

Consider the model theoretic and the natural deduction treatments of the propositional connectives. The two ways of explicating conjunction and disjunction amount to the same thing indeed: if you know the one you can immediately guess the other. Not so with classical negation. The model theoretic definition is in one step: ¬p is true if and only if p is not true. In contrast, natural deduction obtains the same result in three steps. First, elimination and introduction rules for ¬ yield a notion of negation as in minimal logic. Then the rule Ex Falso Sequitur Quodlibet is added to obtain intuitionistic negation, and finally Double Negation Cancellation to obtain classical negation. While it may be a matter of debate which explication is more insightful, it seems clear that the two are intuitively not the same, even though eventually they deliver the same result.

Van BenthemVan Benthem

“Dictum de Omni et Nullo”:

admissible inferences of two kinds:

downward monotonic (substituting stronger predicates for weaker ones),

upward monotonic (substituting weaker predicates for stronger ones).

Conservativity: Q AB iff Q A(BintersectionA)

Toward NL Understanding

Local Textual Inference A measure of understanding a text is the ability to make

inferences based on the information conveyed by it.

Veridicality reasoningDid an event mentioned in the text actually occur?

Temporal reasoningWhen did an event happen? How are events ordered in time?

Spatial reasoningWhere are entities located and along which paths do they

move?

Causality reasoning Enablement, causation, prevention relations between events

Knowledge about words for access to content

The verb “acquire” is a hypernym of the verb “buy”The verbs “get to” and “reach” are synonyms

Inferential properties of “manage”, “fail”, “avert”, “not”

Monotonicity properties of “every”, “a”, “no”, “not”Every (↓) (↑), A (↑) (↑), No(↓) (↓), Not (↓)

Restrictive behavior of adjectival modifiers “small”, “experienced”, “sensitive”

The type of temporal modifiers associated with prepositional phrases headed by “in”, “for”, “through”, or even nothing (e.g. “last week”, “every day”)

Construction of intervals and qualitative relationships between intervals and events based on the meaning of temporal expressions

Local Textual Inference InitiativesLocal Textual Inference Initiatives

PASCAL RTE Challenge (Ido Dagan, Oren Glickman) 2005, 2006

PREMISE

CONCLUSIONTRUE/FALSE

Rome is in Lazio province and Naples is in Campania.

Rome is located in Lazio province.

TRUE ( = entailed by the premise)

Romano Prodi will meet the US President George Bush in his capacity as the president of the European commission.

George Bush is the president of the European commission.

FALSE (= not entailed by the premise)

World knowledge intrusion World knowledge intrusion



FALSE


Romano Prodi is the president of the European commission.

TRUE

G. Karas will meet F. Rakas in his capacity as the president of the European commission.

F. Rakas is the president of the European commission.

TRUE

Inference under a particular construalInference under a particular construal



FALSE (= not entailed by the premise on the correct anaphoric resolution)

G. Karas will meet F. Rakas in his capacity as the president of the European commission.

F. Rakas is the president of the European commission.

TRUE (= entailed by the premise on one anaphoric resolution)

Compositionality: projectivity of Compositionality: projectivity of quantifiers: not every; second argumentquantifiers: not every; second argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

not every ≡ ⊏ ⊐ ⏝ # ⏝ # ≡ ⊐ ⊏ ⏝ ⏝ # #

≣

⊏

⊐

^

|

⏝#



≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

at least two ≡ ⊏ ⊐ # # # # ≡ ⊏ ⊐ # # # #



≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

most ≡ # # # # # # ≡ ⊏ ⊐ | | # #



≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

exactly one ≡ # # # # # # ≡ # # # # # #



≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

all but one ≡ # # # # # # ≡ # # # # # #

somesome

Compositionality: projectivity of Compositionality: projectivity of quantifiers: no; second argumentquantifiers: no; second argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

no ≡ ⊐ ⊏ | # | # ≡ ⊐ ⊏ | # | #

≡⊏

⊐

^

|

⏝#

Compositionality: projectivity of Compositionality: projectivity of quantifiers: every; second argumentquantifiers: every; second argument


≡ ⊏ ⊐ ^ | ⏝ # ≡ ⊏ ⊐ ^ | ⏝ #

every ≡ ⊐ ⊏ | # | # ≡ ⊏ ⊐ | | # #

≡⊏

⊐

^

|

⏝#



Biconditional ≡ # # # # # #



Conditional antecedent

≡ ⊐ ⊏ # # # #

kiss touch⊏If she kissed her, she likes her if she touched her, she likes her⊐human ^ nonhuman



Conditional consequent

≡ ⊏ ⊐ | | # #

kiss touch⊏If he wins I’ll kiss him if he wins I’ll touch him⊏

human ^ nonhumanIf it does this it shows that it is human | if it does this it shows that it is nonhuman

French | GermanIf he wins he gets French wine | if he wins he gets German wine

natural logic? lauri karttunen cleo condoravdi annie zaenen palo alto research center

Documents

syntax slide

semantics modeltheory

domain of natural logic

machineries of model

fullfledged proof theory

parc bridge discussion

correct inferences

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