Natural Logic?
Lauri Karttunen
Cleo Condoravdi
Annie Zaenen
Palo Alto Research Center
Overview
Part 1
Why Natural Logic
MacCartney’s NatLog
Part 2
PARC Bridge
Discussion
Anna Szabolski on the semantic
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 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
Computational Linguistics
Computers don’t have realistic models up to
now; everything is syntax
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
fashion 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?
Long tradition: Aristotle, scholastics, Quine(?)
Wittgenstein(?), Davidson, Parsons, ...
Lakoff
(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 nonsynonymous 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 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
pervasive
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 
we 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 logic
Lexico-syntactic
Incremental
What 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 claims
What 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.
Monotonicity
What 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 Inferences
MacCartney’s Natural Logic (NatLog)
Point of departure: Sanchez Valencia’s
elaborations of Van Benthem’s Natural Logic
Seven relevant relations:
x≡y
x⊏y
x⊐y
x^y
x|y
x‿y
x#y
equivalence
couch ≡ sofa
x=y
forward entailment
crow⊏bird
x⊂y
reverse entailment
Asian⊐Thai
x⊃y
negation
able^unable
x⋂y = 0⋀x⋃y=U
alternation
cat|dog
x⋂y = 0⋀x⋃y≠U
cover
animal‿non-apex⋂y ≠ 0⋀x⋃y=U
independence
hungry#hippo
Table of joins for 7 basic entailment
relations
≡
⊏
⊐
^
|
‿
#
≡
≡
⊏
⊐
^
|
‿
#
⊏
⊏
⊏
≡⊏⊐|#
|
|
⊏^|‿#
⊏|#
⊐
⊐
≡⊏⊐‿#
⊐
‿
⊐^|‿#
‿
⊐‿#
^
^
‿
|
≡
⊐
⊏
#
|
|
⊏^|‿#
|
⊏
≡⊏⊐|#
⊏
⊏|#
‿
‿
‿
⊐^|‿#
⊐
⊐
≡⊏⊐‿#
⊐‿#
#
#
⊏‿#
⊐|#
#
⊐|#
⊏‿#
≡⊏⊐^|‿#
Cases with more than one possibility indicate loss of information.
The join of # and # is totally uninformative.
Entailment relations between expressions
differing in atomic edits (substitution,
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: ⊏ (upwardmonotone contexts are prevalent)
e.g. red car ⊏car
But doesn’t hold for negation, non-intersective
adjectives, implicatives.
Composition
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
connectives
connective
≡
⊏
⊐
^
|
‿
#
Negation (not)
≡
⊐
⊏
^
‿
|
#
Conjunction
(and)/intersectio
n
≡
⊏
⊐
|
|
#
#
Disjunction (or)
≡
⊏
⊐
‿
#
‿
#
Conditional
antecedent
≡
⊐
⊏
#
#
#
#
Conditional
consequent
≡
⊏
⊐
|
|
#
#
Biconditional
≡
#
#
#
#
#
#
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Conjunction
(and)/intersectio
n
≡
⊏
⊐
|
|
#
#
happy ≡ glad
kiss ⊏ touch
human ^ nonhuman
French | German
Metallic ‿ nonferrous
swimming # hungry
kiss and hug ⊏ touch and hug
living human | living nonhuman
French wine | Spanish wine
metallic pipe # nonferrous pipe
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Disjunction (or)
≡
⊏
⊐
‿
#
‿
#
happy ≡ glad
kiss ⊏ touch
human ^ nonhuman
French | German
more that 4 ‿ less than 6
happy or rich ≡ glad or rich
kiss or hug ⊏ touch or hug
human or equine ^ nonhuman or equine
French or Spanish # German or Spanish
3 or more than 4 ‿ 3 or less than 6
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Negation (not)
≡
⊐
⊏
^
‿
|
#
happy ≡ glad
kiss ⊏ touch
human ^ nonhuman
French | German
more that 4 ‿ less than 6
swimming # hungry
not happy ≡ not glad
not kiss ⊐ not touch
not human ^ not nonhuman
not French ‿ not German
not more than 4 | not less than 6
not swimming # not hungry
Compositionality: projectivity of
quantifiers
1st argument
quantifier
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
some
≡
⊏
⊐
⏝
#
⏝
#
≡
⊏
⊐
º
#
⏝
#
no
≡
⊐
⊏
|
#
|
#
≡
⊐
⊏
|
#
|
#
every
≡
⊐
⊏
|
#
|
#
≡
⊏
⊐
|
|
#
#
not every
≡
⊏
⊐
⏝
#
⏝
#
≡
⊐
⊏
⏝
⏝
#
#
at least two
≡
⊏
⊐
#
#
#
#
≡
⊏
⊐
#
#
#
#
most
≡
#
#
#
#
#
#
≡
⊏
⊐
|
|
#
#
exactly one
≡
#
#
#
#
#
#
≡
#
#
#
#
#
#
all but one
≡
#
#
#
#
#
#
≡
#
#
#
#
#
#
Compositionality: projectivity of
quantifiers: some, first argument
1st argument
quantifier
some
≣
⊏
⊐
^
|
⏝
#
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
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
quantifiers: some, second argument
1st argument
quantifier
some
≣
⊏
⊐
^
|
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
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
quantifiers: no; first argument
1st argument
quantifier
no
≡
⊏
⊐
^
|
⏝
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊐
⊏
|
#
|
#
≡
⊐
⊏
|
#
|
#
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
quantifiers: every; first argument
1st argument
quantifier
every
≡
⊏
⊐
^
|
⏝
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊐
⊏
|
#
|
#
≡
⊏
⊐
|
|
#
#
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
quantifiers: not every; first argument
1st argument
quantifier
not every
≡
⊏
⊐
^
|
⏝
#
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
≡
⊐
⊏
⏝
⏝
#
#
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 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)
Factives
Class
Positive ++/-+ forget that
is odd that
Negative +-/-- pretend that
pretend to
Inference Pattern
forget that X ⇝ X, not forget that X ⇝ X
is odd that X ⇝ X, is not odd that X ⇝ X
pretend that X ⇝ not X, not pretend that X ⇝ not X
pretend to X ⇝ not X, not pretend to X ⇝ not X
+
forget that host polarity
pretend that host polarity +
pretend that host polarity
forget that
host polarity
+
+
-
complement polarity
complement polarity
complement polarity
complement polarity
Abraham pretended that Sarah was his sister. ⇝ Sarah was not his sister
Howard did not pretend that it did not happen. ⇝ It happened.
Implicatives
Class
Two-way ++/-- manage to
implicatives +-/-+ fail to
++
Inference Pattern
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
One-way +implicatives
--
be able to
not be able to X ⇝ not X
-+
hesitate to
not hesitate to X ⇝ X
Translating PARC classes into the
MacCartney approach
sign del ins example
implicatives
factives
++/-- ≡
≡
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
#
#
He believed he had won/ he had won
+-/+
+-/-
Neutral
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 smoke
h: We smoked Cuban cigars
i
xi
ei
0
We were not able to
smoke
1
We did not smoke
DEL(perm ⊐
it)
Downward
monotone:⊏
⊏
2
We smoked
DEL(not)
^
Upward monotone: ^
Join of ⊏,^:
|
3
We smoked Cuban
cigars
INS(C.cig
ars)
⊐
Upward monotone: ⊐
Join of |,⊐ :
|
We end up with a contradiction
f(ei) g(xi-1,ei)
h(x0,xi)
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 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 NatLog
NatLog
Symmetrical between t and h
Bottom up
Local edits, more compositional
Surface based
Integrates monotonicity and
implicatives tightly
Bridge
Asymmetrical 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
Anna 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 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 reasoning
Did an event mentioned in the text actually occur?
Temporal reasoning
When did an event happen? How are events ordered in time?
Spatial reasoning
Where 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 Initiatives
PASCAL RTE Challenge (Ido Dagan, Oren Glickman) 2005, 2006
PREMISE
CONCLUSION
TRUE/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
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
Romano Prodi will meet the US President George Bush in his
capacity as the president of the European commission.
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 construal
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 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
quantifiers: not every; second argument
1st argument
quantifier
not every
≣
⊏
⊐
^
|
⏝
#
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
⏝
#
⏝
#
≡
⊐
⊏
⏝
⏝
#
#
Compositionality: projectivity of
quantifiers
1st argument
quantifier
at least two
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
#
#
#
#
≡
⊏
⊐
#
#
#
#
Compositionality: projectivity of
quantifiers
1st argument
quantifier
most
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
#
#
#
#
#
#
≡
⊏
⊐
|
|
#
#
Compositionality: projectivity of
quantifiers
1st argument
quantifier
exactly one
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
#
#
#
#
#
#
≡
#
#
#
#
#
#
Compositionality: projectivity of
quantifiers
1st argument
quantifier
all but one
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
#
#
#
#
#
#
≡
#
#
#
#
#
#
some
Compositionality: projectivity of
quantifiers: no; second argument
1st argument
quantifier
no
≡
⊏
⊐
^
|
⏝
#
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊐
⊏
|
#
|
#
≡
⊐
⊏
|
#
|
#
Compositionality: projectivity of
quantifiers: every; second argument
1st argument
quantifier
every
≡
⊏
⊐
^
|
⏝
#
2nd argument
≡
⊏
⊐
^
|
⏝
#
≡
⊏
⊐
^
|
⏝
#
≡
⊐
⊏
|
#
|
#
≡
⊏
⊐
|
|
#
#
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Biconditional
≡
#
#
#
#
#
#
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Conditional
antecedent
≡
⊐
⊏
#
#
#
#
kiss ⊏ touch
If she kissed her, she likes her ⊐ if she touched her, she likes her
human ^ nonhuman
Composition: projectivity of logical
connectives
connective
≡
⊏
⊐
^
|
‿
#
Conditional
consequent
≡
⊏
⊐
|
|
#
#
kiss ⊏ touch
If he wins I’ll kiss him⊏ if he wins I’ll touch him
human ^ nonhuman
If it does this it shows that it is human | if it does this it shows that it is
nonhuman
French | German
If he wins he gets French wine | if he wins he gets German wine