natural logic - indiana university bloomingtoniulg/moss/overview.pdf · we don’t care if fol...

Natural Logic

Larry Moss, Indiana University

EASLLC, 2014

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This course deals with the relation of logicand natural language

I What does semantics look like when we make inferencethe primary object of study rather than the secondary one?

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This course deals with the relation of logicand natural language

I What does semantics look like when we make inferencethe primary object of study rather than the secondary one?

I What is the relation of logic and semantics?

I What does logic for natural language look likewhen it is done with a minimum of translation?

I Can we build formal tools to simultaneously

— handle inference in interesting fragments— remain on the “good side” of various logical borders:decidability, complexity.

I And will ths be of any interest in semantics?

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A fairly standard view of these matters

I Why does logic enter in to semantics in the first place?

We want to account for natural language inferences such as

Frege’s favorite food was baozi

Frege ate baozi at least once

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The hypothesis and conclusion would berendered in some logical system or other.There would be a background theory (≈ common sense),and then the inference would be modeled either as a semantic fact:

Common sense+Frege’s favorite food was baozi |= Frege ate baozi at least once

or a via a formal deduction:

Common sense+Frege’s favorite food was baozi ` Frege ate baozi at least once

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As an aside:We are going to be vitally interested in the difference betweenthe proof-theoretic concept `and the model-theoretic concept |= in this course!

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Either way, it’s all in one and the same language.3/41

Furthermore

I To carry our this program, it would be advisable to takeas expressive a logical system as possible.

I First-order logic (FOL) is a good starting point,but for many phenomena we’ll need to go further.

I In this regard, FOL is vastly superior totraditional (term) logic.

I Various properties of FOL are interest in this discussion,but only secondarily so.

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What does undecidability have to do with it?

Theorem (Church 1936)

There is no algorithm, which givena finite set Γ of sentences in FOL and another sentence ϕ,decides whether or not Γ |= ϕ.

The same goes for the proof-theoretic notion Γ ` ϕ,since this comes to the same thing,by the Completeness Theorem of FOL.

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This is not really relevant to the semantic enterprise,since we are not taking

Γ |= ϕ

to model what humans actually do.And as for Γ ` ϕ,we don’t care if FOL gives us more power than we actually need.

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FOL in is important also connection with the foundations ofmathematics.

Undecidability is forced on us in that domain:By Godel’s Incompleteness Theorem. the set of true sentences ofmathematicsis not decidable.

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And anyways, what choice to we really have?

One can easily object to the whole enterprise of using FOLin connection with NL inference, on the grounds that FOL cannothandle

I vague words

I intentions of speakers

I ellipsis

I anaphora

I poetic language...

In other words, FOL is too small for the job.

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This course: FOL is also too big!

The point is that for “everyday inference”,a small fragment of FOL should be sufficient.

Also, there is a long tradition in linguistics of dissatisfaction withmodels which are “complete r.e.”and in favor of ones with much less expressive power.

This was once decisive in syntax: the Peters-Ritchie Theorem.

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This course: FOL is also too big!

The point is that for “everyday inference”,a small fragment of FOL should be sufficient.

Also, there is a long tradition in linguistics of dissatisfaction withmodels which are “complete r.e.”and in favor of ones with much less expressive power.

This was once decisive in syntax: the Peters-Ritchie Theorem.

You decide

Consider three activites:

A mathematics: prove the Pythagorean Theorem a2 + b2 = c2.B syntax: parse John knows his mother saw him at her house.C semantics: tell whether a hearer of Nian the Horrible Monster

should infer that the monster liked the color red.

A: mathematics B: syntax

Where would you put C: semantics?

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The perspective of this course

Following a lot of work in subjects like

linguisticsartificial intelligencecognitive science

we aim to formalize using the lightest tools possible.

(This contrasts with what is done in foundations of mathematics,where one typically uses whatever tools are needed to get the jobdone.)

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Another way to make this point

In logic, we have a trade-off between expressivity and tractability:

I propositional logic is not very expressive,but it is decidable and has a complete proof system.

I first-order logic is yet more expressive,it is undecidable but has a complete proof system.

I second-order logic is very expressive,it is undecidable and it is not even clear what a completeproof system would be.

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Natural logic: what it’s all about

Program

Show that significant parts of NL inference can be carried outin decidable logical systems,preferably in “light” systems.

Raise the question of how much semantics can be donein decidable logics.

To axiomatize as much as possible,because the resulting logical systems are likely to be interesting.

To ask how much of language could have been done if thetraditional logicianshad the mathematical tools to go further than they were able to.

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Examples of inferences which we will see inthis course

These are the basic data that the course will account for

First, a few examples from the classical syllogistic:

All men are mortal Socrates is a manSocrates is mortal (1)

All auctioneers are curmudgeons No bartenders are curmudgeons

No auctioneers are bartenders(2)

Our first unit is on the basics of syllogistic logic.

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A look at syllogistic logic

Our “syntax” of sentences will give us

All X are YSome X are YNo X are Y

We adopt the evident semantics.

We craft a logical system which has formal proofsusing our syntax of sentences and nothing else.

After this, we want to extend the idea of syllogistic logic.

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Syllogistic Logic of all and some

Syntax: All p are q, Some p are q

Semantics: A model M is a set M,and for each noun p we have an interpretation [[p]] ⊆ M.

M |= All p are q iff [[p]] ⊆ [[q]]M |= Some p are q iff [[p]] ∩ [[q]] 6= ∅

Proof system:

All p are p

All p are n All n are q

All p are q

Some p are q

Some q are p

Some p are q

Some p are p

All q are n Some p are q

Some p are n

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Semantic and proof-theoretic notions

If Γ is a set of formulas, we write M |= Γ if for all ϕ ∈ Γ, M |= ϕ.

Γ |= ϕ means that every M |= Γ also has M |= ϕ.

A proof tree over Γ is a finite tree Twhose nodes are labeled with sentencesand each node is either an element of Γ,or comes from its parent(s) by an application of one of the rules.

Γ ` S means that there is a proof tree T for over Γwhose root is labeled S .

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How it all works

English:If there is an n, and if all ns are ps and also qs, then some p are q.

Semantic assertion:Some n are n, All n are p, All n are q |= Some p are q.

Proof-theoretic assertion:Some n are n, All n are p, All n are q ` Some p are q.

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How it all works

English:If there is an n, and if all ns are ps and also qs, then some p are q.This is something we could check againsthuman intuition and performance.

Semantic assertion:Some n are n, All n are p, All n are q |= Some p are q.The reasoning here would be a mathematical proof.

Proof-theoretic assertion:Some n are n, All n are p, All n are q ` Some p are q.The proof tree is

All n are q

All n are p Some n are n

Some n are p

Some p are n

Some p are q

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Example of a conclusion which doesn’t follow

All frogs are reptiles.All frogs are animals.

All reptiles are animals.

We can take U = {1, 2, 3, 4, 5, 6}.[[F ]] = {1, 2},[[R]] = {1, 2, 3, 4}.[[A]] = {1, 2, 4, 5, 6}.In this context, the assumptions are true but the conculsion is false.So the argument is invalid.

All frogs are reptiles,All frogs are animals 6|= All reptiles are animals.

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The connection

Completeness Theorem

Γ |= ϕ iff Γ ` ϕ

References to related work: Lukasiewicz 1951, Westerstahl 1989.

All the logical systems in this course are complete.If you follow most of the details, you’ll learn a lot of technicalmaterial.

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Add noun-level negation

Let us add complemented atoms p on top of S,with interpretation is via set complement.

So we have

All p are qSome p are qAll p are q ≡ No p are qSome p are q ≡ Some p aren’t q

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The logical system

All p are p

Some p are q

Some p are p

Some p are q

Some q are p

All p are n All n are q

All p are q

All n are p Some n are q

Some p are q

All q are q

All q are pZero

All q are q

All p are qOne

All p are q

All q are pAntitone Some p are p

ϕ Ex falso quodlibet

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A formal proof tree in our S†

Let Γ be

{All q are p,All q are p,All q are n, Some p are n}

Here is a proof tree showing Γ ` Some p are q:

All q are p

All p are q All q are p

All p are p

All q are p

All q are n

All n are qSome m are nSome n are n

Some n are q

Some q are q

Some q are p

Some p are q

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Some Comments

The rules below seem much less intuitive than the others:

All q are q

All q are pZero

All q are q

All p are qOne

Typically, the logics in this subject contain a few strange rules.

Sometimes this appears to me to be a nuisance,sometimes a delight.

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Ex Falso Quodlibet vs. Reductio ad Absurdum

Some p are pϕ Ex falso quodlibet

To state Reduction ad Absurdum, we need to define negations:

sentence ϕ negation ϕ

All p are q Some p are q

Some p are q All p are q

α α⊥ ⊥I

[ϕ]....⊥ϕ RAA

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More examples

Next, some examples from the extended syllogistic logics:

Some dog sees some cat

Some cat is seen by some dog (3)

Bao is seen and heard by every student Amina is a student

Amina sees Bao (4)

All skunks are mammalsAll who fear all who respect all skunks fear all who respect all mammals

(5)

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More on the “skunks and mammals” example

All skunks are mammalsAll who fear all who respect all skunks fear all who respect all mammals

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We first note the following antitonicity principle:

All skunks are mammalsAll who respect all mammals respect all skunks

This and related principles are going to occupy us for an entirelecture.

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And we can apply antitonicity twice!

All skunks are mammalsAll who respect all mammals respect all skunks

All who fear all who respect all skunks fear all who respect all mammals

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Examples of inferences which we will see inthis course

Every giraffe is taller than every gnuSome gnu is taller than every lionSome lion is taller than some zebraEvery giraffe is taller than some zebra

(6)

More students than professors run More professors than deans run

More students than deans run(7)

At most as many xenophobics as yodelers are zookeepersAt most as many zookeepers as alcoholics are yodelersAt most as many yodelers as xenophobics are alcoholics

At most as many zookeepers as alcoholics are xenophobics

(8)

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More reasoning about the sizes of sets

Example

Assume:

1 There are at least as many non-y as y2 There are at least as many non-z as z3 All x are z4 All non-y are z

Then prove from these that All x are non-y.

Here is a formal proof in the logical system which we’ll see:

∀(x , z)

∀(y , z)

∃≥(y , y) ∃≥(z , z)

∃≥(y , z)(Half)

∀(z , y)(Card Mix)

∀(x , y)(Barbara)

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The map

This course will mention many fragments,and to help keep things straight,I’ll draw a lot of maps.

The Aristotle boundary is the dividing line between fragmentswhich are formulated syllogistically and those which are not.Reductio proofs are ok.Infinitely many rules are not.

Individual variables take us beyond the Aristotle boundary.

The Turing boundary is the dividing line between decidable andundecidable fragments.

The Peano-Frege boundary divides the fragments which areformulatedin first-order logic from those which are not.

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Map of Some Natural Logics

Peano-Frege

Aristotle

Church-Turing

S

S†

S≥ S≥ adds |p| ≥ |q|R

RC

RC(tr)

RC(tr , opp)R†

RC†

RC†(tr)

RC†(tr , opp)

FOL

FO2 + trans

FO2

first-order logic

FO2 + “R is trans”

2 variable FO logic

† adds full N-negation

R + relative clauses

R = relational syllogistic

RC + (transitive)

comparative adjs

RC(tr) + opposites

S + full N-negation

S: all/some/no p are q

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More on the content of the course

I have arranged the course material in a number of units:

I overview + examples

I syllogistic logics

I logics and the sizes of sets

I logics with individual variables

I polarity/monotonicity

I another direction: proof-theoretic semantics

I answers to homework + any further topics you ask about

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Frederic Fitch, 1973Natural deduction rules for English, Phil. Studies, 24:2, 89–104.

1 John is a man Hyp

2 Any woman is a mystery to any man Hyp

3 Jane Jane is a woman Hyp

4 Any woman is a mystery to any man R, 2

5 Jane is a mystery to any man Any Elim, 4

6 John is a man R, 1

7 Jane is a mystery to John Any Elim, 6

8 Any woman is a mystery to John Any intro, 3, 7

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Example of where we would want derivationswith variables

All xenophobics see all astronautsAll yodelers see all zookeepersAll non-yodelers see all non-astronautsAll wardens are xenophobics

All wardens see all zookeepers

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1 All xenophobics see all astronauts Hyp

2 All yodelers see all zookeepers Hyp

3 All non-yodelers see all non-astronauts Hyp

4 All wardens are xenophobics Hyp

5 Jane Jane is a warden Hyp

6 All wardens are xenophobics R, 4

7 Jane is a xenophobic All Eliim, 6

8 All xenophobics see all astronauts R, 2

9 Jane sees all astronauts All Elim, 8

10 Jane is a yodeler Hyp

11 Jane sees all zookeepers Easy from 2

12 Jane is not a yodeler Hyp

13 Jane sees all zookeepers See below

14 Jane sees all zookeepers Cases 10-11, 12-13

15 All wardens see all zookeepers All Intro

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1 Jane is not a yodeler Hyp

2 Jane sees all astronauts R, above

3 All non-yodelers see all non-astronauts R, above

4 Jane sees all non-astronauts All Elim, 1, 3

5 Bob Bob is a zookeeper Hyp

6 Bob is astronaut Hyp

7 Jane sees Bob All Elim, 2

8 Bob is not astronaut Hyp

9 Jane sees Bob All Elim, 4

10 Jane sees Bob Cases

11 Jane sees all zookeepers All Intro

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Introduction/Review of MonotonicityPhenomena

Assume thatwaltz ≤ dance ≤ move

andgrizzly ≤ bear ≤ animal

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Assume that we’re talking about a situation where

some bears dance.

Which one would be true?

I some grizzlies danceI some animals dance

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some bears dance.

I some grizzlies dance falseI some animals dance true

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some bears dance.

We writesome bears↑ dance

Now think about these two

I some bears waltzI some bears move

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some bears dance.

We writesome bears↑ dance↑

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Put the arrows on the words bears and dance.

1 Some bears dance.2 No bears dance.3 Not every bear dances.4 John sees every bear.5 Mary sees no bear.6 Most bears dance.7 Any bear in Hawaii would prefer to live in Alaska.8 If any bear dances, Harry will be happy.9 If you play loud enough music, any bear will start to dance.10 Doreen didn’t see any bears dance in her dorm room.

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Further

The answers are:

1 Some bears↑ dance↑.

2 No bears↓ dance↓.

3 Not every bear↑ dances↓.

4 John sees every bear↓.

5 Mary sees no bear↓.

6 Most bears dance↑. (No arrow goes on bears in this one.)

7 Any bear↓ in Hawaii would prefer to live in Alaska.

8 If any bear↓ dances↓, Harry will be happy.

9 If you play loud enough music, any bear↓ will start to dance↑.

10 Doreen didn’t see any bears↓ dance↓ in her dorm room.

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Further

The answers are:1 Some bears↑ dance↑.2 No bears↓ dance↓.3 Not every bear↑ dances↓.4 John sees every bear↓.5 Mary sees no bear↓.6 Most bears dance↑. (No arrow goes on bears in this one.)7 Any bear↓ in Hawaii would prefer to live in Alaska.8 If any bear↓ dances↓, Harry will be happy.9 If you play loud enough music, any bear↓ will start to dance↑.10 Doreen didn’t see any bears↓ dance↓ in her dorm room.

The last examples are important because we wantto account for shallow inferences inmore syntactically realistic fragments.

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An example which we’ll see later in detailThe idea is to meld inference with categorial grammar

Example

Let Γ contain the following inequalities

cat : p ≤ animal : p

every : p−→ (p

+→ t) ≤ most : p·→ (p

+→ t)

Below is a derivation from Γ:

cat ≤ animal

every [animal↓] ≤ every cat(Anti) every ≤ most

every cat ≤ most cat(Point)

every animal ≤ most cat(Trans)

every animal vomits ≤ most cat vomits(Point)

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More on the content of the course

I have arranged the course material in a number of units:

I overview + examples

I syllogistic logics

I logics and the sizes of sets

I logics with individual variables

I polarity/monotonicity

I another direction: proof-theoretic semantics

I answers to homework + any further topics you ask about

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Homework

Problem 1

The semantics of All X are Y in a model is that[[X ]] ⊆ [[Y ]].

What should the semantics be for

All X which are Y are Z?

Problem 2

Is the following valid or not? And why?

All skunks are mammalsAll who respect all skunks respect all mammals (9)

Problem 3

Is the following a sound inference?

Everyone who likes every logician is a logician There is a duck

There is a logician

If so, explain what the duck is doing.If not, give a counter-model.

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Homework, continued

Problem 4

Is the following valid or not? And why?

Everyone likes everyone who likes PatPat likes every clarinetist

Everyone likes everyone who likes everyone who likes every clarinetist(10)

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Objections to keep in mind

If we were to devise a logic of ordinary languagefor direct use on sentences as they come,we would have to complicate our rules of inferencein sundry unilluminating ways.

W. V. O. Quine, Word and Object

I The logical systems that one would getfrom looking at inference involving surface sentenceswould contain many copies of similar-looking rules.

They would miss a lot of generalizations.

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I The systems would contain ‘rules’ thatare more like complex deduction patternsthat need to be framed as rulesonly because one lacks the machinery to break them downinto more manageable sub-deductions.

Moreover, those complex rules would be unilluminating.

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I The systems would lack variables,and thus they would be tedious and inelegant.

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I It would be impossible to handle standard topics such asquantifier-scope ambiguities.

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Replies, with pointers to what we will do

I Similar rules can be succinctly grouped into meta-rules:van Benthem showed how to do this in a general wayin the monotonicity calculus.

I It’s true that the systems have some complex “rules”.Perhaps this could be turned into an advantage,but aiming for a theory of “shallow inference”

I The systems can have variables in some form.But they lead to an interesting issue!

I Quantifier-scope ambiguities can be handled.But all the other similar phenomena aren’t yet touched.

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natural logic - indiana university bloomingtoniulg/moss/overview.pdf · we don’t care if fol...

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