If, not when

Richard Crouch and Valeria de Paiva

Nuance Communications, CA, USA

IMLA – April, 2013

Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Introduction

I Crouch discussed in his thesis (1993) patterns of temporalreference exhibited by conditional and modal sentences inEnglish.

I A Natural Deduction system of verified and unverifiedassertions emerged.

I de Paiva wants to understand what are the salient propertiesof the constructive modal logic that was arrived at.

I Hence this note.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Goal

I Our goal is to describe Crouch’s logic of verified/unverifiedassertions by answering questions like:

1. What is the phenomena in language that motivate the logic?2. The logic has a natural deduction formulation as well as a

possible world semantics shown sound and complete. How dowe motivate these?

3. How do these relate to other models in the literature?4. Which useful properties can we extract from the logic itself?

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Conditional and Modal Sentences

I This work was motivated by the behavior of the past andpresent tenses in (modal and) conditional sentences in English.

I The interactions between time and modality are crucial tounderstanding both.

I Time has an irreducibly modal dimension, while modality hasan irreducibly temporal dimension.

I Our first goal is to describe what the interactions are. Thenwe propose a logic that captures it.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Conditional and Modal Sentences

I Examination of conditional sentences occurring in corpusraises three questions:

I why is it that in modal and conditional contexts, past andpresent tenses can be deictically shifted so that they refer tofuture times?If I smile when I get out, the interview went well.

I why do the past and present tenses behave asymmetrically?I there are strong semantic constraints on the temporal ordering

between eventualities described by the antecedent andconsequent clauses of conditionals. These depend on thetenses of the antecedent and consequent. How exactly?

I Key insight: Two deictic centres are required. a primarycentre, known as the assertion time, and a secondary centre,known as the verification time.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Deictic shift?

I Deictic shift occurs when a tense locates an event as beingpast or present with respect to some time other than thespeech time.

I Often this results in past and present tenses that refer totimes in the future.

I Example: Anna moves to Boston this Sunday.

I The tenses not only serve to describe the way that the worldchanges over time, but also the way that information aboutthe world changes. To account for that we associate with thepast and the present tense a primary and a secondary deicticcentre

I The two deictic centres correspond to times at whichinformational operations of assertion and verification takeplace.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Isn’t this too complicated?

The English construction “if. . . then. . ." can also be

used to express a sort of causal connection between

antecedent and consequent. [..] As a result, many uses of

“if. . . then. . ." in English just aren’t truth functional.

The truth of the whole depends on something more than

the truth values of the parts; it depends on there being

some genuine connection between the subject matter of

the antecedent and the consequent.

Barwise and Etchmendy, Language, Proof and Logic, 2002

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Meaning as the potential to change states of information?

. . . the slogan “You know the meaning of a sentence if

you know the conditions under which it is true” should be

replaced by . . . “You know the meaning of a sentence if

you know the change it brings about in the information

state of anyone who wants to incorporate the piece of

news conveyed by it.”

I On a truth-conditional account, linguistic devices for temporalreference describe how the world changes over time.

I On a information-change account, there is a second level thattemporal reference operates on: constraining the wayinformation changes over time.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Meaning as the potential to change states of information?

I Typically, tenses state a relation between the time someutterance event occurs (the speech time) and the time theevent being described occurs (the event time).

I A new alternative is to centre tenses on the time at which anupdate is made to one’s stock of information, where thisupdate occurs as the result of the utterance of the sentence.

I In most cases the move from speech time to update time willmake no di�erence: normally, the update occurs as soon asthe utterance is made. But not for conditionals and modalsentences.

I Also update time needs to be refined into assertion time andverification time.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Deictic shift?

I Modal and conditional sentences place constraints on the waythat updates may be made in the future.

I It is necessary to decompose update into two operations:assertion and verification.

I Making an assertion adds a piece of information to one’sinformation state.

I However, the assertion does not enjoy first class status until itbecomes verified.

I A modal like will also has the e�ect of making unverifiedassertions.

I If I hear a sound at the door and say That will be the

postman, I am asserting that the postman is at the door butconceding that until I go to the door and pick up the letters onthe doormat, I have no direct evidence to verify this assertion

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Summary of Motivation

I Goal: Account for temporal data in simple conditionalsI Simple past/present tense antecedent (A) or consequent (C)

IIf the vase fell over, it is on the floor.

IIf the vase is on the floor, it feel over.

I Ordering between A and C eventualitiesI

If I smile when I get out the interview went well

IIf the letter arrives tomorrow, it is already in the post

I Relation of A and C eventualities to speech timeThe linguist’s conclusion (after 2500 examples):

I need primary and secondary deictic shifts, assertion andverification times

I Aim: given “If A then C":I (Hypothetical) assertion of A at time of utteranceI If and when the assertion of A is verifiedI You may assert C (which should eventually be verified)

How? Via semantics, possible world semantics11 / 24

Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Intuitionism and Information States

I Intuitionism is about knowledge-values and verificationconditions rather than truth-values and truth conditions.

I Intuitionism denies that there is anything more to truth thanwhat is furnished by verification, and thus identifies truth andverification conditions. ∆ a useful logic of verification.

I Kripke semantics for intuitionism suggests an agent thatextends its knowledge and the universe of objects it knowsabout over the course of time.

I At each moment t the subject has a stock of sentences, ⌃t , ithas established as true and a stock of objects, Dt , it hasencountered or otherwise established as existent.

I The stock of sentences and objects at a time t constitute thesubject’s information state at time t.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Information Models

I As time goes by, the subject finds out more, and adds furthersentences and further objects to its information state.

I There is a natural (partial) order imposed over the subject’spossible information states, reflecting the ways in which thesubject’s information can accumulate.

I In information models, each information state can be seen as alinearly ordered sequence of temporal ‘snapshots’ of the state,where di�erent formulas are forced at di�erent time points.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Information Models

An information model M is a quintupleM = ÈS, ı t, T , Æ, V Í

where S is a set of information states s

ı t is a relation in S ◊ S ◊ T

and is transitive and reflexive over S for any t

T is a set of time instants t

Æ is a (linear) temporal order over T , and

V is a valuation function

The valuation function V is a function from states, times andatomic sentences in some language L onto the (verification) values1 or 0.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Conditions on Information Models

IMonotonicity of direct verification (‘in-state’ monotonicity)For every state s and atomic sentence p of L

t

1

Æ t

2

implies if V (s, t

1

, p) = 1 then V (s, t

2

, p) = 1I

Monotonicity of information growth (‘out-of-state’ monot.)If s

1

ıt s

2

then for atomic sentences p

(a) {p | V (s1

, t, p) = 1} ™ {p | V (s2

, t, p) = 1}(b) {p | ÷t : V (s

1

, t, p) = 1} ™ {p | ÷t : V (s2

, t, p) = 1}I

Convergence of Verification:

If s

1

ı t

1

s

2

ı t

2

s

3

,then there is a time t

3

such that t

3

Ø t

1

, t

3

Ø t

2

and ’t

4

Ø t

3

s

1

ı t

4

s

3

INo Absurdity:

For no s or t is it the case that V (s, t, ‹) = 115 / 24

Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Forcing in Information Models

To specify what is required for a sentence to be verified as true ata time t in a state s we say:

1. s, t |„ p i� V (s, t, p) = 1 if p is atomic2. s, t |„ „ · Â i� s, t |„ „ and s, t |„ Â

3. s, t |„ „ ‚ Â i� s, t |„ „ or s, t |„ Â

4. s, t |„ „ æ Â i� ’t

1

Ø t, s

1

ˆ„,tt1

s : ÷t

2

Ø t

1

such thats

1

, t

2

|„ Â

5. s, t |„ ¬„ i� ’t

1

Ø t, s

1

ˆ„,tt1

s : ÷t

2

Ø t

1

such that s

1

, t

2

|„ ‹6. s, t |„≥ „ i� ’t

1

Ø t : s, t

1

”|„ „

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Forcing in Information Models

I Minimal information extension: s

1

ˆ„,tt1

s i�a) s

1

ˆt1

s

b) s

1

, t

1

|„ „, andc) ” ÷t

2

, s

2

such that t Æ t

2

< t

1

, s ˆt2

s

2

ˆt2

s

1

and s

2

, t

2

|„ „

I if s

1

is a minimal extension of s with respect to „ at time t,then s

2

is the first state extending s that verifies „ at theearliest time t

1

.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Two Negations?

I Two types of negation are defined: ‘out-of-state’ negation, ¬,and ‘in-state’ negation ≥.

I Out-of-state negation says that a sentence will never beverified in any future state at any future time.

I In-state negation says that a sentence will never be verified inthe current state at any future time.

I we can also say that ≥ amounts to a denial of assertion, while¬ amounts to an assertion of denial.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Stable Sentences?

I the forcing relation in intuitionistic logic is monotonic: once asentence is forced in one state, it remains forced in allsubsequent states. This holds for all sentences.

I For information models we need to consider two distinct kindsof monotonicity: in-state monotonicity, and out-of-statemonotonicity.

I In-state monotonicity holds for all sentences. (theorem)I Out-of-state monotonicity holds only for a restricted set of

stable sentences. (theorem)stability was defined for this, but need to show by induction that itwas well-defined...

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Stable Sentences

For the record we define what stable sentences are.I If p is atomic, then p is stable.I If „ and  are stable, then „ · „ and „ ‚  are stable.I „ æ  is stable if  is stable. (Otherwise, it is semi-stable.)I ¬„ is stable.I If „ is stable, then ≥≥ „ is stable.I Anything not classified as stable by the above is unstable.

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Proof System

(Ax)�, „ „ „

·I

� „ „; � „ Â

� „ „ · ·E

� „ „ · Â

� „ „

‚I

� „ „

� „ „ ‚ ‚E

� „ „ ‚ Â; �, „ „ ‰; �, Â „ ‰

� „ ‰

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Proof System

æ I

Stable(�), „ „≥≥ Â

� „ „ æ Âæ E

� „ „; � „ „ æ Â

� „≥≥ Â

¬I

Stable(�), „ „≥≥ ‹

� „ ¬„‹

� „ ‹

� „ „

≥ I

�, „ „ ‹

� „≥ „≥ Ax

� „≥ „‚ ≥≥ „

≥æ� „≥≥ „; � „ „ æ Â

� „≥≥ Â≥≥ ‹

� „≥≥ ‹

� „ ‹

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Theorem:Soundness and Completeness

I The semantic definitions presented are sound and completewith respect to the Natural Deduction in sequent calculusproof system just introduced.

I Ugly?

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Introduction

Motivation

Deictic shift

Semantics

Proof System

Conclusions

Conclusions

I We described a logic of assertions verified and not, with twonegations

I This comes from accounting for temporal properties ofconditionals in English

I The logic is sound and complete with respect to information

models

I Are there proof theoretic properties that we can prove for thissystem?

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