Talk at IMLA 2013, work of Dick Crouch.

1. If, not when
Richard Crouch and Valeria de Paiva
Nuance Communications, CA, USA
IMLA – April, 2013

2. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Introduction
I Crouch discussed in his thesis (1993) patterns of temporal
reference exhibited by conditional and modal sentences in
English.
I A Natural Deduction system of verified and unverified
assertions emerged.
I de Paiva wants to understand what are the salient properties
of the constructive modal logic that was arrived at.
I Hence this note.
2 / 24

3. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Goal
I Our goal is to describe Crouch’s logic of verified/unverified
assertions 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 do
we motivate these?
3. How do these relate to other models in the literature?
4. Which useful properties can we extract from the logic itself?
3 / 24

4. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Conditional and Modal Sentences
I This work was motivated by the behavior of the past and
present tenses in (modal and) conditional sentences in English.
I The interactions between time and modality are crucial to
understanding both.
I Time has an irreducibly modal dimension, while modality has
an irreducibly temporal dimension.
I Our first goal is to describe what the interactions are. Then
we propose a logic that captures it.
4 / 24

5. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Conditional and Modal Sentences
I Examination of conditional sentences occurring in corpus
raises three questions:
I why is it that in modal and conditional contexts, past and
present tenses can be deictically shifted so that they refer to
future 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 and
consequent clauses of conditionals. These depend on the
tenses of the antecedent and consequent. How exactly?
I Key insight: Two deictic centres are required. a primary
centre, known as the assertion time, and a secondary centre,
known as the verification time.
5 / 24

6. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Deictic shift?
I Deictic shift occurs when a tense locates an event as being
past or present with respect to some time other than the
speech time.
I Often this results in past and present tenses that refer to
times in the future.
I Example: Anna moves to Boston this Sunday.
I The tenses not only serve to describe the way that the world
changes over time, but also the way that information about
the world changes. To account for that we associate with the
past and the present tense a primary and a secondary deictic
centre
I The two deictic centres correspond to times at which
informational operations of assertion and verification take
place. 6 / 24

7. 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
7 / 24

8. 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 temporal
reference describe how the world changes over time.
I On a information-change account, there is a second level that
temporal reference operates on: constraining the way
information changes over time.
8 / 24

9. 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 some
utterance event occurs (the speech time) and the time the
event being described occurs (the event time).
I A new alternative is to centre tenses on the time at which an
update is made to one’s stock of information, where this
update occurs as the result of the utterance of the sentence.
I In most cases the move from speech time to update time will
make no di erence: normally, the update occurs as soon as
the utterance is made. But not for conditionals and modal
sentences.
I Also update time needs to be refined into assertion time and
verification time.
9 / 24

10. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Deictic shift?
I Modal and conditional sentences place constraints on the way
that 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’s
information state.
I However, the assertion does not enjoy first class status until it
becomes verified.
I A modal like will also has the e ect of making unverified
assertions.
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 but
conceding that until I go to the door and pick up the letters on
the doormat, I have no direct evidence to verify this assertion 10 / 24

11. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Summary of Motivation
I Goal: Account for temporal data in simple conditionals
I Simple past/present tense antecedent (A) or consequent (C)
I If the vase fell over, it is on the floor.
I If the vase is on the floor, it feel over.
I Ordering between A and C eventualities
I If I smile when I get out the interview went well
I If the letter arrives tomorrow, it is already in the post
I Relation of A and C eventualities to speech time
The linguist’s conclusion (after 2500 examples):
I need primary and secondary deictic shifts, assertion and
verification times
I Aim: given “If A then C":
I (Hypothetical) assertion of A at time of utterance
I If and when the assertion of A is verified
I You may assert C (which should eventually be verified) 11 / 24

12. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Intuitionism and Information States
I Intuitionism is about knowledge-values and verification
conditions rather than truth-values and truth conditions.
I Intuitionism denies that there is anything more to truth than
what is furnished by verification, and thus identifies truth and
verification conditions. ∆ a useful logic of verification.
I Kripke semantics for intuitionism suggests an agent that
extends its knowledge and the universe of objects it knows
about over the course of time.
I At each moment t the subject has a stock of sentences, ⌃t , it
has established as true and a stock of objects, Dt , it has
encountered or otherwise established as existent.
I The stock of sentences and objects at a time t constitute the
subject’s information state at time t.
12 / 24

13. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Information Models
I As time goes by, the subject finds out more, and adds further
sentences and further objects to its information state.
I There is a natural (partial) order imposed over the subject’s
possible information states, reflecting the ways in which the
subject’s information can accumulate.
I In information models, each information state can be seen as a
linearly ordered sequence of temporal ‘snapshots’ of the state,
where di erent formulas are forced at di erent time points.
13 / 24

14. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Information Models
An information model M is a quintuple
M = 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 and
atomic sentences in some language L onto the (verification) values
1 or 0.
14 / 24

15. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Conditions on Information Models
I Monotonicity of direct verification (‘in-state’ monotonicity)
For every state s and atomic sentence p of L
t1 Æ t2 implies if V (s, t1 , p) = 1 then V (s, t2 , p) = 1
I Monotonicity of information growth (‘out-of-state’ monot.)
If s1 ıt s2 then for atomic sentences p
(a) {p | V (s1 , t, p) = 1} ™ {p | V (s2 , t, p) = 1}
(b) {p | ÷t : V (s1 , t, p) = 1} ™ {p | ÷t : V (s2 , t, p) = 1}
I Convergence of Verification:
If s1 ı t1 s2 ı t2 s3 ,
then there is a time t3 such that t3 Ø t1 , t3 Ø t2 and ’t4 Ø t3
s 1 ı t4 s 3
I No Absurdity:
For no s or t is it the case that V (s, t, ‹) = 1
15 / 24

16. 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 at
a time t in a state s we say:
1. s, t |„ p i V (s, t, p) = 1 if p is atomic
2. 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  i ’t1 Ø t, s1 ˆ„,t s : ÷t2 Ø t1 such that
t1
s1 , t2 |„ Â
5. s, t |„ ¬„ i ’t1 Ø t, s1 ˆ„,t s : ÷t2 Ø t1 such that s1 , t2 |„ ‹
t1
6. s, t |„≥ „ i ’t1 Ø t : s, t1 ”|„ „
16 / 24

17. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Forcing in Information Models
I Minimal information extension: s1 ˆ„,t s i
t1
a) s1 ˆt1 s
b) s1 , t1 |„ „, and
c) ” ÷t2 , s2 such that t Æ t2 < t1 , s ˆt2 s2 ˆt2 s1 and s2 , t2 |„ „
I if s1 is a minimal extension of s with respect to „ at time t,
then s2 is the first state extending s that verifies „ at the
earliest time t1 .
17 / 24

18. 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 be
verified in any future state at any future time.
I In-state negation says that a sentence will never be verified in
the 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.
18 / 24

19. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Stable Sentences?
I the forcing relation in intuitionistic logic is monotonic: once a
sentence is forced in one state, it remains forced in all
subsequent states. This holds for all sentences.
I For information models we need to consider two distinct kinds
of monotonicity: in-state monotonicity, and out-of-state
monotonicity.
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 it
was well-defined...
19 / 24

20. 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.
20 / 24

21. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Proof System
(Ax )
,„ „ „
„ „; „Â „„·Â
·I ·E
„„·Â „„
„„ „ „ ‚ Â; , „ „ ‰; , „ ‰
‚I ‚E
„„‚ „‰
21 / 24

22. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Proof System
Stable( ), „ „≥≥  „ „; „„æÂ
æI æE
„„æ „≥≥ Â
Stable( ), „ „≥≥ ‹ „‹
¬I ‹
„ ¬„ „„
,„ „ ‹
≥I ≥ Ax
„≥ „ „≥ „‚ ≥≥ „
„≥≥ „; „ „ æ Â „≥≥ ‹
≥æ ≥≥ ‹
„≥≥ Â „‹
22 / 24

23. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Theorem:Soundness and Completeness
I The semantic definitions presented are sound and complete
with respect to the Natural Deduction in sequent calculus
proof system just introduced.
I Ugly?
23 / 24

24. Introduction
Motivation
Deictic shift
Semantics
Proof System
Conclusions
Conclusions
I We described a logic of assertions verified and not, with two
negations
I This comes from accounting for temporal properties of
conditionals 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 this
system?
24 / 24