If, not when

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Talk at IMLA 2013, work of Dick Crouch.

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If, not when

  1. 1. If, not whenRichard Crouch and Valeria de Paiva Nuance Communications, CA, USA IMLA – April, 2013
  2. 2. Introduction Motivation Deictic shift Semantics Proof System ConclusionsIntroduction 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. 3. Introduction Motivation Deictic shift Semantics Proof System ConclusionsGoal 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. 4. Introduction Motivation Deictic shift Semantics Proof System ConclusionsConditional 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. 5. Introduction Motivation Deictic shift Semantics Proof System ConclusionsConditional 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. 6. Introduction Motivation Deictic shift Semantics Proof System ConclusionsDeictic 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. 7. Introduction Motivation Deictic shift Semantics Proof System ConclusionsIsn’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. 8. Introduction Motivation Deictic shift Semantics Proof System ConclusionsMeaning 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. 9. Introduction Motivation Deictic shift Semantics Proof System ConclusionsMeaning 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. 10. Introduction Motivation Deictic shift Semantics Proof System ConclusionsDeictic 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. 11. Introduction Motivation Deictic shift Semantics Proof System ConclusionsSummary 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. 12. Introduction Motivation Deictic shift Semantics Proof System ConclusionsIntuitionism 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. 13. Introduction Motivation Deictic shift Semantics Proof System ConclusionsInformation 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. 14. Introduction Motivation Deictic shift Semantics Proof System ConclusionsInformation 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. 15. Introduction Motivation Deictic shift Semantics Proof System ConclusionsConditions 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. 16. Introduction Motivation Deictic shift Semantics Proof System ConclusionsForcing 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 |„ „ ·  i s, t |„ „ and s, t |„  3. s, t |„ „ ‚  i s, t |„ „ or s, t |„  4. s, t |„ „ æ  i ’t1 Ø t, s1 ˆ„,t s : ÷t2 Ø t1 such that t1 s1 , t2 |„  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. 17. Introduction Motivation Deictic shift Semantics Proof System ConclusionsForcing 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. 18. Introduction Motivation Deictic shift Semantics Proof System ConclusionsTwo 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. 19. Introduction Motivation Deictic shift Semantics Proof System ConclusionsStable 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. 20. Introduction Motivation Deictic shift Semantics Proof System ConclusionsStable 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. 21. Introduction Motivation Deictic shift Semantics Proof System ConclusionsProof System (Ax ) ,„ „ „ „ „; „ „„·Â ·I ·E „„·Â „„ „„ „ „ ‚ Â; , „ „ ‰; , „ ‰ ‚I ‚E „„‚ „‰ 21 / 24
  22. 22. Introduction Motivation Deictic shift Semantics Proof System ConclusionsProof System Stable( ), „ „≥≥  „ „; „„æ æI æE „„æ „≥≥  Stable( ), „ „≥≥ ‹ „‹ ¬I ‹ „ ¬„ „„ ,„ „ ‹ ≥I ≥ Ax „≥ „ „≥ „‚ ≥≥ „ „≥≥ „; „ „ æ  „≥≥ ‹ ≥æ ≥≥ ‹ „≥≥  „‹ 22 / 24
  23. 23. Introduction Motivation Deictic shift Semantics Proof System ConclusionsTheorem: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. 24. Introduction Motivation Deictic shift Semantics Proof System ConclusionsConclusions 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

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