Tenses and truth-conditions

1. Tenses and truth-conditions Lecture 11 Marie Duží Tenses

2. Sentences in the present, past and future • “Tom is sick”. • “Tom has been sick”. • “Tom has been sick since October 11th 2008”. • “Tom has always been sick”. • “Tom was sick a few years ago”. • “Tom was sick twice in the year 2008”. • “Tom will be sick tomorrow”. These sentences have different truth-conditions and thus should be furnished with different formal meanings. Tenses

3. Temporal and tense logics • There are many variants of temporal logic(a special case of modal logic) with Kripkean possible-world semantics. • Tense logic by Arthur Prior: • Week Past: P(q) – “It has at some time been the case that q” • Strong Past: H(q) – “It has always been the case that q”. • Week Future (“It will at some time be the case”), Strong Future (“It will always be the case”) • + axiomslike “Strong future implies week future”, … Tenses

4. Temporal logic in computer science • ZoharMannaand Amir Pnueliextended Prior’s logic to the system of temporal logic widely used for formal verification of programs • In addition to Prior’s future and past operators Manna and Pnueli have introduced modal operators like Sinceand Untilthat are provably more expressive than ordinary modal operators • usually interpreted by (labelled) transition systems of program states that are pivotal to the operational semantics of programs. Tenses

5. Tenses in natural language • The systems just mentioned suffer a drawback when applied to the semantics of natural language; • The inability to adequately analyse sentences indicating a ‘point of reference + frequency’ referring to the interval(s) when the sentence was or will be true. • “Tom was sick twice in the year 2008”. • “The Mayor of Dunedin was sick throughout the year 2008”. • “Tom will be sick on April 1st, 2009”. • Such sentences come attached with a presupposition under which a sentence is true or false. Tenses

6. Presupposition • Strawson (1952):a presuppositionP of S: • S |= P • S|= P • P is not only implied but also presupposed by S • Hence,ifP then neither S nor S, i.e.,the sentence S is neither true nor false. • Example: • “Tom stopped (did not stop) smoking” |= “Tom smoked (before)” • If Tom never smoked he couldn’t stop doing that; the assumption is neither true nor false. Tenses

7. Schema of an analysis (S / P) • Tichý (1980) applies the singulariser function to a singleton typed as containing a truth-value in order to make the set fail to deliver a truth-value in case the associated presupposition is not satisfied; “The only truth-value b such that P is true and b according as S is the case” • variable b v (truth-values) • wt[0Sing b [Pwt [b = Swt] Tichý’s analysis is not easy to understand; it is analogous to what a computer scientist would call an imperative rather than declarative analysis. • Yet, there is an elegant and simple alternative: “IfP,thenTrueor False according as S is the case,else no value” • wt IfPwtthen Swt else fail Tenses

8. Presupposition of a sentence in simple past / future • The truth-conditions of S depend not only on whether S is the case, but also on the time at which S is evaluated. • wt[0Pastt Swt] = • wt If(Ref_Time t)then Swt else Fail • wt[0Futuret Swt] = • wt If(Ref_Time t)then Swt else Fail ( ,  precedes / suceedes temporally) Tenses

9. Presupposition of a sentence in simple past • “Tom was sick (just) twice in the year 2008”. • The sentence presupposes that the whole year 2008 precedes the time T of evaluation • the truth-conditions are specified as follows: • If the whole year 2008 precedes T, thenTrueor False according as Tom was sick twice in 2008, elsefail (to produce a truth-value). Tenses

10. “Tom was sick throughout the year 2008”. Sense: wt [0Pastt [0Throughwwt [0Sickwt’0Tom]0Y2008]] Presupposition: the whole year 2008 precedes time t (of evaluation)t1[[0Y2008 t1]  [t1 < t]] Frequency modifier: 0Through = w p c t [[c t]pwt]], hence [0Throughw p c] = t [[c t]pwt]]. (c is the point of reference – Y2008) Underlying proposition:wt [0Sickwt0Tom]; [0Throughwwt [0Sickwt’0Tom]0Y2008] = t [[0Y2008 t][0Sickwt0Tom]]. Analysis: wt ift1[[0Y2008 t1]  [t1 < t]] then t [[0Y2008 t] [0Sickwt0Tom]]elseFail. Gloss: In any possible world (w) at any time of evaluation (t) do … We construct a proposition of type  (W  (T  {T,F}))rather than a truth value (all empirical expressions denote intensions). Tenses

11. “Tom was sick twice in the year 2008”. Sense: wt [0Pastt[0Twicewwt [0Sickwt0Tom]0Y2008]] Presupposition: t1[[0Y2008 t1]  [t1 < t]] Frequency modifier and reference point: [0Twicewwt [0Sickwt0Tom]0Y2008] = [0Card d [[d  t [0Sickwt0Tom]]  [d 0Y2008] Φ] = 02] = [0Card d [t [[d t] [0Sickwt0Tom]]] [t [d t] [0Y2008 t]]] = 02] Analysis: wt ift1[[0Y2008 t1]  [t1 < t]] then [0Twicewwt [0Sickwt0Tom]0Y2008]elseFail. Gloss. In any world w at any time t do: If the whole year 2008 temporally precedes t,then check whether Tom was sick just twice in the year 2008,elsefail. Tenses

12. “Tom will be sick the whole day on April 1st 2008”. Sense: wt[0Futuret[0The_wholewwt[0Sickwt0Tom]0April1]] Presupposition: t1[[0April1 t1]  [t1>t]] (t – time of evaluation) Frequency modifier and reference point: [0The_wholewwt[0Sickwt0Tom]0April1] = t’[[0April1 t’]  [0Sickwt’0Tom]] Analysis: wtift1[[0April1 t1]  [t1 > t]] then t’[[0April1 t’]  [0Sickwt’0Tom]]elseFail. Gloss. In any world w at any time t do: If the whole day of April 1st, 2008 temporally succeeds t,then check whether Tom was sick the whole April 1st, 2008,elsefail. What remains? To definethe ‘if-then-else’ function. Tenses

13. If P then C else D • [[PC]  [PD]] is notan adequate analysis, because bothC and Dare executed; if one of them fails, the whole Composition fails. • [[[5=5]  [n=1]]  [[5=5]  [n=[1/0]]]] • The product should be n=1 rather than Fail. • If-then-else: a non-strict function, not complying with the principle of Compositionality? • No satisfactory reason for non-strictness! We need a ‘lazy evaluation’. Tenses

14. If P then C else D • Semantics: a two-phase instruction: • Make a choice between C and D: [0The_only c [[P c=0C]  [P c=0D]]] • Execute the chosen construction c 2[0The_only c [[P c=0C]  [P c=0D]]] The_only is a function ({n}  n) that returns the only construction, the member of a singleton; otherwise fails cna variable ranging over constructions Tenses

15. If Presupposition then C else Fail 2[0The_only c [P c=0C]] Evaluation: • P vTrue. Then c [0True  [c=0C]] = {0C}. Hence 2[0The_only {0C}] = 2[0C] = C. • P vFalse. Then c [0False  [c=0C]] = c 0False ={ }.Hence 2[0The_only {}] is v-improper, fail. • P v improper, fail. Tenses

16. “Tom was sick twice in the year 2008”. Analysis wt ift1[[0Y2008 t1]  [t1 < t]] then [0Twicewwt [0Sickwt0Tom]0Y2008]elsefail. wt 2[0The_only c[t1[[0Y2008 t1]  [t1 < t]] c = 0[0Twicewwt [0Sickwt0Tom]0Y2008]]] Tenses

17. “All John’s children are asleep” • All John‘s children are asleep. • All John‘s children are not asleep. Hence (1), as well as (2) entail • John has children. If John does not have any children, then (1) and (2) are neither true nor false (which does not mean that they are meaningless). Classical translation of (1) into the language of first-order predicate logic: • x [JC(x) S(x)]. But: JC { } is a model ! This formula is true under every interpretation assigning an empty set of individuals to the predicate JC. We need a richer (hyperintensional) logic such as TIL. Tenses

18. “All John’s children are asleep” • IfJohn has childrenthenT or F according as all his children are asleep, elsefail (to produce a truth-value). (1*) wt if[0Exist[0Children_ofwt0John]] then[[0All [0Children_ofwt0John]] 0Sleepwt]elsefail. (1**) 2[0The_only c [[0Exist[0Children_ofwt0John]]c=0[[0All [0Children_ofwt0John]] 0Sleepwt]]] (Anaphora ignored, for the sake of simplicity.) Tenses

19. “All John’s children were asleep last night” The sentence is ambiguous. • Q: What about John’s children?A: They all were (were not) asleep last night.|= John has children now (which is not only implied but also presupposed). • Q: What was going on last night?A: All John’s children were asleep.|= John had children lastnight (which is only implied). Tenses

20. “All John’s children were asleep last night” • Last_Night/(()) function that associates a given time t with a time-interval (that is last night with respect to t); • Underlying proposition p wt [ct*[[ct*]  [[0All [0Children_ofwt*0John]] 0Sleepwt*]]] [0Last_Night t]] Ad (a); presupposition (a) IfJohn has childrenthenT or F according as all his children were asleep last night, elsefail. (a*) wt if[0Exist[0Children_ofwt0John]] then t* [[[0Last_Night t]t*]  [[0All [0Children_ofwt*0John]] 0Sleepwt*]elsefail. Tenses

21. “All John’s children were asleep last night” • Ad (b); mere entailment (commitment) • “Last night John had children and they were all asleep” (b*)wt [t* [[[0Last_Night t]t*]  [[0Exist[0Children_ofwt*0John]]  [[0Exist[0Children_ofwt*0John]]  [[0All [0Children_ofwt*0John]] 0Sleepwt*]] No, it is not true; non-(b*) wt [t* [[[0Last_Night t]t*]  [[0Exist[0Children_ofwt*0John]]  [[0Exist[0Children_ofwt*0John]]  [[0All [0Children_ofwt*0John]] 0Sleepwt*]] (last night either John did not have any children or John had children but not all of them were asleep) Tenses

22. Concluding remarks • Logical analysis cannot disambiguate any sentence, because it presupposes full linguistic competence. • Yet, our fine-grained method can contribute to a language disambiguation by making these hidden features explicit and logically tractable. In case there are more non-equivalent senses of a sentence we furnish the sentence with different TIL constructions. • Having a formal fine-grained encoding of a sense, we can then automatically infer the relevant consequences. Tenses

23. Future … • We are developing the TIL-Script language, a computational variant of TIL,and examine other complex features of natural language. • The complexity of the work going into building a procedural theory of language is almost certain to guarantee that complications we are currently unaware of will crop up. • Yet we are convinced that if any logic can serve to solve such problems, then it must be a logic with hyper-intensional (most probably procedural) semantics, such as TIL. Tenses

24. References • Duží, M. (2008): ‘TIL as the Logic of Communication in a Multi-Agent System’. In Research in Computing Science, vol. 33, pp. 27-40. • Duží, M. (2009): Topic-Focus Articulation from the Semantic Point of View. In CICLing 2009. Ed. Gelbukh Alexander, Berlin Heidelberg: Springer-Verlag LNCS, vol. 5449, pp. 220-232. • Manna, Z., Pnueli, A. (1995): Temporal Verification of Reactive Systems: Safety. Springer-Verlag, New York. • Prior, A.N. (1967): Past, Present and Future. Oxford: Clarendon Press. • Tichý, P. (1980): ‘The logic of temporal discourse’, Linguistics and Philosophy, vol. 3, pp.343-69, reprinted in Tichý (2004), pp. 373–69. • Tichý, P. (2004): Collected Papers in Logic and Philosophy, V. Svoboda, B. Jespersen, C.Cheyne (eds.), Prague: Filosofia, Czech Academy of Sciences, and Dunedin: Universityof Otago Press. Tenses