1 / 41

Transparent intensional logic, -r ule and Compositionality

Transparent intensional logic, -r ule and Compositionality. Marie Duží VSB -Technical University Ostrava http://www.cs.vsb.cz/duzi. Attitude Logic(s). A reliable test on Compositionality Attitudes: Notional Propositional

blythe
Download Presentation

Transparent intensional logic, -r ule and Compositionality

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Transparent intensional logic, -ruleand Compositionality Marie Duží VSB-Technical University Ostrava http://www.cs.vsb.cz/duzi TIL & beta-rule

  2. Attitude Logic(s) • A reliable test on Compositionality • Attitudes: • Notional • Propositional • We are dealing with a fine difference between the meanings of sentences like (P1) Charles believes that the Pope is in danger (P2) Charles believes of the Pope that he is in danger • Some authors even claim that (P1) is ambiguous, that it can be also read as (P2). Rules of substition

  3. Attitude logics and belief sentences In our opinion it is not so. We can, for instance, reasonably say (it may be true) that Charles believes of the Pope that he is not the Pope, whereas the sentence Charles believes that the Pope is not the Pope cannot be true, unless our Charles is completely irrational. The sentences like (P1) and (P2) have different meanings, and their difference consists in using ‘the Pope’ in the de dicto supposition (P1) vs. the de re supposition (P2). The two sentences are neither equivalent, nor is any of them entailed by the other. Rules of substition

  4. Belief sentences in doxastic logics • In the usual notation of doxastic logics the distinction is characterised as the contrast between BCharles D[p] (de dicto) (x) (x = p  BCharles D[x] (de re) • But there are worrisome questions (Hintikka, Sandu 1989): Where does the existential quantifier come from in the de re case? There is no trace of it in the original sentence. How can the two similar sentences be as different in their logical form as they are? • Hintikka, Sandu propose in their (1996) a remedy by means of the Independence Friendly (IF) first-order logic: Rules of substition

  5. Belief sentences in doxastic logics • “Independence Friendly (IF) first-order logic deals with a frequent and important feature of natural language semantics. Without the notion of independence, we cannot fully understand the logic of such concepts as belief, knowledge, questions and answers, or the de dicto vs. de re contrast.” Hintikka, Sandu (1989): Informational Independence as a Semantical Phenomenon. In J.E. Fenstad et el (eds.), Logic, Methodology and Philosophy of Science, Elsevier, Amsterodam 1989, pp. 571-589. Hintikka, Sandu (1996): A revolution in Logic? Nordic Journal of Philosophical Logic, Vol.1, No.2, pp. 169-183. Rules of substition

  6. Belief sentences and IF semantics • Hinttika, Sandu solve the de dicto case as above, and propose the de re solution with the independence indicator ‘/’: BCharles D[p / BCharles] • This is certainly a more plausible analysis, closer to the syntactic form of the original sentence, and the independence indicator indicates the essence of the matter: • There are two independent questions: • ”Who is the pope” and • ”What does Charles think of that person”. Of course, Charles has to have a relation of an ”epistemic intimity” to a certain individual, but he does not have to connect this person with the office of the Pope (only the ascriber must do so). (Chisholm,R.(1976): Knowledge and Belief: ‘De dicto’ and ‘de re’. Philosophical Studies 29 (1976), 1-20.) Rules of substition

  7. Belief sentences and Intensional logics BCharlesD(p) (de dicto) x BCharlesD(x)(p) (de re) But: x BCharlesD(x)(p)  BCharlesD(p) ! (applying the rule of -reduction). What then is the difference between de dicto and de re? Why is it “forbidden” here to perform the fundamental rule of -calculi? Rules of substition

  8. Solomon Feferman (1995): Logic of Definedness Introduces the axioms (λp) for Partial Lambda Calculus as follows (t↓ means - the term t is defined): • λx.t ↓ • (λx.t(x))y  t(y). The axiom (ii) corresponds to the trivial β-reduction, but the limitation on instantiation in PLC restricts its application to: • s↓ (λx.t(x))s  t(s). (but why this restriction?, proof?) Our system (TIL) introduces a generally valid β-reduction for thePartial Higher-Order Hyper-intensional Lambda Calculus. Rules of substition

  9. Transparent Intensional Logic Formally: • The language of TIL constructions can be viewed as a hyper-intensional -calculus operating over partial functions. • “hyper-intensional”: -terms are not interpreted as set-theoretical mappings (”modern functions”) but as algorithmically structured procedures(which produce as an output the (partial) mapping). • Procedures, known as TIL constructions, are objects sui generis: they can be not only used but also mentioned within a theory. TIL & beta-rule

  10. Suppositio (substitution) • A lot of misunderstanding and many paradoxes arise from confusing different ways in which a meaningful expression can be used. • We are going to show that these different ways consist in using and mentioning entities (by means of an expression) • In which way can an entity be used or mentioned? TIL & beta-rule

  11. Using / Mentioning Entities Expression used mentioned to express its meaning: procedure(‘TIL construction’) de dicto / de re mentionedusedto produce a function: mentioned used to point at … Use / Mention

  12. TIL constructions • Abstract procedures, structured from the algorithmic point of view. • structured meanings: Instructions specifying how to arrive at less-structured entities. • Being abstract, they are reachable only via a verbal definition. • The ‘language of constructions’: a modified version of the typed -calculus, where Montague-like -terms denote, not the functions constructed, but the constructions themselves. • Henk Barendregt (1997): -terms denote functions, yet “... in this interpretation the notion of a function is taken to be (hyper-)intensional, i.e., as an algorithm.” • Operate on input objects (of any type, even constructions) and yield as output objects of any type: they realize functions (mappings) Rules of substition

  13. Kinds of constructions • Atomic: do not contain as a used constituent any other construction but themselves (supply objects …) • Variablesx, y, p, c, … v-constructing • Trivialisation of X: 0X • Compound. • Composition[X X1…Xn]: the instruction to applya (partial) functionf(constructed by X) to an argument A (constructed by X1,…,Xn) to obtain the value (if any) of f at A. • -Closure [x1…xn X]: the instruction to abstract over variables in order to obtain a function. • Double execution2X: the instruction to use a higher-order construction X twice over as a constituent. Constructions

  14. TIL Ramified Hierarchy of Types The formal ontology of TIL is bi-dimensional. One dimension is made up of constructions. The other dimension encompasses non-constructions, i.e., partial functions mapping (the Cartesian product of) types to types. Rules of substition

  15. TIL Ramified Hierarchy of Types 1st-order: non-constructionsBase:, , , , partial functions ((())), (()), …, (01…n) 2nd-order:Base: *1 constructions of 1st-order entities, partial functions involving such constructions: (01…n), i = *1 3rd-order:Base: *2, constructions of 2nd-order entities, partial functions involving such constructions: (01…n), i = *2, or *1 And so on, adinfinitum Rules of substition

  16. -intensions; examples • Functions of type () • Usually both modal and temporal parameters: (()) • Abbreviation:  • Propositions /  • (individual) offices /  • Magnitudes /  • Empirical functions (attributes)/() • Attitudes / (n) Rules of substition

  17. Definition Used* vs. Mentioned* Let C be a construction and D a sub-construction of C. Then an occurrence of D is used* as a constituent of C iff: • If D is identical to C (i.e., 0C = 0D) then the occurrence ofD is used* as a constituent of C. • If C is identical to [X1 X2…Xm] and D is identical to one of the constructions X1, X2,…,Xm, then the occurrence of D is used* as a constituent of C. • If C is identical to [x1…xmX] and D is identical to X, then the occurrence of D is used* as a constituent of C. • If C is identical to 1X or 2X and D is identical to X, then the occurrence of D is used* as a constituent of C. • If C is identical to 2X and Xv-constructs a construction Y and D is identical to Y, then the occurrence of D is used* as a constituent of C. • If an occurrence of D is used* as a constituent of an occurrence of C’ and this occurrence of C’ is used* as a constituent of C, then the occurrence of D is used * as a constituent of C. If an occurrence of a sub-construction D of C is not used* as a constituent of C, then the occurrence of D is mentioned* in C. Use / Mention

  18. Definition Used* vs. Mentioned* Let C be a construction and D a sub-construction of C. Then an occurrence of D is mentioned* in C iff it is not necessary to execute D in order to execute C; Otherwise D is used* as a constituentof C. • Makes a fine individuation possible; finer than just an equivalence. Use / Mention

  19. Two kinds of using a construction:de dicto vs. de re supposition. Roughly: C = [… D … ], D  () • D occurs in C with de dicto suppositioniff D is not composed with a construction A  ; • the respective function / ()is just mentioned • D occurs in C with de re suppositioniff D is composed with a construction A  , and D does not occur as a constituent of a de dicto occurrence D’ (de dicto context is dominant); • the respective function / ()is used as a pointer to its actual, current value /  Rules of substition

  20. Contextssuppositio substitution The President of USA knows that John Kerry wanted to become the President of USA. The President of USA is (=) the husband of Laura Bush.  Hence what ? Did John Kerry want to become the husband of Laura Bush? Rules of substition

  21. Contextssuppositio substitution C1 wt [0= [wt [0Preswt0USA]]wt [0Husbandwt0Bush]] extensional context: of using* de re C2 wt [0Wwt0K [wt [0Bwt0K wt [0Pres 0USA]]] ] intensional context: of using* de dicto C3 wt [0Knowwt [wt [0Preswt0USA]]wt 0[wt [0Wwt0K wt [0Bwt0K wt [0Preswt0USA]]]] ] hyper-intensional context: of mentioning*

  22. Using / Mentioning Constructions • Dividing six by three gives two and dividing six by zero is improper. Types: 0, 2, 3, 6 / , Div / (), Improper/ (1)the class of v-improper constructions for all v [[[0Div06 03] = 02]  [0Improper0[0Div06 00]]] used* constituents mentioned* Rules of substition

  23. Using / Mentioning Constructions • There is a number such that dividing any number by it is improper. Types: Div / (), Improper/ (1),,/(()), x, y  . Existsx for ally [0Improper0[0Divx y]]. But x, yoccur in the hyper-intensional context of mention*; they are not free for evaluation or substitution. How to quantify? To this end we use functions Sub and Tr: Sub/ (1111)the mapping which takes a construction C1, variable x, and a construction C2 to the resulting construction C3, where C3 is the result of substituting C1 for x in C2. Tr/ (1)the mapping which takes a number and returns its trivialisation Rules of substition

  24. Using / Mentioning Constructions (*) [0y [0x [0Improper [0Sub [0Tr y]0y’ [0Sub [0Tr x]0x’ 0[0Div x’ y’ ]]]]]]. • Let a valuation v assign 0 to y and 6 to x. Then the sub-construction [0Sub [0Tr y]0y’ [0Sub [0Tr x]0x’ 0[0Div x’ y’ ]]] • v-constructs the construction[0Div 06 00], which belongs to the class Improper. This is true for any valuation v’ that differs from v at most by assigning another number to x. • The construction (*) constructs True. Rules of substition

  25. De dicto / de re supposition • The temperature in Amsterdam equals the temperature in Prague. • The temperature in Amsterdam is increasing.--------------------------------------------------------- • The temperature in Prague is increasing. Types: Temp(erature in …)/(), Amster(dam), Prague/, Increas(ing)/(). • wt [wt [0Tempwt0Amster]wt = (de re)wt [0Tempwt0Prague]wt] • wt [0Increaswt [wt [0Tempwt0Amst]] the magnitude is (de dicto)mentioned. Rules of substition

  26. Rules of Substitution (logic of partial functions !) • “Homogeneous” substitution: no problemLebniz’s law • Used* de re extensional context de re • Used* de dicto  intensional context de dicto • Mentioned* construction  hyperintensional context • Used* constructions – constituents: • De re(extensional) context: [Cx] = [C’y] • co-incidental constructions substitutable • De dicto(intensional context): C = C’ • equivalent constructionssubstitutable • Mentioned* (hyper-intensional) context: 0C = 0C’ • Only identical constructions substitutable

  27. Rules of Substitution (logic of partial functions !) • Heterogeneous substitutions. • Construction of a lower-order into a higher-order context (which is dominant): • We must not carelessly draw a construction D occurring in a lower-order context into a higher-order context. • Why not? The substitution would not be correct even if there is no collision of variables, due to partiality

  28. De re rules • The president of CR is (is not) an economist.  de reThe president of CR exists. • The president of CR is eligible.  de dictoThe president of CR may not exist. • In the de re case there is an existential presupposition, unlike the de dicto case. Rules of substition

  29. Charles believes of the president of CR that he is an economist. Types: Ch/, B/(), Pr(esident of …)/(), CR/, Ec/() Synthesis (h  , a free variablethe meaning of “he”): He is an economist: wt [0Ecwth] v  (anaphora) ThePresident of CR: wt [0Prwt 0CR]  a) The President of CR is believed by Charles to be an economist – the passive variant wt [h [0Bwt0Ch wt [0Ecwth]] wt [0Prwt 0CR]wt] Now, can we perform -reduction ??? Yes, but only the trivial one: wt [0Prwt 0CR]wt | [0Prwt 0CR] Collision of variables? Let us rename them: Rules of substition

  30. Charles believes of the president of CR that he is an economist. -reduction “by name” : wt [h[0Bwt0Ch w’t’ [0Ecw’t’h]] [0Prwt 0CR]] | ???wt [0Bwt0Ch w’t’ [0Ecw’t’ [0Prwt 0CR]]] No collision of variables, But. [h [0Bwt0Ch w’t’ [0Ecw’t’h]] [0Prwt 0CR]]  [0Existwtwt [0Prwt 0CR]] = [0x [x = [0Prwt 0CR]] Unlike the latter. Therefore, don’t perform -reduction (!?!) Rules of substition

  31. Charles believes of the president of CR that he is an economist. b) The direct analysis of the active form, using Tr and Sub. -reduction “by value”: Now we have to substitute for h the construction of the individual (if any) that actually plays the role of the president: • wt [0Belivewt0Charles2[0Sub[0Tr wt [0Prwt0CR]wt] 0h(extens.)0[wt [0Ecwth]]]] (intens.) Rules of substition

  32. 2-phase -reduction: how does it work? wt [0Belwt0Ch 2[0Sub[0Tr wt [0Prwt0CR]wt] 0h 0[wt [0Ecwth]]]] • Let wt [0Prwt0CR]wt be v-improper (the president does not exist). • Then [0Tr wt [0Prwt0CR]wt] is v-improper and • The function Sub does not have an argument to operate on: • [0Sub[0Tr wt [0Prwt0CR]wt] 0h0[wt [0Ecwth]]]v-improper. (And so is the Double execution.) • The so-constructed proposition does not have a truth-value, as it should be (the existential presupposition) Rules of substition

  33. Substitutionby value (-reduction) wt [0Belwt0Ch 2[0Sub[0Tr wt [0Prwt0CR]wt] 0h 0[wt [0Ecwth]]]] • Let wt [0Prwt0CR]wt be v-proper (the president exist). Then • the construction [0Prwt0CR] v-constructs particular individual Y (For instance V. Klaus.) Then • [0Tr wt [0Prwt0CR]wt] v-constructs 0Y, and Sub inserts it for the variable h. • the result is the construction: [wt [0Ecwt0Y]] that is executed (Double execution) in order to construct the proposition that is believed by Charles. Rules of substition

  34. Substitutionby value (-reduction) Type checking: 2[Sub[0Tr[0Prwt0CR]]0h0[wt [0Ecwth]]] (*1)  (*1*1*1*1) *1*1 *1 *1 () 1. step  2. step (if the 1st did not fail): 1[wt [0Ecwt0Y]]  wt [0Belwt0Ch 20[wt [0Ecwt0Y]] 

  35. -reduction, another example (*) [y [0Deg z [0: z y]]0x ] ( = square root) (Deg/(())-a degenerated function) (*n) -reduced “syntactically-by-name”: [0Deg z [0: z 0x]]  [[0Exist x] 0] ??? NO (*v) -reduced “by value”: 2[0Sub[0Tr 0x] 0y 0[0Deg z [0: z y]]] Rules of substition

  36. Valid rule of -reduction (2-phase) Let C(y) be a construction with a free variable y, y  , and let D  . Then [[y C(y)] D]  2[0Sub [0Tr D] 0y 0C(y)] is a valid rule (proof, see above). Rules of substition

  37. Rules of inference:Types: y  β, x , D / (β), [Dx] β, C(y)  α, y C(y)  (αβ), [[y C(y)] [Dx]]  α. Compositionality: [0Improper0[Dx]] | [0Improper 0[[y C(y)] [Dx]]] [0Improper0[Dx]] | [0Improper02[0Sub [0Tr [Dx]] 0y 0C(y)]] [0Proper0[Dx]] |2[0Sub [0Tr [Dx]] 0y 0C(y)] = [[y C(y)] [Dx]] = C(y/[Dx]) Special case: Existential presuppositionde re Exist / (( (β)) )the property of a (β)-function of being defined at a -argument, [Exist x]  ( (β)) [[0Exist x] D] | [0Improper0[[y C(y)] [Dx]]] [[0Exist x] D] | [0Improper 02[0Sub [0Tr [Dx]] 0y 0C(y)]] But not: C(y/[Dx]) | [[0Exist x] D] … Rules of substition

  38. The two “de re principles”: a) existential presupposition Example: [y [0Deg z [0: z y]][0x]] | [[0Exist x] 0] 2[0Sub [0Tr [0x]] 0y 0[0Deg z [0: z y]]] | [[0Exist x] 0] Indeed: The square root does not exist for x < 0; for x < 0 the left-hand side is (v-)improper. If the left-hand side is true or false, then the square root exists and x  0. However, the result of the “syntactical” β-reduction does not meet these rules: [0Deg z [0: z 0x]] and not (for x < 0) [[0Exist x] 0 ]. Rules of substition

  39. The two “de re principles”: b) inter-substitutivity of co-incidentals [Dx] = [D’ ]  [[y C(y)] [Dx]] = [[y C(y)] [D’ ]] = 2[0Sub [0Tr [D’ ]] 0y 0C(y)] Example: The US President is the husband of Laura. The US President is a Republican. Hence: The husband of Laura is a Republican. But not: John Kerry wanted to become the husband of Laura. Rules of substition

  40. Substitutions in general Types: c  n, 2c  , A  , y   a) “by name” (homogeneous substitution): 2[0Sub00A 0c0C(c)] = C(c/0A) 2[0Sub0A 0y0C(y)] = C(y/A) b) “by value” (generally valid, even for heterogeneous substitution): 2[0Sub [0Tr A] 0y0C(y)] = [y [C(y)] A]  C(y/A) Rules of substition

  41. Conclusions • The top-down, fine-grained approach of TIL makes it possible to adequately model structured meanings,and thus: • to formulate meaning-driven (non ad hoc) rules of substitution taking into account the Use/Mention distinction at all levels; • to adhere to Compositionality and anti-contextualism(even in the cases of anaphora, de re attitudes with anaphoric reference, hyper-intensional attitudes, …); • to take into account partiality; • to meet the two de re extensionalprinciples (existential presupposition, inter-substitutivity of co-referentials). Rules of substition

More Related