Semantic Groundedness I: Kripke and Yablo

Transcription

1 Semantic Groundedness I: Kripke and Yablo Jönne Speck 21st April 2012 The full Tarski T-schema leads to paradox. It is consistent, if restricted to sentences without truth predicate but we want more. What shall we do? Txφy Ø φ... which φ? $ '& '% any the right ones? without T consistent, but weak Restrict the T-schema to those instances grounded in nonsemantic truth. Inheritance concept of groundedness (Kripke): learning We start from the nonsemantic truths and falsehoods and ask: What semantic statements do they ground? Dependence concept of groundedness (Yablo) understanding We start from semantic statements and ask: Are they grounded? Yablo s main result: Modulo formalization these concepts are co-extensional. This work has been funded by the European Research Council 1

2 1 A Generalized Theory of Inductive Definitions Definition 1. For any domain U, let J be a monotone operator on sets of U, that is J : PpUq ÞÑ PpUq such that S S 1 ñ JpSq JpS 1 q. In what follows, focus on S PpUq such that S JpSq (S is sound ) For any such inductive space there is a sequence J 0 psq S J α 1 psq JpJ α qpsq J α psq pj β psqq, for α limit. β α This sequence has a least fixed point J 8 psq S. inductively defined by J. We say that S is built up, or Remark 2. It is the soundness of S which ensures that J α J β, for any α β. Remark 3. If x P JpRq, then Yablo calls R sufficient for x. Definition 4. (level) The level of an x P S is the least α such that x P J α psq. x inf tα : x P J α psqu Remark 5. Yablo s terminology here is unconventional. These levels are often called ranks. Inductive definitions allow for defining Inheritance: Definition 6. x P U is grounded in S just in case x P S. Question 7. What is it about J, an arbitrary monotone operator, that makes it formalize the intuitive idea of learning? What s Yablo s goal here: analyzing an intuitive notion, or replacing an informal idea by a rigorous definition? 2

3 Example (see below): Kripke s fixed point truth predicate. jump. Here, J is the Kripke Remark 8. Yablo allows for operators J on a collection of subsets P PpUq such that (i) H P P (ii) For any S 0 P P, S 1 P P,... S α P P, if S 0 S 1... S α then 0 β α S β P P This generalization seems unnecessary, as any J on such a proper subcollection of PpUq can be extended trivially to the whole of PpUq. 2 Yablo s Dependence Theory Let U, S and J be as in the previous section. Definition 9. (Yablo 1982 Def. 5) U U is an S-dependence relation just in case 1. if x P S then Dy : x y, 2. otherwise, a) if DR : x P JpRq then x y just in case y P R, for some R : x P JpRq and all y, b) otherwise x x. Definition 10. ( -path) Given an S-dependence relation, a -path is a sequence of objects xy 0, y 1,...y where for every α, y α y α 1. Let x denote any -path whose first element is x. Question 11. Yablo defines -paths with the additional clause that every element has only finitely many predecessors. Why does Yablo require the -paths to have order type ω? Definition 12. (S ) An object x is grounded in S ( x P S ) if there is an S-dependence relation such that every -path x is finite. 3

4 3 Yablo s Co-Extensionality Proof Remark 13. The following dependence relation plays a prominent role in Yablo s proof. Figuratively speaking, it traces downwards the inductive definition of S. This is achieved by restricting our attention to the sets J α psq, which are unique for any x P J α 1 psq. Definition 14. (Yablo 1982, p. 123) Let S be the S-dependence relation such that 1. if x P S then x y just in case y P J α psq for x α 1, 2. Otherwise, x y just in case y P R for arbitrary R : x P JpRq. Lemma 15. (Yablo 1982 Prop. 7) For any x P S every S -path x is finite; hence, x P S. 3.1 Dependence Trees Definition 16. (Yablo 1982 Def. 8) For each x P U, let px α q be an infinite sequence of occurrences of x. Let O be the set of occurrences for every x P U. Lemma 17. For each x P U and S-dependence relation, induces a tree T px, q with root x on a subset of the occurrences O. Proof. Recall that a tree is a partially ordered set pn, q, such that for every x P N, ty : y xu is well-ordered by N. Given some x 0 and, Yablo defines a relation T on occurrences (definition 8, p. 124). (a) Du P O : x 0 T u, (b) Du P O : ut x 0, z κ P O : ut z κ only if y z if ut z κ then β κ, v, w P O : ut w and vt w only if u v. Let y z if y is a T -ancestral of z, and N ty : x 0 yu. Clause (d) of Yablo s definition makes a linear ordering. By the clauses (a) and (b), this linear ordering has a least element: x 0. Hence, pn, q is a tree. 4

5 Lemma 18. Given x and, any two T px, q are isomorphic. Proof. Let T px, q and T 1 px, q be two trees. Of course, we have a function from O to the field of such that y α T z β just in case y z. Moreover, by clause (c), each is equipped with a function f from the field of to the occurrences O such that y z just in case f pyqt f pzq. Composed, these two functions provide an isomorphism from T px, q onto T 1 px, q. y α T z β just in case y z just in case fpyqt fpzq. Remark 19. We need multiple occurrences of the objects in U since there may be distinct x, y such that there is a z P S X R, where JpSq Q x and JpRq Q y. If every branch of T px, q is finite, we say that the tree terminates. Lemma 20. x P S just in case there is some S-dependence relation such that T px, q terminates. Proof. By clause (c) above, there is, every branch of T px, q, a bijection from the occurrences in this branch onto a -path x. Hence, every branch x is finite just in case every path of T px, q is, just in case T px, q terminates. For any terminating tree, Yablo defines a natural assignment of ranks to its nodes (def. 9, p. 125). Definition 21. (rank) For an S-dependence relation and a terminating tree T px, q, define the rank Rpuq of a node u as follows Rpuq 0 if u P S, Rpuq 1 if u terminal node but not u P S, Rpuq maxtrpvq 1 : u directly above vu. The rank of a tree we identify with the rank of its root. The rank of an object x P S is the rank of the shortest tree T px, q such that the S-dependence relation grounds x: Rpxq inf trpt px, q : every -path x finiteu Lemma 22. If x P S then x P S. 5

6 Proof. (Sketch) Assume x P S and let be the S-dependence relation such that every x is finite. By lemma 20, the tree T px, q terminates, which allows us to reason by induction on the rank of its nodes, and establish that every node in T px, q is (an occurrence of an object) in the least fixed point of J, hence in S. For details, see Yablo p Theorem 23. The inheritance concept of groundedness and the dependence concept of groundedness coincide. For any J, S S. Proof. By lemmata 22 and 15. Theorem 24. For any x P S, its rank coincides with its level: inf tα : x P J α psqu x Rpxq inf trpt px, q : every -path x finiteu Proof. (ñ) By lemma 22 we know that the trees T px, S q terminate. Based on the definition of S we show by induction that for every such tree, RpT px, q x. Hence Rpxq x. (ð) Since we have set Rpuq 1 for terminal nodes, tree induction shows that for any S-dependence relation, RpT px, qq x. 4 Semantic Groundedness: Inheritance and Dependence Let L be the language of arithmetic, and LrT s L Y t T }. Definition 25. U k txφ, vy : φ P LrT s and v P tt, fuu Definition 26. Let NpSq be the LrT s-structure which for L coincides with the standard model of arithmetic N pω, 0,, q, and assigns to T the extension tφ : xφ, ty P Su and the anti-extension tφ : xφ, fy P Su. Definition 27. (Jump operator) Let J k : PpU k q ÞÑ PpU k q such that JkpSq txφ, ty : NpSq ( sk φu Y txφ, fy : NpSq ( sk φu for ( sk the Strong Kleene evaluation scheme. Lemma 28. J k is monotone on PpUq. 6

7 Proof. It is well-known that if S S 1 PpUq then tφ : NpSq ( sk φu tφ : NpS 1 q ( sk φu. Hence J k psq J k ps 1 q. Also compare the Basic Lemma from handout 4, PPP truth seminar. This allows us to apply the theory of inductive definition ( 1) and observe that Proposition 29. (Kripke 1975) The least fixed point of J k aka S aka the set of grounded semantic facts models arithmetic plus a type-free, partial truth predicate. Compare structure NrT l, F l s from theorem 5, handout 4, PPP truth seminar. Moreover, we can apply Yablo s theory of dependence ( 2), and know by theorem 23 that the set S we obtain is just another characterization of S. Question 30. On p. 128, Yablo writes (my emphasis):... we will settle in advance on Kleene s strong scheme, although the bulk of what follows will go through on any monotone scheme. Isn t he overly cautious? Any monotone logic will give rise to a monotone jump operator, and nothing more is needed for applying Yablo s apparatus. Remark 31. (Level-seeking) In his [1975, fn. 10], Kripke suggests that one advantage of his theory over the Tarskian hierarchy is that sentences seek their own level. Yablo submits ( 9) that his dependence-style characterization clarifies this feature. By definition 21 and theorem 24, the level is determined by the smallest tree: In rough terms, each sentence is given the chance to get its truth value in the easiest possible way.[yablo, 1982, p. 132] 5 Depending vs Being About 5.1 Two Notions of Dependence Definition 32. (strong dependence, fn. 16) x P U strongly depends on y P U if for all S U, if x P JpSq then y P S. For example, the truth of Tx0 1 1y strongly depends on x0 1 1, ty. Yablo notes that many truths and falsehoods do not depend on any fact strongly. For example: xdxt x, ty. Therefore, he prefers the following, weaker concept. 7

8 Definition 33. (potential dependence, fn. 16) x P U potentially depends on y P U if there is some S U : S Q y such that JpSq Q x but JpSztyuq S x. In particular, the truth or falsity of a sentence φ depends on the semantic facts xψ, vy P S such that J k psq Q xφ, vy P S but J k psztxψ, vyuq S xφ, vy P S For example, xdxt x, ty depends on x0 1 1, ty merely potentially. Question 34. y potentially grounds y if y is a necessary part of a set sufficient for x. Does this suggest a connection between grounding and INUS-conditions (Mackie 1965)? Interesting: Metaphysicists have a notion of grounding as an analogue of causation. Definition 35. y is an INUS condition for x if y is an insufficient but necessary part of a larger condition for x which itself is sufficient but unnecessary. Interestingly, not all dependence relations express potential dependence. The existential quantifier in clause (2) of definition 9 allows x to stand in a dependence relation to redundant objects. However, we may remove all garbage until we arrive at a minimal dependence relation: Definition 36. (strict dependence relations) An S-dependence relation is strict just in case there is no 1 ˆ such that 1 is an S-dependence relation. For strict S-dependence relations, and x R S, x y just in case there is an S Q y such that JpSq Q x and there is no S 1 ˆ S, S 1 Q y, such that JpS 1 q Q x (if there were such an S 1, we could define a proper subrelation of ). Hence, strict S-dependence relations express dependence in the sense of definition Dependence vs Aboutness Given the framework of section 4, dependence is a relation between semantic facts, i.e. pairs xφ, vy. But, we easily obtain sentence-dependence as follows: Definition 37. (sentence dependence) φ depends on ψ if xφ, vy depends on xψ, wy, for some v, w P tt, fu. Also, let s identify the rank of a sentence φ with the rank of xφ, vy P S Remark 38. Yablo does not specify his notion of aboutness. But his example on p. 131, bottom, suggests a close connection to Herzberger s notion of domain: 8

9 For a simple sentence whose main verb is intransitive, the domain comprises everything that satisfies its underlying subject term. [...] More complex sentences [...] will have correspondingly complex domains. [Herzberger, 1970, p. 147] In particular, a quantified sentences is not about the whole domain, but only about the range of the restricted quantifier. For example, Everything Nixon said about Watergate is false is not about everything, but about Nixon s statements about Watergate. A sentence does not depend on everything that it is about: Let N be the syntactic predicate... contains the numeral 0. Then, DxpṆ x ^ T xq is about any sentence containing the numeral 0. For example, it is about But it does not depend on it: it depends on (and other truths). In general, Proposition 39. The rank of an object x P U is determined by strict dependence relations: Rpxq inf trpt px, qq : strict u. Proof. Recall from def. 21 that Rpxq inf trpt px, q : every -path x finiteu and assume that Rpxq is determined by a non-strict dependence relation 0, i.e. Rpxq RpT px, 0 qq inf trpt px, q : every -path x finiteu. Then 0 has a proper subrelation 1 which is an S-dependence relation. Since by assumption, every 0 -path x is finite, every x is finite, too. Notice that by lemma 18, for any x, T px, 1 q is isomorphic to a subtree of T px, 0 q. Hence there is a tree T px, 1 q of rank less than T px, 0 q, which contradicts the assumption that RpT px, 0 qq=inf trpt px, qq : every -path x finiteu. It is in this sense, Yablo submits, that the rank of a sentence is determined by what it depends on, not by what it is about. 6 Discussion Question 40. Do Yablo s remarks on p. 122, such as Grounding is not so much like covering all the bases as like having a leg to stand on 9

10 suggest that groundedness is a liberalization of predicativity? Recall the fact that sentences don t depend on everything they are about. Compare also Yablo s remarks about type theory on p Question 41. If groundedness is having just one leg to stand on, then why does definition 12 require all -paths to be finite? Question 42. Yablo submits that neither the inheritance nor the dependence approach has an obvious claim to intuitive or formal primacy over the other. [p. 134] But at least formally, the definition of dependence relations relies on J. Does this mean that we can understand S without understanding S, but not the other way around? Question 43. In footnote 3 Yablo writes... formally analogous inheritance-style characterizations can be drawn almost at random from the plentiful stock of inductive definitions currently in use in most parts of mathematics...[yablo, 1982, p. 135] Are they meant to be cases of groundedness? defined? Is groundedness just being inductively References [Herzberger, 1970] Herzberger, H. G. (1970). Journal of Philosophy, 67: Paradoxes of Grounding in Semantics. [Yablo, 1982] Yablo, S. (1982). Grounding, Dependence and Paradox. Journal of Philosophical Logic, 11(1):