Grounded abstraction

Transcription

1 Grounded abstraction Øystein Linnebo Mini-seminar on groundedness. Winter term Session 3 1 The bad company problem (Σ) α = β α β (HP) (V) (H 2 P) #F = #G F G ˆx.F x = ˆx.Gx x(f x Gx) R = S R S 2 How groundedness might help What extensions there are depends on what concepts there are. What concepts there are depends on what extensions there are. A heavy-handed proposal Require that an abstraction principle be predicative in the sense that its RHS not quantify over the sort of objects to which its LHS refers. A more general idea Abstraction as a process by which more and more abstracts are introduced. Each step of the process presupposes only what was available at the previous step Old news? Asymmetry The right-hand side of an abstraction principle must not presuppose any of the objects to which the left-hand side refers. Quantification incurs presupposition (QIP) A quantified statement presupposes every object in the range of its quantifiers. Conclusion Impredicative abstraction principles are impermissible. 1

2 Can we resist QIP by analyzing presupposition? Following Kripke, we consider monotonic operators J J(X) = Y formally represents the philosophical idea that the elements of Y depend on X, or X suffices for each element of Y, or the elements of Y presuppose at most X. Need to spell out clearly what philosophical idea J is meant to track. Two key questions Q1. What are the sets on which J operates (e.g. sets of abstracta, (representations of) identity facts, etc.)? Q2. What is the operator J (e.g. strong Kleene, supervaluational, or Leitgeb-style)? 3 Option 1: Leitgeb on grounded abstraction (Leitgeb, ming) Consider an abstraction principle: nx.f x = nx.gx Φ[F, G] We wish to restrict this to the instances that are grounded (in a sense to be articulated). Presumably, this means accept only instances nx.φ(x) = nx.ψ(x) Φ[φ/F, ψ/g] whose RHS is grounded. The ingredients A domain Dom of pre-objects. A domain Con (Dom) of [pre-]concepts on these pre-objects. A [pre-] abstraction operator N : Con Dom, which is a bijection. The recipe Build up identity and distinctness facts on Dom by iterating a Leitgeb-jump Questions How to choose Dom? Which of the subsets of Dom to admit into Con? How to choose N? How to advance from groundedness relative to the three ingredients to an account of acceptable instances? 2

3 4 Option 2: Term models for abstraction Cf. (Horsten and Leitgeb, 2009) and (Horsten and Linnebo, 2011), but also (Horsten, 2010) Let T the a set of appropriate abstraction terms (i.e. terms of the form x.φ(x)). Can we in a grounded way build up an equivalence relation R on T such that T/R is a model for (Σ)? Philosophical idea: stepwise determination of facts about identity and distinctness. Negative result This works only when the background second-order logic is predicative. (Horsten and Linnebo, 2011), cf. (Dummett, 1991) Circularity worry But the account will then fail to provide determinate truth-conditions, since the truth-conditions for identity statements will be given in terms of quantified statements and the truth-conditions of quantified statements in terms of the identity statements. (Fine, 2002, p. 88) Groundedness as a response At the outset of our logical process no T -identities are assumed We consider all the possible ways of identifying objects in accordance with (Σ), while respecting the identity and distinctness facts that we have already fixed If there are objects that are identified or distinguished by the criterion on all of these permissible ways, then these identity and distinctness facts can at most presuppose those already fixed. So we add these identity and distinctness facts to those we have fixed ( = supervaluation idea) The identities and differences that are fixed in this way are grounded. Positive results When implemented, this addresses the circularity worry. In fact, this yields a natural model for BLV with predicative comprehension. (Horsten and Linnebo, 2011) 3

4 5 Option 3: Identity presupposition Suppose we are allowed to presuppose that the relevant objects exist but not their identities. (What follows is an absolute version of a construction done relative to set theory in (Fine, 2005) and (Linnebo, 2006).) We don t presuppose the identities of some objects if each is put in an opaque box of the same kind. Suppose we have revealed the identities of the elements of I D. Say that a concept F on D has support in I iff for each π s.t. π I = id I, we have: x ( F (π(x)) F (x) ) More generally, a relation R (of any order) has support in I if for any π s.t. π I = id I, we have π (R) = R, where π is the induced permutation. If we abstract only on concepts that have support in I under equivalence relations that have support in I, then we do not presuppose the identities of any objects outside of I. Positive result If the cardinality of D is strongly inaccessible, our procedure yields a standard model for a simple modification of ZFC which admits complements (Forster, 2008). Negative observation The procedure yields no information about the cardinality of D. This has to be given from the outside. 6 Option 4: Dynamic abstraction We build up an ontology in a grounded way; cf. (Linnebo, 2009), (Linnebo, 2010), (Linnebo, ) 1. Entities are introduced successively through a well-ordered series of stages. 2. The introduction of an entity consists in the specification of a (permanent) identity condition using resources available at the relevant stage. 3. Cumulativity. The licence to individuate an object never goes away but can always be exercised at a later stage. 4. Maximality (optional). At every stage we individuate all the entities we are entitled to individuates. 4

5 5. Groundedness. The identity condition for an entity E may only presuppose entities individuated before E. An associated modal logic Each stage is a possible world, which consists of the entities individuated so far, and which specifies how these entities are related. An accessibility relation (w w ) which holds only if w is a (not necessarily proper) extension of w. So is reflexive, anti-symmetric, and transitive. directed (i.e. x y x z w(y w z w)) well-founded. The resulting Kripke-models validate the modal logic S4.2 = S4 + (G): (G) p p. Actualist generality: within a given world Expressed by and Potentialist generality: across all possible worlds We can prove that the complex strings and ( modalized quantifiers ) behave logically just like quantifiers. Potentialist generality: the theorem Say that a formula φ(u) is stable iff the following two conditionals hold: φ(u) φ(u) φ(u) φ(u) If φ is non-modal, its potentialist translation φ is the result of replacing every quantifier by the corresponding modalized quantifier. Theorem 1 Let be provability by, S4.2, and axioms stating that every atomic predicate is stable, but with no higher-order comprehension. Then we have: φ 1,..., φ n ψ iff φ 1,..., φ n ψ. 5

6 Example: Dynamic Basic Law V Start with a plural version of BLV: {u u xx} = {u u yy} u(u xx u yy) where u xx means that u is one of the objects xx. Then, factor this into a criterion of existence and a criterion of identity. uu x Set(x, uu) Set(x, uu) Set(y, vv) [x = y z(z uu z vv)] Finally, consider the potentialist translations: (V ) xx y Set(y, xx) (V =) Set(x, uu) Set(y, vv) [x = y z(z uu z vv)] What pluralities are there? The easy question Any formula φ(u) actually defines a plurality: (Comp) xx u[u xx φ(u)] The hard question (Linnebo, 2010) When could there be a plurality that is necessarily defined by φ (u)? (Comp ) xx u[u xx φ (u)] Answer to the hard question A plurality has the same elements in every possible world. So φ (u) must apply to the same objects in every possible world. An application to set theory Theorem 2 ((Linnebo, )) Assume the modal logic S4.2 + maximality trans-world extensionality principles for pluralities and sets xx y Set(y, xx) Then we can interpret Zermelo set theory minus Infinity and Foundation. 6

7 Theorem 3 Assume further that every possibility witnessed by the potential universe is witnessed by some possible world: (Refl 0 ) (Refl) φ φ x(φ φ) (Refl 0 ) allows us to interpret Infinity, and (Refl), also Replacement. 7 (Inevitably biased) concluding remarks Dynamic abstraction is the best approach to grounded abstraction Good motivation based on the idea of groundedness. Yields strong theories (unlike Option 2 (and 1?)) Generates, rather than assumes, the domain of objects (unlike Option 3 (and 1?), although this criticism may not apply to version relative to set theory) References Dummett, M. (1991). Frege: Philosophy of Mathematics. Harvard University Press, Cambridge, MA. Fine, K. (2002). The Limits of Abstraction. Oxford University Press, Oxford. Fine, K. (2005). Class and Membership. Journal of Philosophy, 102(11): Forster, T. (2008). The iterative conception of set. Review of Symbolic Logic, 1(1): Horsten, L. (2010). Impredicative identity criteria. Philosophy and Phenomenological Research, 80(2): Horsten, L. and Leitgeb, H. (2009). How abstraction works. In Hieke, A. and Leitgeb, H., editors, Reduction and Elimination in Philosophy and the Sciences, pages Ontos Verlag, Frankfurt. Horsten, L. and Linnebo, Ø. (2011). Term models for abstraction principles. Unpublished manuscript. Leitgeb, H. (Forthcoming). Abstraction grounded. In Ebert, P. and Rossberg, M., editors, Abstractionism in Mathematics: Status Belli. Linnebo, Ø. The Potential Hierarchy of Sets. Unpublished manuscript. Linnebo, Ø. (2006). Sets, Properties, and Unrestricted Quantification. In Rayo, A. and Uzquiano, G., editors, Absolute Generality, pages Oxford University Press, Oxford. 7

8 Linnebo, Ø. (2009). Bad Company Tamed. Synthese, 170(3): Linnebo, Ø. (2010). Pluralities and Sets. Journal of Philosophy, 107(3):