Bolzano and Impeccable Explanation: The State of the Art A RIANNA B - PDF document

Bolzano and Impeccable Explanation: The State of the Art A RIANNA B ETTI & P AULINE VAN W IERST Logic, Language and Computation , ILLC, 25 November 2013 Handout The Claim  Bolzano’s grounding ( Abfolge ) is [6] All propositions of S are known to be true. A non-fundamental proposition is known to the explication of a traditional notion of scientific explanation within a millennia-old ideal of science. be true through its proof in S. (I) Bolzano ’s tremendous step forward in this [7] All concepts or terms of S are adequately known. A non-fundamental concept is tradition consists in his explicit attempts at explicating a notion of impeccable explanation, i.e. a adequately known through its composition notion of (formal or logical) consequence which is (or definition). also explanatory (call it deducibility + ). (II) These  attempts seem frustrated by one counterexample Bolzanian (conceptual) sciences as grounding by Bolzano himself from axiomatic ethics to the rooted trees of trees match this ideal effect that deducibility is not necessary for grounding; however, what the counterexample  ,  -  really shows is a matter of contention. (III) We The tentative definition: Impeccable Explanation? claim that a crucial role in Bolzano’s reasoning is “[ Grounding is ] that ordering of truths which allows played by his specific take on the Ought-Implies- us to deduce from the smallest number of simple Can principle. In sum, for conceptual sciences, premises the largest possible number of the barring a certain difficulty in ethics, grounding is remaining truths as conclusions” (WL §221). deducibility + .   I The Beyträge (1810) The Classical Model (or Ideal) of Science Five copulas A proper science S satisfies the following conditions 1. A is a kind of B [necessary judgements] (de Jong & Betti 2010): 2. A can be B [possibility judgements] 3. A ought to do B [prescriptive judgements] ( 1 ) All propositions and all concepts (or terms) 4. I perceive X [actual/empirical judgements] of S concern a specific set of objects or are 5. A is probably B [probability judgements]. about a certain domain of being(s). (2a) There are in S a number of so-called Four new inference rules fundamental concepts (or terms). [ eigentliche Schlüsse ] of ‘impeccable explanation ’ for (2b) All other concepts (or terms) occurring in S mathematics [not complete; wanting]  are composed of (or are definable from) these fundamental concepts (or terms). Diaries: rules for 3. (Blok 2013) [3] [4] (3a) There are in S a number of so-called Corollaries fundamental propositions. (3b) All other propositions of S follow from or (GrBD-1) If [A  B] is an axiom ( Grundsatz ), then are grounded in (or are provable or A and B are (absolutely) simple demonstrable from) these fundamental (GrBD-2) If C is a(n) (absolutely) simple concept, propositions. then for some B: [B  C] is an axiom ( 4 ) All propositions of S are true. (5) All propositions of S are universal and  necessary in some sense or another. Six (pure & impure) maxims of grounding (Roski 2013) 1

(1) [no increase in] generality ergo (2) [no increase in] simplicity deducibility is not a necessary condition for (3) uniqueness grounding and grounding cannot be defined as a [(4) axiomness: some grounds have no grounds] special kind of derivability. (5) asymmetry (6) deductive economy What is going on here?  ,  : [1], [2],  III The Open Question B’s r ock-bottom & problematic convictions about ethics Is deducibility a necessary condition for grounding?  Ad [Step 1] II › There is a (single) axiom [ Grundwahrheit ] of The puzzling argument from WL§200 Ethics. [GA 2 B 15: 248;  -  ] Ad [Step 2]  [Step 1] › Ought-Implies-Can reads ‘can grounds ought’ Take a practical truth of the form ( p ) one ought to do a certain thing ( G ) One ought to do A provided is grounded by ( q ) one is able to do that certain thing which grounds impeccably all other practical truths such as ‘One ought not to lie’ (i.e. its consequence Ad [Step 3] contains all the practical truths.). › Practical (moral) propositions [containing Sollen ] eg. Bolzano’s ‘highest moral law’: You should are undeducible from a collection of purely promote happiness (GA 2 B 15: 131) theoretical proposition [ not containing Sollen ], i.e. [Step 2] E +  If A were impossible, there could be no duty to do (P1) Stopping eating animals is possible A . (P2) Stopping eating animals promotes happiness (P3) One ought to promote happiness Therefore, ( Bz ) ( G ) has a (partial) ground in the theoretical (C) One ought (to) stop(ping) eating truth animals ( E ) A is possible. compare So, ( G )’s complete ground, ( E +), includes ( E ).  But (P1) Promoting happiness is possible [Step 3] (P2) [ ????] ought [????] [ vs Step 1] no inference rule allows us to infer ( G ) from (C) One ought to promote happiness ( E +): none of the truths in ( E +) can contain the concept of Sollen (otherwise, it would be a [5] practical truth: impossible - we supposed that all [Undefinability of Sollen : Primitive concepts of practical truths are included in the consequence of (proper) lower sciences must be undefinable ( G )) from concepts of higher sciences.] ergo [6] we have grounding between undeducible [NB! attempt at embedding ‘possibility’ in the (collections of) propositions (( E +) and ( G )). axiom!] 2

Definitions & quotes Es gibt keinen Übergang aus theoretischen Wahrheiten auf praktische, [1] und aus praktischen auf theoretische. Der oberste Satz, aus welchem alles Exact deducibility ( WL , §155.26) Sollen gefolgert wird, muß selbst ein Sollen enthalten. Aus einem  is exactly deducible from  with respect to v bloßen Seyn oder Müssen folgt kein Sollen , ohne daß ein andres Sollen schon voraus gesetzt wird. Und eben so kann aus keinem Sollen ein Seyn iff oder Müssen gefolgert werden. Aus lauter theoretischen Sätzen kömt a)  is deducible from  with respect to v man nie auf einen praktischen zu folgern . (RW, our emphasis ) b) if X is a proposition or a part of a proposition contained in  , then  is [6] not deducible from the collection of premises that results from  after Always choose from all actions that are possible for you the one which, all removing X . consequences considered, most advances the welfare of the whole, in whatever parts [2] (RW I, 236; cf. §447, WL IV 119) A sufficient condition for grounding: Bibliography “ If [(i)] a proposition M stands to other propositions A, B, C, … in the Betti, Arianna. 2010. Explanation in metaphysics and Bolzano’s theory of relation of exact deducibility […] with respect to ideas i, j, ..., if, ground and consequence. Logique et analyse , 211 , 281-316. moreover, [(ii)] propositions A, B, C, ... and M are the simplest Bolzano, Bernard. 2009. Gesamtausgabe. Reihe II: Nachlaß. B. propositions among those equivalent to them, and [(iii)] if none of A, B, Wissenschaftliche Tagebücher. Band 15 Philosophische Tagebücher 1803- C, … is more complex than M , then we may assume that M stands to A, 1810. Zweiter Teil. Herausgegeben von Jan Berg. B, C, … in a true relation of ground and consequence […].” ( WL , Blok, Johan. 2013. The Highest Moral Law as an a priori Synthetic §221.7) Principle. Chapter 6 of Bolzano’s Theory of Grounding and the Classical [3] Model of Science . Dissertation, University of Groningen. Mancosu, Paolo. 2013 . “Explanation in Mathematics.” The Stanford Encyclopedia of Philosophy , http://j.mp/1b0f7ez Roski, Stefan. 2013. Bolzano’s Theory of Grounding and the Classical Model of Science. Dissertation, Vrije Universiteit Amsterdam. Rumberg, Antje. 2013. Bolzano’s Concept of Grounding ( Abfolge ) against the Background of Normal Proofs. The Review of Symbolic Logic 6: [4] 424-59. weil es einen Grundsatz der Pflichtsurtheile geben muß; dieser um Van Wierst, Pauline. 2013. Salva Veritate - A master thesis on Bolzanian einfache Begriffe zu erhalten, muß den Begriff des Sollens in der Copula analyticity and computational methods within philosophical research. haben. (GA 2 B 15: 226). MA Thesis, Vrije Universiteit Amsterdam. [5]