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

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Bolzano and Impeccable Explanation: The State of the Art

ARIANNA BETTI & PAULINE VAN WIERST Logic, Language and Computation, ILLC, 25 November 2013 Handout The Claim  Bolzano’s grounding (Abfolge) is the explication of a traditional notion of scientific explanation within a millennia-old ideal of science. (I) Bolzano’s tremendous step forward in this tradition consists in his explicit attempts at explicating a notion of impeccable explanation, i.e. a notion of (formal or logical) consequence which is also explanatory (call it deducibility+). (II) These attempts seem frustrated by one counterexample by Bolzano himself from axiomatic ethics to the effect that deducibility is not necessary for grounding; however, what the counterexample really shows is a matter of contention. (III) We claim that a crucial role in Bolzano’s reasoning is played by his specific take on the Ought-Implies- Can principle. In sum, for conceptual sciences, barring a certain difficulty in ethics, grounding is deducibility+.  I The Classical Model (or Ideal) of Science A proper science S satisfies the following conditions (de Jong & Betti 2010): (1) All propositions and all concepts (or terms)

• f S concern a specific set of objects or are

about a certain domain of being(s). (2a) There are in S a number of so-called fundamental concepts (or terms). (2b) All other concepts (or terms) occurring in S are composed of (or are definable from) these fundamental concepts (or terms). (3a) There are in S a number of so-called fundamental propositions. (3b) All other propositions of S follow from or are grounded in (or are provable or demonstrable from) these fundamental propositions. (4) All propositions of S are true. (5) All propositions of S are universal and necessary in some sense or another. [6] All propositions of S are known to be true. A non-fundamental proposition is known to be true through its proof in S. [7] All concepts or terms of S are adequately

• known. A non-fundamental concept is

adequately known through its composition (or definition).  Bolzanian (conceptual) sciences as grounding rooted trees of trees match this ideal , - The tentative definition: Impeccable Explanation? “[Grounding is] that ordering of truths which allows us to deduce from the smallest number of simple premises the largest possible number of the remaining truths as conclusions” (WL §221).  The Beyträge (1810) Five copulas

• 1. A is a kind of B [necessary judgements]
• 2. A can be B [possibility judgements]
• 3. A ought to do B [prescriptive judgements]
• 4. I perceive X [actual/empirical judgements]
• 5. A is probably B [probability judgements].

Four new inference rules [eigentliche Schlüsse] of ‘impeccable explanation’ for mathematics [not complete; wanting]  Diaries: rules for 3. (Blok 2013) [3] [4] Corollaries (GrBD-1) If [A  B] is an axiom (Grundsatz), then A and B are (absolutely) simple (GrBD-2) If C is a(n) (absolutely) simple concept, then for some B: [B  C] is an axiom  Six (pure & impure) maxims of grounding (Roski 2013)

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(1) [no increase in] generality (2) [no increase in] simplicity (3) uniqueness [(4) axiomness: some grounds have no grounds] (5) asymmetry (6) deductive economy , : [1], [2],  The Open Question Is deducibility a necessary condition for grounding?  II The puzzling argument from WL§200 [GA 2 B 15: 248; -] [Step 1] Take a practical truth of the form (G) One ought to do A which grounds impeccably all other practical truths such as ‘One ought not to lie’ (i.e. its consequence contains all the practical truths.).

• eg. Bolzano’s ‘highest moral law’: You should

promote happiness (GA 2 B 15: 131) [Step 2] Therefore, (Bz) (G) has a (partial) ground in the theoretical truth (E) A is possible. So, (G)’s complete ground, (E+), includes (E). But [Step 3] no inference rule allows us to infer (G) from (E+): none of the truths in (E+) can contain the concept of Sollen (otherwise, it would be a practical truth: impossible - we supposed that all practical truths are included in the consequence of (G)) ergo we have grounding between undeducible (collections of) propositions ((E+) and (G)). ergo deducibility is not a necessary condition for grounding and grounding cannot be defined as a special kind of derivability. What is going on here? III B’s rock-bottom & problematic convictions about ethics Ad [Step 1] › There is a (single) axiom [Grundwahrheit] of Ethics. Ad [Step 2] › Ought-Implies-Can reads ‘can grounds ought’ (p) one ought to do a certain thing provided is grounded by (q) one is able to do that certain thing Ad [Step 3] › Practical (moral) propositions [containing Sollen] are undeducible from a collection of purely theoretical proposition [not containing Sollen], i.e. E+  (P1) Stopping eating animals is possible (P2) Stopping eating animals promotes happiness (P3) One ought to promote happiness (C) One ought (to) stop(ping) eating animals compare  (P1) Promoting happiness is possible (P2) [????] ought [????] [vs Step 1] (C) One ought to promote happiness [5] [Undefinability of Sollen: Primitive concepts of (proper) lower sciences must be undefinable from concepts of higher sciences.] [6] [NB! attempt at embedding ‘possibility’ in the axiom!] If A were impossible, there could be no duty to do A.

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Definitions & quotes [1] Exact deducibility (WL, §155.26)  is exactly deducible from  with respect to v iff a)  is deducible from  with respect to v b) if X is a proposition or a part of a proposition contained in , then  is not deducible from the collection of premises that results from  after removing X. [2] A sufficient condition for grounding: “If [(i)] a proposition M stands to other propositions A, B, C, … in the relation of exact deducibility […] with respect to ideas i, j, ..., if, moreover, [(ii)] propositions A, B, C, ... and M are the simplest propositions among those equivalent to them, and [(iii)] if none of A, B, C, … is more complex than M, then we may assume that M stands to A, B, C, … in a true relation of ground and consequence […].” (WL, §221.7) [3] [4] weil es einen Grundsatz der Pflichtsurtheile geben muß; dieser um einfache Begriffe zu erhalten, muß den Begriff des Sollens in der Copula

• haben. (GA 2 B 15: 226).

[5] Es gibt keinen Übergang aus theoretischen Wahrheiten auf praktische, und aus praktischen auf theoretische. Der oberste Satz, aus welchem alles Sollen gefolgert wird, muß selbst ein Sollen enthalten. Aus einem 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

• der Müssen gefolgert werden. Aus lauter theoretischen Sätzen kömt

man nie auf einen praktischen zu folgern. (RW, our emphasis) [6] Always choose from all actions that are possible for you the one which, all consequences considered, most advances the welfare of the whole, in whatever parts (RW I, 236; cf. §447, WL IV 119) Bibliography Betti, Arianna. 2010. Explanation in metaphysics and Bolzano’s theory of ground and consequence. Logique et analyse, 211, 281-316. Bolzano, Bernard. 2009. Gesamtausgabe. Reihe II: Nachlaß. B. Wissenschaftliche Tagebücher. Band 15 Philosophische Tagebücher 1803-

• 1810. Zweiter Teil. Herausgegeben von Jan Berg.

Blok, Johan. 2013. The Highest Moral Law as an a priori Synthetic

• Principle. Chapter 6 of Bolzano’s Theory of Grounding and the Classical

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: 424-59. Van Wierst, Pauline. 2013. Salva Veritate - A master thesis on Bolzanian analyticity and computational methods within philosophical research. MA Thesis, Vrije Universiteit Amsterdam.