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On ground and consequence

  • S.I.: Ground, Essence, Modality
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Abstract

What does it mean that some proposition follows from others? The standard way of spelling out the notion proceeds in modal terms: x follows from y iff necessarily, if y is true, so is x. But although this yields a useful and manageable account of consequence, it fails to capture certain aspects of our pre-theoretical understanding of consequence. In this paper, an alternative notion of logical consequence, based on the idea of grounding, is developed.

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Notes

  1. See e.g. Fox (1987) on truthmaking, Barcan Marcus (1967: p. 93f.) on essence, and Simons (1987: ch. 8) on dependence.

  2. Compare Schnieder (2006b).

  3. On essence, see Fine (1994). On dependence, see Fine (1995), Lowe (1998), Correia (2005), and Schnieder (2006a). On truth-making, see e.g. Rodriguez-Pereyra (2006) and Schnieder (2006b).

  4. Such a modal definition occurs as commonly in textbooks of formal logic (e.g. Bostock 1997: p. 5) as in informal introductions to logic (e.g. Cornman et al. 1992: p. 8).

  5. In particular, Tarski’s model-theoretical account of consequence is often regarded as such an attempt. Etchemendy (1990) initiated a debate about whether the model-theoretic account can capture the modal notion. In the course of the debate, it has also been questioned whether Tarski really intended his account to model a modal notion of consequence (see e.g. Gómez-Torrente 2009: Sect. 4).

  6. See e.g. Forbes (1994: p. 3).

  7. Incidentally, the modal explication of truth-making is commonly traced back to Fox (1987: p. 189), who indeed first analyzes truth-making in terms of entailment (aka consequence), and then starts to switch between talk of entailment and necessitation, relying on a modal account of entailment.

  8. See e.g. Fine (2012b) and Rosen (2010).

  9. See e.g. Correia (2005) on dependence and Schnieder (2006b) on truth-making.

  10. For further details on theories of grounding, the reader must be referred to the existing literature; see e.g. the introduction in Correia and Schnieder (2012), or Fine (2012b), whose basic framework is employed here.

  11. Some authors distinguish between different kinds of grounding, e.g. between metaphysical, natural, and normative grounding; the question then arises whether they can be defined in terms of a generic notion of grounding together with some differentiating clauses (Fine 2012b: p. 39f.). I want to stay as neutral on this matter as possible; if pressed, I would say I am working with the notion of metaphysical grounding (which in my view would comprise logical grounding).

  12. While for Fine (2012b), grounding is metaphysical explanation, for Schaffer (2012) it is related to metaphysical explanation in the way in which causation is related to causal explanation.

  13. On different uses of ‘because’, see Schnieder (2011: p. 447f.).

  14. See e.g. Fine (2012b), Litland (forthcoming), Rosen (2010), and Schnieder (2011).

  15. Some worries have been raised about the irreflexivity (Jenkins 2011) and transitivity (Schaffer 2012) of grounding. For responses, see Litland (2013), Raven (2013), Krämer and Roski (2017). Moreover, Fine (2010) describes puzzles of ground that could be resolved by giving up irreflexivity, though there are other options (for related cases, see Krämer 2013 and Correia 2014: p. 54f.). The issue cannot be discussed here; we will simply adopt an irreflexive and transitive notion.

  16. For a possible exception to the rule consider the conjunction of a fact with itself; here the conjunct may count as a full ground of the conjunction.

  17. For more on the logic of grounding used in the paper, see e.g. Fine (2012b) or Schnieder (2011).

  18. See Correia (2017). He contrasts the representational conception of grounding with a worldly conception, on which its relata are subject to a more coarse-grained individuation.

  19. A similar notion of consequence can be found in Bolzano (Bolzano 1837, vol. I: §198). The only difference is that Bolzano would require all the premises to ground a consequence; so unlike the notion above, his notion is not monotonic. Note that Bolzano distinguished between consequence (‘Abfolge’) and what he called deducibility (‘Ableitbarkeit’), where this latter notion resembles Tarski’s notion of consequence (and thereby, to some extent, the notion of modal consequence). On Bolzano’s notions see e.g. Morscher (2008) or Roski (2017); on Bolzano’s notion of deducibility, see also Siebel (2002).

  20. Two notes: First, one can equally well work under the hypothesis that the premises are grounds, as under the hypothesis that they are true. However, I prefer to use the hypothesis that the premises are grounded, in preparation for my formal framework in Sect. 5. Second, Fine (2012b: p. 49f.) suggests that apart from the factive, there is also a non-factive notion of ground (such a notion is explored in Correia 2014). If one accepts Fine’s suggestion, one could also rely on the non-factive notion of ground in order to turn DC.1 into a definition of a non-factive notion of consequence.

  21. As to relatives of my account, web consequence is similar to Fine’s (2012a: p. 235f.) notion of inexact consequence (though Fine does not motivate his account in the same way), a predecessor of which is van Fraassen (1969). After presenting a model-theoretic framework for web consequence (Sect. 5), I briefly discuss some differences between web and inexact consequence (Sect. 6.2).

  22. As noted earlier, it may be debatable whether the pretheoretical notion of consequence is reflexive or not. I concentrate on a reflexive notion since it is more agreeable to contemporary logicians. But my approach can easily be modified so as to get an irreflexive notion instead.

  23. See Priest (1997).

  24. The present paper thereby constitutes a partial response to Wilson’s (2014) contention that there is ‘no work for a theory of grounding’: The notion of grounding can be fruitfully employed in definitions of a range of important philosophical notions (see also Correia 2005; Schnieder 2006a, b). Therefore grounding is a useful philosophical tool, pace Wilson. (This line of argument is developed in more detail in Schnieder, forthcoming.).

  25. Two small notes: (i) Unlike grounding proper, thin grounding is not asymmetrical but only anti-symmetrical; but since that property will play no role in the model-theory, it was omitted from the recursive definition. (ii) G.3 is partly redundant: it already follows from the definition of ‘G’ that if φ⋟ψ, then Gφ. The redundancy was only kept to make it apparent how the clause relates to the factivity of grounding as introduced earlier.

  26. While G.4 to G.8 correspond to what Fine (2012) calls introduction rules for grounding, the downward clauses G.4D to G.8D correspond to what he calls elimination rules. In the present framework, one can also obtain stronger elimination rules than G.4D to G.8D, by adding exclusivity clauses: G.4D, for instance, could be strengthened into claiming that every ground of φ∨ψ is a thin ground of one of its disjuncts (this rule then comes close to Fine’s own rule—2012: 64). But such stronger elimination rules are inessential for my approach (on this point, see also below, Sect. 6.2). They hold in the current framework because the current models are designed to deal with logical grounding relations only: they only recognize grounding relations that hold in virtue of the meaning of the logical constants. But in principle one can, without doing damage to the notion of web consequence, expand the grounding structures and allow the introduction of other, non-logical grounding relations; in such a frame, the stronger elimination rules fail.

  27. A proof is omitted for reasons of space, as it can straightforwardly be derived from Schnieder (2011: p. 456f.) (Schnieder proves only a version of the above claim restricted to formulas which are tautological, but the proof directly carries over to any formula which is grounded).

  28. Due to the format of the paper, the issue of disjunctive syllogism cannot be discussed in more detail; for further discussion, see e.g. Mares (2004: ch. 10).

  29. For a concise presentation of FDE see Priest (2001: ch. 8).

  30. See Schnieder (2011: p. 455ff.) for a general proof that given the introduction rules for grounding, truth-functional compounds are always grounded in the contained literals. This suffices for the notion of web consequence to work properly.

  31. See Aristotle’s Posterior Analytics I.13.

  32. See. e.g., Correia (2005: ch. 3.2) and Schnieder (2006b) on truth-making, or Correia (2005) and Schnieder (2006a) on existential dependence.

  33. For comments and discussion, I would like to thank Fabrice Correia, Kit Fine, Manuel-García Carpintero, Hannes Leitgeb, Max Kölbel, Stephan Krämer, Jon Litland, Stefan Roski, Moritz Schulz, three anonymous referees for this journal, and the audiences at research colloquia in Bristol, Gothenburg, Helsinki, and Munich, and at the following workshops and conferences: Metaphysics and Logic, East and West (Shanghai), Language and World (Tokyo), 5th International Symposium on Philosophy of Language and Metaphysics (Rio de Janeiro); Explaining Without Causes (Cologne 2013); Explanations in Metaphysics (Neuchâtel), IIPMW (Barcelona). For financial support I would like to thank the BWF Hamburg (Grant “Welt der Gründe”), the DFG (Maimonides Centre for Advanced Studies, FOR 2311), and the SNF (project: Grounding Metaphysics, Science, and Logic, CRSII1 147685/1). And last but not least, I would like to thank Tuomas Tahko for his kind patience and encouragement.

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    Benjamin Schnieder

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Schnieder, B. On ground and consequence. Synthese 198 (Suppl 6), 1335–1363 (2021). https://doi.org/10.1007/s11229-018-02012-9

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