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On constructing a logic for the notion of complete and immediate formal grounding


In Poggiolesi (2016b) we have introduced a rigorous definition of the notion of complete and immediate formal grounding; in the present paper our aim is to construct a logic for the notion of complete and immediate formal grounding based on that definition. Our logic will have the form of a calculus of natural deduction, will be proved to be sound and complete and will allow us to have fine-grained grounding principles.

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  1. 1.

    Analogous considerations hold for the grounding principles governing negation of conjunction.

  2. 2.

    Even if it is not spelled out in these terms, a similar idea can be found in Fine (2010).

  3. 3.

    In a recent paper Rumberg (2013) Rumberg has put forward the view according to which normal derivations bear a strong resemblance to Bolzano’s grounding-trees for conceptual truths. Since Bolzanian grounding trees correspond to proofs-why, the links between the present paper and Rumberg’s work seem to be deep and interesting. However, since such a comparison would be quite long and beyond the scope of this work, we leave it for future research.

  4. 4.

    As we will explain in Sect. 2, after Definition 4.2, the connective \(\triangleright \) does not entirely match the natural language term ‘because’. Rather it is supposed to capture those uses of ‘because’ that correspond to the concept of grounding.

  5. 5.

    This is formalized, for example in classical propositional logic, by the deduction theorem.

  6. 6.

    We omit a comparison with the logic introduced in Fine (2012b), since, contrary to ours, it is a purely structural logic for grounding.

  7. 7.

    We believe this point to be directly linked with the previous one.

  8. 8.

    The distribution principle has been questioned in the literature on grounding, e.g. see Krämer and Roski (2015). Note also that in a recent paper Correia (2016), Correia proposes a novel logic describing factual equivalence.

  9. 9.

    We work with multisets of formulas rather than with sets of formulas because we need to take into account the number of occurrences of each formula of M.

  10. 10.

    The symbol \(\circ \) \(\in \) \(\{\wedge , \vee \}\).

  11. 11.

    Note that we use the same symbol for the relation of complete and immediate formal explanation and the relation of complete and immediate formal grounding, introduced in Definition 3.7. Though this might at first appear confusing, both notions are syntactic and will be proved to be equivalent, so we prefer to leave the notation as it is. This is analogous to what happens in Hilbert systems and Gentzen systems: they share the same symbol for derivability.


  1. Aristotle, (1994). Posterior analytics (2nd ed.). Oxford: Oxford University Press.

  2. Bliss, R., & Trogdon, K. (2016). Metaphysical grounding. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (pp. 1–42).

  3. Bolzano, B. (2015). Theory of science. Oxford: Oxford University Press.

  4. Correia, F. (2010). Grounding and truth-functions. Logique et Analyse, 53(211), 79–251.

  5. Correia, F. (2014). Logical grounds. Review of Symbolic Logic, 7(1), 31–59.

  6. Correia, F. (2016). On the logic of factual equivalence. Review of Symbolic Logic, 9(1), 103–122.

  7. Correia, F., & Schnieder, B. (2012). Grounding: An opinionated introduction. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding (pp. 1–36). Cambridge: Cambridge University Press.

  8. Fine, K. (2010). Some puzzles of ground. Notre Dame Journal of Formal Logic, 51(1), 97–118.

  9. Fine, K. (2012). Guide to ground. In F. Correia & B. Schnieder (Eds.), Metaphysical grounding (pp. 37–80). Cambridge: Cambridge University Press.

  10. Fine, K. (2012). The pure logic of ground. Review of Symbolic Logic, 25(1), 1–25.

  11. Krämer, S., & Roski, S. (2015). A note on the logic of worldly ground. Thought, 4, 59–68.

  12. Poggiolesi, F. (2010). Getzen calculi for modal propositional logic. Dordrecht: Springer.

  13. Poggiolesi, F. (2016a). A critical overview of the most recent logics of grounding. In F. Boccuni & A. Sereni (Eds.), Objectivity, realism and proof. Boston studies in the philosophy and history of science (pp. 1–18).

  14. Poggiolesi, F. (2016b). On defining the notion of complete and immediate formal grounding. Synthese, 193: 3147–3167.

  15. Rumberg, A. (2013). Bolzano’s theory of grounding against the background of normal proofs. Review of Symbolic Logic, 6(3), 424–459.

  16. Schaffer, J. (2009). On what grounds what. In Metametaphysics (pp. 347–383).

  17. Schnieder, B. (2010). A puzzle about ‘because’. Logique et Analyse, 53(4), 317–343.

  18. Schnieder, B. (2011). A logic for ‘because’. The Review of Symbolic Logic, 4(03), 445–465.

  19. Sebestik, J. (1992). Logique et mathematique chez Bernard Bolzano. Paris: J. Vrin.

  20. Tatzel, A. (2002). Bolzano’s theory of ground and consequence. Notre Dame Journal of Formal Logic, 43(1), 1–25.

  21. Troelstra, A. S., & Schwichtenberg, H. (1996). Basic proof theory. Cambridge: Cambridge University Press.

  22. van Dalen, D. (1991). Logic and structure. Dordrecht: Springer.

  23. Wansing, H. (1994). Sequent calculi for normal modal propositional logics. Journal of Logic and Computation, 4, 125–142.

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I wish to thank Brian Hill for having corrected the English of the manuscript but also for several precious comments and suggestions. I would also like to thank the anonymous referees for their deep, clarifying and smart remarks.

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Correspondence to Francesca Poggiolesi.

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Poggiolesi, F. On constructing a logic for the notion of complete and immediate formal grounding. Synthese 195, 1231–1254 (2018). https://doi.org/10.1007/s11229-016-1265-z

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  • Grounding
  • derivability
  • logic