Abstract
The aim of this paper is to provide a definition of the the notion of complete and immediate formal grounding through the concepts of derivability and complexity. It will be shown that this definition yields a subtle and precise analysis of the concept of grounding in several paradigmatic cases.
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Notes
For the sake of clarity, let us note that grounding might also be seen as a sentential operator, rather than a relation. For a detailed discussion of these issues see Correia and Schnieder (2012).
We work with multisets rather than with sets because, as will become evident later on, we need to take into account the number of occurrences of each formula of M.
For a detailed analysis of Bolzano’s complexity constraint see Rumberg (2013).
From now on \(\circ \) \(=\) \(\wedge , \vee \).
We omit the formal definition of this notion for the sake of brevity.
The relation \(\cong \) as defined only captures some aspects of the informal notion of “being about the same thing”. A more thorough attempt at capturing this notion would perhaps add an item stating that formulas of the form A, \(A\wedge A\), \(A\wedge A\wedge A\), ... , as well as formulas of the form A, \(A\vee A\), \(A\vee A\vee A\), ..., fall under the relation \(\cong \), since they clearly concern the same issue. Doing so would not be difficult, and involve minimal changes in the discussion and points made below. However, we prefer not to do it for the following reason: formulas of the form \(A\wedge A\), \(A\wedge A\wedge A\) \(\dots \), as well as formulas of the form \(A\vee A\), \(A\vee A\vee A\), \(\dots , \) are rather peculiar formulas, not very significant in the framework of a theory of grounding. Therefore, adding to Definition 4.6 an item concerning them would just burden the definition without providing any new significant insight.
In Bolzano a similar idea of condition can be found, see Bolzano (2014, Sect. 222, p. 389).
Let us note that in Correia (2014) we can find a similar idea for the logic of conceptual grounding.
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Acknowledgments
I wish to thank Brian Hill for the patience and the care that he has dedicated to this paper and for his hints on the notion of g-complexity. I also wish to thank the two anonymous referees for their insightful remarks that have helped me to improve the paper.
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Poggiolesi, F. On defining the notion of complete and immediate formal grounding. Synthese 193, 3147–3167 (2016). https://doi.org/10.1007/s11229-015-0923-x
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DOI: https://doi.org/10.1007/s11229-015-0923-x