Abstract
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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Notes
Analogous considerations hold for the grounding principles governing negation of conjunction.
Even if it is not spelled out in these terms, a similar idea can be found in Fine (2010).
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.
This is formalized, for example in classical propositional logic, by the deduction theorem.
We omit a comparison with the logic introduced in Fine (2012b), since, contrary to ours, it is a purely structural logic for grounding.
We believe this point to be directly linked with the previous one.
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.
The symbol \(\circ \) \(\in \) \(\{\wedge , \vee \}\).
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.
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Acknowledgements
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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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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DOI: https://doi.org/10.1007/s11229-016-1265-z