Abstract
I give a semantic characterisation of a system for the logic of grounding similar to the system introduced by Kit Fine in his “Guide to Ground”, as well as a semantic characterisation of a variant of that system which excludes the possibility of what Fine calls ‘zero-grounding’.
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20 April 2017
An erratum to this article has been published.
Notes
It may be objected that calling Fine’s conception of grounding representational is at odds with his claim that “insofar as [grounding] is regarded as a relation, it should be seen to hold between entities of the same type and, insofar as a choice needs to be made, these entities should probably be taken to be worldly entities, such as facts, rather than representational entities, such as propositions” (p. 43). I should stress that the conception I am alluding to here is the one that is at work in the systematic rules of inference he introduces, as opposed to the conception at work in the semantics he puts forward. As we will see in the next section, the semantics is ill-suited for the characterisation of the notion intended to be captured by the rules—it turns out that the semantics is more suited to characterise a worldly notion of grounding (a version of the semantics characterises my 2010 system; see Fine [16]). If Fine’s claim is understood as concerning the notion characterised by the semantics, then I do not have anything against it. But I disagree with the claim if it is understood as concerning the notion intended to be captured by the systematic rules: I do not see how ϕ and ϕ ∧ ϕ, ϕ ∨ ϕ and ¬¬ϕ could fail to express the same worldly fact (if they express any such facts at all).
Since I will exclusively focus on metaphysical grounding, I shall henceforth systematically omit the qualification ‘metaphysical’.
The only other systematic studies I know of are [1] and [28], but their target notions are different from mine. Batchelor and Schnieder both focus on a one-to-one notion of partial grounding, while the notion of primary interest to me is a notion of full grounding which is many-to-one (several items can jointly ground a further item) in an irreducible way (in particular, as Fine ([15], p. 50) argues, it cannot be defined in terms of a partial notion).
The informal readings of the ground-theoretic expressions given above—which Fine takes on board—are somewhat improper, since he takes these expressions to be sentential operators rather than predicates.
See the references in footnote 2.
Notational conventions regarding unions which are standard in sequent calculi are used here: ‘ Δ, Γ’ is used for ‘\({\Delta }\cup {\Gamma }\)’ and ‘ ϕ, Γ’ for ‘\(\{\phi \}\cup {\Gamma }\)’. I will use these conventions throughout the paper.
As previously stressed, he also introduces principles concerning the quantifiers and lambda-abstracts, but they are out of the scope of this paper.
Each introduction rule states that given certain truths, we can infer that these truths ground a further truth. This conditional form is required if we take grounding to be factive: the unconditional rule \(\frac {}{\phi , \psi < \phi \wedge \psi }\), for instance, would not be acceptable since it would allow us to derive any instance of ϕ,¬ϕ < ϕ ∧¬ϕ from zero premisses, which is at odds with the factivity of grounding. Fine formulates the introduction rules in unconditional form, but conditionalisation is implicit.
Fine mentions two further introduction rules, namely
$$\frac{\phi \quad \psi}{\phi, \psi < \phi\vee\psi}\qquad \frac{\neg\phi \quad \neg \psi}{\neg\phi, \neg \psi < \neg(\phi\wedge\psi)} $$but they follow from the corresponding simpler rules thanks to the fact that < obeys Amalgamation.
The \(\bigvee \) notation used to express disjunctive conclusions is mine, not Fine’s.
Interestingly, the same conclusion can be drawn if we start with the following alternative characterisation of ground-theoretic equivalence:
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\(\phi \doteq \psi \) iff (i) for all Δ, \({\Delta }\leqslant \phi \) iff \({\Delta }\leqslant \psi \), and (ii) for all Δ and χ, \(\phi ,{\Delta } \leqslant \chi \) iff \(\psi ,{\Delta } \leqslant \chi \).
Thanks to an anonymous referee for pointing this out to me.
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I am happy to translate ‘ ϕ 1,ϕ 2,... < ϕ’ understood non-factively into the English ‘its being true that ϕ 1, ϕ 2, ... makes it true that ϕ’, but I know that some speakers would deny that the latter sentence can be understood in a non-factive way. Another suggestion is to translate the sentence into ‘its being true that ϕ 1, ϕ 2, ... would make it true that ϕ’, but with the caveat that ‘would’ should be understood as tolerant toward counteractual, and even counterpossible, antecedents.
Such notions of equivalence are currently gaining importance in the philosophical literature, as e.g. Agustín Rayo’s [25] book-length study of “just is” statements illustrates. Yet the views about their properties vary considerably. For instance, Rayo is happy with the view that ‘For it to be the case that ϕ is for it to be the case that ψ’ holds iff ‘Necessarily, ϕ iff ψ’ does. By contrast, neither propositional equivalence as understood here nor factual equivalence as understood in my 2010 paper is coextensive with necessary equivalence: both are much more fine-grained than the latter.
It is crucial here that these stated connections concern the truth of propositions rather than the obtaining of states of affairs, where the latter are understood as worldly items. Since e.g. the state of affairs that ϕ and the state of affairs that ϕ ∨ ϕ are one and the same, it cannot be the case that the state of affairs that ϕ ∨ ϕ obtains in virtue of the fact that the state of affairs that ϕ obtains.
I am here again using the standard notational conventions for set-theoretic unions alluded to in footnote 8.
Had the language allowed to build formulas from the set of sequents using operators of conjunction and disjunction applicable to possibly infinite sequences of formulas, we could have formulated all the rules in the more standard format of rules with zero or more formulas as premisses and one formula as a conclusion. The unusual format of the rules of the system to be formulated is a consequence of the scarcity of the language.
There are redundancies I did not bother to eliminate, e.g. given the commutativity rule for ∨ and R-Substitution, having only one of the two ∨-introduction rules would be sufficient.
I do not have firm thoughts about the issue. See [22] for a defense of the idea, but with a peculiar conception of grounding in the background.
References
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Acknowledgments
An earlier version of this paper was discussed at an eidos meeting in April 2014 at the University of Geneva, and I presented other versions of the paper at three workshops: Necessary Connections, Glasgow, May 2014, Recent Work on the Logic of Ground, Oslo, June 2014, and Logical and Metaphysical Perspectives on Grounding, Osnabrück, September 2015. I am grateful to the respective audiences, and also to two anonymous referees of this journal, for helpful comments and criticisms. This work was carried out while I was in charge of the Swiss National Science Foundation projects CRSII1-147685, 100012-150289, BSCGI0-157792 and 100012-159472, and of a module of the H2020 project MSCA-ITN-2015-675415. During that period, I was also a member of the Spanish Ministry of Economy and Competitiveness project FFI2012-35026.
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The original version of this article has been revised. The mistakes and corrections are described in the following list: 1) Footnotes 1and 2 were incorrectly assigned. The first two footnotes found on the first page should have been footnote 1and 2, respectively. Citation of these footnotes should have been found in the first paragraph of the main text. The problem has been fixed, and all the remaining footnotes have been correctly re-numbered. 2) There was an incorrect renumbering and internal referencing of the Definitions, Facts, Propositions and Theorems. These problems have been fixed.
An erratum to this article is available at https://doi.org/10.1007/s10992-017-9435-0.
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Correia, F. An Impure Logic of Representational Grounding. J Philos Logic 46, 507–538 (2017). https://doi.org/10.1007/s10992-016-9409-7
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DOI: https://doi.org/10.1007/s10992-016-9409-7