Abstract
I develop a graph-theoretic model theory for pure and iterative grounding logics.
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Notes
Being ungrounded is a property which cannot be expressed within pure logics of ground.
See for example Correira’s provision in [3], p. 536, where he provides a version the Nonzero rule given here.
Sometimes reflexivity is replaced with the stronger condition that for all X and a, X,a →Aa. But clearly in the presence of monotonicity this stronger condition is just equivalent to reflexivity.
As reflexivity and monotonicity are only needed to provide the strengthened reflexivity principle discussed in the last footnote, we can infer that if A is Tarski-cuttable and X,a →Aa for all X and a, then A is cuttable.
References
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Acknowledgements
Thanks to Timothy Williamson, A. C. Paseau, and Tuomas Tahko for reading earlier versions of this paper. Thanks also to an anonymous reviewer for their many helpful comments, including first seeing the existence of informal failures of certain rules in iterative calculi which I now treat in Section 4.1, and also the initial suggestion to use either a regular or strongly inaccessible cardinal bound on sequent-length to avoid such failures.
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This work was supported by an AHRC OOCDTP scholarship.
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Appendices
Appendix A: Zero-Grounding
All of the logics considered in the main discussion allow for the possibility of so-called ‘zero-grounding’. Zero-grounding—or being grounded in the empty set of grounds—is supposed to be distinguished from being ungrounded; zero-grounded things are grounded, even though there are no things which serve as their groundsFootnote 1. Some have had doubts regarding the intelligibility of this notion, and have wanted to prohibit it from their logics of ground in some way or other.Footnote 2 One way of doing this would be grammatical; we could simply rule out \(\varnothing < \phi \) as grammatically ill-formed by requiring of sequents that they have a nonempty set of operands on the left. Another way, which we will show here, is to introduce a rule which encodes the idea that, though well-formed and present in the language, statements of zero-ground are always false.
We may add this rule to each of the \({\mathscr{L}}_<\)-calculi introduced in §2 to yield new, stronger calculi. I will refer to these as S+ where S is the \({\mathscr{L}}_<\)-calculus to which the rule is added. We now establish soundness and completeness for these zero-free calculi (Fig. 3).
Definition A.1
G is zero-free or ‘nonzero’ iff \(\varnothing \not <_{G} a\) for all a ∈|G|.
We let ‘Nz’ stand for the class of zero-free grounding structures. Then,
Proposition A.2
For any \({\Sigma }\subseteq {\mathscr{L}}_<\) and \(\sigma \in {\mathscr{L}}_<\):
-
(i)
\( {\Sigma }\vdash _{\mathsf {C}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Nz}}\sigma \).
-
(ii)
\( {\Sigma }\vdash _{\mathsf {I}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Irr, Nz}}\sigma \).
-
(iii)
\( {\Sigma }\vdash _{\mathsf {A}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Add, Nz}}\sigma \).
-
(iv)
\( {\Sigma }\vdash _{\mathsf {G}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Irr, Add, Nz}}\sigma \).
-
(v)
\( {\Sigma }\vdash _{\mathsf {M}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Mon, Nz}}\sigma \).
-
(vi)
\( {\Sigma }\vdash _{\mathsf {R}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Ref, Nz}}\sigma \).
-
(vii)
\( {\Sigma }\vdash _{\mathsf {T}^{\text {+}}}\sigma \Leftrightarrow {\Sigma }\vDash _{\text {Ref, Mon, Nz}}\sigma \).
Proof
(Soundness/Left to Right) Except for Nonzero, the rules of these systems are unchanged, as are the arguments that these rules preserve truth over the respective classes of models. So it suffices to note that Nonzero vacuously preserves truth over models with zero-free structures: at such models \(\varnothing < \phi \) is never true, and so if such a sequent were true, any sequent would be true. (Completeness/Right to Left) It suffices to show for each calculus S that for all S-consistent Σ, GΣ,S must be in the relevant class of structures. The arguments in each case are as in Proposition 2.8, with the addition that if Σ is S+-consistent, then for all ϕ we must have \({\Sigma }\not \vdash _{\mathsf {S}^{\text {+}}}\varnothing <\phi \). Thus for all ϕ we must have \(\varnothing \not <_{G_{\Sigma , \mathsf {S}^{\text {+}}}}\phi \), meaning that \(G_{\Sigma , \mathsf {S}^{\text {+}}}\) is zero-free. □
The naming conventions and Propositions from Section 4 allow each of these new calculi to be extended to \({\mathscr{L}}^{\prime }_<\) calculi, which then receive soundness and completeness results with respect to the obvious classes of iterative grounding structures. For example, \(\mathsf {C}^{+}_{\text {Int}}\) is the result of adding Nonzero to CInt, and can be shown sound and complete with respect to the class of internal iterative grounding structures whose underlying grounding structure is in Nz.
Appendix B: Cut Properties in Directed Semi-Hypergraphs
The only property that we have presumed to hold of ground throughout our discussion is cuttability—a many-one analogue of the transitivity property in binary relations. This assumption was implemented by requiring in the definition of grounding structures that they be cuttable. Here we are interested in the definition of a grounding structure, and in particular the kind of cut property used in this definition.
1.1 B.1 Finite Cut
As mentioned above, other authors have made use of the term ‘grounding structure’, and have wanted to include a cut property in the definition of such structures. However, our notion of grounding structure differs from these writers in two ways. Firstly, our definition is more purely mathematical; both Dixon’s grounding structures and Rabin and Rabern’s grounding structures appeal to real grounding relations between their inhabitants, and posit these inhabitants to be real relata of ground (facts or whatever). While we allow that real grounding relata can serve for the domain of a grounding structure and that a real grounding relation can serve for the relation of a grounding structure, this is not essential to grounding structures as such. For us: any nonempty set can serve as the domain for a grounding structure and any cuttable relation can serve as the relation of a grounding structure. The second way in which our grounding structures differ from the structures of these previous writers is the choice of cut property itself. These authors each settle for the following property as part of the definition of a grounding structure.
Definition B.1
A is finitely cuttable iff: if X,b →Aa and Y →Ab, then X,Y →Aa.
In particular, for Dixon, as for Rabin and Rabern, the real grounding relation is only posited to be finitely cuttable (though in both cases it is also posited to be irreflexive). Now, it is evident just from the definitions that every cuttable structure A is also finitely cuttable. To show that the cuttability property of our grounding structures is strictly stronger than finite cuttability then, it suffices to show:
Proposition B.2
Some finitely cuttable directed semi-hypergraphs are not cuttable.
Proof
Let |A| = {0, 1, 2,...,ω} be the set of ordinals from 0 to ω inclusive. Then where < is the normal ordering on ordinals, let →A be the relation on |A| such that for all \(X\subseteq \vert A\vert \) and α ∈|A|, we have X →Aα if and only if (i) β < α for all β ∈ X and (ii) if α = ω, then X is infinite. This structure is not cuttable. For we have {0, 1, 2,...}→Aω, and additionally {0}→An for every finite n ≥ 1; if A were cuttable we would thus have {0}→Aω, which is not so. However, the structure is finitely cuttable. To see this, suppose we have X →Aα and Y →Aβ for all β ∈ X. Either α is a finite ordinal or it is ω. Where α is finite, then we immediately have X,Y →Aα since by hypothesis every β ∈ X is less than α and every γ ∈ Y is less than every β ∈ X, hence also less than α. If α = ω, then by (ii), X must be infinite, in which case X ∪ Y is infinite. But again, by hypothesis every β ∈ X is less than α and every γ ∈ Y is less than every β ∈ X, hence also less than ω. Hence we have X,Y →Aω. So in call cases, finite cut is respected.□
Finitary cut is therefore a strict weakening of the cut property we have specified for our grounding structures; I think that it is clearly an unmotivated weakening, such that it is preferable to adopt the stronger condition in defining grounding structures. Whatever reasons we have for thinking that ground is cuttable do not seem restricted to the finitary.
1.2 B.2 Tarskian Cut
Earlier I claimed that our grounding structures are essentially a generalised kind of abstract consequence structure—ones which relax the conditions of monotonicity and reflexivity. Abstract consequence structures in this way form a subset of grounding structures, on our understanding. In particular they are the reflexive and monotonic grounding structures. Here I will show that this is indeed the case. This is necessary since, often, the definition of an abstract consequence structure is introduced using a different cut property to that we specified for grounding structures.
Definition B.3
An abstract consequence structure is a directed semi-hypergraph A such that for all \(X, Y\subseteq \vert A\vert \) and a ∈|A|:
If X →Aa and Y →Ab for all b ∈ X, then Y →Aa.
a →Aa.
If X →Aa and \(X\subseteq Y\), then Y →Aa.
The second and third of these conditions are just the reflexivity and monotonicity conditions as defined for grounding structures in Section 2 above.Footnote 3 The first is the alternate cut property just mentioned. I will name this condition by itself:
Definition B.4
A is Tarski-cuttable iff for all \(X, Y\subseteq \vert A\vert \) and a ∈|A|, if X →Aa and Y →Ab for all b ∈ X, then Y →Aa.
Like finite cuttability, Tarksi-cuttability is a strictly weaker condition on structures than cuttability. We can see this with the following two propositions.
Proposition B.5
All cuttable directed semi-hypergraphs are Tarski-cuttable.
Proof
Suppose that A is cuttable. Then also suppose that X → a and Y → b for all b ∈ X. Let X be indexed by I so that X = {bi}i∈I and for all i let Yi := Y. Then \(\varnothing , \{b_{i}\}_{i\in I}\to a\) and Yi → bi for all i. Hence by Cut, we have \(\varnothing , \bigcup _{i\in I} Y_{i}\to a\). Since \(\bigcup _{i\in I}Y_{i} = Y\), we have Y → a. Since X, a and Y were chosen arbitrarily, it follows that A is Tarski-cuttable. □
Proposition B.6
Some Tarski-cuttable directed semi-hypergraphs are not cuttable.
Proof
Let |A| = {a,b,c,d,e} and let →A be the relation on |A| such that b,c →Aa and d,e →Ab. The resulting structure A is vacuously Tarski-cuttable, since there are no x,X, and Y such that X →Ax and Y →Ay for all y ∈ X. But the structure is not cuttable, since if it were we would also have d,e,c →Aa, which is not so. □
However, in the presence of monotonicity, as is the case in the definition of abstract consequence structures, cuttability and Tarski-cuttability are equivalent:
Proposition B.7
If A is monotonic and reflexive, then A is cuttable if it is Tarski-cuttable.
Proof
Suppose that A is monotonic, reflexive and Tarski-cuttable. Then also suppose that X,{bi}i∈I → a and that Yi → bi for all i ∈ I. Now given reflexivity and monotonicity, we have that \(X, \bigcup _{i\in I}Y_{i}\to _{A} y\) for every y in every Yi. So by Tarski-cuttability we have \(X, \bigcup _{i\in I}Y_{i}\to _{A} b_{i}\) for every i, as well as \(X, \bigcup _{i\in I}Y_{i}\to _{A} x\) for every x ∈ X. Hence by Tarski-cuttability we have \(X, \bigcup _{i\in I}Y_{i}\to _{A} a\). Hence A is cuttableFootnote 4. □
In the absence of monotonicity and/or reflexivity, the weakness of Tarski-cuttability as a property of grounding is shown in that it does not even guarantee the transitivity of partial grounding in the way that cuttability and finite cuttabilty do. To make this thought precise, I introduce the following notation.
Definition B.8
Where A is a directed semi-hypergraph, and a,b ∈|A|, we may write \(a \rightharpoondown _{A} b\) just in case there is some \(X\subseteq \vert A\vert \) such that X,a →Ab.
The relation \(\rightharpoondown _{A}\) clearly corresponds to the notion of partial grounding introduced in the first part of Section 3. And as there, we can see that cuttability guarantees that this derivative binary relation is transitive. In fact, even finite cuttability suffices.
Proposition B.9
If A is finitely cuttable, then \(\rightharpoondown _{A}\) is transitive.
Proof
If \(a\rightharpoondown _{A} b\) and \(b\rightharpoondown _{A} c\), then there are X,Y such that X,a →Ab and Y,b →Ac. So by finite cuttability we have Y,X,a →Ac and so \(a\rightharpoondown _{A}c\). □
On the other hand, Tarski-cuttability is not sufficient in this sense:
Proposition B.10
There are Tarski-cuttable A where \(\rightharpoondown _{A}\) is non-transitive.
Proof
Take A from Proposition B.6. Here we have \(d\rightharpoondown _{A} b\) and \(b\rightharpoondown _{A} a\) but . □
I take this to be another good reason to prefer our own notion of cuttability in thinking about ground. However, conceivably, those who dislike transitivity of partial ground may see the adoption of Tarksian cut as a way of retaining a cut property for grounding while accommodating certain influential objections to the transitivity of partial ground known in the literature (i.e. those of [13]). I would not endorse this move, since I do not think that the counterexamples in question succeed in undermining the transitivity of ≺, but my reasons for thinking this are besides the present topic.
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Bevan, M. The Metalogic of Ground: Pure and Iterative Systems. J Philos Logic 52, 609–641 (2023). https://doi.org/10.1007/s10992-022-09681-5
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DOI: https://doi.org/10.1007/s10992-022-09681-5