Difference-making grounds

Abstract

We define a notion of difference-making for partial grounds of a fact in rough analogy to existing notions of difference-making for causes of an event. Using orthodox assumptions about ground, we show that it induces a non-trivial division with examples of partial grounds on both sides. We then demonstrate the theoretical fruitfulness of the notion by applying it to the analysis of a certain kind of putative counter-example to the transitivity of ground recently described by Jonathan Schaffer. First, we show that our conceptual apparatus of difference-making enables us to give a much clearer description than Schaffer does of what makes the relevant instances of transitivity appear problematic. Second, we suggest that difference-making is best seen as a mark of good grounding-based explanations rather than a necessary condition on grounding, and argue that this enables us to deal with the counter-example in a satisfactory way. Along the way, we show that Schaffer’s own proposal for salvaging a form of transitivity by moving to a contrastive conception of ground is unsuccessful. We conclude by sketching some natural strategies for extending our proposal to a more comprehensive account of grounding-based explanations.

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Notes

  1. 1.

    A picture roughly like that to be described is proposed by Schaffer (2016) and Audi (2012a, b). See also Schnieder (2015, 2010).

  2. 2.

    In the debate about causal explanation, a view like this is endorsed, for example, by Strevens (2008), Woodward (2003) and Ruben (2012).

  3. 3.

    See Strevens (2008), in particular ch. 2. Criteria of difference-making play an important role in the debate on causation in general—sometimes not quite with the same role they have in Strevens; cf. e.g. Lewis (1973, 160f), and List and Menzies (2009). We will briefly come back to this in Sect. 7 below.

  4. 4.

    This is one way to understand Lewis’s view on the matter; cf. Lewis (1986a).

  5. 5.

    For reasons of space, we have to assume a basic familiarity with the notion of grounding. For general introductory overviews see, e.g., Correia and Schnieder (2012) as well as Trogdon (2013).

  6. 6.

    Some authors prefer to think of ground as expressed by a sentential connective, and deny that there is, strictly speaking, a relation of grounding obtaining between facts. We make this assumption purely for ease of expression; our arguments as well as the idea of grounding as explanation-backing may be easily transposed to the alternative setting. For discussion of the matter, see (Correia and Schnieder 2012, 10ff).

  7. 7.

    The symbolism is taken over from Fine (2012b). Note, though, that our understanding of the symbolism differs from Fine’s in two respects. First, Fine prefers not to think of ground as a relation between facts and hence does not use the symbolism to abbreviate fact-talk. Second, Fine uses \(\prec\) for what he calls strict partial ground, which is defined in terms of his notion of weak ground. The notion we express by \(\prec\), which is more common in the current debate, is what Fine calls partial strict ground and writes \(\prec ^*\). Under Fine’s semantics, partial strict ground is strictly stronger than strict partial ground (cf. Fine 2012b, 4).

  8. 8.

    Although we prefer to allow for scenarios including mere states of affairs, this is not necessary for our purposes, so readers with ontological qualms about such entities need not be concerned.

  9. 9.

    The matter is taken up in the appendix.

  10. 10.

    We are then not strictly speaking comparing an actual scenario with a non-actual one, as Strevens would have us do. We could do so, however, by replacing f with some non-obtaining state of affairs. But since this produces exactly the same results as our simpler procedure, we stick to the latter. Note also that Strevens speaks of the actual scenario, whereas we have many, namely any collection of facts. We see no attractive way of amending our procedure to appeal to just one actual scenario. Finally, many people’s first idea for cashing out talk of comparison of actual scenarios and non-actual versions of them will be in terms of possible worlds and counterfactuals. But this is by no means mandatory; indeed, Strevens’ own account also does not explicate his talk of comparison of actual and nonactual scenarios in terms of counterfactuals and possible worlds (cf. Strevens 2008, 111ff). For our purposes, the present way of cashing Strevens’ idea out is much more fruitful. We shall have some use for counterfactuals in considering alternative possibilities later on, though (cf. Sect. 4).

  11. 11.

    There is an obvious strengthening of this notion of a difference-making partial ground, on which it is required that for all full grounds that include the fact that P, removing that fact yields a collection that does not contain a full ground anymore. However, given the common assumption that if \(\Gamma < P\) and \(\varDelta < P\), then \(\Gamma \cup \varDelta < P\), this would imply that any fact which has several full grounds—which, at least assuming transitivity, is true of the vast majority of facts—has no difference-making partial grounds. So the resulting notion of difference-making would not be very useful.

  12. 12.

    Note that (Df. \(\prec _D\)) has the consequence that any difference-making partial ground is a partial ground, as one would have hoped. For suppose S contains a full ground of Q, but \(S{\setminus}\{\)the fact that \(P\}\) does not. Let \(\Gamma \subseteq S\) be such that \(\Gamma < Q\). Then \(\Gamma\) is not a subset of \(S{\setminus} \{\)the fact that \(P\}\). It follows that the fact that P is a member of \(\Gamma\), and hence that \(P \prec Q\).

  13. 13.

    In Krämer (2016), a similar notion of something getting us closer to the truth of a proposition is employed in formulating a criterion of evidential relevance. Given that the notions of relevance and difference-making seem to be very closely related, this may provide some additional motivation for our approach to difference-making, and hints at the possibility of a unified account of difference-making and relevance.

  14. 14.

    Fine (2012a, 47f) suggests that some facts may be zero-grounded, which is supposed to amount to being grounded by the empty set of facts, and distinguished from being ungrounded. If so, then it may be that not every case in which a single fact fully grounds another gives rise to a case of difference-making partial grounding. However, zero-grounding, if there is such a thing, is supposed to be a feature of only a rather special and rare sort of fact, so our general point is not threatened. For simplicity, we tacitly exclude the possibility of zero-grounding in what follows.

  15. 15.

    When we speak of standard assumptions in the (propositional) logic of ground, we mean assumptions that are explicitly endorsed in both Correia (2010) and Fine (2012a), which are so far the only reasonably developed systems for the propositional logic of ground. We give more precise versions of our informal arguments here by reference to these systems in the appendix.

  16. 16.

    The exact condition of independence required varies slightly with the details of the logic of ground assumed. Roughy speaking, it is sufficient to choose facts with disparate subject matters, such as that this ball is red and that that chair is brown. Details are given in the appendix.

  17. 17.

    We have in mind Schaffer’s case of the dented sphere. His particular example suffers from some special problems not affecting the example we describe below, which is why we prefer to focus on our case. Schaffer describes two more putative counter-examples to transitivity, which we do not discuss here. For criticism of the examples, see e.g. Litland (2013).

  18. 18.

    Although this reading is not mandatory, it seems quite natural. As we have already mentioned, it is quite common to explicate difference-making in terms of counterfactual dependence in something like this way. (For a classic statement of this intuition in the case of causal difference-making, see Lewis (1986b, 161–162).) More or less every way of understanding the counterfactual in (ii) gives quite a plausible claim, moreover. For example, it is true that if it were not the case that the ball is green in spot s, the ball would still be largely-red.

  19. 19.

    Schaffer’s remark in his (2016, 31) that in the case of a disjunction being grounded by its true disjuncts ‘one loses counterfactual dependence due to grounding overdetermination’ is suggestive of the idea that failures of counterfactual dependence, at least with respect to the grounding of disjunctions, always result from grounding overdetermination.

  20. 20.

    More precisely, we let \(<_T\) be the closure of < under the principle Cut: If \(\Gamma < P\) and \(P, \varDelta < Q\), then \(\Gamma , \varDelta < Q\). We then let \(\prec _T\) be the partial cousin of \(<_T\): \(P \prec _T Q\) iff \(\Gamma <_T Q\) for some \(\Gamma\) with \(P \in \Gamma\).

  21. 21.

    An alternative way to capture the two-faced nature of the relevant kind of ground is by appeal to a suitable non-factive understanding of ground (cf. Fine 2012a, 48ff). Writing \(\prec _{T0}\) for non-factive partial t-ground, we would then count a fact P a double agent wrt Q iff \(P \prec _T Q\) and \(P \prec _{T0} \lnot Q\). This option promises to yield more satisfactory results in the case of necessarily obtaining groundees Q, for which the present proposal counts any partial ground a double agent, assuming the orthodox view that counterfactuals with impossible antecedents are vacuously true. However, since the existence of a clear non-factive understanding of ground may be doubted and since the counterfactual serves our present purposes well enough, we here stick to the counterfactual version.

  22. 22.

    Again, the logic of ground would allow us to systematically produce examples of double agents given suitably independent PQ. For we then have \(P \prec_T (P \wedge Q) \vee (\lnot P \wedge Q)\). Now if the groundee had been false, Q would have been false, and we may assume that P would still have been true. But then we would get that \(P \prec_T \lnot (\lnot P \wedge Q)\), and since \(\lnot (\lnot P \wedge Q) \prec_T \lnot ((P \wedge Q) \vee (\lnot P \wedge Q))\), by transitivity, \(P \prec_T \lnot ((P \wedge Q) \vee (\lnot P \wedge Q))\), as required for P’s being a double agent.

  23. 23.

    Given suitably independent PQ, a logical example is given by the true four-way disjunction \(R := (P \wedge Q) \vee (\lnot P \wedge Q) \vee (\lnot P \wedge \lnot Q) \vee Q\). Crucially, if R were false, P would be required to render false the third disjunct of R, and thereby would come out a difference-making ground of \(\lnot R\).

  24. 24.

    Note that this is our preferred view, but it is not fully mandatory for the arguments in this section. In particular, it would suffice if difference-making was merely a minimal condition for good explanations. The odd ring of the because-claims under discussion could then be blamed on their expressing a particularly bad explanation due to being utterly uninformative and misleading.

  25. 25.

    It might be objected that many grounding claims sound plausible even though they, too, are badly suited to remove any relevant epistemic predicament. For instance, it does not sound implausible—or at least not as implausible as (6)—to say that \(P, Q < P \wedge Q\). But would anyone who is puzzled over why \(P \wedge Q\) obtains be helped by pointing out that this is because P and Q obtain? In response, we wish to highlight a significant disanalogy. The problem in this last case is one of triviality. Any typical epistemic predicament with respect to \(P \wedge Q\) will extend to P and/or Q, so the envisaged explanation appeals to facts for which the hearer is also in need of an explanation. The problem in our case of non-difference-making grounding, however, is of a quite different structure. For partially explaining the ball’s being largely-red by citing its green spot is bad even if the hearer is not also puzzled over the green spot. Here the problem with the tie appealed to in the explanation is not that it is obvious and thus uninformative, but rather that it has the wrong strength and/or direction, as it were.

  26. 26.

    See Owens (1992, 16ff), Heslow (1981).

  27. 27.

    Note that this picture also yields a very strong and very literal connection between grounding and difference-making, in that it portrays ground simply as the making of one difference by others.

  28. 28.

    Cf. Schaffer (2016, 68).

  29. 29.

    We can, at this point, stay agnostic with respect to the issue whether being a strong difference-maker merely determines pragmatic acceptability of certain because-claims, or whether it captures an objective criterion of explanatory relevance.

  30. 30.

    See e.g. Strevens (2008, §3.5), Sartorio (2005, 75), and cf. Yablo (1992, §4).

  31. 31.

    Of course the specific weight is causally responsible for specific features of the window’s breaking (for how it breaks exactly) in a way the determinable weight is not. Strevens assumes, however, a scenario wherein the explanandum in question is not the window’s breaking-in-a-highly-specific-way, but rather its simply breaking. This seems plausible: in any ordinary sense of ‘explanation’, there is a wide array of explanations of the occurrence of some given event, where highly specific details of how the event occurred are simply beside the question. Compare on this also Schaffer (2012, 135).

  32. 32.

    Whether the ground-theoretic assumption holds depends on some subtle details of one’s theory of ground and of how the property of being signal-coloured is conceived. For the sake of argument, however, we grant the ground-theoretic assumption to our opponent.

  33. 33.

    See Strevens (2008, 101ff). We adopt the terminology of a cohesive difference-maker from him.

  34. 34.

    Note that in the debate on the metaphysics of causation, notions of difference-making that follow an idea analogous to the one we have been developing in this section have been suggested as necessary conditions for being a causal influence (often under the label ‘proportionality’); cf. e.g. Yablo (1992, 273ff), and Sartorio (2005). For reasons outlined earlier (Sect. 5), we prefer to view cohesive difference-making as a condition (or a good-making feature) for explanations, rather than as a condition for the obtaining of the respective explanation-backing relation. A view congenial to ours is that of Weslake (2013, §6) who defends the idea that proportionality is a dimension of explanatory value.

  35. 35.

    The fineness of grain of a conception of ground is a matter of the conditions under which sentences are substitutable salva veritate in the scope of a grounding-operator. More coarse-grained conceptions allow more substitutions, more fine-grained conceptions less.

  36. 36.

    Fine in Fine (2012a) actually speaks of facts rather than states. Since we have used ‘fact’ for the relata of the grounding relation facts, to avoid terminological confusion, we use ‘state’ instead. (Note that in other versions of his truthmaker semantics, Fine also appeals to states, and then typically allows non-actual and indeed often impossible states alongside the actual, obtaining ones. So we should perhaps emphasize again that we here restrict ‘state’ to actual, obtaining states.)

  37. 37.

    Fusion is assumed to be associative, so that \(\bigsqcup P_0 \sqcup \bigsqcup P_1 \sqcup \ldots = \bigsqcup (P_0 \cup P_1 \cup \ldots )\).

    Use of this fact will often be tacit in what follows.

  38. 38.

    With respect to (Df. \(\prec\)), it should be noted that this definition of \(\prec\) is faithful to our above understanding of partial ground as applying to the parts of a strict ground, and thereby deviates from Fine’s definition of \(\prec\) by the condition that \(P \preceq Q\) and \(Q \not \preceq P\).

  39. 39.

    Since fusion is defined for the empty set of states, we always have a minimal state which is part of every state, called the nullstate. The relation of overlap must therefore be taken to require sharing of non-null parts, otherwise it trivializes.

  40. 40.

    More formally, let \(\Gamma\) be indexed by an index set I, so that \(\Gamma = \{P_i: i \in I\}\). Then \(\bigwedge \Gamma = \{\bigsqcup \{f(i): i \in I\}: f \in \Pi \{P_i: i \in I\}\}\).

  41. 41.

    Here is a counter-example. Suppose that \(P = \{s\}\), \(Q = \{t\}\), where t and s do not overlap. Then \(Q \vee (P \wedge Q)\) = \(\{t, s \sqcup t\}\). Now suppose \(R = \{u\}\) with u strictly between t and \(s \sqcup t\). Note that \(s \sqcup u = s \sqcup t\). Now consider \(\Gamma = \{R\}\). Then \(\Gamma , P \le Q \vee (P \wedge Q)\). But as before, \(Q \vee (P \wedge Q) \not \preceq P\), and likewise \(Q \vee (P \wedge Q) \not \preceq R\), since u is a proper part of \(s \sqcup t\). So \(\Gamma , P < Q \vee (P \wedge Q)\). But \(R \not \subseteq Q \vee (P \wedge Q)\), so \(R \not \preceq Q \vee (P \wedge Q)\), and a fortiori, \(R {\nless} Q \vee (P \wedge Q)\).

  42. 42.

    Note that if we were to assume that every proposition P is convex in the sense that \(u \in P\) whenever \(s, t \in P\), s is part of u, and u is part of t, this case cannot obtain. This is significant given that according to Fine (ms, 11), grounding defined as above on convex propositions coincides with ground as per the logic of Correia (2010). So it would appear that with respect to Correia’s logic, the argument for P being a non-difference-making partial ground of \(Q \vee (P \wedge Q)\) indeed goes through in its original form. For criticism of the convexity constraint and the corresponding principle in Correia’s logic, see Krämer and Roski (2015) and Correia (2016).

  43. 43.

    We write \(\prec ^{sp}\) for the Finean notion of strict partial ground, defined as non-mutual weak partial ground. It should be emphasized that Fine puts forth all these rules with a reading of \(\prec\) as strict partial ground in mind. However, in the cases in which we substitute the notion of partial strict ground, i.e. Sub(</\(\prec\)) and Sub(\(\prec\)/\(\preceq\)), it is clear that they retain all of their plausibility under this reinterpretation.

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Acknowledgments

We would like to thank our fellow phlox research-group members Michael Clark and Martin Lipman as well as Neil McDonnell for their comments on an earlier version of the paper. We'd also like to thank Ansten Klev and an anonymous referee of this journal for their comments. Stephan Krämer’s work on this paper has been funded by the Deutsche Forschungsgemeinschaft (Grant KR 4516/1-1); Stefan Roski’s work has been funded by the Behörde für Wissenschaft und Forschung Hamburg (Grant: Welt der Gründe). Both of us gratefully acknowledge the support.

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Appendix 1: Logical cases of non-difference-making

Appendix 1: Logical cases of non-difference-making

We have claimed above that plausible assumptions concerning the logic of ground allow us to identify a systematic way of producing instances of non-difference-making ground given a pair of suitably independent facts to start with. This appendix develops our above, rough and informal argument for this claim in more detail.

The required logical assumptions concern the pure and the propositional logic of ground. Unfortunately, there is currently no standard, fully worked out propositional logic of ground. Rather, what we have is: (I) a system of natural inference rules that are plausible relative to a very fine-grained conception of ground, proposed by Kit Fine, but with as yet no adequate semantics, and acknowledged as incomplete with respect to any plausible understanding of ground (cf. Fine 2012a, 67); (II) a natural truthmaker-semantics, also proposed in Fine (2012a), which yields a logic adequate to a much more coarse-grained conception of ground; (III) a logical system proposed by Correia (2010), addressing again a coarse-grained conception of ground, proven sound and complete for a corresponding algebraic semantics.Footnote 35 An additional difficulty is that the languages of these logical systems are expressively too weak to state anything like the quantified claim that for every \(\Gamma\), if \(\Gamma , P < Q\), then \(\Gamma ^0 < Q\) for some \(\Gamma ^0 \subseteq \Gamma\) with \(P \notin \Gamma ^0\).

Nevertheless, we believe a useful case for our claim can be made. To this end, we shall do two things. Firstly, we discuss a sharpening of our above informal argument in the context of the truthmaker-semantics for ground. We suggest that in this context, it is best to work with a slightly more refined conception of what it is for \(\Gamma\) to contain a full ground of some fact than the simple set-theoretic conception employed above. It can then be shown that for suitably independent, true P and Q, the informal argument given above goes through. It would take a lot of work to reconstruct the argument within the logic of (III), so we choose not to do so here. However, it is known that the logic obtained on the semantics in (II) is very close to that in (III) (cf. Fine ms, 11), and it is clear that a version of our argument can be given for Correia’s system, too. Secondly, we discuss our informal argument in the context of the system (I). We show that given an additional rule that turns out valid on both of the only two semantics known to us that validate the others of Fine’s rules, our argument goes through given very modest assumptions of independence of P and Q. Without the additional rule, a move parallel to that made before of refining the relevant notion of containment will secure our result.

Appendix 1.1: The coarse-grained framework

In his truthmaker-semantics for ground, Fine associates with each sentence A a set of states that verify A and a set of states that falsify A. States are here thought of as obtaining. Since no sentence is both true and false and no sentence is neither, exactly one of the two sets of states associated with a given sentence is empty.Footnote 36 For our purposes, it is easiest to reason directly about the sets of states and forget about the sentences to which they are assigned. We may then think of a non-empty set of states as a true proposition which is verified by exactly its members. Following Fine, we assume that (i) whenever there are states \(s, t, u, \ldots\), there is also their fusion \(\bigsqcup \{s, t, u, \ldots \} = s \sqcup t \sqcup u \sqcup \ldots\),Footnote 37 and that (ii) any proposition P is closed under non-empty fusions, so that \(\bigsqcup P^\prime \in P\) whenever \(\emptyset \subset P^\prime \subseteq P\).

The full and partial notions of (strict) ground are defined by Fine in terms of prior notions of full and partial weak ground. The definitions are as follows:Footnote 38

(Df. \(\le\)):

\(P_1, P_2, \ldots \le Q =_{\text {df.}}s_1 \sqcup s_2 \sqcup \ldots \in Q\) whenever \(s_1 \in P_1, s_1 \in P_2, \ldots\)

(Df. \(\preceq\)):

\(P \preceq Q =_{\text {df.}}\Gamma , P \le Q\) for some set of propositions \(\Gamma\)

(Df. \({<}\)):

\(P_1, P_2, \ldots < Q =_{\text {df.}}P_1, P_2, \ldots \le Q\) and \(Q \not \preceq P_i\) for all i

(Df. \(\prec\)):

\(P \prec Q =_{\text {df.}}\Gamma , P < Q\) for some set of propositions \(\Gamma\)

Now say that a state s is part of a state t iff for some state \(s^\prime\), \(t = s \sqcup s^\prime\), and say that two states s and t overlap iff they share some (non-nullFootnote 39) part.

The conjunction \(P \wedge Q\) of two propositions is the set \(\{s \sqcup t: s \in P\) and \(t \in Q\}\), and the disjunction \(P \vee Q\) is the set \(P \cup Q \cup (P \wedge Q)\). Now consider any truths PQ which are independent in the sense that \(\bigsqcup P\) and \(\bigsqcup Q\) do not overlap. We may now show that

  1. (A)

    \(P \prec Q \vee (P \wedge Q)\)

This is immediate by definition given that \(P, Q < Q \vee (P \wedge Q)\), which may be established as follows. Suppose \(s \in P\) and \(t \in Q\). Then \(s \sqcup t \in P \wedge Q\), and hence \(s \sqcup t \in Q \vee (P \wedge Q)\). So \(P, Q \le Q \vee (P \wedge Q)\). Now consider \(\bigsqcup (Q \vee (P \wedge Q)) = \bigsqcup P \sqcup \bigsqcup Q\). Since \(\bigsqcup P\) and \(\bigsqcup Q\) do not overlap, \(\bigsqcup P \sqcup \bigsqcup Q\) is not a part of \(\bigsqcup P\) or \(\bigsqcup Q\). But it is easy to verify that \(R \preceq P\) only if \(\bigsqcup R\) is part of \(\bigsqcup P\), so it follows that \(Q \vee (P \wedge Q) \not \preceq P\) and that \(Q \vee (P \wedge Q) \not \preceq Q\). Hence \(P, Q < Q \vee (P \wedge Q)\).

Now suppose that some scenario S which includes the fact that P contains a scenario which is a strict full ground of \(Q \vee (P \wedge Q)\). To show that P is not a difference-maker, we need to show that \(S{\setminus} \{\)the fact that \(P\}\) still contains a strict full ground of P. If this fails for any scenario, it fails for a scenario which is a strict full ground of \(Q \vee (P \wedge Q)\), so we may restrict attention to such scenarios. Suppose, therefore, that

  1. (H)

    \(\Gamma , P < Q \vee (P \wedge Q)\)

Writing \(\bigwedge \Gamma\) for the conjunction of all the members of \(\Gamma\),Footnote 40 it follows that every verifier of \(\bigwedge \Gamma \wedge P\) is a verifier of \(Q \vee (P \wedge Q)\). Since any such verifier has a part that verifies P, by the assumption of independence, no such verifier is a verifier of Q, and hence it must be a verifier of \(P \wedge Q\). But since no verifier of P is part of a verifier of Q, it follows that every verifier of \(\bigwedge (\Gamma {\setminus} \{P\})\) must have a part that verifies Q.

But does it follow that \(\Gamma {\setminus} \{P\}\) contains a strict full ground of \(Q \vee (P \wedge Q)\)? If a scenario contains another just in case the latter is a subset of the former, it does not.Footnote 41 In a rough approximation, the problem is that although some subset of \(\Gamma {\setminus} \{P\}\) must be strong enough to establish Q, it may be that every such subset is, as it were, strictly between Q and \(P \wedge Q\), and thereby fail to ground \(Q \vee (P \wedge Q)\).Footnote 42

However, intuitively, what this shows is not that P may after all be a difference-maker with respect to \(Q \vee (P \wedge Q)\), but that the set-theoretic interpretation of the containment of one scenario in another is unsatisfactory. For in the intuitively relevant sense, since every verifier of \(\bigwedge (\Gamma {\setminus} \{P\})\) contains a verifier of Q, \(\Gamma {\setminus} \{P\}\) contains \(\{Q\}\), and thereby a strict full ground of \(Q \vee (P \wedge Q)\). So what we should do is refine the conception of containment appealed to in the definition of difference-making. We shall therefore say that a scenario \(\Gamma\) contains a scenario \(\Gamma ^0\) just in case every verifier of \(\bigwedge \Gamma\) has a part that verifies \(\bigwedge \Gamma ^0\). Then since \(\Gamma {\setminus} \{P\}\) contains \(\{Q\}\) whenever \(\Gamma , P < Q \vee (P \wedge Q)\), P comes out a non-difference-making partial ground of \(Q \vee (P \wedge Q)\).

Appendix 1.2: The fine-grained framework

We use the following rules:Footnote 43

Sub(\({<}/\prec\)):

From \(\Gamma , A < B\) infer \(A \prec B\)

Sub(\({<}\)/\(\le\)):

From \(\Gamma < B\) infer \(\Gamma \le B\)

Sub(\(\le\)/\(\preceq\)):

From \(\Gamma , A \le B\) infer \(A \preceq B\)

Sub(\(\prec\)/\(\preceq\)):

From \(A \prec B\) infer \(A \preceq B\)

Trans(\({<}\)/\({<}\)):

From \(A < B\) and \(B < C\) infer \(A < C\)

Trans(\(\le\)/\({<}\)):

From \(\Gamma \le A\) and \(A < B\), infer \(\Gamma < B\)

Trans(\(\prec ^{sp}\)/\(\preceq\)):

From \(A \prec ^{sp} B\) and \(B \preceq C\) infer \(A \prec ^{sp} C\)

Irr(\(\prec ^{sp}\)):

From \(A \prec ^{sp} A\), infer \(\bot\)

Rev Sub(\(\le\)/\({<}\)):

From \(A_1, A_2, \ldots \le B\) and \(A_1 \prec ^{sp} B\), \(A_2 \prec ^{sp} B\), \(\ldots\), infer \(A_1, A_2, \ldots < B\)

(\(\wedge\)I):

From A and B, infer \(A, B < A \wedge B\)

(\(\vee\)I):

From A, infer \(A < A \vee B\) or \(B < A \vee B\)

(\(\vee\)E):

From \(\Gamma < A \vee B\), infer that either \(\Gamma \le A\), or \(\Gamma \le B\), or for some \(\Gamma _A, \Gamma _B\) with \(\Gamma = \Gamma _A \cup \Gamma _B\): \(\Gamma _A \le A\) and \(\Gamma _B \le B\)

(\(\wedge\)E):

From \(\Gamma < A \wedge B\) infer that for some \(\Gamma _A, \Gamma _B\) with \(\Gamma = \Gamma _A \cup \Gamma _B\): \(\Gamma _A \le A\) and \(\Gamma _B \le B\)

We make the following assumptions.

  1. (A1)

    P

  2. (A2)

    Q

  3. (A3)

    \(P \not \preceq Q\)

Using the rules (\(\wedge\)I), (\(\vee\)I), Trans(<), and Sub(</\(\prec\)) we obtain

  1. (1)

    \(P \prec Q \vee (P \wedge Q)\)

Now suppose

  1. (S)

    \(\Gamma , P < Q \vee (P \wedge Q)\)

By (\(\vee\)E), we may infer from (S) that one of the following three claims holds:

  1. (i)

    \(\Gamma , P \le Q\)

  2. (ii)

    \(\Gamma , P \le P \wedge Q\)

  3. (iii)

    For some \(\varDelta _1, \varDelta _2\) with \(\varDelta = \varDelta _1 \cup \varDelta _2\): \(\varDelta _1 \le Q\) and \(\varDelta _2 \le P \wedge Q\)

But from (i), it follows by Sub(\(\le\)/\(\preceq\)) that \(P \preceq Q\), contrary to our assumption.

Now suppose (iii), and let \(\varDelta _1 \le Q\). Note that \(P \notin \varDelta _1\), for otherwise again \(P \preceq Q\), contrary to our assumption. So \(\varDelta _1\) is a subset of \(\Gamma \cup \{P\}\), not including P, with \(\varDelta _1 \le Q\). Using that \(Q < Q \vee (P \wedge Q)\) as well as Trans(\(\le\)/<), we obtain that \(\varDelta _1 < Q \vee (P \wedge Q)\). So \(\varDelta _1\) is the required witness for the claim that P is a non-difference-making partial ground of \(Q \vee (P \wedge Q)\) for the case that (iii) holds.

Suppose finally that (ii) holds. Then by Sub(\(\le\)/\(\preceq\)), we have \(R \preceq P \wedge Q\) for all \(R \in \Gamma , P\). Since any relationship of weak partial ground is either mutual or a relationship of strict partial ground, it follows by Rev Sub(\(\le\)/<) that one of the following two conditions holds:

  1. (a)

    \(\Gamma , P < P \wedge Q\)

  2. (b)

    \(P \wedge Q \preceq R\) for some \(R \in \Gamma , P\)

If (a), then by (\(\wedge\)E) we obtain that for some subset \(\varDelta _1\) of \(\Gamma , P\), we have \(\varDelta _1 \le Q\). Note that \(P \notin \varDelta _1\) for otherwise \(P \preceq Q\), contrary to our assumption. So \(\varDelta _1\) is a subset of \(\Gamma \cup \{P\}\), not including P, with \(\varDelta _1 \le Q\), and thereby as before, \(\varDelta _1 < Q \vee (P \wedge Q)\). So \(\varDelta _1\) is the required witness for the claim that P is a non-difference-making partial ground of \(Q \vee (P \wedge Q)\) for the case that (a) holds.

If (b) holds, then we have for some \(R \in \Gamma \cup \{P\}\) both \(P \wedge Q \preceq R\) and \(R \preceq P \wedge Q\). It is tempting to infer from this that \(P \wedge Q\) and R are mutual weak full grounds, as per the rule:

Rev Sub(\(\preceq\)/\(\le\)):

From \(A \preceq B\) and \(B \preceq A\), infer \(A \le B\)

And on the only proposals for a semantics for a logic of ground incorporating Fine’s rules—in as yet unpublished work by Krämer (ms) and Correia (ms)—this inference indeed comes out valid. But then \(R \le P \wedge Q\), and since \(P \wedge Q < Q \vee (P \wedge Q)\) we obtain \(R < Q \vee (P \wedge Q)\). Now it can be shown that \(R \ne P\). For otherwise we obtain \(P \le P \wedge Q\). But \(P \wedge Q \not \preceq P\), for otherwise we obtain by Trans(\(\prec ^{sp}\)/\(\preceq\)) that \(P \prec ^{sp} P\), and thus \(\bot\) by Irr(\(\prec ^{sp}\)). It follows by Rev Sub(\(\le\)/<) that \(P < P \wedge Q\). But then by the elimination rule (\(\wedge\)E), \(P \le Q\) and hence \(P \preceq Q\), contrary to our assumption. So \(P \notin \{R\}\), and hence \(\{R\}\) is our required witness for the claim that P is a non-difference-making partial ground of \(Q \vee (P \wedge Q)\) for the case that (iii) holds.

Now it may be that there could also be a plausible semantics for the kind of logic of ground Fine proposes which delivers natural counter-models to Rev Sub(\(\preceq\)/\(\le\)). In that case R would again, roughly speaking, have to lie between Q and \(P \wedge Q\) in strength. But then similar means to those proposed above would give a natural way to salvage our case by refining the conception of containment to fit the relevant sense of ‘between’.

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Krämer, S., Roski, S. Difference-making grounds. Philos Stud 174, 1191–1215 (2017). https://doi.org/10.1007/s11098-016-0749-5

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Keywords

  • Grounding
  • Causation
  • Explanation
  • Difference-making
  • Transitivity