Towards a theory of ground-theoretic content

Abstract

A lot of research has recently been done on the topic of ground, and in particular on the logic of ground. According to a broad consensus in that debate, ground is hyperintensional in the sense that even logically equivalent truths may differ with respect to what grounds them, and what they ground. This renders pressing the question of what we may take to be the ground-theoretic content of a true statement, i.e. that aspect of the statement’s overall content to which ground is sensitive. I propose a novel answer to this question, namely that ground tracks how, rather than just by what, a statement is made true. I develop that answer in the form of a formal theory of ground-theoretic content and show how the resulting framework may be used to articulate plausible theories of ground, including in particular a popular account of the grounds of truth-functionally complex truths that has proved difficult to accommodate on alternative views of content.

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Notes

  1. 1.

    The pioneering contributions initiating this debate were Batchelor (2010), Correia (2010, 2014a), Fine (2010, 2012c, b), Rosen (2010) and Schnieder (2011). More recent work includes Correia (2014b), deRosset (2013), deRosset (2014), Krämer (2013), Krämer and Roski (2015), Krämer and Roski (2016), Litland (2013, 2016a) and Poggiolesi (2015).

  2. 2.

    A word on notation. I shall be somewhat sloppy in my use of the letters ‘P’, ‘Q’, etc., in that I sometimes use them as schematic sentence letters, and sometimes as variables ranging over contents that may be assigned to sentences.

  3. 3.

    The operator option is chosen, for example, by Fine (2012b), Correia (2010) and Schnieder (2011). The predicate option is preferred by Rosen (2010), as well as Schaffer (2009). (The latter is something of an outlier in the current debate, though, in that he takes ground to relate not just truths, but objects of any kind. His conception of ground will not be canvassed in this paper.)

  4. 4.

    I assume here that ground is sensitive only to differences between sentences that concern content. This is not obvious prior to investigation; it may be that the best account of the distinctions drawn by ground sees (some of) them as purely syntactic. For the purposes of this paper, the assumption has the status of a working hypothesis. That is, I propose that we try and see if we can make sufficiently fine-grained distinctions pertaining to content to capture the distinctions drawn by ground. (One potentially problematic kind of case arises in connection with conceptual analyses. It might be suggested that if Alice is a vixen, say, then this is so because Alice is a female fox, where the ‘because’ indicates grounding, and that nevertheless ‘Alice is a vixen’ and ‘Alice is a female fox’ are exactly alike in content. For discussion, see Schnieder (2010). Thanks to an anonymous referee for highlighting the relevance of these cases.)

  5. 5.

    Note that the distinction, if it can be made, may also be transposed to the sentential-operator setting, where it turns into a distinction between worldly and representational conceptions of ground-theoretic content; cf. Correia (2010, p. 257).

  6. 6.

    It may be objected that I treat ground, in effect, as a relation between ground-theoretic contents, and since contents are representational entities, this commits me to a representational conception of ground. If this were so, the same reasoning would reveal Correia’s ostensively worldly conception of ground as representational, for he, too, treats ground in effect as a relation between what he calls the worldly contents of sentences. In his intended sense, then, a content may be worldly and thus non-representational. This is one of the reasons why I do not find this way of making the distinction very helpful.

  7. 7.

    This kind of view is endorsed, for example, by Fine (2012b), and in related forms by Rosen (2010), Schnieder (2011) and Correia (2014b).

  8. 8.

    Cf. Correia (2010, p. 267f)—it is plausible that a number of other cases should then also be disallowed; cf. ibid, p. 269.

  9. 9.

    Although Fine does not say so, the definitions should be read as restricted to truths, otherwise \(P_1,\,P_2,\, \ldots \le Q\) will vacuously hold whenever one of \(P_1,\, P_2, \ldots \) is false.

  10. 10.

    Note that I am relying here on the assumption, noted in Footnote 4 above, that ground is sensitive only to differences in content. If this assumption were given up, and ground seen as sensitive to the linguistic guises of contents, then the above line of reasoning could be resisted. Thanks here to an anonymous referee.

  11. 11.

    Cf. Schnieder (2008, p. 665f). As Schnieder points out elsewhere (cf. his 2011, p. 450f) similar cautionary remarks are in order when locutions such as ‘because’ are used to convey relationships of full ground.

  12. 12.

    This illustrates that Fine’s notion of verification is non-monotonic in the sense that a fact may fail to verify a proposition even though it has a part which verifies the proposition. For this reason, Fine sometimes describes this notion of verification, which he also calls exact verification, as imposing a requirement of holistic relevance: for a fact to verify a given proposition, it must not contain any part which is irrelevant to the truth of the proposition (cf. Fine 2012a, p. 234; 2014, p. 551f et passim).

  13. 13.

    The non-distributive notion of verification has also been put to use in Litland (2016b) in developing a logic of a many-many notion of grounding, on which what is grounded is irreducibly a collection of truths or facts.

  14. 14.

    The case of conjunction makes especially clear why the ‘full’ interpretation of ‘by’ must be assumed. For suppose that every fact s that verifies P also verifies Q and therefore \(P \wedge Q\). By ordinary standards, it might then be true to say that every verifier s of P verifies \(P \wedge Q\) by (among other things) verifying P. But it would not therefore be true to say that \(P < P \wedge Q\). We avoid this result if we read ‘by’ as requiring fullness. For on such a reading, it is false that in general every verifier of P verifies \(P \wedge Q\) by verifying P, since verifying P is usually only part of what is required for verifying \(P \wedge Q\).

  15. 15.

    That this kind of asymmetric account should be accepted is a surprising result; independently of the connection to the introduction principles, it might have seemed more natural to endorse one of the symmetric accounts. So the question arises whether there might be independent philosophical reasons for endorsing the asymmetric account. Although the matter calls for a much more extended discussion than I can offer here, it may be worth mentioning one possible source of independent motivation. I have in mind the kind of asymmetric account of truth and falsity that is endorsed, for example, by Williamson (1994, p. 188), which can be captured by the following principles:

    1. (T)

      If a proposition says that P, then it is true iff P.

    2. (F)

      If a proposition says that P, then it is false iff \(\lnot P\).

    This account immediately ties both the truth of a proposition saying that \(\lnot P\) and the falsity of a proposition saying that P to the same thing: it being the case that \(\lnot P\). But it does not in the same way tie the falsity of a proposition saying that \(\lnot P\) and the truth of a proposition saying that P to the same thing. Rather, the first is tied, in the first instance, to it being the case that \(\lnot \lnot P\), and the second to it being the case that P. This asymmetry is at least strongly reminiscent of the identification of the falsification of P with the verification of \(\lnot P\), in the absence of the identification of the verification of P with the falsification of \(\lnot P\). As such, it may perhaps provide an independent basis for the latter.

  16. 16.

    The possibility of this simplification was suggested to me by Kit Fine.

  17. 17.

    On the idea of a non-factive notion of ground, cf. e.g. Fine (2012b, p. 48ff) and Correia(2014b, p. 36).

  18. 18.

    A similar move may be considered for the direct modes. Specifically, we may follow (cf. Fine 2014, p. 557f, 2016, p. 8), and replace the appeal to the notion of a fact by an appeal to a broader notion of a state, which is like that of a fact except in that a state need not be actual, and indeed need not even be possible. There is then no obstacle to assuming every proposition to have at least one verifier and at least one falsifier. However, since ground, on my account, is determined purely by the presence or absence of indirect modes in a given proposition, this extension of the conception of verifiers is not required for our construction to work as intended. To the extent that non-actual and impossible modes are less problematic than non-actual and impossible states, it is an advantage of my framework that it can accommodate non-factive ground using less worrisome resources than are required on Fine’s approach. (It should be noted, though, that if we do not allow non-actual and impossible states, then the presence of a mode m in a proposition P does not in general represent exactly that every verifier of the propositions corresponding to m thereby verifies P. For the latter condition will be vacuously satisfied for any non-actual mode. The presence of m in P should then simply be understood to represent that verifying the propositions corresponding to m is a way to verify P—though it may be logically impossible to verify P in this way.)

  19. 19.

    For present purposes, we may restrict attention to concatenations of an at most countable sequence of sequences.

  20. 20.

    There may be purposes for which a putative ‘nullmode’ corresponding to \(\langle \rangle \) may be useful. If a mode is counted actual iff all propositions in the corresponding sequence are true, then the nullmode would automatically be actual and hence every proposition containing it trivially true. It would thereby have a similar profile to the nullfact (or nullstate) in Fine’s framework, which is the fusion of the empty set of facts (or states), and part of every fact (state). The most obvious application of the nullmode would be to capture Fine’s idea that some truths may be zero-grounded, where this is supposed to be distinct from being ungrounded; cf. Fine (2012b, p. 47f).

  21. 21.

    It may be worth pointing out that for cardinality reasons, in a constrained mode-space, V will be undefined for most sequences. Thanks here to an anonymous referee.

  22. 22.

    I borrow the labels from the corresponding inference rules in Fine (2012c)’s pure logic of ground. – Here and in what follows, I adopt the familiar convention of writing \(\varGamma \cup \{Q\}\) as \(\varGamma ,\, Q\) as well as \(\varGamma \cup \varDelta \) as \(\varGamma , \,\varDelta \), and similarly in other cases.

  23. 23.

    This parallels Fine’s account of the truthmakers of disjunctions which include not only the verifiers of the disjuncts, but also any fusions of these.

  24. 24.

    Note that absent any assumptions to the effect that P cannot be verified in part by verifying P, there is no guarantee that \(\uparrow P \ne P\). – One might wonder whether an argument is not needed for the claim that there always exists such a proposition as \(\uparrow P\). Formally, the assumption that P is raisable, in conjunction with the closure of the set of modes under fusion, makes sure that a suitable proposition always exists. But we may then ask for a defence of this assumption; what justifies disregarding propositions that are not raisable? The simplest answer is perhaps this. For any legitimate proposition \({\mathbf {P}}\), it should be possible to form its double negation \(\lnot \lnot {\mathbf {P}}\). Given the proposed account of ground and negation, \((\lnot \lnot {\mathbf {P}})^+\) relates to \({\mathbf {P}}^+\) exactly so that \(\uparrow ({\mathbf {P}}^+) = (\lnot \lnot {\mathbf {P}})^+\) (similarly for \((\lnot \lnot {\mathbf {P}})^-\)). So whenever \(\uparrow P\) does not exist, P cannot occur within a legitimate bilateral content and may for that reason be discarded. We can perhaps also argue for the raisability of legitimate propositions on independent grounds. For it seems that if P is a legitimate (unilateral) proposition, then there is such a thing as (non-factively) verifying P – if no sense can be made of the idea of P being verified, there is something incoherent about P. But if there is such a thing as verifying P, then it seems we can ask what can be done by verifying P. Now this question is about a way, or mode of doing something, namely the mode of doing it by verifying P. So this mode should be taken to exist. But since this mode is just the mode \(V\langle P\rangle \), it follows that P is raisable.

  25. 25.

    A proof of this result is given in the Appendix.

  26. 26.

    The elimination rules are perhaps more controversial than the introduction rules; for instance, it might be suggested that \(P \vee \lnot P\) is not only grounded by the weak grounds of its true disjunct, but also by the laws of logic, on some suitable construal of that phrase (this idea is also mentioned, but not endorsed in Schnieder 2011, p. 457f). So it may be worth noting that the elimination rules may be invalidated in a natural way without the introduction rules thereby also becoming invalid. All we need to do is to drop the requirement that the mode-space be constrained. (Whether and how the suggestion that the laws of logic ground \(P \vee \lnot P\) could be implemented within the overall framework developed here is harder to answer, and depends strongly on how exactly that view is spelt out.) Thanks here to an anonymous referee.

  27. 27.

    It might also be possible to achieve the same result not by changing the definition of disjunction, but by revising the definition of grounding, giving up on the tight connection that every mode of verification corresponds to an instance of grounding. – Note that the wish to reject \(P < P \vee P\) is not the only possible motivation for wanting to restrict the disjunction principle that \(P < P \vee Q\); some of the possible responses to the puzzles of ground presented in Fine (2010) also involve such a restriction. With respect to these views, the same comments apply: as long as the relevant restrictions can be captured within our framework, the views can be accommodated. Thanks to an anonymous referee for bringing up the matter of the puzzles of ground.

  28. 28.

    Since this is the standard view, more interesting than listing sources for it is to give some sources where the principle has been called into question. One context in which self-grounding has been considered a possibility is that of the paradoxes of ground described in Fine (2010) and Krämer (2013); see in particular (Correia 2014b, Sect. 7). Different kinds of doubts about irreflexivity are raised in Jenkins (2011).

  29. 29.

    Throughout this section, by ‘content’ and ‘proposition’ I shall mean unilateral content.

  30. 30.

    For the first principle, assume \(\varGamma ,\, P < P\). Then \(\varGamma ,\, P\) is a ground-set of some derivative mode \(m \in P\), so for some sequence of propositions \(\gamma \) with \(V(\gamma ) = m\), \(\varGamma ,\, P\) is the set underlying \(\gamma \). But then P is an element of \(\gamma \), contrary to our assumption. The second principle follows by the definition of \(\prec \). The other directions are equally straightforward.

  31. 31.

    Assume \(\varGamma \le P\) and \(Q \prec P\) for all \(Q \in \varGamma \). Note that \(P \notin \varGamma \), for otherwise \(P \prec P\), contradicting irreflexivity. But then neither \(\varGamma = \{P\}\) nor \(\{P\} \subset \varGamma \), and therefore \(\varGamma < P\).

  32. 32.

    For the first principle, assume \(\varGamma ,\, P \le P\) for non-empty \(\varGamma \). By irreflexivity, \(\varGamma ,\, P \not < P\). There remain two cases. (i) \(\varGamma \cup \{P\} = \{P\}\). Then \(\varGamma = \{P\}\), and hence \(\varGamma \le P\). (ii) \((\varGamma \cup \{P\})\setminus \{P\} \le P\). If \(P \in \varGamma \), then \(\varGamma \cup \{P\} = \varGamma \) and hence by assumption \(\varGamma \le P\). If \(P \notin \varGamma \), then \((\varGamma \cup \{P\})\setminus \{P\} = \varGamma \) and hence again, \(\varGamma \le P\). The second principle is immediate given asymmetry and the definition of \(\preceq \).

  33. 33.

    Again, it may be more interesting to mention dissenters. The most influential arguments against transitivity are perhaps those given in Schaffer (2012, p. 126ff). For replies, see Litland (2013), Raven (2013) and Krämer and Roski (2016).

  34. 34.

    We might perhaps also consider allowing the replacement of only some occurrences of \(\langle P\rangle \), which would create some additional complications because what modes can be obtained by transitivity from a mode can then not be read off from the corresponding ground-set.

  35. 35.

    It is immediate that Transitivity(<) will hold exactly for the propositions satisfying this constraint.

  36. 36.

    The proofs are in the appendix.

  37. 37.

    As we have seen, however, they satisfy some additional principles as well.

  38. 38.

    Further structural principles about ground might of course be considered, and implemented in the form of suitable constraints on propositions. The most significant ones among them may be the various versions of the claim that ground is well-founded. For an illuminating discussion of the various possible interpretations of the claim, see Dixon (2016). See Litland (2016a) for an argument for an instance of non-well-founded grounding.

  39. 39.

    I borrow the term from Fine (2012b, pp. 63, 67) who does not explicitly define it but seems to use it in at least roughly the same sense.

  40. 40.

    For assume \({\mathbf {P}}\approx {\mathbf {Q}}\). Note that \({\mathbf {P}}< \lnot \lnot {\mathbf {P}}\), so \({\mathbf {Q}}< \lnot \lnot {\mathbf {P}}\), so \({\mathbf {Q}}\le {\mathbf {P}}\), and hence either \({\mathbf {Q}}< {\mathbf {P}}\) or \({\mathbf {Q}}^+ = {\mathbf {P}}^+\). So if \({\mathbf {Q}}^+ \ne {\mathbf {P}}^+\), it follows that \({\mathbf {Q}}< {\mathbf {P}}\). By \({\mathbf {P}}\approx {\mathbf {Q}}\), we may infer \({\mathbf {P}}< {\mathbf {P}}\), contrary to the assumption of irreflexivity.

  41. 41.

    The crucial observation is that if modes are extensional, the operation of fusion on the modes is idempotent, i.e. \(m \sqcup m = m\). This renders the operations on closed contents of \(\sqcup \) and \(+\) idempotent, from which the above identities follow straightforwardly.

  42. 42.

    Note that somewhat unusally, our operation of fusion applies to sequences of modes and is thereby potentially sensitive to order and repetition of the modes being fused.

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Acknowledgments

The research for this paper was funded by the Deutsche Forschungsgemeinschaft (Grant KR 4516/1-1). I gratefully acknowledge the support. Earlier versions of the material were presented at a research colloquium at the University of Hamburg, at a Senior Seminar at the Université Aix-Marseille, at the Workshop On Ground and Consequence in Gothenburg, at the conference Truth and Grounds in Ascona, and at the Mathematical Logic seminar of the Department of Mathematics at the University of Hamburg. I thank all members of the audiences for their comments and criticisms, and in particular Jon Litland as well as my fellow phlox members Stefan Roski and Benjamin Schnieder for their extensive and especially useful feedback. Special thanks are also due to Nick Haverkamp for very valuable comments on some of the more technical material, to Louis deRosset for a helpful email exchange on the topic, and to my wife for advice on presentational matters, as well as for a great deal of patience. The biggest debt of gratitude I owe to Kit Fine for many comments, suggestions, and discussions of the topic and of the material on which this paper is based. Finally, two anonymous referees for this journal have provided comments that have led to a number of improvements, and I am very grateful for their efforts.

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Appendix

Appendix

We begin with the definition of a mode-space.

Definition 1

(Mode-spaces) A mode-space is a pair \(\langle M,\,V\rangle \) such that

  1. 1.

    M is non-empty

  2. 2.

    V is a non-empty, partial function taking non-empty sequences of non-empty subsets of M into members of M

  3. 3.

    the domain of V is closed under non-empty subsequences and countable concatenations of sequences

  4. 4.

    \(V(\gamma _1\,^\frown \gamma _2\,^\frown \ldots ) = V(\delta _1^\frown \delta _2^\frown \ldots )\) whenever \(V(\gamma _1) = V(\delta _1), V(\gamma _2) = V(\delta _2), \ldots \)

Given a fixed mode-space \(\langle M,\, V\rangle \), we call a mode any member of M. Any mode which is the value of V for some argument is called derivative, every other mode is called fundamental. We write \(M^D\) (\(M^F\)) for the set of derivative (fundamental) modes. Any subset of M will be called a content, and their set will be denoted by \({\mathscr {C}}\). The contents containing only derivative modes will themselves be called derivative, and the other contents will be called fundamental. We write \({\mathscr {C}}^D\) (\({\mathscr {C}}^F\)) for the set of derivative (fundamental) contents. Any sequence for which V is defined is called a via-sequence. Since the domain of V is closed under non-empty subsequences, for any content P that is an element of some via-sequence, there is also the via-sequence \(\langle P\rangle \) corresponding to the mode of verifying via P. We shall call any such content raisable and denote their set by \({\mathscr {R}}\). We call a ground-set of a derivative mode m any set of contents that underlies a content-sequence \(\gamma \) with \(V(\gamma ) = m\).

We say that a derivative mode m is a fusion of the sequence of derivative modes \(\langle m_1,\, m_2,\, \ldots \rangle \) iff there are via-sequences \(\gamma _1,\, \gamma _2, \ldots \) such that \(V(\gamma _1) = m_1,~V(\gamma _2) = m_2,~\ldots \), and \(m = V(\gamma _1\,^\frown \gamma _2\,^\frown \ldots )\).Footnote 42 By the third constraint on mode-spaces, fusions will always exist, and by the fourth constraint, they will be unique. For the sequence of derivative modes \(\langle m_1,\, m_2,\, \ldots \rangle \), we write its fusion as \(\bigsqcup \langle m_1, \,m_2,\, \ldots \rangle \) and also as \(m_1 \sqcup m_2 \sqcup \ldots \) We note a first central lemma (the proof is elementary):

Lemma 1

If \(\varGamma _1,\, \varGamma _2, \ldots \) are ground-sets of \(m_1,\, m_2, \ldots \), respectively, then \(\varGamma _1 \cup \varGamma _2 \cup \ldots \) is a ground-set of \(m_1 \sqcup m_2 \sqcup \ldots \)

For convenience, we repeat the definitions of the grounding relationships and of the truth-functions.

Definition 2

Let \(\varGamma \subseteq {\mathscr {C}}\) and \(P \in {\mathscr {C}}\). Then

figurec

As before, we write \(\varGamma < \{P_1,\, P_2,\, \ldots \}\) to abbreviate that for some \(\varGamma _1, \,\varGamma _2,\, \ldots \), \(\varGamma = \varGamma _1 \cup \varGamma _2 \cup \ldots \) and \(\varGamma _1< P_1,~\varGamma _2 < P_2,~\ldots \), and similarly for the case of \(\varGamma \le \{P_1,\, P_2,\, \ldots \}\).

Definition 3

For \(P,\,Q \in {\mathscr {C}}^D\):

figured

Definition 4

For \(P, Q \in {\mathscr {R}}\):

figuree

We shall mainly be interested in mode-spaces in which the set of raisable contents is closed under these operations.

Definition 5

A mode-space is called complete iff \(P \wedge Q \in {\mathscr {R}}\), \(P \vee Q \in {\mathscr {R}}\), and \(\uparrow P \in {\mathscr {R}}\) whenever \(P,\,Q \in {\mathscr {R}}\).

Lemma 2

(Introduction Lemma) In any complete mode-space, for \(\varGamma \subseteq {\mathscr {R}}\) and \(P, \,Q \in {\mathscr {R}}\):

  1. 1.

    If \(\varGamma \le P\), then \(\varGamma < \uparrow P\).

  2. 2.

    If \(\varGamma < \{P,\, Q\}\), then \(\varGamma < P \sqcup Q\).

  3. 3.

    If \(\varGamma < P\) or \(\varGamma < Q\) or \(\varGamma < \{P,\, Q\}\), then \(\varGamma < P + Q\)

  4. 4.

    If \(\varGamma \le \{P,\, Q\}\), then \(\varGamma < P \wedge Q\)

  5. 5.

    If \(\varGamma \le P\) or \(\varGamma \le Q\) or \(\varGamma < \{P,\, Q\}\), then \(\varGamma < P \vee Q\).

Proof

For 1: Suppose \(\varGamma \le P\). Then either (i) \(\varGamma = \{P\}\), or (ii) \(\varGamma < P\), or (iii) \(\varGamma \setminus \{P\} < P\). Suppose (i). By definition of \(\uparrow \), \(V\langle P\rangle \in \uparrow P\). Since \(\{P\}\) is the set underlying \(\langle P\rangle \), it is a ground-set of \(V\langle P\rangle \), so \(\varGamma < P\). Suppose (ii). Then \(\varGamma \) is a ground-set of some mode \(m \in P\). But then by definition of \(\uparrow \), it follows that \(m \in \,\uparrow P\), and hence \(\varGamma < P\). Suppose (iii). Then for some \(m \in P\), \(\varGamma \setminus \{P\}\) is a ground-set of m. But then by definition of \(\uparrow \), it follows that \(V\langle P\rangle \sqcup m \in \,\uparrow P\). By Lemma 1, \(\varGamma \setminus \{P\} \cup \{P\} = \varGamma \) is a ground-set of \(V\langle P\rangle \sqcup m\), hence \(\varGamma < P\).

For 2: Suppose \(\varGamma < \{P,\, Q\}\). Let \(\varGamma _P < P\) and \(\varGamma _Q < Q\) with \(\varGamma _P \cup \varGamma _Q = \varGamma \). Then \(\varGamma _P\) is a ground-set of some mode \(m_P \in P\) and \(\varGamma _Q\) is a ground-set of some mode \(m_Q \in Q\). By definition of \(\sqcup \), \(m_P \sqcup m_Q \in P \sqcup Q\), and by Lemma 1, \(\varGamma _P \cup \varGamma _Q = \varGamma \) is a ground-set of \(m_P \sqcup m_Q\), hence \(\varGamma < P \sqcup Q\).

For 3: Suppose \(\varGamma < P\). Then there is a mode \(m \in P\) of which \(\varGamma \) is a ground-set. By definition of \(+\), the same mode is included in \(P + Q\), hence \(\varGamma < P + Q\). Suppose \(\varGamma < Q\). Then by the same reasoning, \(\varGamma < P + Q\). Finally, suppose \(\varGamma < \{P,\, Q\}\). Then by part 2., \(\varGamma < P \sqcup Q\), and hence by definition of \(+\), \(\varGamma < P + Q\).

For 4: Suppose \(\varGamma \le \{P,\, Q\}\). Let \(\varGamma _P \le P\) and \(\varGamma _Q \le Q\) with \(\varGamma _P \cup \varGamma _Q = \varGamma \). By part 1., \(\varGamma _P < \uparrow P\) and \(\varGamma _Q < \uparrow Q\), hence \(\varGamma < \{\uparrow P, \,\uparrow Q\}\), and by part 2., \(\varGamma < P \wedge Q\).

For 5: If \(\varGamma \le P\) or \(\varGamma \le Q\), then by part 2., \(\varGamma < \uparrow P\) or \(\varGamma <\uparrow Q\). If \(\varGamma \le \{P,\, Q\}\), then by the reasoning in part 4., \(\varGamma < \{\uparrow P,\, \uparrow Q\}\). So by part 3., \(\varGamma < P \vee Q\). \(\square \)

Definition 6

A mode-space is called constrained iff \(V(\gamma ) = V(\delta )\) only if the same ground-set corresponds to \(\gamma \) and \(\delta \).

In a constrained mode-space, every derivative mode m corresponds to a unique ground-set, which we denote by |m|.

Lemma 3

(Elimination Lemma) In any constrained and complete mode-space, for \(\varGamma \subseteq {\mathscr {R}}\) and \(P,\, Q \in {\mathscr {R}}\):

  1. 1.

    If \(\varGamma < \,\uparrow P\), then \(\varGamma \le P\)

  2. 2.

    If \(\varGamma < P \sqcup Q\), then \(\varGamma < \{P,\, Q\}\)

  3. 3.

    If \(\varGamma < P + Q\), then \(\varGamma < P\) or \(\varGamma < Q\) or \(\varGamma < \{P,\, Q\}\)

  4. 4.

    If \(\varGamma < P \wedge Q\), then \(\varGamma \le \{P,\, Q\}\)

  5. 5.

    If \(\varGamma < P \vee Q\), then \(\varGamma \le P\) or \(\varGamma \le Q\) or \(\varGamma \le \{P,\, Q\}\)

Proof

For 1: Suppose \(\varGamma < \,\uparrow P\). Let \(m \in \,\uparrow P\) with \(\varGamma = |m|\). By definition of \(\uparrow \), there are three cases. (i) \(m \in P\cap M^D\). Then \(\varGamma < P\) and hence \(\varGamma \le P\). (ii) \(m = V\langle P\rangle \). Then \(\varGamma = |V\langle P\rangle | = \{P\}\), so again \(\varGamma \le P\). (iii) \(m = V\langle P\rangle \sqcup n\) for some \(n \in P\cap M^D\). Since \(n \in P\), \(|n| < P\). By Lemma 1, \(\varGamma = |n| \cup \{P\}\). Then either \(|n| = \varGamma \) or \(|n| = \varGamma \setminus \{P\}\), so either \(\varGamma < P\) or \(\varGamma \setminus \{P\} < P\), and hence \(\varGamma \le P\).

For 2: Suppose \(\varGamma < P \sqcup Q\). Let \(|m| = \varGamma \) and \(m \in P \sqcup Q\). Then \(m = m_P \sqcup m_Q\) for some \(m_P \in P\) and \(m_Q \in Q\). So \(|m_P| < P\) and \(|m_Q| < Q\). But \(\varGamma = |m_P| \cup |m_Q|\), and hence \(\varGamma < \{P,\, Q\}\).

For 3: Suppose \(\varGamma < P + Q\). Then it is immediate from the definition of \(+\) that \(\varGamma < P\) or \(\varGamma < Q\) or \(\varGamma < P \sqcup Q\), which by part 2. implies \(\varGamma < \{P,\, Q\}\).

For 4: Suppose \(\varGamma < P \wedge Q\). By part 2., \(\varGamma < \{\uparrow P,\, \uparrow Q\}\). So let \(\varGamma _P < \uparrow P\) and \(\varGamma _Q < \uparrow Q\) with \(\varGamma = \varGamma _P \cup \varGamma _Q\). By part 1., \(\varGamma _P \le P\) and \(\varGamma _Q \le Q\), hence \(\varGamma \le \{P,\, Q\}\).

For 5: Suppose \(\varGamma < P \vee Q\). By part 3., there are three cases. (i) \(\varGamma < \uparrow P\). Then by part 1., \(\varGamma \le P\). (ii) \(\varGamma < \uparrow Q\). Then by part 1. again, \(\varGamma \le Q\). (iii) \(\varGamma \le \{\uparrow P,\, \uparrow Q\}\). Then by the reasoning in part 4., \(\varGamma \le \{P,\, Q\}\). \(\square \)

We move on to the case of bilateral contents. We define the truth-functional operations on bilateral contents as well as the notion of ground-theoretic equivalence (\(\approx \)).

Definition 7

For \({\mathbf {P}},\, {\mathbf {Q}}\in {\mathscr {R}}\times {\mathscr {R}}\):

figuref

Definition 8

For \({\mathbf {P}},\, {\mathbf {Q}}\in {\mathscr {R}}\times {\mathscr {R}}\): \({\mathbf {P}}\approx {\mathbf {Q}}:\leftrightarrow \) for all \({\varvec{\varGamma }}\): \({\varvec{\varGamma }}< {\mathbf {P}}\) iff \({\varvec{\varGamma }}< {\mathbf {Q}}\), and for all \({\varvec{\varDelta }}\) and \({\mathbf {R}}\): \({\varvec{\varDelta }}, {\mathbf {P}}< {\mathbf {R}}\) iff \({\varvec{\varDelta }}, {\mathbf {Q}}< {\mathbf {R}}\).

Recall that ground between bilateral contents is defined simply as ground between the positive components. As a result, bilateral contents will be ground-theoretically equivalent (written \(\approx \)) provided their positive components are the same.

Lemma 4

(DeMorgan) For \({\mathbf {P}},\, {\mathbf {Q}}\in {\mathscr {R}}\times {\mathscr {R}}\):

  1. 1.

    \(\lnot ({\mathbf {P}}\wedge {\mathbf {Q}}) \approx \lnot {\mathbf {P}}\vee \lnot {\mathbf {Q}}\)

  2. 2.

    \(\lnot ({\mathbf {P}}\vee {\mathbf {Q}}) \approx \lnot {\mathbf {P}}\wedge \lnot {\mathbf {Q}}\)

Proof

By application of the definitions.

For 1: \((\lnot ({\mathbf {P}}\wedge {\mathbf {Q}}))^+ = ({\mathbf {P}}\wedge {\mathbf {Q}})^- = {\mathbf {P}}^- \vee {\mathbf {Q}}^- = (\lnot {\mathbf {P}})^+ \vee (\lnot {\mathbf {Q}})^+ = (\lnot {\mathbf {P}}\vee \lnot {\mathbf {Q}})^+\)

For 2: \((\lnot ({\mathbf {P}}\vee {\mathbf {Q}}))^+ = ({\mathbf {P}}\vee {\mathbf {Q}})^- = {\mathbf {P}}^- \wedge {\mathbf {Q}}^- = (\lnot {\mathbf {P}})^+ \wedge (\lnot {\mathbf {Q}})^+ = (\lnot {\mathbf {P}}\wedge \lnot {\mathbf {Q}})^+\) \(\square \)

As an immediate consequence of the definition of ground on bilateral contents and the previous lemmata, we obtain

Theorem 5

(Truth-functions and ground, introduction) In any complete mode-space, for \({\varvec{\varGamma }}\subseteq {\mathscr {R}}\times {\mathscr {R}}\) and \({\mathbf {P}},\, {\mathbf {Q}}\in {\mathscr {R}}\times {\mathscr {R}}\):

  1. 1.

    If \({\varvec{\varGamma }}\le \{{\mathbf {P}},\, {\mathbf {Q}}\}\), then \({\varvec{\varGamma }}< {\mathbf {P}}\wedge {\mathbf {Q}}\)

  2. 2.

    If \({\varvec{\varGamma }}\le {\mathbf {P}}\) or \({\varvec{\varGamma }}\le {\mathbf {Q}}\) or \({\varvec{\varGamma }}< \{{\mathbf {P}},\, {\mathbf {Q}}\}\), then \({\varvec{\varGamma }}< {\mathbf {P}}\vee {\mathbf {Q}}\).

  3. 3.

    If \({\varvec{\varGamma }}\le {\mathbf {P}}\), then \({\varvec{\varGamma }}< \lnot \lnot {\mathbf {P}}\)

  4. 4.

    If \({\varvec{\varGamma }}\le \lnot {\mathbf {P}}\) or \({\varvec{\varGamma }}\le \lnot {\mathbf {Q}}\) or \({\varvec{\varGamma }}\le \{\lnot {\mathbf {P}},\, \lnot {\mathbf {Q}}\}\), then \({\varvec{\varGamma }}< \lnot ({\mathbf {P}}\wedge {\mathbf {Q}})\)

  5. 5.

    If \({\varvec{\varGamma }}\le \{\lnot {\mathbf {P}},\, \lnot {\mathbf {Q}}\}\), then \({\varvec{\varGamma }}< \lnot ({\mathbf {P}}\vee {\mathbf {Q}})\)

Theorem 6

(Truth-functions and ground, elimination) In any complete and constrained mode-space, for \({\varvec{\varGamma }}\subseteq {\mathscr {R}}\times {\mathscr {R}}\) and \({\mathbf {P}},\, {\mathbf {Q}}\in {\mathscr {R}}\times {\mathscr {R}}\):

  1. 1.

    If \({\varvec{\varGamma }}< {\mathbf {P}}\wedge {\mathbf {Q}}\), then \({\varvec{\varGamma }}\le \{{\mathbf {P}},\, {\mathbf {Q}}\}\)

  2. 2.

    If \({\varvec{\varGamma }}< {\mathbf {P}}\vee {\mathbf {Q}}\), then \({\varvec{\varGamma }}\le {\mathbf {P}}\) or \({\varvec{\varGamma }}\le {\mathbf {Q}}\) or \({\varvec{\varGamma }}\le \{{\mathbf {P}},\, {\mathbf {Q}}\}\)

  3. 3.

    If \({\varvec{\varGamma }}< \lnot \lnot {\mathbf {P}}\), then \({\varvec{\varGamma }}\le {\mathbf {P}}\)

  4. 4.

    If \({\varvec{\varGamma }}< \lnot ({\mathbf {P}}\wedge {\mathbf {Q}})\), then \({\varvec{\varGamma }}\le \lnot {\mathbf {P}}\) or \({\varvec{\varGamma }}\le \lnot {\mathbf {Q}}\) or \(\varGamma \le \{\lnot {\mathbf {P}},\, \lnot {\mathbf {Q}}\}\)

  5. 5.

    If \({\varvec{\varGamma }}< \lnot ({\mathbf {P}}\vee {\mathbf {Q}})\), then \({\varvec{\varGamma }}\le \{\lnot {\mathbf {P}},\, \lnot {\mathbf {Q}}\}\)

We turn now to the structural properties of ground.

Definition 9

A proposition P is

  • irreflexive iff: P does not occur in \(\gamma \) whenever \(V(\gamma ) \in P\),

  • closed iff: P contains a mode with ground-set \(\varGamma _1 \cup \varGamma _2 \cup \ldots \) whenever P contains modes with ground-sets \(\varGamma _1, \,\varGamma _2, \ldots \),

  • transitive iff: P includes a mode with ground-set \(\varGamma ,\, \varDelta \) whenever P includes a mode with ground-set \(\varDelta ,\, Q\) and Q includes a mode with ground-set \(\varGamma \).

  • normal iff: irreflexive, closed, and transitive.

Theorem 7

(Structural Principles for Normal Propositions) In any constrained mode-space, for normal propositions \(P,\, P_1,\, P_2,\, \ldots ,\, Q,\, R\) and sets of normal propositions \(\varGamma ,\, \varGamma _1,\, \varGamma _2,\, \ldots ,\, \varDelta \):

  1. 1.

    \(P \not \prec P\)

  2. 2.

    If \(\varGamma _1 < P\), \(\varGamma _2 <P\), ..., then \(\varGamma _1,\, \varGamma _2,\, \ldots < P\).

  3. 3.

    If \(\varGamma \le P\) and \(Q \prec P\) for all \(Q \in \varGamma \), then \(\varGamma < P\).

  4. 4.

    If \(\varGamma ,\, P \le P\), then \(\varGamma \le P\).

  5. 5.

    If \(\varGamma _1 \le P,~\varGamma _2 \le P,~\ldots \), then \(\varGamma _1 \cup \varGamma _2 \cup \ldots \le P\).

  6. 6.

    If \(\varGamma < P\) and \(\varDelta ,\, P < Q\), then \(\varGamma ,\, \varDelta < Q\).

  7. 7.

    If \(\varGamma _1 \le P_1\), \(\varGamma _2 \le P_2\), ..., and \(P_1,\, P_2,\, \ldots \le Q\) then \(\varGamma _1,\, \varGamma _2,\, \ldots \le Q\)

  8. 8.

    If \(P \preceq Q\) and \(Q \prec R\) then \(P \prec R\)

  9. 9.

    If \(P \prec Q\) and \(Q \preceq R\) then \(P \prec R\)

  10. 10.

    If \(P \preceq Q\) and \(Q \preceq R\) then \(P \preceq R\)

Proof

1–4., and 6. were established in Sect. 5.4 above.

For 5., suppose \(\varGamma _1 \le P,~\varGamma _2 \le P, \ldots \) Consider all the \(\varGamma _i\) which are distinct from \(\{P\}\). By definition of \(\le \) and 1., \(\varGamma _i\setminus \{P\} < P\) for any such \(\varGamma _i\). If there are any such \(\varGamma _i\), let \(\varGamma ^\prime \) be the result of removing P from their union. By 2, \(\varGamma ^\prime < P\), so \(\varGamma _1 \cup \varGamma _2 \cup \ldots = \varGamma ^\prime \cup \{P\} \le P\). If there are no such \(\varGamma _i\), then \(\varGamma _1 \cup \varGamma _2 \cup \ldots = \{P\} \le P\).

For 7., suppose \(\varGamma _1 \le P_1\), \(\varGamma _2 \le P_2\), ..., and \(P_1,\, P_2,\, \ldots \le Q\). We confine ourselves to showing that \(\varGamma _1 \cup \{P_2,\, \ldots \} \le Q\) follows. By applying the same reasoning repeatedly and making use of the reflexivity of \(\le \), we may establish the desired conclusion. Now either (a) \(\varGamma _1 = \{P_1\}\), or (b) \(\varGamma _1 < P_1\), or (c) \(\{P_1\} \subset \varGamma _1\) and \(\varGamma _1\setminus \{P_1\} < P_1\). If (a), then our intended result \(\varGamma _1 \cup \{P_2,\, \ldots \} \le Q\) follows immediately. Suppose that (b). Then if (b1) \(P_1 = Q\), we have \(\varGamma _1 < Q\) and hence \(\varGamma _1 \le Q\). Moreover, by 4., we have \(\{P_2,\, \ldots \} \le Q\). So by 5., \(\varGamma _1 \cup \{P_2,\, \ldots \} \le Q\) follows. But if (b2) \(P_1 \ne Q\), then \(P_1 \in \{P_1,\, P_2,\, \ldots \}\setminus \{Q\}\) and \(\{P_1,\, P_2,\, \ldots \}\setminus \{Q\} < Q\), so by 6., \(\varGamma _1 \cup \{P_2,\, \ldots \}\setminus \{Q\} < Q\), hence \(\varGamma _1 \cup \{P_2,\, \ldots \} \le Q\). Suppose finally that (c). Then if (c1) \(P_1 = Q\), \(\varGamma _1 \le Q\), and so \(\varGamma _1 \cup \{P_2,\, \ldots \} \le Q\) follows as in case (b1). But if (c2) \(P_1 \ne Q\), then by similar reasoning as in case (b2), \(\varGamma _1\setminus \{P_1\} \cup \{P_1,\, P_2,\, \ldots \} = \varGamma _1 \cup \{P_2, \ldots \} \le Q\).

For 8., suppose \(P \preceq Q\) and \(Q \prec R\). If \(P = Q\), then \(P \prec R\) follows immediately. So suppose \(P \ne Q\), and hence \(P \prec Q\). Let \(m \in Q\) be such that \(P \in |m|\), and let \(n \in R\) be such that \(Q \in |n|\). Then \(|m| \cup \{P\} < Q\) and \(|n| \cup \{Q\} < R\), so by 6, \(|n| \cup |m| \cup \{P\} < R\), so \(P \prec R\).

For 9., suppose \(P \prec Q\) and \(Q \preceq R\). If \(Q = R\), then \(P \prec R\) follows immediately. So suppose \(Q \ne R\), and hence \(Q \prec R\). Then by the same reasoning as before, \(P \prec R\).

For 10., suppose \(P \preceq Q\) and \(Q \preceq R\). If either \(P \prec Q\) or \(Q \prec R\), then it follows by the previous results that \(P \prec R\), and hence \(P \preceq R\). If neither \(P \prec Q\) nor \(Q \prec R\), then \(P = Q\) and \(Q = R\), hence \(P = R\), and thus again \(P \preceq R\). \(\square \)

Together with the Subsumption principles and the Identity principle \(P \le P\) for weak ground established in Sect. 5.2, parts 1, 3, 7–10 correspond to the basic rules of the logic proposed in Fine (2012c), which is thereby shown to be sound with respect to the class of normal propositions in a constrained mode-space.

We call a bilateral content irreflexive, closed, transitive, or normal, just in case its positive component has the relevant property. We wish then to show that any truth-functional combinations of normal propositions are themselves normal. By the definitions of the truth-functional operations, it suffices to show that for unilateral contents, normality is preserved under \(\wedge , \vee \), and \(\uparrow \).

Theorem 8

For \(P,\, Q \in {\mathscr {R}}\), in some constrained mode-space: If P and Q are normal, then so are \(P \wedge Q\), \(P \vee Q\), and \(\uparrow P\).

Proof

By reducing failures for \(P \wedge Q\), \(P \vee Q\), \(\uparrow P\) of the characteristic principles of irreflexivity, amalgamation, and transitivity (1, 2 and 6 in Theorem 7) to failures of the corresponding principles for P or Q. We give the proof for \(P \wedge Q\); the other cases may be established by parallel means.

For irreflexivity, suppose that \(P \wedge Q \prec P \wedge Q\). Then for some \(\varGamma \): \(\varGamma ,\, P \wedge Q < P \wedge Q\). Then by the Elimination Lemma, \(\varGamma ,\, P \wedge Q \le \{P,\, Q\}\), and hence either (i) \(P \wedge Q \preceq P\) or (ii) \(P \wedge Q \preceq Q\). Suppose (i). Then since \(P \prec P \wedge Q\), it follows by transitivity of P that \(P \prec P\), in contradiction to P’s irreflexivity.

For amalgamation, suppose that \(\varGamma _1 < P \wedge Q\), \(\varGamma _2 < P \wedge Q\), .... For each \(\varGamma _i\): \(\varGamma _i \le \{P,\, Q\}\). So let \(\varGamma _{i_P} \le P\) and \(\varGamma _{i_Q} \le Q\) with \(\varGamma _i = \varGamma _{i_P} \cup \varGamma _{i_Q}\) for all i. Then by weak ground amalgamation for P and Q, \(\varGamma _{1_P},\, \varGamma _{2_P},\, \ldots \le P\), and \(\varGamma _{1_Q},\, \varGamma _{2_Q},\, \ldots \le Q\), so \(\varGamma _1,\, \varGamma _2,\, \ldots \le \{P,\, Q\}\), and hence \(\varGamma _1,\, \varGamma _2,\, \ldots < P \wedge Q\).

For transitivity, suppose that \(\varDelta ,\, R < P \wedge Q\) and \(\varGamma < R\). Then \(\varDelta ,\, R \le \{P,\, Q\}\). So we may write \(\varDelta ,\, R\) as the union of weak full grounds of respectively P and Q. R will be a member of at least one of these. By replacing R with \(\varGamma \), using the transitivity of P and Q, we may infer \(\varGamma ,\, \varDelta < \{P,\, Q\}\), and hence \(\varGamma ,\, \varDelta < P \wedge Q\). \(\square \)

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Krämer, S. Towards a theory of ground-theoretic content. Synthese 195, 785–814 (2018). https://doi.org/10.1007/s11229-016-1242-6

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Keywords

  • Ground
  • Content
  • Logic of ground
  • Truthmaking