We show that any predicational theory of partial ground that
          extends a standard theory of syntax and that proves some
          commonly accepted principles for partial ground is inconsistent.
          We suggest a way to obtain a consistent predicational theory of
          ground

Korbmacher, Johannes

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We show that any predicational theory of partial ground that extends a standard theory of syntax and that proves some commonly accepted principles for partial ground is inconsistent. We suggest a way to obtain a consistent predicational theory of ground

Korbmacher, J.

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Yet Another Puzzle of Ground

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Yet Another Puzzle of Ground∗Johannes KorbmacherAbstractWe show that any predicational theory of partial ground thatextends a standard theory of syntax and that proves somecommonly accepted principles for partial ground is inconsistent.We suggest a way to obtain a consistent predicational theory ofground.Keywords: grounding, standard theory of groundFine defines ground as “the relation of one truth holding in virtue ofothers” [7, p. 1].1 Given this definition, it is natural to think that weshould formulate axiomatic first-order theories of ground, which formal-ize ground by means of a relational ground predicate of true sentences.Call such theories predicational theories of ground. Predicational theoriesof ground contrast with operational theories of ground, which formalizeground by means of a sentential ground operator [3, p. 253–54, 5, p. 46–47]. So far, most theories of ground in the literature are operationaltheories of ground. But there are at least three theoretical reasons fordeveloping predicational theories of ground:1. Quantification: Predicational theories of ground have greater ex-pressive strength than operational theories of ground. In partic-ular, using a ground predicate, we can formalize ground-theoreticprinciples involving quantification over truths in a natural way.Take, for example, the intuitively plausible claim that every truthis either fundamental or grounded in some other truths. We canstraightforwardly formalize this claim using a ground predicate andfirst-order quantification over truths, but using a ground operatorthis is impossible. Without the use of non-classical devices, such∗This paper won the SOPhiA 2015 Best Paper Award at the Salzburg Conferencefor Young Analytic Philosophy, September 2–4, 2015.Kriterion – Journal of Philosophy, 2015, 29(2): 1–10.http://www.kriterion-journal-of-philosophy.org c© 2015 The author2 KRITERION – Journal of Philosophy, 2015, 29(2): 1–10as propositional quantification, it is impossible to formalize thenested universal and existential quantification over truths in theprinciple. Using a ground predicate, in contrast, we can formal-ize the principle comfortably in the purview of classical first-orderlogic.2. Truth and Modality : Predicational theories of ground allow us tostudy ground in the same context as truth and modality. It is gen-erally accepted that truth should be treated as a predicate of sen-tences, and it has recently been suggested to extend this approachto modality as well [10, 13, 8]. There is an obvious connection be-tween ground and truth, since ground is a relation among truths.But there is also a close connection between ground and modality,since ground is usually assumed to imply necessary consequence: ifa truth holds in virtue of some other truths, then the former truthshould be a necessary consequence of the latter truths [5, p. 38–39].Both of these connections are most naturally studied using pred-icational theories of ground: by combining predicational theoriesof ground with predicational theories of truth and modality.3. Models: Predicational theories of ground allow us to discover andto study models of ground using classic model-theoretic methods.It is currently an open problem to provide a semantics for theimpure logic of ground developed by Fine [5, p. 58–71]. This logicis formulated using a ground operator, but once we translate itinto a predicational theory of ground and show its consistency, wecan rely on model-theoretic theorems to establish the existence offirst-order models. Once we know that such models exist, we canstudy them using methods of model theory. This should provide uswith new insights into the semantics of the impure logic of ground.But predicational theories of ground face a paradox of self-reference,similar to the well-known paradoxes of self-reference that arise in predi-cational theories of truth and modality. In this paper, I shall prove thispoint for predicational theories of partial ground in particular. This isthe relation of one truth holding partially in virtue of another truth—the relation of one truth “helping” to ground another truth [5, p. 50].I show that any predicational theory of partial ground that extends astandard theory of syntax and that proves some commonly acceptedprinciples for partial ground is inconsistent. Fine [6] and Kra¨mer [11]present puzzles about the irreflexivity of partial ground: the principlethat no truth partly grounds itself. They show that certain intuitivelyJohannes Korbmacher: Yet Another Puzzle of Ground 3plausible principles of logic and metaphysics lead to counterexamples tothe irreflexivity of ground. I add yet another puzzle of ground to themix. The new puzzle does not mention the irreflexivity of ground ormetaphysical principles unrelated to ground, thus it is genuinely differ-ent from the previously known paradoxes.To formulate a predicational theory of partial ground, we first need atheory of syntax that allows us to talk about sentences.2 It is well-knownthat we can develop such a theory in any sufficiently strong backgroundtheory, like Robinson arithmetic for example. For the present purpose,however, our concrete choice of background theory does not matter. Allthat matters is that our background theory Θ satisfies the following threeminimal syntax conditions:3 The first condition is that Θ proves thatwe have a unique name pϕq for every sentence ϕ in the sense that for allsentences ϕ and ψ, Θ ` pϕq = pψq only if ϕ = ψ. The second conditionis that Θ proves that we have a function symbol ∨ that represents thesyntactic operation ∨ of disjunction in the sense that for all sentences ϕand ψ, Θ ` pϕq ∨ pψq = pϕ ∨ ψq. And the third condition is that Θproves the diagonal lemma. Informally, this lemma states that for everycondition on sentences there is a sentence that is provably equivalentto the condition holding of itself. More precisely, if ϕ(x) is a formulawith exactly one free variable, then there exists a sentence δ such thatΘ ` δ ↔ ϕ(pδq). Note that any standard background theory of syntax,such as Robinson arithmetic, satisfies all three of our minimal syntaxconditions.Next, we need a way of representing partial ground. For this purpose,we use the relational predicate x/y. For sentences ϕ and ψ, we informallyread the atomic formula pϕq/pψq as saying that the truth of ϕ partiallygrounds the truth of ψ. For a negated atomic formula of the form ¬(pϕq/pψq) we also write pϕq 6 pψq, which we correspondingly read as sayingthat the truth of ϕ does not even partially ground the truth of ψ.Philosophers have laid down various principles for partial ground [cf.17, 5, 7], but it is already sufficient for a predicational theory of partialground to be inconsistent that it proves two widely accepted principles.Let Θ now be a predicational theory of partial ground that satisfiesthe minimal syntax conditions. The first of our two principles followsdirectly from partial ground being a relation of true sentences: If thetruth of one sentence partially grounds the truth of another, then bothsentences should be true. This principle is known as the “factivity ofground” and is generally accepted in the literature on ground [6, p. 100,1, § 3].4 We get the condition on Θ that for all sentences ϕ and ψ:4 KRITERION – Journal of Philosophy, 2015, 29(2): 1–10(FactL): Θ ` pϕq / pψq→ ϕ(FactR): Θ ` pϕq / pψq→ ψThe second principle concerns the interaction of partial ground and dis-junction: Given that partial ground is the relation of one truth holdingpartially in virtue of another, if a disjunction is true, then its truth shouldbe partially grounded in each of its true disjuncts. Also this principleis generally accepted in the literature on ground [6, p. 101, 17, p. 117].From this, we get the condition on Θ that for all sentences ϕ and ψ:(∨1): Θ ` ϕ→ pϕq / pϕ ∨ ψq(∨2): Θ ` ψ → pψq / pϕ ∨ ψqThe minimal syntax conditions and the conditions concerning partialground all may seem fairly uncontroversial when viewed individually. Soit may be somewhat surprising to learn that there can be no consistentpredicational theory of partial ground that satisfies all of them:Theorem (Inconsistency Theorem). Any theory Θ that satisfies theminimal syntax conditions, (FactL/R), and (∨1/2) is inconsistent.Proof. Let ϕ(x) be the formula x 6 x ∨ x. By the diagonal lemma,there is a sentence δ such that Θ ` δ ↔ pδq 6 pδq ∨ pδq. Intuitively,this is a sentence which “says of itself” that it does not partially groundits own disjunction. By the second minimality condition, we have thatΘ ` pδq ∨ pδq = pδ ∨ δq. From this and Θ ` δ ↔ pδq 6 pδq ∨ pδq, weget that Θ ` δ ↔ pδq 6 pδ ∨ δq by the substitutivity of identicals. Thissplits up into the following two conditions:(a) Θ ` δ → pδq 6 pδ ∨ δq(b) Θ ` pδq 6 pδ ∨ δq→ δWe get finally the following argument:1. Θ ` (pδq / pδ ∨ δq → δ) → ((δ → pδq 6 pδ ∨ δq) → pδq 6 pδ ∨ δq)(Tautology†)2. Θ ` pδq / pδ ∨ δq→ δ (FactL)3. Θ ` (δ → pδq 6 pδ ∨ δq) → pδq 6 pδ ∨ δq (1, 2: MP)4. Θ ` δ → pδq 6 pδ ∨ δq (a)5. Θ ` pδq 6 pδ ∨ δq (3, 4: MP)6. Θ ` pδq 6 pδ ∨ δq→ δ (b)7. Θ ` δ (5, 6: MP)Johannes Korbmacher: Yet Another Puzzle of Ground 58. Θ ` δ → pδq / pδ ∨ δq (∨1)9. Θ ` pδq / pδ ∨ δq (7,8: MP)10. Θ ` ⊥ (5,9: ⊥)(†) : Note that every sentence of the form (ϕ→ ψ) → ((ψ → ¬ϕ) → ¬ϕ)is a classical tautology and that theories prove all classical tautologies.The inconsistency theorem is very similar to Tarski’s theorem aboutpredicational theories of truth [18] and Montague’s theorem about pred-icational theories of modality [15] in that it is, essentially, a paradox ofself-reference. From a technical perspective, it should in fact not be sur-prising that we get such a theorem after all: Combining self-reference viathe diagonal lemma with principles like (FactL/R) that allow us to pusha sentence outside the scope of a predicate and principles like (∨1/2) thatallow us to push a sentence into the scope of a predicate is recipe fordisaster.5 But from a philosophical perspective, there is a lesson to belearned: We already know that we cannot understand ground simply interms of truth and modality [5], but ground behaves syntactically toomuch like a combination of truth and modality to escape inconsistencywhen paired with self-reference.Three natural ways in which we could try to block the inconsis-tency theorem suggest themselves: First, we could try to rule out self-referential sentences of ground like the one used in the proof of theinconsistency theorem. Second, we could try to restrict the principlesof partial ground used in the proof of the inconsistency theorem. Andthird, we could try to formulate a non-standard logic of ground thatdoes not sanction the logical principles used in the proof of the incon-sistency theorem. The analogy between the inconsistency theorem andthe theorems of Tarski and Montague suggests a terminology for theseapproaches. Analogously to predicational theories of truth [9] and predi-cational theories of modality [8], we get: typed theories of partial ground,which avoid paradox by putting type-restrictions on the relation of par-tial ground, effectively ruling out self-referential sentences like the onein the proof; untyped theories of partial ground, which avoid paradoxby restricting the principles of partial ground; and finally non-classicaltheories of partial ground, which avoid paradox (or: triviality) by aban-doning classical logic in favor of alternative logics.Untyped theories of partial ground are particularly appealing, be-cause considerations of ground are already part of intuitively appealingapproach to untyped theories of truth. On an influential view about6 KRITERION – Journal of Philosophy, 2015, 29(2): 1–10predicational theories of truth, self-referential sentences are ungroundedand this is the reason some self-referential sentences lead to inconsistency[12, 14]. This leads to the idea that we should restrict the principles oftruth to their grounded instances.6 Carrying this idea from theories oftruth over to predicational theories of ground, we arrive at the condi-tion that the principles of partial ground apply if and only if the truthsinvolved are themselves grounded. There is a straightforward way offormulating the desired restriction on the principles of partial groundalready in the language of partial ground. We can express that a sen-tence ϕ is grounded by the formula ∃x(x / pϕq) and we can express thata sentence ϕ is ungrounded by the formula ¬∃x(x / pϕq). The desiredrestriction on (∨1/2) then amounts to saying that for all predicationaltheories of ground Θ and for all sentences ϕ and ψ:(∨∗1): Θ ` ∃x(x / pϕq) ↔ pϕq / pϕ ∨ ψq(∨∗2): Θ ` ∃x(x / pψq) ↔ pψq / pϕ ∨ ψqEvery predicational theory of partial ground that satisfies the minimalsyntax conditions, (FactL/R), and the new conditions (∨∗1/2), proves thatthe paradoxical sentence in the proof of the theorem is ungrounded:Observation. Let Θ be a predicational theory of ground that satisfiesthe minimal syntax conditions, (FactL/R), and (∨∗1/2). By the diagonallemma and the same reasoning as in the proof of the theorem, we get asentence δ such that:Θ ` δ ↔ pδq 6 pδ ∨ δq.But we can show that:Θ ` ¬∃x(x / pδq).Proof. By applying (∨∗1) to δ, we get that:Θ ` ∃x(x / pδq) ↔ pδq / pδ ∨ δqWe only need the “left-to-right direction” of this biconditional for ourproof, which we can obtain via ↔-Elimination:Θ ` ∃x(x / pδq) → pδq / pδ ∨ δqStarting from there, we get the following argument:1. Θ ` ∃x(x / pδq) → pδq / pδ ∨ δq2. Θ ` pδq / pδ ∨ δq→ δ (FactL)Johannes Korbmacher: Yet Another Puzzle of Ground 73. Θ ` ∃x(x / pδq) → δ (1,2: MP)4. Θ ` δ ↔ pδq 6 pδ ∨ δq (Diagonal Lemma)5. Θ ` ∃x(x / pδq) → pδq 6 pδ ∨ δq (3, 4: ↔-Elim)6. Θ ` ∃x(x / pδq) → ⊥ (1, 5: ⊥-Intro)7. Θ ` ¬∃x(x / pδq) (6: ¬-Intro)This result should make us optimistic about the prospects for a untypedtheory of partial ground. Moreover, we could add such a untyped theoryof partial ground “on top” of untyped predicational theories of truth andmodality. I conjecture that an interesting, consistent, untyped predica-tional theory of ground can be developed in this way.AcknowledgementsI would like to thank Albert J. J. Anglberger, Hannes Leitgeb, ThomasSchindler, and Ole Thomassen Hjortland for helpful comments and sug-gestions.Notes1 For (opinionated) introductions to ground, see [4, 5]. For an overview of therecent literature on ground, see [2, 19, 16].2 A sentence is a formula without any free variables.3 A theory is a set of formulas that is closed under derivability: a set of formulasΘ is a theory iff (if and only if) for all formulas ϕ, if Θ ` ϕ, then ϕ ∈ Θ.4 There are notions of ground in the literature that violate the factivity of ground[cf. 5, p. 48–50]. According to such non-factive notions, ground is a relation onsentences regardless of their truth value. Although non-factive notions of groundmake for an interesting theoretical possibility, in this paper we shall deal onlywith the standard factive notion of ground, which satisfies the factivity of ground.5 It should be clear at this point that not much depends on the concrete condition(∨1/2)—the paradox is not a paradox of disjunction. All that matters is thatour predicational theory of partial ground proves a principle to the effect thatany true sentence partially grounds some other sentence. We could give thefollowing variant of the inconsistency theorem: If Θ satisfies (FactL/R) and eitherΘ ` ϕ→ ∃x(pϕq / x) or Θ ` ϕ→ ∃x(x / pϕq), then Θ is inconsistent. I leave thedetails of the proof to the interested reader.6 The concept of ground used in the context of theories of truth is not exactly thesame as the concept of ground discussed in this paper. For example, the notionof dependence defined by Leitgeb [14] is reflexive, whereas (partial) ground isstandardly taken to be irreflexive. The point here is that there is a striking8 KRITERION – Journal of Philosophy, 2015, 29(2): 1–10analogy between the two concepts and that ideas that work for the one may verywell work for the other.Johannes KorbmacherMCMP, LMU MunichGeschwister-Scholl-Platz 180539 Munich, Germany<jkorbmacher@gmail.com><http://jkorbmacher.com/>References[1] Ricki Bliss and Kelly Trogdon. “Metaphysical Grounding”. In: TheStanford Encyclopedia of Philosophy. Ed. by Edward N. Zalta.Winter 2014. 2014.[2] Michael Clark and David Liggins. “Recent Work on Grounding”.In: Analysis 72.4 (2012), pp. 812–23.[3] Fabrice Correia. “Grounding and Truth-Functions”. In: Logique etAnalyse 53.211 (2010), pp. 251–79.[4] Fabrice Correia and Benjamin Schnieder. “Grounding. An Opini-ated Introduction”. In: Metaphysical Grounding: Understandingthe Structure of Reality. Ed. by Fabrice Correia and BenjaminSchnieder. Cambridge, UK: Cambridge University Press, 2012,pp. 1–36.[5] Kit Fine. “Guide to Ground”. In: Metaphysical Grounding: Un-derstanding the Structure of Reality. Ed. by Fabrice Correia andBenjamin Schnieder. Cambridge, UK: Cambridge University Press,2012, pp. 37–80.[6] Kit Fine. “Some Puzzles of Ground”. In: Notre Dame Journal ofFormal Logic 51.1 (2010), pp. 97–118.[7] Kit Fine. “The Pure Logic of Ground”. In: Review of SymbolicLogic 5.1 (2012), pp. 1–25.Johannes Korbmacher: Yet Another Puzzle of Ground 9[8] V. Halbach and P. Welch. “Necessities and Necessary Truths: AProlegomenon to the Use of Modal Logic in the Analysis of Inten-sional Notions”. In: Mind 118.469 (2009), pp. 71–100.[9] Volker Halbach. Axiomatic Theories of Truth. Cambridge, UK:Cambridge University Press, 2011.[10] Volker Halbach, Hannes Leitgeb, and Philip Welch. “Possible-Worlds Semantics for Modal Notions Conceived as Predicates”.In: Journal of Philosophical Logic 32.2 (2003), pp. 179–223.[11] Stephan Kra¨mer. “A Simpler Puzzle of Ground”. In: Thought: AJournal of Philosophy 2.2 (2013), pp. 85–89.[12] Saul A. Kripke. “Outline of a Theory of Truth”. In: Journal ofPhilosophy 72.19 (1975), pp. 690–716.[13] Hannes Leitgeb. “Towards a Logic Of Type-Free Modality andTruth”. In: Logic Colloquium 2005: Proceedings of the Annual Eu-ropean Summer Meeting of the Association for Symbolic Logic,Held in Athens, Greece, July 28-August 3, 2005. Ed. by CostasDimitracopoulos. Cambridge University Press, 2008, pp. 28–68.[14] Hannes Leitgeb. “What Truth Depends On”. In: Journal of Philo-sophical Logic 34.2 (2005), pp. 155–92.[15] Richard Montague. “Syntactical Treatment of Modality, withCorollaries on Reflection Principles and Finite Axiomatizability”.In: Acta Philosophica Fennica 16 (1963), pp. 153–67.[16] Michael J. Raven. “Ground”. In: Philosophy Compass 10.5 (2015),pp. 322–333.[17] Gideon Rosen. “Metaphysical Dependence: Grounding and Re-duction”. In: Modality: Metaphysics, Logic, and Epistemology. Ed.by Bob Hale and Aviv Hoffmann. Oxford University Press, 2010,pp. 109–36.[18] Alfred Tarski. “Der Wahrheitsbegriff in den FormalisiertenSprachen”. In: Studia Philosophica 1 (1936), pp. 261–405.[19] Kelly Trogdon. “An Introduction to Grounding”. In: Varieties ofDependence. Ontological Dependence, Grounding, Supervenience,Repsonse-Dependence. Ed. by Miguel Holtje, Benjamin Schnieder,and Alex Steinberg. Mu¨nchen: Philosophia Verlag, 2013, pp. 97–122.

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