¹ I have many people to thank because the history of this paper goes far back. In my talks ‘Foundation and Paradox’ in May 2015 (Logica 2015, Hejnice) and ‘Metaphysical Foundationalism and the Paradoxes of Non-Well-Foundedness’ in September 2015 (Congress on Analytical Philosophy, Osnabrück), I presented the results about the semantic and set theoretical paradoxes as puzzles of grounding that are now recounted in sections 4 and 5. In January 2016, I completed a first draft of the full paper, which I then circulated among a small number of people and later used as the basis of my talk ‘Paradox and the Structure of Explanation’ in May 2019 (ExLog2019, Louvain-la-Neuve). I am grateful to all these audiences for many helpful comments, as well as to the following individuals: Daniel Boyd, Scott Dixon, Kit Fine, Stephan Krämer, Tobias Martin, Peter Milne, Chris Scambler, Niko Strobach, Graham Priest, and Yale Weiss.

² Another important source of our model-theoretic approach to truth is Gödel's first incompleteness theorem. Although not itself paradoxical, it is still close enough to the Liar paradox. For if – per impossibile – arithmetic were complete (in a semantic sense), then provability would be equivalent to truth, and a Gödel sentence would not only claim its own unprovability in an indirect way, but also claim its own untruth, albeit in an even more mediated way.

³ Cf., e.g.: Salmon, Wesley C.: Zeno's Paradoxes (Indianapolis: Hackett, 2001)Google Scholar.

⁴ Fine, Kit: ‘Some Puzzles of Ground’, Notre Dame Journal of Formal Logic 51 (2010), 97–118CrossRefGoogle Scholar; Krämer, Stephan: ‘A Simpler Puzzle of Ground’, Thought 2 (2013), 85–59Google Scholar; Korbmacher, Johannes: ‘Yet Another Puzzle of Ground’, Kriterion – Journal of Philosophy 29.2 (2015), 1–10Google Scholar; see Stephan Krämer: ‘Logical Puzzles of Ground’, to appear in: Mike Raven (ed.): Routledge Handbook of Metaphysical Grounding [forthcoming]. Johannes Korbmacher studies a paradox of self-reference that does employ the grounding predicate. However, it makes crucial use of the factive aspect of the notion of grounding in question (as has been observed by Stephan Krämer), and should therefore be seen as a variant of the Liar paradox, not a genuine paradox of grounding.

⁵ For laudable exceptions, cf. Woods, Jack: ‘Emptying a Paradox of Ground’, Journal of Philosophical Logic 47.4 (2018), 631–648CrossRefGoogle Scholar and (in a way) Litland, Jon Erving: ‘Grounding, Explanation, and the Limit of Internality’, Philosophical Review 124.4 (2015), 481–532CrossRefGoogle Scholar. In a number of presentations under the title ‘Ground and Paradox’, Boris Kment has reported work that is probably closest to the project of the present paper. As far as I know (from one of these presentations), the main overlap concerns the diagnosis that the semantic and set theoretic paradoxes give rise to further puzzles of ground (cf. section 5). From this shared observation, however, Kment appears to draw quite different conclusions. Cf. my methodological considerations in section 11.

⁶ Cameron, Ross P.: ‘Turtles All the Way Down: Regress, Priority and Fundamentality’, The Philosophical Quarterly 58 (2008), 1–14Google Scholar; Schaffer, Jonathan: ‘Monism: The Priority of the Whole’, Philosophical Review 119.1 (2010), 37CrossRefGoogle Scholar; Paseau, Alexander: ‘Defining Ultimate Ontological Basis and the Fundamental Layer’, The Philosophical Quarterly 60 (2010), 169–175CrossRefGoogle Scholar; Bliss, Ricki: ‘Viciousness and the Structure of Reality’, Philosophical Studies 166.2 (2013), 399–418CrossRefGoogle Scholar; Tahko, Tuomas: ‘Boring Infinite Descent’, Metaphilosophy 45.2 (2014), 257–269CrossRefGoogle Scholar; Gabriel Oak Rabin / Brian Rabern: ‘Well Founding Grounding Grounding’, Journal of Philosophical Logic (2015), 1–31; Jon Erving Litland: ‘An Infinitely Descending Chain of Ground Without a Lower Bound’, Philosophical Studies (2015), 1–9; and T. Scott Dixon: ‘What Is the Well-Foundedness of Grounding?’, Mind, doi:10.1093/ mind/fzv112 (first published online: February 28, 2016).

⁷ Kit Fine: ‘Guide to Ground’, in: Correia, Fabrice / Schneider, Benjamin (eds.): Metaphysical Grounding: Understanding the Structure of Reality (Cambridge: Cambridge University Press, 2012), 37–80CrossRefGoogle Scholar.

⁸ Correia, Fabrice / Schneider, Benjamin: ‘Grounding: An Opinionated Introduction’, in: Correia, Fabrice / Schneider, Benjamin (eds.): Metaphysical Grounding: Understanding the Structure of Reality (Cambridge: Cambridge University Press, 2012), 1CrossRefGoogle Scholar.

⁹ Ricki Bliss / Kelly Trogdon: ‘Metaphysical Grounding’, in: Edward N. Zalta (ed.): The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), URL = http://plato.stanford.edu/archives/win2014/entries/grounding/.

¹⁰ Op. cit. note 8, 3; my emphasis.

¹¹ Op. cit. note 8, 10–12.

¹² ‘<’ is a sentential operator of the object language, in contrast to the logic of ground of Fine's ‘Guide to Ground’ (op. cit. note 7), where it is a meta language symbol, in the same way as the turnstile ‘⊨’.

¹³ One might be tempted to see ‘x predicationally grounds y’ just as a paraphrase of the converse ‘y depends ontologically on x’. But the relation between grounding and ontological dependence is more complicated than that; cf., e.g., Benjamin Schneider: ‘Grounding and Dependence’, Synthese, doi: 10.1007/s11229-017-1378-z (first published online: May 18, 2017).

¹⁴ Here my terminology is not standard. Others talk of an ‘operation’ (e.g., Litland in op. cit note 5) or ‘operationalism’ (T. Scott Dixon: ‘Upward Grounding’, Philosophy and Phenomenological Research (2017), doi: 10.1111/phpr.12366). But this might be misleading because what the sentential operator of grounding expresses has nothing to do with either the set theoretical notion of an operation as a certain kind of mapping nor the everyday notion of an operation as a certain kind of action.

¹⁵ Cf. Schaffer op. cit. note 6, 35; going back to Aristotle's Categories, i.e., Aristotle, : ‘Categories’, in: Barnes, Jonathan (ed.): The Complete Works of Aristotle. Vol. 1 (Princeton: Princeton University Press, 1984), 22Google Scholar.

¹⁶ Schaffer op. cit. note 6, 35; Paseau op. cit. note 6, 169; and Fine, Kit: ‘Essence and Modality’, Philosophical Perspectives 8 (1994), 4–5CrossRefGoogle Scholar.

¹⁷ In order to do justice to everyone I should add that with regard to predicational grounding, there is no consensus about which sorts of objects it can relate. While Jonathan Schaffer does not appear to accept any restriction (crucially, he discusses whether a physical object is grounded in its parts or vice versa; Schaffer op. cit. note 6), others hold that the relata of a claim of predicational grounding must be facts or propositions, so that the only substantial difference to the corresponding claim of operational grounding would be grammatical. In some places I will also talk as if I made no restrictions for the possible relata of predicational grounding. But this way of speaking can be made acceptable to everyone, because whoever is unhappy with understanding this as a primitive claim is welcome to analyze it (away) in terms of operational grounding and a (non-trivial) existence predicate, according to the following definition of a suitably wide notion of predicational grounding: x ◃ y ⇔_defE!(x) < E!(y). I would like to thank Stephan Krämer for convincing me of both the importance of this problem and the viability of a solution in terms of existence.

¹⁸ In the next three paragraphs I use the distinctions of Fine op. cit. note 7, 46–54 to set out a spectrum of options, but I follow my own preferences in picking from them.

¹⁹ From here on, I will speak of grounding only in terms of a relation that holds among objects – taking the above table to establish that whatever structural properties we require of predicational grounding can be translated into corresponding requirements on operational grounding.

²⁰ Jenkins, Carrie S.: ‘Is Metaphysical Dependence Irreflexive?’, The Monist 94.2 (2011), 267–276CrossRefGoogle Scholar; Rodriguez-Pereyra, Gonzalo: ‘Grounding is Not a Strict Order’, Journal of the American Philosophical Association 1.3 (2015), 517–534CrossRefGoogle Scholar; Woods op. cit. note 5.

²¹ E.g., cf. again Rabin / Rabern op. cit. note 6.

²² See Krämer op. cit. note 4 for an up-to-date overview.

²³ Fine op. cit. note 4.

²⁴ Fine op. cit. note 4, 98.

²⁵ Fine op. cit. note 4, 97.

²⁶ Cf. the seminal paper by David Lewis, with its praise of consistent time travel stories by Robert Heinlein. Lewis, David: ‘The Paradoxes of Time Travel’, American Philosophical Quarterly 13.2 (1976), 145–152Google Scholar; Heinlein, Robert A.: ‘By His Bootstraps’, in: Heinlein, Robert A.: The Menace from Earth (New York: Signet, 1959[1941]), 39–88Google Scholar; and Heinlein, Robert A.: ‘“– All You Zombies –”’, in: Heinlein, Robert A.: The Unpleasant Profession of Jonathan Hoag (New York: Ace, 1959), 138–151Google Scholar. Today, we might want to add the Doctor Who episode ‘Blink’ as another example for a well-crafted Predestination paradox.

²⁷ Eldridge-Smith, Peter: ‘Paradoxes and Hypodoxes of Time Travel’, in: Jones, J. Lloyd / Campbell, P. / Wylie, P. (eds.): Art and Time (Melbourne: Australian Scholarly Publishing, 2007), 172–189Google Scholar.

²⁸ Cf., e.g.: Schaffer, Jonathan: ‘Grounding in the Image of Causation’, Philosophical Studies 173.1 (2016), 49–100CrossRefGoogle Scholar.

²⁹ Heinlein op. cit. note 26.

³⁰ For a fuller version of this argument for the claim that every instance of Curry's paradox is paradoxical, cf. Martin Pleitz: ‘Curry's Paradox and the Inclosure Schema’, in: Pavel Arazim / Michal Dančák (eds.): The Logica Yearbook 2014 (London: College Publications, 2015), section 4.

³¹ Carroll, Lewis: ‘What the Tortoise Said to Achilles’, Mind 4.14 (1895), 278–280CrossRefGoogle Scholar.

³² Armstrong, David M.: A World of States of Affairs (Cambridge: Cambridge University Press, 1997), 157–158CrossRefGoogle Scholar; cf. Cameron op. cit. note 6, 1–3.

³³ Here we skid over technical details regarding the different adicity and (perhaps) different level of the various notions of instantiation involved.

³⁴ Dixon op. cit. note 14.

³⁵ Note the similar use of ‘upward’ and ‘downward’ in Dixon op. cit. note 14 – despite the slightly different view about foundationality in Dixon op. cit. note 6.

³⁶ The turtle at ground zero might dissent, and complain (with more justification than Carroll's Tortoise) that it feels flattened by the weight on its back.

³⁷ As mentioned earlier, some of the work reported here has been done independently by Boris Kment.

³⁸ Recall that we work with a non-factive notion of grounding.

³⁹ The principle (Grounding and ¬¬-intro) is endorsed in Fine op. cit. note 7, 63. It may well be contested by those who think so classically about negation that they believe that in a logical context, ¬¬ϕ just is ϕ. Thanks go to Peter Milne for raising this point. However, as the phrase ‘just is’ indicates, in that case a stronger principle than (G¬¬) would hold, stating some kind of grounding equivalence between ¬¬ϕ and ϕ, thus licensing the intersubstitutability of ¬¬ϕ and ϕ in grounding contexts and providing an alternative way of deriving line (4) of the reasoning above.

⁴⁰ We use underlining as an operator that forms a term that refers to the underlined expression. For our reasoning we assume that this sort of term is robust in the sense that we can substitute one such term for another such term if they are co-extensional. In particular, to derive line (1) and (2) from (GT⁺) and (GT⁻), respectively, we assume that the fact that λ = ¬True(λ) licenses that we substitute ‘λ’ for ‘¬True(λ)’. As the grounding operator ‘<’ creates a hyperintensional context, this is a substantial assumption. I would like to thank Stephan Krämer for alerting me to this.

⁴¹ Paseau op. cit. note 6, 169; cf. Fine op. cit. note 16, 4–5.

⁴² Because of the irreflexivity of elementhood, every set would satisfy the condition that x ∉ x. Thus if the Russell set existed, it would be the Universal set. But in this scenario, there cannot be a Universal set, again because of the requirement of irreflexivity.

⁴³ Yablo, Stephen: ‘Paradox without Self-Reference’, Analysis 53 (1993), 251–252CrossRefGoogle Scholar; Goldstein, Laurence: ‘A Yabloesque Paradox in Set Theory’, Analysis 54.4 (1994), 223–227CrossRefGoogle Scholar; cf. Roy T. Cook: The Yablo Paradox. An Essay on Circularity (Oxford: Oxford University Press, 2014)Google Scholar.

⁴⁴ Here the symbol ‘>’ has it usual meaning ‘is greater than’, and has nothing to do with grounding.

⁴⁵ The symbol ‘▹’ is of course the converse of ‘◃’, and can be read as ‘… is grounded in …’.

⁴⁶ Recall that we work with a non-factive notion of grounding.

⁴⁷ Priest, Graham: ‘The Structure of the Paradoxes of Self Reference’, Mind 103(409) (1994), 25–34CrossRefGoogle Scholar.

⁴⁸ Kripke, Saul: ‘Outline of a Theory of Truth’, Journal of Philosophy 72 (1975), 690–716CrossRefGoogle Scholar; Maudlin, Tim: Truth and Paradox: Solving the Riddles (Oxford: Oxford University Press, 2004)CrossRefGoogle Scholar.

⁴⁹ Strengthening this condition to ‘a < b’ would lead to a gunky model.

⁵⁰ Schaffer op. cit. note 6.

⁵¹ Rabin / Rabern op. cit. note 6; Litland op. cit. note 6; Dixon op. cit. note 6.

⁵² Given a grounding structure (O, ◃), a subset C of O is a chain if and only if for all x, y ∈C either x ◃ y or x = y or y ◃ x. A chain C is downward maximal if and only if there is no z ∈ O with z ∉ C such that z ◃ x for every x ∈ C (cf. Dixon op. cit. note 6, 15).

⁵³ Here, again, the symbol ‘>’ has it usual meaning ‘is greater than’.

⁵⁴ I give the proof of the paradoxicality of this ω + 1-sequence, because as far as I know it has not yet been considered.

⁵⁵ Adapted from Dixon op. cit. note 6, 11.

⁵⁶ Here I re-use a term from Woods op. cit. note 5 with a somewhat different sense.

⁵⁷ Friends of predicational grounding can phrase this condition entirely without operational grounding if they have a fact-term-forming operator ‘[…]’ and allow for conjunctive facts, in the following way: ‘[x ◃ y ∧ y ◃ z] ◃ [x ◃ z]’.

⁵⁸ This can be made more precise in graph theoretical terms.

⁵⁹ Cf., however, the methodological reflection in section 11.

⁶⁰ It will also allow for the converse of (Holism), according to which the whole is grounded in its parts.

⁶¹ To those who ask how there could be results in both directions, here is a short answer: Reflective equilibrium. Methodologically, we need not dig deeper on this occasion, because we shall be quite cautious about how much we can learn from ground for paradox.

⁶² Priest op. cit. note 47.

⁶³ I will admit that I have felt the allure of the overly optimistic reaction myself at one point. And from how I understand it, the proposal that Boris Kment is developing on the basis of the observation that the paradoxes are puzzles of ground is in the same vicinity.

⁶⁴ For a fuller version of this consideration about reductio and paradox, cf. Martin Pleitz: Logic, Language, and the Liar Paradox (Münster: Mentis, 2018), 242–244.

⁶⁵ One promising way to go would be too argue for a uniform non-existence solution, according to which the purported paradoxical objects (hypersets, self-referential expressions, Yablo sequences . . . ) do not exist. For this, however, we would need independent justification. With regard to the semantic paradoxes, I have made a detailed proposal in my Logic, Language, and the Liar Paradox; op. cit. note 64, 255–629.

⁶⁶ If we were inclined to increase the rhetoric, we might vary Hilbert's famous slogan about knowledge in mathematics and proclaim about paradox: ‘We must solve! We will solve!’

⁶⁷ Cf., e.g., Leitgeb, Hannes: ‘What Theories of Truth Should be Like (but Cannot be)’, Philosophy Compass 2.2 (2007), 276–290CrossRefGoogle Scholar, with its telling title.

⁶⁸ In a similar vein, I would see the examples given in Dixon op. cit. note 6 and Litland op. cit. note 5 and op. cit. note 6 for circular or infinitely descending grounding chains not as casting doubt on intuitive structural claims about grounding, but as posing further puzzles that call for a solution. In the case of the infinite disjunctions discussed by Dixon, this would likely concern which infinitary extensions of our languages are legitimate, and for the examples discussed by Litland, which make essential use of truth ascriptions, it would likely need to be in line with the correct solution to the Liar paradox – whatever that might turn out to be.