Benjamin Schnieder

It is often assumed there are two sorts of mathematical proofs: some only show that a theorem holds, while others show why it holds. Yet, some argue mathematical explanation is impossible, citing (i) explanatory realism—explanations rely on objective dependencies—and (ii) the nature of mathematical facts, which allegedly lack such dependencies. We rebut this argument, then show that explanatory realism combined with grounding and essence allows for mathematical explanation. Finally, we argue that the real issue touches scope: grounding logics exclude many explanatory proofs. Our paper thus defends mathematical explanation’s possibility and calls for better alignment with actual practice.

Contemporary ontologies often assign states-of-affairs a central role in explaining semantic content, causation, and truth. This paper examines how Bernard Bolzano’s ontology—built around substances, adherences, and propositions in themselves—successfully fulfils these explanatory roles without invoking states-of-affairs, conceived as concrete complexes of objects and universals. By closely analyzing Bolzano’s original texts, including key aspects previously overlooked, the paper reconstructs his alternative machinery and highlights its ongoing systematic relevance. It also revisits the wrong historical narrative that credits Bolzano with pioneering the notion of states-of-affairs, arguing that such attributions rest on a misreading of his ontology. Reappraising Bolzano’s rigorous approach not only corrects the historical record but also offers fresh insight into contemporary debates on ontology, grounding, and truth.

This paper reassesses Kant's early treatment of the Principle of Sufficient Reason (PSR) in the Nova dilucidatio, situating it historically and in contemporary grounding debates. First, I lay out Kant's conceptual framework—his definition of a sufficient reason, and his distinction between antecedently and consequently determining reasons—relating it to Wolff and Crusius and to contemporary grounding theory. I argue that the framework yields valuable insights but also harbours instructive confusions. Second, I reconstruct Kant's distinction between two versions of the PSR and two of his proofs—the arguments from indeterminacy and from exclusion. I show that the arguments fail, chiefly by conflating epistemic with objective notions of reasons and by sliding from entailment to grounding. Third, I underscore Kant's emphasis on the groundedness of truth while separating his Aristotle‐inspired insight—that p is true because p—from any full PSR, and I use this distinction to resist a recent defence of the PSR by Della Rocca.

This chapter contains translations of texts in which Bolzano develops and applies his mature theory of grounding. The heart of the chapter are excerpts from Bolzano’s main work _Theory of Science_, where he gives the most comprehensive account of his conception of grounding and discusses the role of grounding in several philosophical areas ranging from the methodology of science, to the philosophy of causation, to metaphysics. In addition to that, the chapter includes a brief excerpt from a survey of Bolzano’s _Theory of Science_ and his _Theory of the Science of Religion_, in which Bolzano gives a succinct summary of central ideas of his mature theory. Finally, the chapter includes the complete translation of Bolzano’s essay _What Is Philosophy?_ which defends a ground-theoretical conception of philosophy.

This chapter contains translations of excerpts from three early texts by Bolzano (written around 1810): ▪ the 1810 booklet _Contributions to a Better-Grounded Presentation of Mathematics_; ▪ the unpublished manuscript _Aetiology_; ▪ the unpublished manuscript _General Mathematics_. Topic-wise, the texts centre around scientific method and the foundations of mathematics. In the _Contributions_ Bolzano discusses grounding in connection with three topics: the classification of mathematical disciplines, the notion of ground-revealing proofs and its relation to syllogistic inferences, and finally the notion of axioms or basic propositions. In the _Aetiology_ Bolzano presents his first attempt at a rigorous, axiomatic theory of grounding and relates the concept of grounding to several concepts and issues in mathematics. In the part of _General Mathematics_ we present here, Bolzano critically assesses the Principle of Sufficient Reason and some related metaphysical issues.

In his ontology Bolzano uses the notion of grounding to make claims about the dependent and independent existence of entities. In particular, he argues that there must be a _fundamental_ object (in Bolzano’s terminology: an _unconditioned_ object) whose existence is not grounded in the existence of any other object. The argument is a philosophical gem, whether or not one endorses it in its entirety. Since the conception of grounding Bolzano works with is remarkably modern in spirit, his argument can be treated and discussed like a contribution to the contemporary debate about grounding and fundamentality. This paper reconstructs Bolzano’s argument, explains its Leibnizian roots, and puts the argument under scrutiny.

In der Athanasia skizziert Bernard Bolzano seine bikategoriale Ontologie des Wirklichen wie folgt: 'Alles, was ist, d. h. in Wirklichkeit besteht, [...] gehört zu einer von folgenden zwei Arten: es ist und besteht entweder an etwas Anderem, als Beschaffenheit desselben, oder es ist nicht eine bloße Beschaffenheit an etwas Anderem, sondern bestehet, wie man zu sagen pflegt, für sich. [...] Die Wirklichkeiten der ersteren Art pflegen die Weltweisen mit einem lateinischen Wort auch Adhärenzen, jene der letzten aber Substanzen zu nennen.' Benjamin Schnieder entfaltet in seiner Untersuchung die in diesem Zitat in prägnanter Kürze umrissene Ontologie: Bolzanosche Adhärenzen sind partikularisierte Eigenschaften, Entitäten, denen gerade in den vergangenen zwanzig Jahren großes Interesse in der analytischen Ontologie entgegengebracht wurde. Das interessanteste Ergebnis besteht im Nachweis, daß Bolzano Adhärenzen neben einer kausalen Rolle auch eine semantische Rolle spielen läßt, was einen bisher unbeachteten Zug seines Systems darstellt. Wirkliche Dinge, welche keine Adhärenzen sind, nennt Bolzano Substanzen. Neben dieser offiziellen Definition des Substanzbegriffs finden sich bei Bolzano auch Anklänge an eine klassische Unabhängigkeitsdefinition von Substanzen. Es wird sowohl gezeigt, wieso er gut daran tat, nicht auf die definitorische Karte der Tradition zu setzen, als auch auf interne Probleme seines eigenen Ansatzes hingewiesen - freilich nicht, ohne dabei auch mögliche Auswege und Reaktionen anzubieten.

Aristotle famously observed: That you are pale is true because you are pale—but not vice versa. This insight plays an important role in contemporary debates about truth by expressing truth’s dependency on being. However, how to account for this insight remains controversial. This paper employs the logic of grounding to derive Aristotle’s insight rigorously. The derivation requires a specific truth theory as its starting point. Here, Wolfgang Künne’s Modest Conception of Truth is chosen, but it can also be adapted to other theories of truth. As a by-product, this analysis uncovers a previously unnoticed problem in Künne’s theory—one for which I also propose a solution.

Three traditional philosophical issues that van Inwagen discusses in his metaphysical works are the Principle of Sufficient Reason, the question of why there is something rather than nothing, and the question of whether free will is compatible with determinism. The three topics are connected by a conceptual tie: the notion of a ground. In this essay, it is argued that van Inwagen’s take on the three topics, ingenious as it otherwise is, suffers from an inadequate conception of the underlying notion of a ground.

This survey paper pursues three aims. The first aim is to sketch the overall role that grounding plays in Bolzano’s philosophy and to indicate some historical roots of his views, namely Aristotle’s conception of scientific proofs and the rationalist’s debate about the Principle of Sufficient Reason. The second aim is to provide readers coming from the current debate about metaphysical grounding with an accessible overview of Bolzano’s mature conception of grounding. The third and final aim of the paper is to introduce elements and terminology from Bolzano’s conceptual and ontological framework that are required to understand his theory of grounding.

reality is a complex affair. It comprises a huge variety of different elements. Importantly, though, reality is not a mere aggregate of its elements but rather a structured whole or system whose building blocks are not all on the same level. Instead, they form hierarchical networks ordered by relations of priority. In such networks, derivative aspects of reality obtain in virtue of their grounds, that is, in virtue of more fundamental aspects of reality that are prior to them. This picture of reality as a structured whole is currently enjoying a renaissance in the works of philosophers such as Kit Fine, Jonathan Schaffer, and many others. But far from being a new picture, it has been widely endorsed throughout the...

In describing and classifying things we often rely on their modal characteristics. We will in general not have a satisfactory account of the nature and character of an object, unless we specify at least partly how the thing might be or cannot be, and also how it might have been or could not have been. In his contribution to the Second Jerusalem Philosophical Encounter,1 Strawson addressed the issue of how to understand such ascriptions of modal characteristics. Although his paper is terse and provocative, and develops an intriguing account of modal predications, it has never received much attention in the philosophical literature. Recently, the issues dealt with in Strawson’s paper have become the subject of a widespread debate. Hence, we think it worthwhile to put Strawson’s account under closer scrutiny. In what follows, we first discuss his account of present tense ‘might’-statements, then his account of past perfect ‘might’-statements, and finally some essentialist remarks that he makes in his paper. That the discussion will be rather critical for the most part (though not exclusively so) should not belie the originality and inherent value of his pioneering approach.

In my paper I am concerned with Peter van Inwagen's Consequence Argument. I focus on its probably best known version. In this form it crucially employs the notion of rendering a proposition false, anotion that has never been made sufficiently clear. The main aim of my paper is to shed light on thisnotion. The explications offered so far in thedebate all are based on modal concepts. Iargue that for sufficient results a ``stronger'', hyper-intensional concept is needed, namely the concept expressed by the word ``because''. I show that my analysis is superior to the prior ones. On the basis of this analysis I further explain why van Inwagen''s argument fails.

The main question of this paper is how to understand the notion of a truth-maker. In section 1, I show that the identification of truth-making with necessitation cannot capture the pretheoretic understanding of notions such as ‘x makes something true’. In section 2, I examine Barry Smith’s reaction to this problem: he defines truth-making as the combination of necessitation and projection. I focus on the formal part of Smith’s account, which is shown to yield undesired results. However, in section 3, I present an alternative account of projection, which fares better and can fruitfully be employed to circumvent the problems raised in section 1. Unfortunately, the account still has to face some troublemakers, as I show in the final section. I conclude, therefore, with a pessimistic view on the project of defining truth-making via necessitation and projection.

This volume contains fifteen essays in honour of Wolfgang Künne. The essays deal with issues from the philosophy of language and logic, broadly conceived. They cover topics ranging from truth, reference, and the ontology of abstract objects, to action, intentionality, and speech acts. By taking into account the works of early analytic philosophers—including Bolzano, Frege, Peirce, Husserl, and Wittgenstein—they foster our understanding of the history of the ideas discussed, while at the same time contributing to the systematic debate. The collection also includes an up-to-date bibliography of Künne's published work.

Wenn Plato weise ist, dann gibt es mindestens dreierlei: Plato, die Weisheit und Platos Weisheit. Dies sind beispielhafte Vertreter von drei ontologischen Kategorien (Substanz, universelle Eigenschaft, partikularisierte Eigenschaft), die grundlegend für unsere Weltorientierung sind. Als Beitrag zur deskriptiven Metaphysik im Sinne P.F. Strawsons entwickelt Schnieder eine Konzeption dieser Kategorien und ihres Zusammenhangs. Er expliziert und verteidigt klassische Ideen der Philosophiegeschichte mit analytischer Methodik und unter Einbezug von sprachphilosophischen, modallogischen und mereologischen Erwägungen. Unter anderem argumentiert er, dass Substanzen aufgrund ihrer existentiellen Unabhängigkeit eine ontologische Sonderstellung zukommt. Indem es auf das erneut erwachsene Interesse an der Metaphysik reagiert und Grundzüge der aktuellen Debatten voraussetzungsarm erläutert, kann das Buch auch als allgemeine kritische Einführung in das Thema dienen.