Julio De Rizzo

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.

A state is luminous if and only if, whenever one is in it, one is in a position to know that one is. A state is bright if and only if, whenever one is in it, one is in a position to believe that one is. Beliefs have long been regarded, both historically and from a contemporary perspective, as luminous and bright. This paper evaluates Timothy Williamson’s influential anti-luminosity arguments as they apply to the luminosity and brightness of belief. While his margin-for-error argument may effectively challenge the luminosity of knowledge, it cannot be straightforwardly extended to undermine either the luminosity or the brightness of belief. Some authors responded to the more general anti-luminosity argument based on a constitutive connection between states and attitudes. An influential reply on Williamson’s behalf by Srinavasan in terms of degrees of confidence opens way to the claim that there is a constitutive connection between confidence and belief. The more general argument, in the specific case of belief, can then be resisted by drawing on independently defensible views of how we form beliefs about our own beliefs.

A growing body of scholarship examines whether Husserl’s account of perception should be read as disjunctivist or conjunctivist. Focusing on the Logical Investigations, this paper defends a conciliatory interpretation: Husserl’s view incorporates elements of both. At the level of descriptive psychology, perceptual acts are individuated by their intentional essence—quality and matter—supporting conjunctivism. Yet Husserlian epistemology, centered on the notion of fulfillment, requires object-dependent intuitive contents for perceptual knowledge, which marks an essential difference between veridical and non-veridical cases, thereby aligning with disjunctivism. Drawing on recent work in the philosophy of perception, I resolve this tension by distinguishing between coarse-grained content relevant to psychological description and explanation and fine-grained representational content, which incorporates context-bound elements relevant for perceptual knowledge. I argue that this strategy not only proves useful in the contemporary context, but also finds support within Husserl’s own approach to perception and perceptual knowledge in the Logical Investigations.

Laws play some role in explanations: at the very least, they somehow connect what is explained, or the explanandum, to what explains, or the explanans. Thus, thermodynamical laws connect the match's being struck and its lightning, so that the former causes the latter; and laws about set formation connect Socrates' existence with {Socrates}'s existence, so that the former grounds the latter. But is there more to the explanatory role of laws? A natural proposal, which finds considerable support in the literature, is that laws simply are reasons why. That is, besides connecting the explanandum with a suitable ground or cause, they take part, together with these, in explaining why the explanandum obtains. In this paper, I present two arguments that show that this proposal cannot hold in general. They each point to one unacceptable consequence: a suspicious infinite extendibility of explanantia; and a failure of the irreflexivity of “because.”

There are limits to how things might have been. Yet tiny permissible differences between possibilities add up to large ones that exceed those limits. This tension lies at the heart of the so-called tolerance puzzle. In this paper, I propose a novel response grounded in the idea that metaphysical modality is perspectival. The perspectivalist framework is used to dissolve the puzzle and is contrasted with two alternatives: one that posits a plenitude of finely individuated entities, and another that appeals to a double relativization of modal claims—both to a world considered as actual and to a world considered as counterfactual.

Two influential objections to deflationism about truth question its ability to explain the role of true beliefs in successful actions; and to account for general compositional principles linking truth to complex sentences governed by truth-functional connectives. In this paper, I address recent formulations of these objections by Will Gamester and Richard Heck. My responses draw on recent work on explanation, grounding, and the logic of “because”. However, each response leaves a residual concern for deflationists that these strategies alone cannot fully resolve. In the final section, I propose a view I call "Aristotelian Deflationism," which incorporates specific “because” principles relating to truth as an alternative to standard instances of the T-schema. While not deflationist in the strictest sense, I argue that this approach offers compelling ways to address both the primary objections and the residual concerns effectively.

A common way of clarifying the notion of ground is by way of examples from logic: thus a conjunction is grounded in both of its conjuncts; a disjunction in each of its true disjuncts; a double negation in its negatum; and so on. Developing a semantics that accommodates these logical examples in full generality turned out to be a difficult task. In this paper, I develop a novel approach that substitutes fusion for a more discerning relation of combination between states in the framework of truth-maker semantics.

Is the macro grounded in the micro? That is, is every truth about a macroscopic object fully grounded in a truth wholly about its microscopic parts? In a recent interesting paper, Martin Glazier argued for a negative answer. Following him, call the position that the macro is grounded in the micro ‘priority micro pluralism’ (‘pluralism’ for short). In this discussion note, I propose a way out for the pluralist. In brief, it consists in the recognition that some metaphysical positions, including pluralism, are best expressed by fundamentally de dicto grounding claims. In that these claims have a distinctive modal force, this recognition has far-reaching consequences that go beyond a defence of pluralism.

A traditional account of coincidences has it that two facts are coincidental whenever they are not related as cause and effect and do not have a common cause. A recent contribution by Lando : 132–151, 2017) showed that this account is mistaken. In this paper, I argue against two alternative accounts of coincidences, one suggested by Lando, and another by Bhogal : 677–694, 2020), and defend a third one in their place. In short, I propose that how explanatory links relate to non-coincidental facts in explanation is what drives a wedge between coincidences and non-coincidences. This proposal is not susceptible to the worries I raise, and is more general, since it is not restricted to coincidences and non-coincidences involving physical facts.

I provide a case-by-case definition of essential truths based on the notions of metaphysical necessity and ontological dependence. Relying on suggestions in the literature, I adopt a definition of the latter notion in terms of the notion of ground. The resulting account is adequate in the sense that it is not subject to Kit Fine’s famous counterexamples to the purely modal account of essence. In addition, it provides us with a novel conception of truths pertaining to the essence of objects, which might help to dispel doubts on the legitimacy of the notion of essence itself.

Few if any distinctions are more easily recognisable and assented to than that between _objects_, that is, things which are some ways, and that which they are, that is, _ways for objects to be_ (‘ways of being’ for short). In this paper I present an argument designed to show that this distinction is indeterminate in the sense that the truth-conditions of predicational sentences leave open what should count as an object and a way of being. The bulk of the argument is inspired by the celebrated permutation argument advanced by Quine, Wallace, Putnam and others.

A natural proposal for the grounds of negative existential truths, such as that Vulcan does not exist, states that these truths are grounded in the totality truth affirming the existence of every existent thing together with the truth that they are all. In this paper I will put forward three objections to straightforward formulations of this idea, and argue that a change in the usual grammar of grounding claims, allowing for pluralities of sentences to express not only grounds, but also groundees, is effective in making the account immune to the objections raised.

Many philosophers have shown sympathy to the thought that reality is fundamentally positive. Julio De Rizzo formulates this idea precisely by means of the notion of grounding, and examines how the resulting thesis fares with respect to three much discussed classes of negative truths, namely that of negative predications, that of negative causal reports, and that of negative existential truths. By shedding light on the issues advocates of the thesis have to deal with, this work shows the positivist account to be a tenable position in metaphysics.

This paper is a response to McKenzie (2017). I argue that the case she presents is not a genuine counterexample to the thesis she labels Brute Fundamentalism. My response consists of two main points. First, that the support she presents for considering her case a metaphysical explanation is misguided. Second, that there are principled reasons for doubting that partial explanations in Hempel’s sense, of which her case is an instance, are genuinely explanatory in the first place. Thus McKenzie’s attack on Brute Fundamentalism fails.