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We introduce non-associative skew Laurent polynomial rings and characterize when they are simple. Thereby, we generalize results by Jordan, Voskoglou, and Nystedt and Öinert.
Young children often lack words for what they want to talk about. To fill the gaps in their lexicon, they coin new words. They rely on compounding and derivation to do this. This means identifying and analyzing parts of words – roots or stems, and affixes – and learning their meanings, as well as which combinations are possible. Some languages favor compounding and some derivation in word formation. Children are sensitive to which options are the most productive and adopt those first. Two-year-olds offer analyses of word meanings, as in running-stick (I run with it) or high-chair (it is high), and provide analyses of novel compounds where they take account of language structure (head noun first in Hebrew, second in English). They also analyze derived forms with agentive endings. They start to produce novel words from as young as age two, whether compounds in Germanic languages, or derived forms in Romance and Semitic. They begin with simple forms (minimal or no change to the root), advance to compound or derived word forms that are transparent in meaning, and opt for the most productive options in the adult language, with the goal of finding the right words to convey the child-speaker’s meaning.
Despite its apparent complexity, our world seems to be governed by simple laws of physics. This volume provides a philosophical introduction to such laws. I explain how they are connected to some of the central issues in philosophy, such as ontology, possibility, explanation, induction, counterfactuals, time, determinism, and fundamentality. I suggest that laws are fundamental facts that govern the world by constraining its physical possibilities. I examine three hallmarks of laws-simplicity, exactness, and objectivity-and discuss whether and how they may be associated with laws of physics.
We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.
I explore the promise of Beall's proposal for a long-standing challenge for traditional theology. I first offer a sketch of the problem and a brief overview of some of the more common responses to it. I then show how Beall's proposal holds initial promise; following this I highlight some concerns and raise some questions.
Beall's original understanding of the nature of the divine allows for contradictory statements to be true of God, by assuming that parts of reality, such as the Trinity, are ‘glutty’ (namely, what we can say about them is both true and false). Is the divine is the only glutty part of reality, and if so, why? Furthermore, does the glutty nature of the divine undermine its simplicity? Beall argues that God is not mereologically complex, but on his account God is logically and hence, it appears, metaphysically complex.
We generalise the properties $\mathsf {OP}$, $\mathsf {IP}$, k-$\mathsf {TP}$, $\mathsf {TP}_{1}$, k-$\mathsf {TP}_{2}$, $\mathsf {SOP}_{1}$, $\mathsf {SOP}_{2}$, and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$, analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.
‘God is simple’, wrote Messiaen; here is an aesthetic model for this composer – sometimes called a theologian – that is present in diverse aspects of his music and that receives attention in this chapter. Simplicity is revealed in different forms and techniques, and in his ‘naïveté. It is so engraved in the composer’s works that it becomes almost a caricature. This chapter examines these ideals in Messiaen’s thought, character, and in the critical reception of his music.
Tertullian provides evidence in several writings addressed to catechumens of the ways in which Christian contestations about ritual related to knowledge of God. Against what he describes as the obfuscations of heretical and pagan ritual, Tertullian emphasizes the simplicity of Christian ritual as a fitting mode for expressing true divine power. This chapter focuses on De spectaculis, De oratione, De baptismo, and Tertullian’s appeals to the Rule of Faith.
Jeremiah Carey presents a version of panentheism which he attributes to Gregory Palamas, as well as other Greek patristic thinkers. The Greek tradition, he alleges, is more open to panentheistic metaphysics than the Latin. Palamas, for instance, hold that God's energies are participable, even if God's essence is not. Carey uses Palamas' metaphysics to sketch an account on which divine energies are the forms of created substances, and argues that it is open to Orthodox Christians to affirm that God is in all things as their formal cause. I argue that Carey's reading is premised on a superficial examination of the patristic literature. More importantly, Palamas' metaphysics is opposed to that of Carey, since Palamas' distinction aims to uphold the view that created persons are only contingent participants in God. On this, Palamas and the Latins are in complete accord. In conclusion, I propose that panentheistic metaphysics begins from a false dilemma.
Jacques Derrida is one of the most controversial philosophers of the twentieth century, who is hailed by his followers as a genius, derided by his detractors as a charlatan. His work continues to be a source of often inordinate praise and blame. How does Derrida provoke such violent reactions? What is ‘deconstruction’, his most famous technique? And is there something in his work that can be useful to even the most hostile of his critics?
A meta-theology makes claims about the structure of theological claims: it identifies a single, fundamental claim about God, and shows how other theological claims are derivable from the fundamental claim. In his book Depicting Deity and other articles, Jon Kvanvig has identified three distinct meta-theologies: Creator Theology, Perfect Being Theology, and Worship-worthiness Theology. In this article, we argue that the medieval Islamic philosopher Avicenna's views about God have the structure of a meta-theology, and that it is distinct from the three projects Kvanvig identifies. This view is Necessary Existent Theology.
A common argument put forth by naturalists (including the prominent philosopher Graham Oppy) in support of naturalism as a worldview over theism, is to claim that naturalism is a simpler hypothesis. Theism posits the existence of everything that naturalism does, plus the existence of a theistic realm. Thus, all things being equal, via Ockham's Razor, naturalism should be preferred to theism. In this essay, we argue that the Classical Theist need not worry about the naturalist's Simplicity argument. Specifically, we argue that, the one holding to a scholastic metaphysics (i.e., potency-act distinction, participatory metaphysics, and existence-in-degree), in the end, will be the one with the simpler worldview.
This chapter completes our critical exploration of Popper’s key work, the Logic of Scientific Discovery and how it applies to corpus linguistics. In this chapter we address the question of how easily linguistics may be viewed as a science, in Popper’s terms. We also consider important critiques of Popper’s work and use those to both clarify and, where necessary, adapt the framework.
Although it is helpful to appreciate the general nature of explanations, we might reasonably want more than this. As this book is part of the Understanding Life series, we may expect to delve into details about kinds of explanations that are specific to the life sciences.
It is widely held that science is a (if not the) primary source of our knowledge of the world around us. Further, most accept that scientific knowledge is the best confirmed and well-supported kind of knowledge that we have of the world. But, how do scientific explanations lead to scientific knowledge? The short answer is that they do so via a method known as “inference to the best explanation” (IBE), sometimes called “abduction.” Before we get into the details of IBE, let’s take a quick look at an obvious way that scientific explanations give us scientific knowledge.
A general way of appreciating some of the main ideas of the previous chapter is to recognize that explanations aim at providing understanding. Scientists and philosophers agree that understanding is a (if not the) primary epistemic goal of scientific inquiry. Both explanation and prediction tend to be closely related to understanding. We want explanations in science because we want to understand why the world is as it is and how things happen. And, once we understand various phenomena, we can make accurate predictions about them. One simple, and widely accepted, way of assessing the quality of a given explanation is to look at the understanding it provides. Roughly, the better an explanation, the more understanding that explanation (if true) would provide. As philosopher Peter Lipton explained, the explanation that is the best is simply the explanation that, if true, would provide the deepest understanding of the phenomena being explained. That being said, some worry because it seems that we might misjudge how well we understand something.
In this chapter we’re looking at the relation between scientific explanations and predictions. It is tempting to think that the only difference between explanations and predictions is that one looks back and tells us how or why things happened as they did, and the other looks forward and tells us how or why certain things will (or are likely to) happen. This thought can seem particularly plausible when we consider that in many cases a good scientific hypothesis will both explain phenomena and allow us to make accurate predictions. Despite its initial plausibility, the idea that explanation and prediction are symmetrical is mistaken. The way to see this is to take a look at a particular theory of scientific explanation that entails this relationship between explanation and prediction. The particular theory of scientific explanation in question, the covering law model, which we discussed in Chapter 2, is false. One of the reasons that this theory of explanation fails helps illustrate the fact that explanation and prediction are not symmetrical.
Explanation is central to our lives, in general. We seem to have an innate (or nearly so) drive to explain and seek explanations. When our favorite app is not working, we want to know why, and we want to know how to fix it. When trying to understand why people engage in an odd behavior – refusing to wear masks during the COVID-19 pandemic, say – we want an explanation. What reasons do they have for doing something that seems so clearly misguided? Why are they resistant to expert advice on the issue? Ultimately, we seek explanations to help us understand and navigate the world around us.