26 ideas
| 13734 | Modern Quinean metaphysics is about what exists, but Aristotelian metaphysics asks about grounding [Schaffer,J] |
| Full Idea: On the now dominant Quinean view, metaphysics is about what there is (such as properties, meanings and numbers). I will argue for the revival of a more traditional Aristotelian view, on which metaphysics is about what grounds what. | |
| From: Jonathan Schaffer (On What Grounds What [2009], Intro) | |
| A reaction: I find that an enormously helpful distinction, and support the Aristotelian view. Schaffer's general line is that what exists is fairly uncontroversial and dull, but the interesting truths about the world emerge when we grasp its structure. |
| 13751 | If you tore the metaphysics out of philosophy, the whole enterprise would collapse [Schaffer,J] |
| Full Idea: Traditional metaphysics is so tightly woven into the fabric of philosophy that it cannot be torn out without the whole tapestry unravelling. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.3) | |
| A reaction: I often wonder why the opponents of metaphysics still continue to do philosophy. I don't see how you address questions of ethics, or philosophy of mathematics (etc) without coming up against highly general and abstract over-questions. |
| 13743 | We should not multiply basic entities, but we can have as many derivative entities as we like [Schaffer,J] |
| Full Idea: Occam's Razor should only be understood to concern substances: do not multiply basic entities without necessity. There is no problem with the multiplication of derivative entities - they are an 'ontological free lunch'. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.1) | |
| A reaction: The phrase 'ontological free lunch' comes from Armstrong. This is probably what Occam meant. A few extra specks of dust, or even a few more numbers (thank you, Cantor!) don't seem to challenge the principle. |
| 10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
| Full Idea: The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], n 3) |
| 10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
| Full Idea: Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
| Full Idea: Some authors argue that second-order logic (with standard semantics) is not logic at all, but is a rather obscure form of mathematics. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) |
| 10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
| Full Idea: If the goal of logical study is to present a canon of inference, a calculus which codifies correct inference patterns, then second-order logic is a non-starter. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be because it is not 'complete'. However, moves like plural quantification seem aimed at capturing ordinary language inferences, so the difficulty is only that there isn't a precise 'calculus'. |
| 10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
| Full Idea: Informally, logical consequence is sometimes defined in terms of the meanings of a certain collection of terms, the so-called 'logical terminology'. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: This seems to be a compositional account, where we build a full account from an account of the atomic bits, perhaps presented as truth-tables. |
| 10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
| Full Idea: Second-order variables can range over properties, sets, or relations on the items in the domain-of-discourse, or over functions from the domain itself. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
| Full Idea: Upward Löwenheim-Skolem: if a set of first-order formulas is satisfied by a domain of at least the natural numbers, then it is satisfied by a model of at least some infinite cardinal. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
| Full Idea: Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.3.2) |
| 10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
| Full Idea: The Löwenheim-Skolem theorem is usually taken as a sort of defect (often thought to be inevitable) of the first-order logic. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.4) | |
| A reaction: [He is quoting Wang 1974 p.154] |
| 10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
| Full Idea: Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| Full Idea: Full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.1) | |
| A reaction: [he credits Cowles for this remark] Having an unworkable model theory sounds pretty serious to me, as I'm not inclined to be interested in languages which don't produce models of some sort. Surely models are the whole point? |
| 13741 | If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J] |
| Full Idea: We can automatically infer 'there are roses' from 'there are red roses' (with no shift in the meaning of 'roses'). Likewise one can automatically infer 'there are numbers' from 'there are prime numbers'. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.1) | |
| A reaction: He similarly observes that the atheist's 'God is a fictional character' implies 'there are fictional characters'. Schaffer is not committing to a strong platonism with his claim - merely that the existence of numbers is hardly worth disputing. |
| 13748 | Grounding is unanalysable and primitive, and is the basic structuring concept in metaphysics [Schaffer,J] |
| Full Idea: Grounding should be taken as primitive, as per the neo-Aristotelian approach. Grounding is an unanalyzable but needed notion - it is the primitive structuring conception of metaphysics. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.2) | |
| A reaction: [he cites K.Fine 1991] I find that this simple claim clarifies the discussions of Kit Fine, where you are not always quite sure what the game is. I agree fully with it. It makes metaphysics interesting, where cataloguing entities is boring. |
| 13747 | Supervenience is just modal correlation [Schaffer,J] |
| Full Idea: Supervenience is mere modal correlation. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.2) |
| 13744 | The cosmos is the only fundamental entity, from which all else exists by abstraction [Schaffer,J] |
| Full Idea: My preferred view is that there is only one fundamental entity - the whole concrete cosmos - from which all else exists by abstraction. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.1) | |
| A reaction: This looks to me like weak anti-realism - that there are no natural 'joints' in nature - but I don't think Schaffer intends that. I take the joints to be fundamentals, which necessitates that the cosmos has parts. His 'abstraction' is clearly a process. |
| 13739 | Maybe categories are just the different ways that things depend on basic substances [Schaffer,J] |
| Full Idea: Maybe the categories are determined by the different grounding relations, ..so that categories just are the ways things depend on substances. ...Categories are places in the dependence ordering. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 1.3) |
| 10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
| Full Idea: In studying second-order logic one can think of relations and functions as extensional or intensional, or one can leave it open. Little turns on this here, and so words like 'property', 'class', and 'set' are used interchangeably. | |
| From: Stewart Shapiro (Higher-Order Logic [2001], 2.2.1) | |
| A reaction: Important. Students of the metaphysics of properties, who arrive with limited experience of logic, are bewildered by this attitude. Note that the metaphysics is left wide open, so never let logicians hijack the metaphysical problem of properties. |
| 13742 | There exist heaps with no integral unity, so we should accept arbitrary composites in the same way [Schaffer,J] |
| Full Idea: I am happy to accept universal composition, on the grounds that there are heaps, piles etc with no integral unity, and that arbitrary composites are no less unified than heaps. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.1 n11) | |
| A reaction: The metaphysical focus is then placed on what constitutes 'integral unity', which is precisely the question which most interested Aristotle. Clearly if there is nothing more to an entity than its components, scattering them isn't destruction. |
| 13752 | The notion of 'grounding' can explain integrated wholes in a way that mere aggregates can't [Schaffer,J] |
| Full Idea: The notion of grounding my capture a crucial mereological distinction (missing from classical mereology) between an integrated whole with genuine unity, and a mere aggregate. x is an integrated whole if it grounds its proper parts. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 3.1) | |
| A reaction: That gives a nice theoretical notion, but if you remove each of the proper parts, does x remain? Is it a bare particular? I take it that it will have to be an abstract principle, the one Aristotle was aiming at with his notion of 'form'. Schaffer agrees. |
| 13749 | Belief in impossible worlds may require dialetheism [Schaffer,J] |
| Full Idea: One motivation for dialetheism is the view that there are impossible worlds. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.3) |
| 13740 | 'Moorean certainties' are more credible than any sceptical argument [Schaffer,J] |
| Full Idea: A 'Moorean certainty' is when something is more credible than any philosopher's argument to the contrary. | |
| From: Jonathan Schaffer (On What Grounds What [2009], 2.1) | |
| A reaction: The reference is to G.E. Moore's famous claim that the existence of his hand is more certain than standard sceptical arguments. It sounds empiricist, but they might be parallel rational truths, of basic logic or arithmetic. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |