36 ideas
| 9408 | Science studies phenomena, but only metaphysics tells us what exists [Mumford] |
| Full Idea: Science deals with the phenomena, ..but it is metaphysics, and only metaphysics, that tells us what ultimately exists. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.2) |
| 9429 | Many forms of reasoning, such as extrapolation and analogy, are useful but deductively invalid [Mumford] |
| Full Idea: There are many forms of reasoning - extrapolation, interpolation, and other arguments from analogy - that are useful but deductively invalid. | |
| From: Stephen Mumford (Laws in Nature [2004], 04.4) | |
| A reaction: [He cites Molnar for this] |
| 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? |
| 9427 | For Humeans the world is a world primarily of events [Mumford] |
| Full Idea: For Humeans the world is a world primarily of events. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.6) |
| 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. |
| 9446 | Properties are just natural clusters of powers [Mumford] |
| Full Idea: The view of properties I find most attractive is one in which they are natural clusters of, and exhausted by, powers (plus other connections to other properties). | |
| From: Stephen Mumford (Laws in Nature [2004], 10.6) |
| 9435 | A 'porridge' nominalist thinks we just divide reality in any way that suits us [Mumford] |
| Full Idea: A 'porridge' nominalist denies natural kinds, and thinks there are no objective divisions in reality, so concepts or words can be used by a community to divide the world up in any way that suits their purposes. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.3) |
| 9447 | If properties are clusters of powers, this can explain why properties resemble in degrees [Mumford] |
| Full Idea: If a cluster of ten powers exhausts property F, and property G differs in respect of just one power, this might explain why properties can resemble other properties and in different degrees. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.6) | |
| A reaction: I love this. The most intractable problem about properties and universals is that of abstract reference - pink resembles red more than pink resembles green. If colours are clusters of powers, red and pink share nine out of ten of them. |
| 12248 | How can we show that a universally possessed property is an essential property? [Mumford] |
| Full Idea: Essentialists fail to show how we ascend from being a property universally possessed, by all kind members, to the status of being an essential property. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.5) | |
| A reaction: This is precisely where my proposal comes in - the essential properties, as opposed to the accidentaly universals, are those which explain the nature and behaviour of each kind of thing (and each individual thing). |
| 9430 | Singular causes, and identities, might be necessary without falling under a law [Mumford] |
| Full Idea: One might have a singularist view of causation in which a cause necessitates its effect, but they need not be subsumed under a law, ..and there are identities which are metaphysically necessary without being laws of nature. | |
| From: Stephen Mumford (Laws in Nature [2004], 04.5) |
| 9445 | We can give up the counterfactual account if we take causal language at face value [Mumford] |
| Full Idea: If we take causal language at face value and give up reducing causal concepts to non-causal, non-modal concepts, we can give up the counterfactual dependence account. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.5) |
| 9443 | It is only properties which are the source of necessity in the world [Mumford] |
| Full Idea: If laws do not give the world necessity, what does? I argue the positive case for it being properties, and properties alone, that do the job (so we might call them 'modal properties'). | |
| From: Stephen Mumford (Laws in Nature [2004], 10.1) |
| 9444 | There are four candidates for the logical form of law statements [Mumford] |
| Full Idea: The contenders for the logical form of a law statement are 1) a universally quantified conditional, 2) a second-order relation between first-order universals, 3) a functional equivalence, and 4) a dispositional characteristic of a natural kind. | |
| From: Stephen Mumford (Laws in Nature [2004], 10.3) |
| 9431 | Pure regularities are rare, usually only found in idealized conditions [Mumford] |
| Full Idea: Pure regularities are not nearly as common as might have been thought, and are usually only to be found in simplified or idealized conditions. | |
| From: Stephen Mumford (Laws in Nature [2004], 05.3) | |
| A reaction: [He cites Nancy Cartwright 1999 for this view] |
| 9441 | Regularity laws don't explain, because they have no governing role [Mumford] |
| Full Idea: A regularity-law does not explain its instances, because such laws play no role in determining or governing their instances. | |
| From: Stephen Mumford (Laws in Nature [2004], 09.7) | |
| A reaction: Good. It has always seemed to me entirely vacuous to explain an event simply by saying that it falls under some law. |
| 9415 | Would it count as a regularity if the only five As were also B? [Mumford] |
| Full Idea: While it might be true that for all x, if Ax then Bx, would we really want to count it as a genuine regularity in nature if only five things were A (and all five were also B)? | |
| From: Stephen Mumford (Laws in Nature [2004], 03.3) |
| 9416 | Regularities are more likely with few instances, and guaranteed with no instances! [Mumford] |
| Full Idea: It seems that the fewer the instances, the more likely it is that there be a regularity, ..and if there are no cases at all, and no S is P, that is a regularity. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.3) | |
| A reaction: [He attributes the second point to Molnar] |
| 9422 | If the best system describes a nomological system, the laws are in nature, not in the description [Mumford] |
| Full Idea: If the world really does have its own nomological structure, that a systematization merely describes, why are the laws not to be equated with the nomological structure itself, rather than with the system that describes it? | |
| From: Stephen Mumford (Laws in Nature [2004], 03.4) |
| 9421 | The best systems theory says regularities derive from laws, rather than constituting them [Mumford] |
| Full Idea: The best systems theory (of Mill-Ramsey-Lewis) says that laws are not seen as regularities but, rather, as those things from which regularities - or rather, the whole world history including the regularities and everything else - can be derived. | |
| From: Stephen Mumford (Laws in Nature [2004], 03.4) | |
| A reaction: Put this way, the theory invites questions about ontology. Regularities are just patterns in physical reality, but axioms are propositions. So are they just features of human thought, or do these axioms actuallyr reside in reality. Too weak or too strong. |
| 9432 | Laws of nature are necessary relations between universal properties, rather than about particulars [Mumford] |
| Full Idea: The core of the Dretske-Tooley-Armstrong view of the late 70s is that we have a law of nature when we have a relation of natural necessitation between universals. ..The innovation was that laws are about properties, and only indirectly about particulars. | |
| From: Stephen Mumford (Laws in Nature [2004], 06.2) | |
| A reaction: It sounds as if we should then be able to know the laws of nature a priori, since that was Russell's 1912 definition of a priori knowledge. |
| 9433 | If laws can be uninstantiated, this favours the view of them as connecting universals [Mumford] |
| Full Idea: If there are laws that are instantiated in no particulars, then this would seem to favour the theory that laws connect universals rather than particulars. | |
| From: Stephen Mumford (Laws in Nature [2004], 06.4) | |
| A reaction: There is a dispute here between the Platonic view of uninstantiated universals (Tooley) and the Aristotelian instantiated view (Armstrong). Mumford and I prefer the dispositional account. |
| 9434 | Laws of nature are just the possession of essential properties by natural kinds [Mumford] |
| Full Idea: If dispositional essentialism is granted, then there is a law of nature wherever there is an essential property of a natural kind; laws are just the havings of essential properties by natural kinds. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.2) | |
| A reaction: [He is expounding Ellis's view] |
| 9437 | To distinguish accidental from essential properties, we must include possible members of kinds [Mumford] |
| Full Idea: Where properties are possessed by all kind members, we must distinguish the accidental from essential ones by considering all actual and possible kind members. | |
| From: Stephen Mumford (Laws in Nature [2004], 07.5) | |
| A reaction: This is why we must treat possibilities as features of the actual world. |
| 9439 | The Central Dilemma is how to explain an internal or external view of laws which govern [Mumford] |
| Full Idea: The Central Dilemma about laws of nature is that, if they have some governing role, then they must be internal or external to the things governed, and it is hard to give a plausible account of either view. | |
| From: Stephen Mumford (Laws in Nature [2004], 09.2) | |
| A reaction: This dilemma is the basis of Mumford's total rejection of 'laws of nature'. I think I agree. |
| 9412 | You only need laws if you (erroneously) think the world is otherwise inert [Mumford] |
| Full Idea: Laws are a solution to a problem that was misconceived. Only if you think that the world would be otherwise inactive or inanimate, do you have the need to add laws to your ontology. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.5) | |
| A reaction: This is a bold and extreme view - and I agree with it. I consider laws to be quite a useful concept when discussing nature, but they are not part of the ontology, and they don't do any work. They are metaphysically hopeless. |
| 9411 | There are no laws of nature in Aristotle; they became standard with Descartes and Newton [Mumford] |
| Full Idea: Laws do not appear in Aristotle's metaphysics, and it wasn't until Descartes and Newton that laws entered the intellectual mainstream. | |
| From: Stephen Mumford (Laws in Nature [2004], 01.5) | |
| A reaction: Cf. Idea 5470. |