31 ideas
| 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) |
| 21495 | Theoretical and practical politics are both concerned with the best lives for individuals [Russell] |
| Full Idea: Political ideals must be based upon ideals for the individual life. The aim of politics should be to make the lives of individuals as good as possible. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Russell floats between socialism and anarchism, but this foundational remark is classic liberalism. |
| 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? |
| 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. |
| 16078 | Clay is intrinsically and atomically the same as statue (and that lacks 'modal properties') [Rudder Baker] |
| Full Idea: Arguments for statue being the clay are: that the clay is intrinsically like the statue, that the clay has the same atoms as the statue', that objects don't have modal properties such as being necessarily F, and the reference of 'property' changes. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], II) | |
| A reaction: [my summary of the arguments she identifies - see text for details] Rudder Baker attempts to refute all four of these arguments, in defence of constitution as different from identity. |
| 16077 | The clay is not a statue - it borrows that property from the statue it constitutes [Rudder Baker] |
| Full Idea: I argue that a lump of clay borrows the property of being a statue from the statue. The lump is a statue because, and only because, there is something that the lump constitutes that is a statue. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], n9) | |
| A reaction: It is skating on very thin metaphysical ice to introduce the concept of 'borrowing' a property. I've spent the last ten minutes trying to 'borrow' some properties, but without luck. |
| 16080 | Is it possible for two things that are identical to become two separate things? [Rudder Baker] |
| Full Idea: A strong intuition shared by many philosophers is that some things that are in fact identical might not have been identical. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: This flies in the face of the Kripkean view that if Hesperus=Phosphorus then the identity is necessary. I don't think I have an intuition that some given thing might have been two things - indeed the thought seems totally weird. Amoeba? Statue/clay? |
| 16076 | Constitution is not identity, as consideration of essential predicates shows [Rudder Baker] |
| Full Idea: I want to resuscitate an essentialist argument against the view that constitution is identity, of the form 'x is essentially F, y is not essentially F, so x is not y'. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], Intro) | |
| A reaction: The point is that x might be essentially F and y only accidentally F. Thus a statue is essentially so, but a lump if clay is not essentially a statue. Another case where 'necessary' would do instead of 'essentially'. |
| 16081 | The constitution view gives a unified account of the relation of persons/bodies, statues/bronze etc [Rudder Baker] |
| Full Idea: Constitution-without-identity is superior to constitution-as-identity in that it provides a unified view of the relation between persons and bodies, statues and pieces of bronze, and so on. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: I have a problem with the intrinsic dualism of this whole picture. Clay needs shape, statues need matter - there aren't two 'things' here which have a 'relation'. |
| 16082 | Statues essentially have relational properties lacked by lumps [Rudder Baker] |
| Full Idea: The statue has relational properties which the lump of clay does not have essentially. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], V) | |
| A reaction: She has in mind relations to the community of artistic life. I don't think this is convincing. Is something only a statue if it is validated by an artistic community? That sounds like relative identity, which she doesn't like. |
| 21517 | Individuals need creativity, reverence for others, and self-respect [Russell] |
| Full Idea: What we shall desire for individuals is now clear: strong creative impulses, overpowering and absorbing the instinct of possession; reverence for others; respect for the fundamental creative impulses in ourselves. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Interesting that when Russell focuses on morality, he turns to virtues, rather than to rules. He uses 'reverence' where I would favour 'respect'. His concept of creativity is broad, and does not just concern art etc. |
| 21523 | We would not want UK affairs to be settled by a world parliament [Russell] |
| Full Idea: We should none of us like the affairs of Great Britain to be settled by a parliament of the world. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: The UK is currently (Dec 2018) living with a plan to quit Europe, mainly on the grounds that a European parliament has some authority over Britain. In every country resentment of the government increases with distance from the capital city. |
| 21522 | Democracy is inadequate without a great deal of devolution [Russell] |
| Full Idea: Democracy is not at all an adequate device unless it is accompanied by a very great amount of devolution. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: This whole book of Russell's is an appeal for the devolution of power, and for workplace democracy. |
| 21521 | Anarchy does not maximise liberty [Russell] |
| Full Idea: The greatest degree of liberty is not secured by anarchy. ...[22] The results of anarchy between states should suffice to persuade us that anarchism has no solution to offer for the evils of the world. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: I've heard Russell described as an anarchist, but this clearly wasn't true in 1917. Presumably liberty has to be protected. That we were watching anarchy between states in 1917 is a vivid observation. |
| 21528 | Groups should be autonomous, with a neutral authority as arbitrator [Russell] |
| Full Idea: For maximum freedom with minimum force: Autonomy within each politically important group, and a neutral authority for deciding questions involving relations between groups. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: This is workplace democracy, and also considerable self-rule amongst minority groups such as religions. |
| 21527 | On every new question the majority is always wrong at first [Russell] |
| Full Idea: It is a mistake to suppose that the majority is necessarily right. On every new question the majority is always wrong at first. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: Sounds like bitter experience. This is a good argument for taking time over decisions, and (topical) for a second referendum some time after the first one (if you must have a referendum). |
| 21526 | Unfortunately ordinary voters can't detect insincerity [Russell] |
| Full Idea: It is a painful fact that the ordinary voter, at any rate in England, is quite blind to insincerity. | |
| From: Bertrand Russell (Political Ideals [1917], 3) | |
| A reaction: Gor blimey yes! Well said, Bertie. Even in the age of television, when you can examine them in closeup, people seem to confuse superficial charm with genuine positive convictions. Why are people better at detecting it in private life? |
| 21525 | When the state is the only employer, there is no refuge from the prejudices of other people [Russell] |
| Full Idea: Under state socialism ...where the State is the only employer, there is no refuge from its prejudices such as may now accidentally arise through the differing opinions of men. | |
| From: Bertrand Russell (Political Ideals [1917], 2) | |
| A reaction: There is also a strong likelihood in full state socialism that the state will control housing as well as employment. This hadn't come to pass in 1917. |
| 21518 | Men unite in pursuit of material things, and idealise greed as part of group loyalty [Russell] |
| Full Idea: Men combine in groups to attain more strength in the scramble for material goods, and loyalty to the group spreads a halo of quasi-idealism round the central impulse of greed. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: See the 'greed is good' speech in the film 'Wall Street'. This sounds like a description of the USA, but Russell was very much in England at this stage. |
| 21519 | We need security and liberty, and then encouragement of creativity [Russell] |
| Full Idea: Security and liberty are only the negative conditions for good political institutions. When they have been won, we need also the positive condition: encouragement of creative energy. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: This sounds like some sort of liberal socialism. The nearest connection I can see is to the 'capabilities approach' of Martha Nussbaum. How do you intervene to encourage creativity? |
| 21524 | The right to own land gives a legal right to a permanent income [Russell] |
| Full Idea: There are many ways of becoming rich without contributing anything to the wealth of the community. Ownership of land or capital, whether acquired or inherited, gives a legal right to a permanent income. | |
| From: Bertrand Russell (Political Ideals [1917], 2) | |
| A reaction: I suspect that in the past land was the main source of this right, but now it is more likely to be capital. Land carries obligations of some sort, so income from capital is more fun. |
| 21520 | That our heaven is a dull place reflects the misery of excessive work in life [Russell] |
| Full Idea: It is a sad evidence of the weariness mankind has suffered from excessive toil that his heavens have usually been places where nothing ever happened or changed. | |
| From: Bertrand Russell (Political Ideals [1917], 1) | |
| A reaction: Has any religion got an idea of heaven as a place full of lively activity and creative problem-solving? That is what suits us best. |