| 22873 | Liberalism should improve the system, and not just ameliorate it [Dewey] |
| Full Idea: Liberalism must become radical in the sense that, instead of using social power to ameliorate the evil consequences of the existing system, it shall use social power to change the system. | |
| From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:287), quoted by David Hildebrand - Dewey 4 'Dewey' | |
| A reaction: Conservative liberals ask what people want, and try to give it to them. Radical liberals ask what people actually need, and try to make it possible. The latter is bound to be a bit paternalistic, but will probably create a better world. |
| 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. |
| 18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
| Full Idea: The natural understanding of first-order logic is that in writing down first-order schemata we are implicitly asserting their validity, that is, making second-order assertions. ...Thus even quantification theory involves reference to classes. | |
| From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
| A reaction: If, as a nominalist, you totally rejected classes, presumably you would get by in first-order logic somehow. To say 'there are no classes so there is no logical validity' sounds bonkers. |
| 10767 | Elementary logic is complete, but cannot capture mathematics [Tharp] |
| Full Idea: Elementary logic cannot characterize the usual mathematical structures, but seems to be distinguished by its completeness. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 13840 | First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking] |
| Full Idea: First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem. | |
| From: Ian Hacking (What is Logic? [1979], §13) |
| 13844 | A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking] |
| Full Idea: Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification. | |
| From: Ian Hacking (What is Logic? [1979], §13) | |
| A reaction: See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours. |
| 18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
| Full Idea: First-order logic is not decidable. That is, there is no test which can be applied to any arbitrary formula of that logic and which will tell one whether the formula is or is not valid (as proved by Church in 1936). | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) |
| 18109 | The completeness of first-order logic implies its compactness [Bostock] |
| Full Idea: From the fact that the usual rules for first-level logic are complete (as proved by Gödel 1930), it follows that this logic is 'compact'. | |
| From: David Bostock (Philosophy of Mathematics [2009], 5.5) | |
| A reaction: The point is that the completeness requires finite proofs. |
| 10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
| Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!). | |
| From: Wilfrid Hodges (Model Theory [2005], 3) | |
| A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy! |
| 19315 | In quantified language the components of complex sentences may not be sentences [Kirkham] |
| Full Idea: In a quantified language it is possible to build new sentences by combining two expressions neither of which is itself a sentence. | |
| From: Richard L. Kirkham (Theories of Truth: a Critical Introduction [1992], 5.4) | |
| A reaction: In propositional logic the components are other sentences, so the truth value can be given by their separate truth-values, through truth tables. Kirkham is explaining the task which Tarski faced. Truth-values are not just compositional. |
| 17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
| Full Idea: First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.411-2) | |
| A reaction: He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1). |
| 13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
| Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
| A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment. |
| 13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
| Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5) | |
| A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on. |
| 13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
| Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1) |
| 13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
| Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another. | |
| From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1) | |
| A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic. |
| 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) |
| 10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
| Full Idea: A formula of a first-order language is a 'sentence' just in case it has no free variables. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) |
| 10986 | Not all validity is captured in first-order logic [Read] |
| Full Idea: We must recognise that first-order classical logic is inadequate to describe all valid consequences, that is, all cases in which it is impossible for the premisses to be true and the conclusion false. | |
| From: Stephen Read (Thinking About Logic [1995], Ch.2) | |
| A reaction: This is despite the fact that first-order logic is 'complete', in the sense that its own truths are all provable. |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic. |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable. | |
| From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3) | |
| A reaction: Peter Smith calls the stronger version 'negation completeness'. |
| 20100 | Classical liberalism seeks freedom of opinion, of private life, of expression, and of property [Micklethwait/Wooldridge] |
| Full Idea: The classical liberals agreed on a basic list of freedoms: of opinion (including religion), of private life, of expression, and of property | |
| From: Micklethwait,J/Wooldridge,A (The Fourth Revolution [2014], 9) | |
| A reaction: Mill is main articulator of this. Modern neo-liberals focus on economic freedom. Neither of them seem to make freedom of opportunity central, though I suspect our modern Liberal Party would. |
| 21122 | Liberal Nationalism says welfare states and democracy needed a shared sense of nationality [Shorten] |
| Full Idea: The Liberal Nationalist argument is that if we want to have welfare states or vibrant democracies, then we will need the kind of solidarity that shared nationality fosters. …Unwelcome democratic decisions are more acceptable when made by co-nationals. | |
| From: Andrew Shorten (Contemporary Political Theory [2016], 02) | |
| A reaction: We've just experienced this with Brexit (2016), where perfectly sensible decisions were being made in Brussels, but the popular press whipped up hostility because the British had a restricted role in the decisions. Prefer our idiots to their sages. |
| 22609 | Philosophers accepted first-order logic, because they took science to be descriptive, not explanatory [Ingthorsson] |
| Full Idea: First-order predicate logic was accepted so easily by the philosophical community …because philosophy was already geared toward a neo-Humean view of both science and philosophy as primarily descriptive rather than explanatory. | |
| From: R.D. Ingthorsson (A Powerful Particulars View of Causation [2021], 1.8) | |
| A reaction: The point, I think, is that explanatory thinking needs second-order logic, where the properties (or powers) are players in the game, and not just adjuncts of the catalogue of objects. I find this idea mind-expanding. (That's a good thing). |