50 ideas
| 10757 | Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg] |
| Full Idea: Henkin semantics (for second-order logic) specifies a second domain of predicates and relations for the upper case constants and variables. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This second domain is restricted to predicates and relations which are actually instantiated in the model. Second-order logic is complete with this semantics. Cf. Idea 10756. |
| 10759 | There are at least seven possible systems of semantics for second-order logic [Rossberg] |
| Full Idea: In addition to standard and Henkin semantics for second-order logic, one might also employ substitutional or game-theoretical or topological semantics, or Boolos's plural interpretation, or even a semantics inspired by Lesniewski. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This is helpful in seeing the full picture of what is going on in these logical systems. |
| 10751 | Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg] |
| Full Idea: Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1) | |
| A reaction: The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics. |
| 10753 | Logical consequence is intuitively semantic, and captured by model theory [Rossberg] |
| Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation. |
| 10752 | Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg] |
| Full Idea: Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another. |
| 10754 | In proof-theory, logical form is shown by the logical constants [Rossberg] |
| Full Idea: A proof-theorist could insist that the logical form of a sentence is exhibited by the logical constants that it contains. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2) | |
| A reaction: You have to first get to the formal logical constants, rather than the natural language ones. E.g. what is the truth table for 'but'? There is also the matter of the quantifiers and the domain, and distinguishing real objects and predicates from bogus. |
| 10756 | A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg] |
| Full Idea: A standard model is a set of objects called the 'domain', and an interpretation function, assigning objects in the domain to names, subsets to predicate letters, subsets of the Cartesian product of the domain with itself to binary relation symbols etc. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: The model actually specifies which objects have which predicates, and which objects are in which relations. Tarski's account of truth in terms of 'satisfaction' seems to be just a description of those pre-decided facts. |
| 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |
| Full Idea: A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'. |
| 10761 | Completeness can always be achieved by cunning model-design [Rossberg] |
| Full Idea: All that should be required to get a semantics relative to which a given deductive system is complete is a sufficiently cunning model-theorist. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §5) |
| 10755 | A deductive system is only incomplete with respect to a formal semantics [Rossberg] |
| Full Idea: No deductive system is semantically incomplete in and of itself; rather a deductive system is incomplete with respect to a specified formal semantics. | |
| From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3) | |
| A reaction: This important point indicates that a system might be complete with one semantics and incomplete with another. E.g. second-order logic can be made complete by employing a 'Henkin semantics'. |
| 7222 | It is a crime for someone with a violent disposition to get drunk [Mill] |
| Full Idea: The making himself drunk, in a person whom drunkenness excites to do harm to others, is a crime against others. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This principle (based on knowing your own dispositions) is a very good account of the ethics drunkenness. We have a moral duty to know and remember our own dispositions. Violent people should avoid arguments as well as alcohol. |
| 7214 | Ethics rests on utility, which is the permanent progressive interests of people [Mill] |
| Full Idea: I regard utility as the ultimate appeal on all ethical questions; but it must be utility in the largest sense, grounded on the permanent interests of a man as a progressive being. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: Mill, writing in praise of personal liberty, is desperate to introduce a paternalistic element into his politics, and the 'maximisation of happiness' will justify such paternalism, while his basic liberal principle (Idea 7211) won't. Mill's Dilemma. |
| 22673 | Wherever there is a small community, the association of the people is natural [Tocqueville] |
| Full Idea: The village or township is the only association which is so perfectly natural that, wherever a number of men are collected, it seems to constitute itself. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: Seems like a chicken and egg issue. I would have thought that association precedes the development of a village. |
| 22676 | The people are just individuals, and only present themselves as united to foreigners [Tocqueville] |
| Full Idea: The people in themselves are only individuals; and the special reason why they need to be united under one government is that they may appear to advantage before foreigners. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: I take this to be an observation on 1830s America, rather than a universal truth. It fits modern western societies rather well though. |
| 22679 | Vast empires are bad for well-being and freedom, though they may promote glory [Tocqueville] |
| Full Idea: Nothing is more opposed to the well-being and the freedom of men than vast empires. …But there is a love of glory in those who regard the applause of a great people as a worthy reward. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Presumably the main the problem is the central dominance over distant colonies. There may also be some freedom in being distant from the centres, especially in 1830. The Wild West. |
| 22680 | People would be much happier and freer in small nations [Tocqueville] |
| Full Idea: If none but small nations existed, I do not doubt that mankind would be more happy and more free. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: In modern times many small states have appeared in Europe (in the Balkans and on the Baltic), and it looks to me a good thing. The prospect of Scottish independence may currently be looming, and De Tocqueville would approve. |
| 7212 | Individuals have sovereignty over their own bodies and minds [Mill] |
| Full Idea: Over himself, over his own body and mind, the individual is sovereign. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: If I should not even think about evil deeds, then neither should you. I would prevent you if I could. I would prevent you from drinking yourself to death, if I could. It is just that intrusions into private lives leads to greater trouble. |
| 7210 | The will of the people is that of the largest or most active part of the people [Mill] |
| Full Idea: The will of the people practically means the will of the most numerous or the most active part of the people. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: Hence the nicely coined modern phrase 'the silent majority', on whose behalf certain politicians, usually conservative, offer to speak. It is unlikely that the silent majority are actually deeply opposed to the views of the very active part. |
| 22675 | In American judges rule according to the Constitution, not the law [Tocqueville] |
| Full Idea: The Americans have acknowledged the right of judges to found their decisions on the Constitution, rather than on the laws. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.05) | |
| A reaction: Obviously the Constitution is one short document, so the details must be enshrined in the laws (which presumably defer to the Constitution). |
| 22677 | A monarchical family is always deeply concerned with the interests of the state [Tocqueville] |
| Full Idea: The advantages of a monarchy are that the private interests of a family are connected with the interests of the state, …and at least there is always someone available to conduct the affairs of a monarchy. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: The second one is not much of a reason! The same defence can be given for the dominance of the Mafia. His defences are deliberately feeble, I suspect. England had plenty of monarchs who showed limited interest. |
| 7227 | It is evil to give a government any more power than is necessary [Mill] |
| Full Idea: Government interference should be restricted because of the great evil of adding unnecessarily to its power. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This would need justification, because it might be replied that individuals should not have unnecessary power either. The main problem is that governments have armies, police and money. |
| 22683 | Despots like to see their own regulations ignored, by themselves and their agents [Tocqueville] |
| Full Idea: In despotic states the sovereign is so much attached to his power that he dislikes the constraints even of his own regulations, and likes to see his agents acting irregularly. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: A nice observation. What would Machiavelli say? At least the citizens can see where the real power resides. |
| 22669 | Aristocracy is constituted by inherited landed property [Tocqueville] |
| Full Idea: Land is the basis of an aristocracy; …it is by landed property handed down from generation to generation that an aristocracy is constituted. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: Presumably there can be aristocrats by mere royal patronage, who have perhaps gambled away their land. They need protection by the other aristocrats. |
| 7228 | Individuals often do things better than governments [Mill] |
| Full Idea: Government power should be restricted because things are often done better by individuals. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This contains some truth, but it is obvious that innumerable things can be done better by governments, and also (and more importantly) that innumerable other good things might be done by governments which individuals can't be bothered to do. |
| 22674 | In Europe it is thought that local government is best handled centrally [Tocqueville] |
| Full Idea: The partisans of centralisation in Europe are wont to maintain that the government can administer the affairs of each locality better than the citizens can do it for themselves. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: In the modern UK we have lots of local government, which is thoroughly starved of funds by the central government. He is contrasting it with the strong local system in the U.S. |
| 7230 | Aim for the maximum dissemination of power consistent with efficiency [Mill] |
| Full Idea: The safest practical ideal is to aim for the greatest dissemination of power consistent with efficiency. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a very nice principle, which I would think desirable within an institution as well as on the scale of the state. I am becoming a fan of Mill's politics. I still say that freedom is an overrated virtue, so efficiency must be underrated. |
| 20515 | Maximise happiness by an area of strict privacy, and an area of utilitarian interventions [Mill, by Wolff,J] |
| Full Idea: For Mill the greatest happiness will be achieved by giving people a private sphere of interests where no intervention is permitted, while allowing a public sphere where intervention is possible, but only on utilitarian grounds. | |
| From: report of John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Liberty' | |
| A reaction: This is probably standard liberal practice nowadays. Freely consenting adult sexual activity is agreed to be wholly private. At least some lip-service is paid to increasing happiness when government intervenes. |
| 7229 | People who transact their own business will also have the initiative to control their government [Mill] |
| Full Idea: A people accustomed to transacting their own business is certain to be free; it will never let itself be enslaved by any man or body of men because these are able to seize and pull the reins of the central administration. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: He makes reference to Americans. This is an important idea, because it shows that democratic control is not just a matter of elections (which can be abolished or suborned), but is also a characteristic of a certain way of life. |
| 22678 | An election, and its lead up time, are always a national crisis [Tocqueville] |
| Full Idea: The period which immediately precedes an election, and that during which the election is taking place, must always be considered as a national crisis. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Rousseau said something similar. Election day in modern Britain is very peaceful and civilised, but it used to be chaotic. The weeks preceding it are invariably nasty. |
| 22682 | Universal suffrage is no guarantee of wise choices [Tocqueville] |
| Full Idea: Universal suffrage is by no means a guarantee of the wisdom of the popular choice. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: This was precisely Plato's fear about democracy. There seems no way at all of preventing the people from electing representatives on superficial grounds of personality. |
| 7211 | Prevention of harm to others is the only justification for exercising power over people [Mill] |
| Full Idea: The only purpose for which power can be rightfully exercised over any member of a civilised community, against his will, is to prevent harm to others; his own good, either physical or moral, is not a sufficient warrant. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: This is the key idea in Mill's liberalism, though he goes on to offer some qualifications of this absolute prohibition. I don't disagree with this principle, but there may be a lot more indirect harm than we realise (eg. in allowing liberal sex or drugs). |
| 7231 | The worth of a State, in the long run, is the worth of the individuals composing it [Mill] |
| Full Idea: The worth of a State, in the long run, is the worth of the individuals composing it. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a key idea of liberalism, opposed to any idea that we should abandon our own value to that of our state. I agree, but communitarians can subscribe to this too, while disagreeing that maximum freedom is the strategy to follow. |
| 7217 | The main argument for freedom is that interference with it is usually misguided [Mill] |
| Full Idea: The strongest of all the arguments against the interference of the public with purely personal conduct is that, when it does interfere, the odds are that it interferes wrongly, and in the wrong place. | |
| From: John Stuart Mill (On Liberty [1857], Ch.4) | |
| A reaction: This is also a well known objection to capital punishment. Generalised, well established, legal interferences are perhaps more likely to get it right than ad hoc decisions about individuals by individual officials. |
| 22670 | Slavery undermines the morals and energy of a society [Tocqueville] |
| Full Idea: Slavery dishonours labour; it introduces idleness into society, and with idleness, ignorance and pride, luxury and distress. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: A pretty feeble reason (in the 1830s) for disliking slavery. He seems only concerned with the adverse effects on the slave-owning society, and shows no interest in the slaves themselves. |
| 22681 | The liberty of the press is more valuable for what it prevents than what it promotes [Tocqueville] |
| Full Idea: I approve of the liberty of the press from a consideration more of the evils it prevents than of the advantages it ensures. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.10) | |
| A reaction: He accepts the freedom of the press as inevitable in a democracy, but he found U.S. newspapers to be nearly as bad then as they are now. |
| 7213 | Liberty arises at the point where people can freely and equally discuss things [Mill] |
| Full Idea: Liberty, as a principle, has no application to any state of things anterior to the time when mankind have become capable of being improved by free and equal discussion. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: There is a Victorian (and Enlightenment) optimism here which a glimpse of the freedoms of the early twenty-first century might dampen. I doubt if Mill expected British tabloid newspapers, or porn on cable TV. Education and freedom connect. |
| 20516 | Mill defends freedom as increasing happiness, but maybe it is an intrinsic good [Wolff,J on Mill] |
| Full Idea: Mill has presented liberty as instrumentally valuable, as a way of achieving the greatest possible happiness in society. But perhaps he should have argued that liberty is an intrinsic good, good in itself. | |
| From: comment on John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Intrinsic' | |
| A reaction: If freedom is intrinsically good, does this leave us (as Wolff warned earlier) unable to defend its value? Freedom isn't an intrinsic good for infants, so why should it be so for adults? Good because it brings happiness, or fulfils our nature? |
| 20517 | Utilitarianism values liberty, but guides us on which ones we should have or not have [Mill, by Wolff,J] |
| Full Idea: Utilitarianism provides an account of what liberties we should and should not have. Mill argues we should be free to compete in trade, but not to use another's property without consent. Thus he sets limits to liberty, while paying it great respect. | |
| From: report of John Stuart Mill (On Liberty [1857]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Intrinsic' |
| 7215 | True freedom is pursuing our own good, while not impeding others [Mill] |
| Full Idea: The only freedom which deserves the name, is that of pursuing our own good in our own way, so long as we do not attempt to deprive others of theirs, or impede their efforts to obtain it. | |
| From: John Stuart Mill (On Liberty [1857], Ch.1) | |
| A reaction: This principle will probably lead up a Prisoner's Dilemma cul-de-sac. The only freedom which deserves the name is the collective agreed freedom of a whole community to live well, when citizens volunteer to restrict their individual freedoms. |
| 7220 | Restraint for its own sake is an evil [Mill] |
| Full Idea: All restraint, qua restraint, is an evil. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: The ultimate justification for this is (presumably) utilitarian, but that would mean that there was nothing wrong with restraint if the person did not mind, or was not aware of the restraint. What is intrinsically wrong with restraint? |
| 7218 | Individuals are not accountable for actions which only concern themselves [Mill] |
| Full Idea: My first maxim is that the individual is not accountable to society for his actions, in so far as these concern the interests of no person but himself. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is a key idea of liberalism, and one which communitarians have doubts about (because it is almost impossible to perform an action which is of no interest, in the short or long term, to others). I share these doubts. |
| 7221 | Blocking entry to an unsafe bridge does not infringe liberty, since no one wants unsafe bridges [Mill] |
| Full Idea: An official could turn a person back from an unsafe bridge without infringeing their liberty; for liberty consists in doing what one desires, and he does not desire to fall into the river. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: Seems fair enough, but it justifies paternalist interference. The tricky one is where the official and the citizen disagree over what the citizen 'truly' desires. Asking people may involve too much time, but it could also involve too much effort. |
| 7223 | Pimping and running a gambling-house are on the border between toleration and restraint [Mill] |
| Full Idea: A person being free to be a pimp, or to keep a gambling-house, lies on the exact boundary line between two principles, of toleration and of restraint. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: Nothing illuminates a philosopher's principles more than for them to specify cases that lie on their borderlines. Both professions seem, unfortunately, to lead people into worse activities, such as violent bullying, or theft. Tricky.. |
| 22672 | It is admirable to elevate the humble to the level of the great, but the opposite is depraved [Tocqueville] |
| Full Idea: One manly and lawful passion for equality elevates the humble to the rank of the great. But there exists also a depraved taste for equality, which impels the weak to attempt to lower the powerful to their own level. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: There is a distinction in modern political rhetoric between 'levelling down' and 'levelling up'. Since levelling down is just destructive, and levelling up is unaffordable, it seems obvious that true equality needs to be a compromise. |
| 22671 | Equality can only be established by equal rights for all (or no rights for anyone) [Tocqueville] |
| Full Idea: I know of only two methods of establishing equality in the political world; rights must be given to every citizen, or none at all to anyone. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: We may have a vague concept of 'natural' rights, but primarily they are a tool of social engineering. You could grant equal rights on inheritance, for example, which turn out in practice to hugely favour the rich. |
| 7219 | Society can punish actions which it believes to be prejudicial to others [Mill] |
| Full Idea: My second maxim is that for actions that are prejudicial to the interests of others, the individual is accountable, and subject to social or legal punishment, if society believes that this is requisite for its protection. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: (wording compressed). The trouble with this would seem to be the possible disagreement between the individual and the society over whether the actions actually are prejudicial to others. It would justify a conservative society in being repressive. |
| 7226 | Benefits performed by individuals, not by government, help also to educate them [Mill] |
| Full Idea: It is often desirable that beneficial things should be done by individuals, rather than by the government, as a means to their own mental education. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This raises the important danger, which even those on the political left must acknowledge, of the 'nanny state'. It offers a nicely paternalistic, and even patronising reason for giving people freedom, just as a parent might to a child. |
| 7224 | We need individual opinions and conduct, and State education is a means to prevent that [Mill] |
| Full Idea: Individuality of character, and diversity in opinions and modes of conduct, involves diversity of education; a general State education is a mere contrivance for moulding people to be exactly like one another. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This strikes me as being particularly true with the advent in Britain of the National Curriculum in the early 1990s. However, if there is a pressure towards conformity in state education, private education is dominated by class and money. |
| 7225 | It is a crime to create a being who lacks the ordinary chances of a desirable existence [Mill] |
| Full Idea: To bestow a life on someone which may be either a curse or a blessing, unless the being on whom it is to be bestowed will have at least the ordinary chances of a desirable existence, is a crime against that being. | |
| From: John Stuart Mill (On Liberty [1857], Ch.5) | |
| A reaction: This is the standard utilitarian attitude to engendering people. I think I have to agree. It is no argument against this to say that we value people with poor life prospects, once they have arrived. Altruism towards children may disguise selfish parents. |
| 7216 | The ethics of the Gospel has been supplemented by barbarous Old Testament values [Mill] |
| Full Idea: To extract from the Gospel a body of ethical doctrine, has never been possible withouth eking it out from the Old Testament, that is, from a system elaborate indeed, but in many respects barbarous, and intended only for a barbarous people. | |
| From: John Stuart Mill (On Liberty [1857], Ch.2) | |
| A reaction: 'Barbarous' has a quaint Victorian ring to it, but his point is that the surviving teachings of Jesus are very thin and generalised. Christians would do better to expand their implications, than to borrow from the Old Testament. |