31 ideas
| 10775 | The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp] |
| Full Idea: The main objection to the axiom of choice was that it had to be given by some law or definition, but since sets are arbitrary this seems irrelevant. Formalists consider it meaningless, but set-theorists consider it as true, and practically obvious. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3) |
| 10766 | Logic is either for demonstration, or for characterizing structures [Tharp] |
| Full Idea: One can distinguish at least two quite different senses of logic: as an instrument of demonstration, and perhaps as an instrument for the characterization of structures. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: This is trying to capture the proof-theory and semantic aspects, but merely 'characterizing' something sounds like a rather feeble aspiration for the semantic side of things. Isn't it to do with truth, rather than just rule-following? |
| 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) |
| 10769 | Second-order logic isn't provable, but will express set-theory and classic problems [Tharp] |
| Full Idea: The expressive power of second-order logic is too great to admit a proof procedure, but is adequate to express set-theoretical statements, and open questions such as the continuum hypothesis or the existence of big cardinals are easily stated. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10762 | In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' [Tharp] |
| Full Idea: In sentential logic there is a simple proof that all truth functions, of any number of arguments, are definable from (say) 'not' and 'and'. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §0) | |
| A reaction: The point of 'say' is that it can be got down to two connectives, and these are just the usual preferred pair. |
| 10776 | The main quantifiers extend 'and' and 'or' to infinite domains [Tharp] |
| Full Idea: The symbols ∀ and ∃ may, to start with, be regarded as extrapolations of the truth functional connectives ∧ ('and') and ∨ ('or') to infinite domains. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §5) |
| 17743 | De Morgan introduced a 'universe of discourse', to replace Boole's universe of 'all things' [De Morgan, by Walicki] |
| Full Idea: In 1846 De Morgan introduced the enormously influential notion of a possibly arbitrary and stipulated 'universe of discourse'. It replaced Boole's original - and metaphysically a bit suspect - universe of 'all things'. | |
| From: report of Augustus De Morgan (works [1846]) by Michal Walicki - Introduction to Mathematical Logic History D.1.1 | |
| A reaction: This not only brings formal logic under control, but also reflects normal talk, because there is always an explicit or implicit domain of discourse when we talk. Of virtually any conversation, you can say what it is 'about'. |
| 10774 | There are at least five unorthodox quantifiers that could be used [Tharp] |
| Full Idea: One might add to one's logic an 'uncountable quantifier', or a 'Chang quantifier', or a 'two-argument quantifier', or 'Shelah's quantifier', or 'branching quantifiers'. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §3) | |
| A reaction: [compressed - just listed for reference, if you collect quantifiers, like collecting butterflies] |
| 10777 | Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp] |
| Full Idea: Skolem deduced from the Löwenheim-Skolem theorem that 'the absolutist conceptions of Cantor's theory' are 'illusory'. I think it is clear that this conclusion would not follow even if elementary logic were in some sense the true logic, as Skolem assumed. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §7) | |
| A reaction: [Tharp cites Skolem 1962 p.47] Kit Fine refers to accepters of this scepticism about the arithmetic of infinities as 'Skolemites'. |
| 10773 | The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp] |
| Full Idea: The Löwenheim-Skolem property seems to be undesirable, in that it states a limitation concerning the distinctions the logic is capable of making, such as saying there are uncountably many reals ('Skolem's Paradox'). | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10765 | Soundness would seem to be an essential requirement of a proof procedure [Tharp] |
| Full Idea: Soundness would seem to be an essential requirement of a proof procedure, since there is little point in proving formulas which may turn out to be false under some interpretation. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10763 | Completeness and compactness together give axiomatizability [Tharp] |
| Full Idea: Putting completeness and compactness together, one has axiomatizability. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §1) |
| 10770 | If completeness fails there is no algorithm to list the valid formulas [Tharp] |
| Full Idea: In general, if completeness fails there is no algorithm to list the valid formulas. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: I.e. the theory is not effectively enumerable. |
| 10771 | Compactness is important for major theories which have infinitely many axioms [Tharp] |
| Full Idea: It is strange that compactness is often ignored in discussions of philosophy of logic, since the most important theories have infinitely many axioms. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: An example of infinite axioms is the induction schema in first-order Peano Arithmetic. |
| 10772 | Compactness blocks infinite expansion, and admits non-standard models [Tharp] |
| Full Idea: The compactness condition seems to state some weakness of the logic (as if it were futile to add infinitely many hypotheses). To look at it another way, formalizations of (say) arithmetic will admit of non-standard models. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10764 | A complete logic has an effective enumeration of the valid formulas [Tharp] |
| Full Idea: A complete logic has an effective enumeration of the valid formulas. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) |
| 10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge [Tharp] |
| Full Idea: Despite completeness, the mere existence of an effective enumeration of the valid formulas will not, by itself, provide knowledge. For example, one might be able to prove that there is an effective enumeration, without being able to specify one. | |
| From: Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2) | |
| A reaction: The point is that completeness is supposed to ensure knowledge (of what is valid but unprovable), and completeness entails effective enumerability, but more than the latter is needed to do the key job. |
| 18648 | Freedom to live according to our own conception of the good is the ultimate value [Nozick, by Kymlicka] |
| Full Idea: Nozick says that the freedom to lead our lives in accordance with our own conception of the good is the ultimate value, so important that it cannot be sacrificed for other social ideals (e.g. equality of opportunity). | |
| From: report of Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.2.b.ii | |
| A reaction: Clearly this ultimate value will not apply to children, so this view needs a sharp dislocation between children and adults. But some adults need a lot of looking after. Maybe we ALL need looking after (by one another)? |
| 20585 | If an experience machine gives you any experience you want, should you hook up for life? [Nozick] |
| Full Idea: Suppose there were an experience machine that would give you any experience you desired ...such as writing a great novel, or making a friend, or reading an interesting book. ...Should you plug into this machine for life? | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], 3 'Experience') | |
| A reaction: A classic though experiment which crystalises a major problem with hedonistic utilitarianism. My addition is a machine which maximises the pleasure of my family and friends, to save me the bother of doing it. |
| 18643 | A minimal state should protect, but a state forcing us to do more is unjustified [Nozick] |
| Full Idea: A minimal state, limited to the narrow functions of protection against force, theft, fraud, enforcement of contracts, and so on, is justified; any more extensive state will violate persons' rights not to be forced to do certain things, and is unjustified. | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], Pref) | |
| A reaction: This has some plausibility for a huge modern state, where we don't know one another, but it would be a ridiculous attitude in a traditional village. |
| 18642 | Individual rights are so strong that the state and its officials must be very limited in power [Nozick] |
| Full Idea: Individuals have rights, and there are things no person or group may do to them (without violating their rights). So strong and far-reaching are these rights that they raise the question of what, if anything, the state and its officials may do. | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], Pref) | |
| A reaction: This claim appears to be an axiom, but I'm not sure that the notion of 'rights' make any sense unless someone is granting the rights, where the someone is either a strong individual, or the community (perhaps represented by the state). |
| 18644 | States can't enforce mutual aid on citizens, or interfere for their own good [Nozick] |
| Full Idea: A state may not use its coercive apparatus for the purposes of getting some citizens to aid others, or in order to prohibit activities to people for their own good or protection. | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], Pref) | |
| A reaction: You certainly can't apply these principles to children, so becoming an 'adult' seems to be a very profound step in Nozick's account. At what age must we stop interfering with people for their own good. If the state is prohibited, are neighbours also? |
| 22661 | My Anarchy, State and Utopia neglected our formal social ties and concerns [Nozick on Nozick] |
| Full Idea: The political philosophy represented in Anarchy, State and Utopia ignored the importance of joint and official symbolic statement and expression of our social ties and concern, and hence (I have written) is inadequate. | |
| From: comment on Robert Nozick (Anarchy,State, and Utopia [1974], p.32) by Robert Nozick - The Nature of Rationality p.32 | |
| A reaction: In other words, it was far too individualistic, and neglected community, even though it has become the sacred text for libertarian individualism. Do any Nozick fans care about this recantation? |
| 18641 | If people hold things legitimately, just distribution is simply the result of free exchanges [Nozick, by Kymlicka] |
| Full Idea: If we assume that everyone is entitled to the goods they currently possess (their 'holdings'), then a just distribution is simply whatever distribution results from people's free exchanges. | |
| From: report of Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.1.b | |
| A reaction: If people's current 'legitimate' holdings are hugely unequal, it seems very unlikely that the ensuing exchanges will be 'free' in the way that Nozick envisages. |
| 20539 | Property is legitimate by initial acquisition, voluntary transfer, or rectification of injustice [Nozick, by Swift] |
| Full Idea: Nozick identified three ways in which people can acquire a legitimate property holding: initial acquisition, voluntary transfer, and rectification (of unjust transfers). | |
| From: report of Robert Nozick (Anarchy,State, and Utopia [1974]) by Adam Swift - Political Philosophy (3rd ed) 1 'Nozick' | |
| A reaction: I think it is a delusion to look for justice in the ownership of property. You can't claim justice for buying property if the money to do it was acquired unjustly. And what rights over those who live on the land come with the 'ownership'? |
| 18645 | Nozick assumes initial holdings include property rights, but we can challenge that [Kymlicka on Nozick] |
| Full Idea: Nozick assumes that the initial distribution of holdings includes full property-rights over them, ..but our preferred theory may not involve distributing such particular rights to particular people. ...The legitimacy of such rights is what is in question. | |
| From: comment on Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.1.c | |
| A reaction: [somewhat compressed] All of these political philosophies seem to have questionable values (such as freedom or equality) built into their initial assumptions. |
| 20521 | Can I come to own the sea, by mixing my private tomato juice with it? [Nozick] |
| Full Idea: If I own a can of tomato juice and spill it in the sea so that its molecules mingle evenly throughout the sea, do I thereby come to own the sea? | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], p.175) | |
| A reaction: This is a reductio of Locke's claim that I can own land by 'mixing' my labour with it. At first glance, mixing something with something would seem to have nothing to do with ownership. |
| 18646 | How did the private property get started? If violence was involved, we can redistribute it [Kymlicka on Nozick] |
| Full Idea: How did these natural resources, which were not initially owned by anyone, come to be part of someone's private property? ...The fact that the initial acquisition often involved force means there is no moral objection to redistributing existing wealth. | |
| From: comment on Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.2.b | |
| A reaction: [He cites G.A. Cphen 1988 for the second point] Put like this, Nozick's theory just looks like the sort of propaganda which is typically put out by the winners. Is there an implicit threat of violent resistance in his advocacy of individual rights? |
| 18647 | If property is only initially acquired by denying the rights of others, Nozick can't get started [Kymlicka on Nozick] |
| Full Idea: If there is no way that people can appropriate unowned resources for themselves without denying other people's claim to equal consideration, then Nozick's right of transfer never gets off the ground. | |
| From: comment on Robert Nozick (Anarchy,State, and Utopia [1974]) by Will Kymlicka - Contemporary Political Philosophy (1st edn) 4.2.b.i | |
| A reaction: The actual history of these things is too complex to judge. Early peoples desperately wanted a lord to rule over them, and their lord's ownership of the land implied the people's right to live there. See Anglo-Saxon poetry. |
| 21737 | Unowned things may be permanently acquired, if it doesn't worsen the position of other people [Nozick] |
| Full Idea: One may acquire a permanent bequeathable property right in a previously unowned thing, as long as the position of others no longer at liberty to use the thing is not thereby worsened. | |
| From: Robert Nozick (Anarchy,State, and Utopia [1974], p.178), quoted by G.A. Cohen - Are Freedom and Equality Compatible? 2 | |
| A reaction: Cohen attacks this vigorously. His main point is that Nozick has a very narrow view of what the acquisition should be compared with. There are many alternatives. Does being made unable to improve something 'worsen' a person's condition? |
| 21738 | Maybe land was originally collectively owned, rather than unowned? [Cohen,GA on Nozick] |
| Full Idea: Why should we not regard land as originally collectively owned rather than, as Nozick takes for granted, owned by no one? | |
| From: comment on Robert Nozick (Anarchy,State, and Utopia [1974], p.178) by G.A. Cohen - Are Freedom and Equality Compatible? 2 | |
| A reaction: Did native Americans and Australians collectively own the land? Lots of peoples, I suspect, don't privately own anything, because the very concept has never occured to them (and they have no legal system). |