46 ideas
| 10482 | The logic of ZF is classical first-order predicate logic with identity [Boolos] |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 18192 | Do the Replacement Axioms exceed the iterative conception of sets? [Boolos, by Maddy] |
| Full Idea: For Boolos, the Replacement Axioms go beyond the iterative conception. | |
| From: report of George Boolos (The iterative conception of Set [1971]) by Penelope Maddy - Naturalism in Mathematics I.3 |
| 7785 | The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos] |
| Full Idea: We should abandon the idea that the use of plural forms commits us to the existence of sets/classes… Entities are not to be multiplied beyond necessity. There are not two sorts of things in the world, individuals and collections. | |
| From: George Boolos (To be is to be the value of a variable.. [1984]), quoted by Henry Laycock - Object | |
| A reaction: The problem of quantifying over sets is notoriously difficult. Try http://plato.stanford.edu/entries/object/index.html. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |
| 13547 | Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Boolos, by Potter] |
| Full Idea: Weak Limitation of Size: If there are no more Fs than Gs and the Gs form a collection, then Fs form a collection. Strong Limitation of Size: A property F fails to be collectivising iff there are as many Fs as there are objects. | |
| From: report of George Boolos (Iteration Again [1989]) by Michael Potter - Set Theory and Its Philosophy 13.5 |
| 10699 | Does a bowl of Cheerios contain all its sets and subsets? [Boolos] |
| Full Idea: Is there, in addition to the 200 Cheerios in a bowl, also a set of them all? And what about the vast number of subsets of Cheerios? It is haywire to think that when you have some Cheerios you are eating a set. What you are doing is: eating the Cheerios. | |
| From: George Boolos (To be is to be the value of a variable.. [1984], p.72) | |
| A reaction: In my case Boolos is preaching to the converted. I am particularly bewildered by someone (i.e. Quine) who believes that innumerable sets exist while 'having a taste for desert landscapes' in their ontology. |
| 10830 | Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos] |
| Full Idea: The metatheory of second-order logic is hopelessly set-theoretic, and the notion of second-order validity possesses many if not all of the epistemic debilities of the notion of set-theoretic truth. | |
| From: George Boolos (On Second-Order Logic [1975], p.45) | |
| A reaction: Epistemic problems arise when a logic is incomplete, because some of the so-called truths cannot be proved, and hence may be unreachable. This idea indicates Boolos's motivation for developing a theory of plural quantification. |
| 14249 | Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley] |
| Full Idea: Boolos's conception of plural logic is as a reinterpretation of second-order logic. | |
| From: report of George Boolos (On Second-Order Logic [1975]) by Oliver,A/Smiley,T - What are Sets and What are they For? n5 | |
| A reaction: Oliver and Smiley don't accept this view, and champion plural reference differently (as, I think, some kind of metalinguistic device?). |
| 10225 | Monadic second-order logic might be understood in terms of plural quantifiers [Boolos, by Shapiro] |
| Full Idea: Boolos has proposed an alternative understanding of monadic, second-order logic, in terms of plural quantifiers, which many philosophers have found attractive. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984]) by Stewart Shapiro - Philosophy of Mathematics 3.5 |
| 10736 | Boolos showed how plural quantifiers can interpret monadic second-order logic [Boolos, by Linnebo] |
| Full Idea: In an indisputable technical result, Boolos showed how plural quantifiers can be used to interpret monadic second-order logic. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984], Intro) by Øystein Linnebo - Plural Quantification Exposed Intro |
| 10780 | Any sentence of monadic second-order logic can be translated into plural first-order logic [Boolos, by Linnebo] |
| Full Idea: Boolos discovered that any sentence of monadic second-order logic can be translated into plural first-order logic. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984], §1) by Øystein Linnebo - Plural Quantification Exposed p.74 |
| 10829 | A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos] |
| Full Idea: One may be of the opinion that no sentence ought to be considered as a truth of logic if, no matter how it is interpreted, it asserts that there are sets of certain sorts. | |
| From: George Boolos (On Second-Order Logic [1975], p.44) | |
| A reaction: My intuition is that in no way should any proper logic assert the existence of anything at all. Presumably interpretations can assert the existence of numbers or sets, but we should be able to identify something which is 'pure' logic. Natural deduction? |
| 10697 | Identity is clearly a logical concept, and greatly enhances predicate calculus [Boolos] |
| Full Idea: Indispensable to cross-reference, lacking distinctive content, and pervading thought and discourse, 'identity' is without question a logical concept. Adding it to predicate calculus significantly increases the number and variety of inferences possible. | |
| From: George Boolos (To be is to be the value of a variable.. [1984], p.54) | |
| A reaction: It is not at all clear to me that identity is a logical concept. Is 'existence' a logical concept? It seems to fit all of Boolos's criteria? I say that all he really means is that it is basic to thought, but I'm not sure it drives the reasoning process. |
| 10832 | '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos] |
| Full Idea: One may say that '∀x x=x' means 'everything is identical to itself', but one must realise that one's answer has a determinate sense only if the reference (range) of 'everything' is fixed. | |
| From: George Boolos (On Second-Order Logic [1975], p.46) | |
| A reaction: This is the problem now discussed in the recent book 'Absolute Generality', of whether one can quantify without specifying a fixed or limited domain. |
| 13671 | Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology [Boolos, by Shapiro] |
| Full Idea: Boolos proposes that second-order quantifiers be regarded as 'plural quantifiers' are in ordinary language, and has developed a semantics along those lines. In this way they introduce no new ontology. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984]) by Stewart Shapiro - Foundations without Foundationalism 7 n32 | |
| A reaction: This presumably has to treat simple predicates and relations as simply groups of objects, rather than having platonic existence, or something. |
| 10267 | We should understand second-order existential quantifiers as plural quantifiers [Boolos, by Shapiro] |
| Full Idea: Standard second-order existential quantifiers pick out a class or a property, but Boolos suggests that they be understood as a plural quantifier, like 'there are objects' or 'there are people'. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984]) by Stewart Shapiro - Philosophy of Mathematics 7.4 | |
| A reaction: This idea has potential application to mathematics, and Lewis (1991, 1993) 'invokes it to develop an eliminative structuralism' (Shapiro). |
| 10698 | Plural forms have no more ontological commitment than to first-order objects [Boolos] |
| Full Idea: Abandon the idea that use of plural forms must always be understood to commit one to the existence of sets of those things to which the corresponding singular forms apply. | |
| From: George Boolos (To be is to be the value of a variable.. [1984], p.66) | |
| A reaction: It seems to be an open question whether plural quantification is first- or second-order, but it looks as if it is a rewriting of the first-order. |
| 7806 | Boolos invented plural quantification [Boolos, by Benardete,JA] |
| Full Idea: Boolos virtually patented the new device of plural quantification. | |
| From: report of George Boolos (To be is to be the value of a variable.. [1984]) by José A. Benardete - Logic and Ontology | |
| A reaction: This would be 'there are some things such that...' |
| 10834 | Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos] |
| Full Idea: A weak completeness theorem shows that a sentence is provable whenever it is valid; a strong theorem, that a sentence is provable from a set of sentences whenever it is a logical consequence of the set. | |
| From: George Boolos (On Second-Order Logic [1975], p.52) | |
| A reaction: So the weak version says |- φ → |= φ, and the strong versions says Γ |- φ → Γ |= φ. Presumably it is stronger if it can specify the source of the inference. |
| 13841 | Why should compactness be definitive of logic? [Boolos, by Hacking] |
| Full Idea: Boolos asks why on earth compactness, whatever its virtues, should be definitive of logic itself. | |
| From: report of George Boolos (On Second-Order Logic [1975]) by Ian Hacking - What is Logic? §13 |
| 10491 | Infinite natural numbers is as obvious as infinite sentences in English [Boolos] |
| Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless. |
| 10483 | Mathematics and science do not require very high orders of infinity [Boolos] |
| Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.122) | |
| A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology? |
| 10833 | Many concepts can only be expressed by second-order logic [Boolos] |
| Full Idea: The notions of infinity and countability can be characterized by second-order sentences, though not by first-order sentences (as compactness and Skolem-Löwenheim theorems show), .. as well as well-ordering, progression, ancestral and identity. | |
| From: George Boolos (On Second-Order Logic [1975], p.48) |
| 10490 | Mathematics isn't surprising, given that we experience many objects as abstract [Boolos] |
| Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised. |
| 18680 | To avoid misunderstandings supervenience is often expressed negatively: no A-change without B-change [Orsi] |
| Full Idea: It is no part of supervenience that 'if p then q' entails 'if not p then not q'. To avoid such misunderstandings, it is common (though not more accurate) to describe supervenience in negative terms: no difference in A without a difference in B. | |
| From: Francesco Orsi (Value Theory [2015], 5.2) | |
| A reaction: [compressed] In other words it is important to avoid the presupposition that the given supervenience is a two-way relation. The paradigm case of supervenience is stalking. |
| 10700 | First- and second-order quantifiers are two ways of referring to the same things [Boolos] |
| Full Idea: Ontological commitment is carried by first-order quantifiers; a second-order quantifier needn't be taken to be a first-order quantifier in disguise, having special items, collections, as its range. They are two ways of referring to the same things. | |
| From: George Boolos (To be is to be the value of a variable.. [1984], p.72) | |
| A reaction: If second-order quantifiers are just a way of referring, then we can see first-order quantifiers that way too, so we could deny 'objects'. |
| 10488 | It is lunacy to think we only see ink-marks, and not word-types [Boolos] |
| Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words? |
| 10487 | I am a fan of abstract objects, and confident of their existence [Boolos] |
| Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'. |
| 10489 | We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos] |
| Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'? |
| 7295 | Maybe induction is only reliable IF reality is stable [Mitchell,A] |
| Full Idea: Maybe we should say that IF regularities are stable, only then is induction a reliable procedure. | |
| From: Alistair Mitchell (talk [2006]), quoted by PG - Db (ideas) | |
| A reaction: This seems to me a very good proposal. In a wildly unpredictable reality, it is hard to see how anyone could learn from experience, or do any reasoning about the future. Natural stability is the axiom on which induction is built. |
| 8693 | An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect [Boolos] |
| Full Idea: Hume's Principle has a structure Boolos calls an 'abstraction principle'. Within the scope of two universal quantifiers, a biconditional connects an identity between two things and an equivalence relation. It says we don't care about other differences. | |
| From: George Boolos (Is Hume's Principle analytic? [1997]), quoted by Michèle Friend - Introducing the Philosophy of Mathematics 3.7 | |
| A reaction: This seems to be the traditional principle of abstraction by ignoring some properties, but dressed up in the clothes of formal logic. Frege tries to eliminate psychology, but Boolos implies that what we 'care about' is relevant. |
| 18684 | Rather than requiring an action, a reason may 'entice' us, or be 'eligible', or 'justify' it [Orsi] |
| Full Idea: Many have suggested alternative roles or sorts of reasons, which are not mandatory. Dancy says some reasons are 'enticing' rather than peremptory; Raz makes options 'eligible' rather than required; Gert says they justify rather than require action. | |
| From: Francesco Orsi (Value Theory [2015], 6.4) | |
| A reaction: The third option is immediately attractive - but then it would only justify the action because it was a good reason, which would need explaining. 'Enticing' captures the psychology in a nice vague way. |
| 18685 | Final value is favoured for its own sake, and personal value for someone's sake [Orsi] |
| Full Idea: Final value is to be favoured for its own sake; personal value is to be favoured for someone's sake. | |
| From: Francesco Orsi (Value Theory [2015], 7.2) | |
| A reaction: This gives another important dimension for discussions of value. I like the question 'what gives rise to this value?', but we can also ask (given the value) why we should then promote it. Health isn't a final value, and truth isn't a personal value? |
| 18666 | Value-maker concepts (such as courageous or elegant) simultaneously describe and evaluate [Orsi] |
| Full Idea: Examples of value-maker concepts are courageous, honest, cowardly, corrupt, elegant, tacky, melodious, insightful. Employing these concepts normally means both evaluating and describing the thing or person one way or another. | |
| From: Francesco Orsi (Value Theory [2015], 1.2) | |
| A reaction: The point being that they tell you two things - that this thing has a particular value, and also why it has that value. Since I am flirting with the theory that all values must have 'value-makers' this is very interesting. |
| 18667 | The '-able' concepts (like enviable) say this thing deserves a particular response [Orsi] |
| Full Idea: The '-able' concepts, such as valuable, enviable, contemptible, wear on their sleeve the idea that the thing so evaluated merits or is worth a certain attitude or response (of valuing, envying, despising). | |
| From: Francesco Orsi (Value Theory [2015], 1.2) | |
| A reaction: Compare Idea 18666. Hence some concepts point to the source of value in the thing, and others point to the source of the value in the normative attitude of the speaker. Interesting. |
| 18679 | Things are only valuable if something makes it valuable, and we can ask for the reason [Orsi] |
| Full Idea: If a certain object is valuable, then something other than its being valuable must make it so. ...One is always in principle entitled to an answer as to why it is good or bad. | |
| From: Francesco Orsi (Value Theory [2015], 5.2) | |
| A reaction: What Orsi calls the 'chemistry' of value. I am inclined to think that this is the key to a philosophical study of value. Without this assumption the values float free, and we drift into idealised waffle. Note that here he only refers to 'objects'. |
| 18682 | A complex value is not just the sum of the values of the parts [Orsi] |
| Full Idea: The whole 'being pleased by cats being tortured' is definitely not better, and is likely worse, than cats being tortured. So its value cannot result from a sum of the intrinsic values of the parts. | |
| From: Francesco Orsi (Value Theory [2015], 5.3) | |
| A reaction: This example is simplistic. It isn't a matter of just adding 'pleased' and 'tortured'. 'Pleased' doesn't have a standalone value. Only a rather gormless utilitarian would think it was always good if someone was pleased. I suspect values don't sum at all. |
| 18683 | Trichotomy Thesis: comparable values must be better, worse or the same [Orsi] |
| Full Idea: It is natural to assume that if we can compare two objects or states of affairs, X and Y, then X is either better than, or worse than, or as good as Y. This has been called the Trichotomy Thesis. | |
| From: Francesco Orsi (Value Theory [2015], 6.2) | |
| A reaction: This is the obvious starting point for a discussion of the difficult question of the extent to which values can be compared. Orsi says even if there was only one value, like pleasure, it might have incommensurable aspects like duration and intensity. |
| 18686 | The Fitting Attitude view says values are fitting or reasonable, and values are just byproducts [Orsi] |
| Full Idea: The main claims of the Fitting Attitude view of value are Reduction: values such are goodness are reduced to fitting attitudes, having reasons, and Normative Redundancy: goodness provides no reasons for attitudes beyond the thing's features. | |
| From: Francesco Orsi (Value Theory [2015], 8.2) | |
| A reaction: Orsi's book is a sustained defence of this claim. I like the Normative Redundancy idea, but I am less persuaded by the Reduction. |
| 18672 | Values from reasons has the 'wrong kind of reason' problem - admiration arising from fear [Orsi] |
| Full Idea: A support for the fittingness account (against the buck-passing reasons account) is the 'wrong kind of reasons' problem. There are many reasons for positive attitudes towards things which are not good. We might admire a demon because he threatens torture. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: [compressed] I like the Buck-Passing view, but was never going to claim that all reasons for positive attitudes bestow value. I only think that there is no value without a reason |
| 18677 | A thing may have final value, which is still derived from other values, or from relations [Orsi] |
| Full Idea: Many believe that final values can be extrinsic: objects which are valuable for their own sake partly thanks to their relations to other objects. ...This might depend on the value of other things...or an object's relational properties. | |
| From: Francesco Orsi (Value Theory [2015], 2.3) | |
| A reaction: It strikes me that virtually nothing (or even absolutely nothing) has final value in total isolation from other things (Moore's 'isolation test'). Values arise within a tangled network of relations. Your final value is my instrumental value. |
| 18668 | Truths about value entail normative truths about actions or attitudes [Orsi] |
| Full Idea: My guiding assumption is that truths about value, at least, regularly entail normative truths of some sort about actions or attitudes. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: Not quite as clear as it sounds. If I say 'the leaf is green' I presume a belief that it is green, which is an attitude. If I say 'shut the door' that implies an action with no value. One view says that values are entirely normative in this way. |
| 18670 | The Buck-Passing view of normative values says other properties are reasons for the value [Orsi] |
| Full Idea: Version two of the normative view of values is the Buck-Passing account, which says that 'x is good' means 'x has the property of having other properties that provide reasons to favour x'. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: [He cites Scanlon 1998:95-8] I think this is the one to explore. We want values in the world, bridging the supposed 'is-ought gap', and not values that just derive from the way human beings are constituted (and certainly not supernatural values!). |
| 18669 | Values can be normative in the Fitting Attitude account, where 'good' means fitting favouring [Orsi] |
| Full Idea: Version one of the normative view of values is the Fitting Attitude account, which says that 'x is good' means 'it is fitting to respond favourably to (or 'favour') x'. | |
| From: Francesco Orsi (Value Theory [2015], 1.4) | |
| A reaction: Brentano is mentioned. Orsi favours this view. The rival normative view is Scanlon's [1998:95-8] Buck-Passing account, in Idea 18670. I am interested in building a defence of the Buck-Passing account, which seems to suit a naturalistic realist like me. |