17 ideas
| 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. |
| 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. |
| 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 |
| 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. |
| 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...' |
| 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'. |
| 6613 | The natural kinds are objects, processes and properties/relations [Ellis] |
| Full Idea: There are three hierarchies of natural kinds: objects or substances (substantive universals), events or processes (dynamic universals), and properties or relations (tropic universals). | |
| From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
| A reaction: Most interesting here is the identifying of natural kinds with universals, making universals into the families of nature. Universals are high-level sets of natural kinds. To grasp universals you must see patterns, and infer the underlying order. |
| 6616 | Least action is not a causal law, but a 'global law', describing a global essence [Ellis] |
| Full Idea: The principle of least action is not a causal law, but is what I call a 'global law', which describes the essence of the global kind, which every object in the universe necessarily instantiates. | |
| From: Brian Ellis (Katzav on limitations of dispositions [2005]) | |
| A reaction: As a fan of essentialism I find this persuasive. If I inherit part of my essence from being a mammal, I inherit other parts of my essence from being an object, and all objects would share that essence, so it would look like a 'law' for all objects. |
| 6615 | A species requires a genus, and its essence includes the essence of the genus [Ellis] |
| Full Idea: A specific universal can exist only if the generic universal of which it is a species exists, but generic universals don't depend on species; …the essence of any genus is included in its species, but not conversely. | |
| From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
| A reaction: Thus the species 'electron' would be part of the genus 'lepton', or 'human' part of 'mammal'. The point of all this is to show how individual items connect up with the rest of the universe, giving rise to universal laws, such as Least Action. |
| 6614 | A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis] |
| Full Idea: The hierarchy of natural kinds proposed by essentialism may be more elaborate than is strictly required for purposes of ontology, but it is necessary to explain the necessity of the laws of nature, and the universal applicability of global principles. | |
| From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
| A reaction: I am all in favour of elaborating ontology in the name of best explanation. There seem, though, to be some remaining ontological questions at the point where the explanations of essentialism run out. |
| 6612 | Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis] |
| Full Idea: It is objected to dispositionalism that without the principle of least action, or some general principle of equal power, the specific dispositional properties of things could tell us very little about how these things would be disposed to behave. | |
| From: Brian Ellis (Katzav on limitations of dispositions [2005], 90) | |
| A reaction: Ellis attempts to meet this criticism, by placing dispositional properties within a hierarchy of broader properties. There remains a nagging doubt about how essentialism can account for space, time, order, and the existence of essences. |
| 20695 | God's eternal power and deity are clearly seen in what has been created [Paul] |
| Full Idea: From the creation of the world God's invisible nature, namely his eternal power and deity, are clearly perceived in the things that have been made. | |
| From: St Paul (06: Romans [c.55], 19-21), quoted by Brian Davies - Introduction to the Philosophy of Religion | |
| A reaction: St Paul says that for this reason the Gentiles are 'without excuse' for not believing (which means they are in trouble if Christians ever gain political power). Davies says it is unusual to find an argument for God's existence in the Bible. |