41 ideas
| 19275 | You cannot understand what exists without understanding possibility and necessity [Hale] |
| Full Idea: I defend the thesis that questions about what kinds of things there are cannot be properly understood or adequately answered without recourse to considerations about possibility and necessity. | |
| From: Bob Hale (Necessary Beings [2013], Intro) | |
| A reaction: Good. I would say that this is a growing realisation in contemporary philosophy. The issue is focused when we ask what are the limitations of Quine's approach to metaphysics. If you don't see possibilities around you, you are a fool. |
| 19291 | A canonical defintion specifies the type of thing, and what distinguish this specimen [Hale] |
| Full Idea: One might think of a full dress, or canonical, definition as specifying what type of thing it is, and what distinguishes it from everything else within its type. | |
| From: Bob Hale (Necessary Beings [2013], 06.4) | |
| A reaction: Good! At last someone embraces the Aristotelian ideas that definitions are a) quite extensive and detailed (unlike lexicography), and b) they aim to get right down to the individual. In that sense, an essence is captured by a definition. |
| 19297 | The two Barcan principles are easily proved in fairly basic modal logic [Hale] |
| Full Idea: If the Brouwersche principle, p ⊃ □◊p is adjoined to a standard quantified vesion of the weakest modal logic K, then one can prove both the Barcan principle, and its converse. | |
| From: Bob Hale (Necessary Beings [2013], 09.2) | |
| A reaction: The Brouwersche principle (that p implies that p must be possible) sounds reasonable, but the Barcan principles strike me as false, so something has to give. They are theorems of S5. Hale proposes giving up classical logic. |
| 19301 | With a negative free logic, we can dispense with the Barcan formulae [Hale] |
| Full Idea: I reject both Barcan and Converse Barcan by adopting a negative free logic. | |
| From: Bob Hale (Necessary Beings [2013], 11.3) | |
| A reaction: See section 9.2 of Hale's book, where he makes his case. I can't evaluate this bold move, though I don't like the Barcan Formulae. We can anticipate objections to Hale: are you prepared to embrace the unexpected consequences of your new logic? |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 19296 | If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale] |
| Full Idea: Contrary to what Quine supposes, it is neither necessary nor desirable to interpret bound higher-order variables as ranging over sets. Sets are a species of object. They should range over entities of a completely different type: properties and relations. | |
| From: Bob Hale (Necessary Beings [2013], 08.2) | |
| A reaction: This helpfully clarifies something which was confusing me. If sets are objects, then 'second-order' logic just seems to be the same as first-order logic (rather than being 'set theory in disguise'). I quantify over properties, but deny their existence! |
| 19289 | Maybe conventionalism applies to meaning, but not to the truth of propositions expressed [Hale] |
| Full Idea: An old objection to conventionalism claims that it confuses sentences with propositions, confusing what makes sentences mean what they do with what makes them (as propositions) true. | |
| From: Bob Hale (Necessary Beings [2013], 05.2) | |
| A reaction: The conventions would presumably apply to the sentences, but not to the propositions. Since I think that focusing on propositions solves a lot of misunderstandings in modern philosophy, I like the sound of this. |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 19298 | Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems [Hale] |
| Full Idea: In contrast with axiomatic systems, in natural deductions systems of logic neither the premises nor the conclusions of steps in a derivation need themselves be logical truths or theorems of logic. | |
| From: Bob Hale (Necessary Beings [2013], 09.2 n7) | |
| A reaction: Not sure I get that. It can't be that everything in an axiomatic proof has to be a logical truth. How would you prove anything about the world that way? I'm obviously missing something. |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 19295 | Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale] |
| Full Idea: The existence of the natural numbers is not a matter of pure logic - it cannot be proved in pure logic. It can be proved in second-order logic plus Hume's principle. Truths of arithmetic are not logic - they depend on the nature of natural numbers. | |
| From: Bob Hale (Necessary Beings [2013], 07.4) | |
| A reaction: Hume's principles needs entities which can be matched to one another, so a certain ontology is needed to get neo-logicism off the ground. |
| 19281 | Interesting supervenience must characterise the base quite differently from what supervenes on it [Hale] |
| Full Idea: Any intereresting supervenience thesis requires that the class of facts on which the allegedly supervening facts supervene be characterizable independently, without use or presupposition of the notions involved in stating the supervening facts. | |
| From: Bob Hale (Necessary Beings [2013], 03.4.1) | |
| A reaction: There might be intermediate cases here, since having descriptions which are utterly unconnected (at any level) might be rather challenging. |
| 19278 | There is no gap between a fact that p, and it is true that p; so we only have the truth-condtions for p [Hale] |
| Full Idea: There is no clear gap between its being a fact that p and its being true that p, no obvious way to individuate the fact a true statement records other than via that statement's truth-conditions. | |
| From: Bob Hale (Necessary Beings [2013], 03.2) | |
| A reaction: Typical of philosophers of language. The concept of a fact is of something mind-independent; the concept of a truth is of something mind-dependent. They can't therefore be the same thing (by the contrapositive of the indiscernability of identicals!). |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 19302 | If a chair could be made of slightly different material, that could lead to big changes [Hale] |
| Full Idea: How shall we prevent a sorites taking us to the conclusion that a chair might have originated in a completely disjoint lot of wood, or even in some other material altogether? | |
| From: Bob Hale (Necessary Beings [2013], 11.3.7) | |
| A reaction: This seems a good criticism of Kripke's implausible claim that his lectern is necessarily (or essentially) made of the piece of wood it is made of. Could his lectern have had a small piece of plastic inserted in it? |
| 19290 | Absolute necessities are necessarily necessary [Hale] |
| Full Idea: I argue that any absolute necessity is necessarily necessary. | |
| From: Bob Hale (Necessary Beings [2013], 05.5.2) | |
| A reaction: This requires the principle of S4 modal logic, that necessity implies necessary necessity. He argues that S5 is the logical of absolute necessity. |
| 19286 | 'Absolute necessity' is when there is no restriction on the things which necessitate p [Hale] |
| Full Idea: The strength of the claim that p is 'absolutely necessary' derives from the fact that in its expression as a universally quantified counterfactual ('everything will necessitate p'), the quantifier ranges over all propositions whatever. | |
| From: Bob Hale (Necessary Beings [2013], 04.1) | |
| A reaction: Other philosophers don't seem to use the term 'absolute necessity', but it seems a useful concept, in contrast to conditional or local necessities. You can't buy chocolate on the sun. |
| 19288 | Logical and metaphysical necessities differ in their vocabulary, and their underlying entities [Hale] |
| Full Idea: The difference between logical and metaphysical necessities lies, not in the range of possibilities for which they hold, but - at the linguistic level - in the kind of vocabulary essential to their expression, and the kinds of entities that explain them. | |
| From: Bob Hale (Necessary Beings [2013], 04.5) | |
| A reaction: I don't think much of the idea that the difference is just linguistic, and I don't like the idea of 'entities' as grounding them. I see logical necessities as arising from natural deduction rules, and metaphysical ones coming from the nature of reality. |
| 19285 | Logical necessity is something which is true, no matter what else is the case [Hale] |
| Full Idea: We can identify the belief that the proposition that p is logically necessary, where p may be of any logical form, with the belief that, no matter what else was the case, it would be true that p. | |
| From: Bob Hale (Necessary Beings [2013], 04.1) | |
| A reaction: I find this surprising. I take it that logical necessity must be the consequence of logic. That all squares have corners doesn't seem to be a matter of logic. But then he seems to expand logical necessity to include conceptual necessity. Why? |
| 19287 | Maybe each type of logic has its own necessity, gradually becoming broader [Hale] |
| Full Idea: We can distinguish between narrower and broader kinds of logical necessity. There are, for example, the logical necessities of propostional logic, those of first-order logic, and so on. Maybe they are necessities expressed using logical vocabulary. | |
| From: Bob Hale (Necessary Beings [2013], 04.5) | |
| A reaction: Hale goes on to prefer a view that embraces conceptual necessities. I think in philosophy we should designate the necessities according to their sources. This might clarify a currently rather confused situation. First-order includes propositional logic. |
| 19282 | It seems that we cannot show that modal facts depend on non-modal facts [Hale] |
| Full Idea: I think we may conclude that there is no significant version of modal supervenience which both commands acceptance and implies that all modal facts depend asymmetrically on non-modal ones. | |
| From: Bob Hale (Necessary Beings [2013], 03.4.3) | |
| A reaction: This is the conclusion of a sustained and careful discussion, recorded here for interest. I'm inclined to think that there are very few, if any, non-modal facts in the world, if those facts are accurately characterised. |
| 19276 | The big challenge for essentialist views of modality is things having necessary existence [Hale] |
| Full Idea: Whether the essentialist theory can account for all absolute necessities depends in part on whether the theory can explain the necessities of existence (of certain objects, properties and entities). | |
| From: Bob Hale (Necessary Beings [2013], Intro) | |
| A reaction: Hale has a Fregean commitment to all sorts of abstract objects, and then finds difficulty in explaining them from his essentialist viewpoint. His book didn't convince me. I'm more of a nominalist, me, so I sleep better at nights. |
| 19293 | Essentialism doesn't explain necessity reductively; it explains all necessities in terms of a few basic natures [Hale] |
| Full Idea: The point of the essentialist theory is not to provide a reductive explanation of necessities. It is, rather, to locate a base class of necessities - those which directly reflect the natures of things - in terms of which the remainder may be explained. | |
| From: Bob Hale (Necessary Beings [2013], 06.6) | |
| A reaction: My picture is of most of the necessities being directly explained by the natures of things, rather than a small core of natures generating all the derived ones. All the necessities of squares derive from the nature of the square. |
| 19294 | If necessity derives from essences, how do we explain the necessary existence of essences? [Hale] |
| Full Idea: If the essentialist theory of necessity is to be adequate, it must be able to explain how the existence of certain objects - such as the natural numbers - can itself be absolutely necessary. | |
| From: Bob Hale (Necessary Beings [2013], 07.1) | |
| A reaction: Hale and his neo-logicist pals think that numbers are 'objects', and they necessarily exist, so he obviously has a problem. I don't see any alternative for essentialists to treating the existing (and possible) natures as brute facts. |
| 19279 | What are these worlds, that being true in all of them makes something necessary? [Hale] |
| Full Idea: We need an explanation of what worlds are that makes clear why being true at all of them should be necessary and sufficient for being necessary (and true at one of them suffices for being possible). | |
| From: Bob Hale (Necessary Beings [2013], 03.3.2) | |
| A reaction: Hale is introducing combinatorial accounts of worlds, as one possible answer to this. Hale observes that all the worlds might be identical to our world. It is always assumed that the worlds are hugely varied. But maybe worlds are constrained. |
| 19299 | Possible worlds make every proposition true or false, which endorses classical logic [Hale] |
| Full Idea: The standard conception of worlds incorporates the assumption of bivalence - every proposition is either true or false. But it is infelicitous to build into one's basic semantic machinery a principle endorsing classical logic against its rivals. | |
| From: Bob Hale (Necessary Beings [2013], 10.3) | |
| A reaction: No wonder Dummett (with his intuitionist logic) immediately spurned possible worlds. This objection must be central to many recent thinkers who have begun to doubt possible worlds. I heard Kit Fine say 'always kick possible worlds where you can'. |
| 19300 | The molecules may explain the water, but they are not what 'water' means [Hale] |
| Full Idea: What it is to be (pure) water is to be explained in terms of being composed of H2O molecules, but this is not what the word 'water' means. | |
| From: Bob Hale (Necessary Beings [2013], 11.2) | |
| A reaction: Hale says when the real and verbal definitions match, we can know the essence a priori. If they come apart, presumably we need a posteriori research. Interesting. It is certainly dubious to say a stuff-word means its chemical composition. |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |