1992 | Hat-Tricks and Heaps |
p.2 | 9117 | The smallest heap has four objects: three on the bottom, one on the top | |
Full Idea: Hart argues that the smallest heap consists of four objects: three on the bottom, one on the top. | |||
From: report of William D. Hart (Hat-Tricks and Heaps [1992]) by Roy Sorensen - Vagueness and Contradiction Intro | |||
A reaction: If the objects were rough bolders, you could get away with two on the bottom. He's wrong. No one would accept as a 'heap' four minute grains barely visible to the naked eye. No one would describe such a group of items in a supermarket as a heap. |
2010 | The Evolution of Logic |
p.22 | 13456 | Set theory articulates the concept of order (through relations) | |
Full Idea: It is set theory, and more specifically the theory of relations, that articulates order. | |||
From: William D. Hart (The Evolution of Logic [2010]) | |||
A reaction: It would seem that we mainly need set theory in order to talk accurately about order, and about infinity. The two come together in the study of the ordinal numbers. |
1 | p.4 | 13442 | Without the empty set we could not form a∩b without checking that a and b meet |
Full Idea: Without the empty set, disjoint sets would have no intersection, and we could not form a∩b without checking that a and b meet. This is an example of the utility of the empty set. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: A novice might plausibly ask why there should be an intersection for every pair of sets, if they have nothing in common except for containing this little puff of nothingness. But then what do novices know? |
1 | p.4 | 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension |
Full Idea: 'Comprehension' is the assumption that every predicate has an extension. Naïve set theory is the theory whose axioms are extensionality and comprehension, and comprehension is thought to be its naivety. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: This doesn't, of course, mean that there couldn't be a more modest version of comprehension. The notorious difficulty come with the discovery of self-referring predicates which can't possibly have extensions. |
1 | p.5 | 13443 | ∈ relates across layers, while ⊆ relates within layers |
Full Idea: ∈ relates across layers (Plato is a member of his unit set and the set of people), while ⊆ relates within layers (the singleton of Plato is a subset of the set of people). This distinction only became clear in the 19th century. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: Getting these two clear may be the most important distinction needed to understand how set theory works. |
1 | p.18 | 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry |
Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) |
1 | p.23 | 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers |
Full Idea: A total order 'well-orders' its field just in case any nonempty subset B of its field has an R-least member, that is, there is a b in B such that for any a in B different from b, b bears R to a. So less-than well-orders natural numbers, but not integers. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: The natural numbers have a starting point, but the integers are infinite in both directions. In plain English, an order is 'well-ordered' if there is a starting point. |
1 | p.23 | 13458 | A partial ordering becomes 'total' if any two members of its field are comparable |
Full Idea: A partial ordering is a 'total ordering' just in case any two members of its field are comparable, that is, either a is R to b, or b is R to a, or a is b. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: See Idea 13457 for 'partial ordering'. The three conditions are known as the 'trichotomy' condition. |
1 | p.23 | 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets |
Full Idea: We say that a binary relation R 'partially orders' a field A just in case R is irreflexive (so that nothing bears R to itself) and transitive. When the set is {a,b}, its subsets {a} and {b} are incomparable in a partial ordering. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) |
1 | p.26 | 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers |
Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) |
1 | p.27 | 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) |
Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) |
1 | p.27 | 13461 | We can choose from finite and evident sets, but not from infinite opaque ones |
Full Idea: When a set is finite, we can prove it has a choice function (∀x x∈A → f(x)∈A), but we need an axiom when A is infinite and the members opaque. From infinite shoes we can pick a left one, but from socks we need the axiom of choice. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: The socks example in from Russell 1919:126. |
1 | p.27 | 13462 | With the Axiom of Choice every set can be well-ordered |
Full Idea: It follows from the Axiom of Choice that every set can be well-ordered. | |||
From: William D. Hart (The Evolution of Logic [2010], 1) | |||
A reaction: For 'well-ordered' see Idea 13460. Every set can be ordered with a least member. |
10 | p.268 | 13515 | To study abstract problems, some knowledge of set theory is essential |
Full Idea: By now, no education in abstract pursuits is adequate without some familiarity with sets. | |||
From: William D. Hart (The Evolution of Logic [2010], 10) | |||
A reaction: A heart-sinking observation for those who aspire to study metaphysics and modality. The question is, what will count as 'some' familiarity? Are only professional logicians now allowed to be proper philosophers? |
10 | p.270 | 13516 | If we accept that V=L, it seems to settle all the open questions of set theory |
Full Idea: It has been said (by Burt Dreben) that the only reason set theorists do not generally buy the view that V = L is that it would put them out of business by settling their open questions. | |||
From: William D. Hart (The Evolution of Logic [2010], 10) | |||
A reaction: Hart says V=L breaks with the interative conception of sets at level ω+1, which is countable is the constructible view, but has continuum many in the cumulative (iterative) hierarch. The constructible V=L view is anti-platonist. |
2 | p.31 | 13466 | We are all post-Kantians, because he set the current agenda for philosophy |
Full Idea: We are all post-Kantians, ...because Kant set an agenda for philosophy that we are still working through. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: Hart says that the main agenda is set by Kant's desire to defend the principle of sufficient reason against Hume's attack on causation. I would take it more generally to be the assessment of metaphysics, and of a priori knowledge. |
2 | p.36 | 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' |
Full Idea: It is an ancient and honourable view that truth is correspondence to fact; Tarski showed us how to do without facts here. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: This is a very interesting spin on Tarski, who certainly seems to endorse the correspondence theory, even while apparently inventing a new 'semantic' theory of truth. It is controversial how far Tarski's theory really is a 'correspondence' theory. |
2 | p.41 | 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic |
Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) |
2 | p.44 | 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several |
Full Idea: There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels). | |||
From: William D. Hart (The Evolution of Logic [2010], 2) |
2 | p.47 | 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest |
Full Idea: A Fregean concept is a function that assigns to each object a truth value. So instead of the colour green, the concept GREEN assigns truth to each green thing, but falsity to anything else. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: This would seem to immediately hit the renate/cordate problem, if there was a world in which all and only the green things happened to be square. How could Frege then distinguish the green from the square? Compare Idea 8245. |
2 | p.53 | 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori |
Full Idea: In the case of the parallels postulate, Euclid's fifth axiom (the whole is greater than the part), and comprehension, saying was believing for a while, but what was said was false. This should make a shrewd philosopher sceptical about a priori knowledge. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: Euclid's fifth is challenged by infinite numbers, and comprehension is challenged by Russell's paradox. I can't see a defender of the a priori being greatly worried about these cases. No one ever said we would be right - in doing arithmetic, for example. |
2 | p.53 | 13477 | The problems are the monuments of philosophy |
Full Idea: The real monuments of philosophy are its problems. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: Presumably he means '....rather than its solutions'. No other subject would be very happy with that sort of claim. Compare Idea 8243. A complaint against analytic philosophy is that it has achieved no consensus at all. |
2 | p.58 | 13481 | Maybe sets should be rethought in terms of the even more basic categories |
Full Idea: Some have claimed that sets should be rethought in terms of still more basic things, categories. | |||
From: William D. Hart (The Evolution of Logic [2010], 2) | |||
A reaction: [He cites F.William Lawvere 1966] It appears to the the context of foundations for mathematics that he has in mind. |
3 | p.59 | 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals |
Full Idea: The Burali-Forti Paradox was a crisis for Cantor's theory of ordinal numbers. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.63 | 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that |
Full Idea: Berry's Paradox: by the least number principle 'the least number denoted by no description of fewer than 79 letters' exists, but we just referred to it using a description of 77 letters. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) | |||
A reaction: I struggle with this. If I refer to 'an object to which no human being could possibly refer', have I just referred to something? Graham Priest likes this sort of idea. |
3 | p.71 | 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' |
Full Idea: Jespersen calls a noun a mass word when it has no plural, does not take numerical adjectives, and does not take 'fewer'. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) | |||
A reaction: Jespersen was a great linguistics expert. |
3 | p.74 | 13491 | The axiom of infinity with separation gives a least limit ordinal ω |
Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.74 | 13490 | Von Neumann defines α<β as α∈β |
Full Idea: One of the glories of Von Neumann's theory of numbers is to define α < β to mean that α ∈ β. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.75 | 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton |
Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.79 | 13493 | In the modern view, foundation is the heart of the way to do set theory |
Full Idea: In the second half of the twentieth century there emerged the opinion that foundation is the heart of the way to do set theory. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) | |||
A reaction: It is foundation which is the central axiom of the iterative conception of sets, where each level of sets is built on previous levels, and they are all 'well-founded'. |
3 | p.80 | 13495 | Foundation Axiom: an nonempty set has a member disjoint from it |
Full Idea: The usual statement of Foundation is that any nonempty set has a member disjoint from it. This phrasing is ordinal-free and closer to the primitives of ZFC. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.80 | 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains |
Full Idea: That the iterative sets suffice for most of ZFC does not show they are necessary, nor is it evident that the set of operations has no fixed points (as 0 is a fixed point for square-of), and no infinitely descending chains (like negative integers). | |||
From: William D. Hart (The Evolution of Logic [2010], 3) | |||
A reaction: People don't seem to worry that they aren't 'necessary', and further measures are possible to block infinitely descending chains. |
3 | p.80 | 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset |
Full Idea: First-order logic is 'compact', which means that any logical consequence of a set (finite or infinite) of first-order sentences is a logical consequence of a finite subset of those sentences. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
3 | p.88 | 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe |
Full Idea: The theory of types is a thing of the past. There is now nothing to choose between ZFC and NBG (Neumann-Bernays-Gödel). NF (Quine's) is a more specialized taste, but is a place to look if you want the universe. | |||
From: William D. Hart (The Evolution of Logic [2010], 3) |
4 | p.90 | 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent |
Full Idea: A 'conditional proof' licenses inferences to a conditional from a deduction of its consequent from its antecedent. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: That is, a proof can be enshrined in an arrow. |
4 | p.96 | 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' |
Full Idea: When a quantifier is attached to a variable, as in '∃(y)....', then it should be read as 'There exists an individual, call it y, such that....'. One should not read it as 'There exists a y such that...', which would attach predicate to quantifier. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: The point is to make clear that in classical logic the predicates attach to the objects, and not to some formal component like a quantifier. |
4 | p.101 | 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth |
Full Idea: In any first-order language, there are infinitely many T-sentences. Since definitions should be finite, the agglomeration of all the T-sentences is not a definition of truth. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: This may be a warning shot aimed at Davidson's extensive use of Tarski's formal account in his own views on meaning in natural language. |
4 | p.107 | 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do |
Full Idea: We explain truth for sentences in terms of satisfaction of formulae. The crux here is that for a sentence, either all sequences satisfy it or none do (with no middle ground). For formulae, some sequences may satisfy it and others not. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: This is the hardest part of Tarski's theory of truth to grasp. |
4 | p.108 | 13505 | Model theory studies how set theory can model sets of sentences |
Full Idea: Modern model theory investigates which set theoretic structures are models for which collections of sentences. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: So first you must choose your set theory (see Idea 13497). Then you presumably look at how to formalise sentences, and then look at the really tricky ones, many of which will involve various degrees of infinity. |
4 | p.111 | 13506 | The universal quantifier can't really mean 'all', because there is no universal set |
Full Idea: All the main set theories deny that there is a set of which everything is a member. No interpretation has a domain with everything in it. So the universal quantifier never gets to mean everything all at once; 'all' does not mean all. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: Could you have an 'uncompleted' universal set, in the spirit of uncompleted infinities? In ordinary English we can talk about 'absolutely everything' - we just can't define a set of everything. Must we 'define' our domain? |
4 | p.122 | 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar |
Full Idea: In effect, the machinery introduced to solve the liar can always be rejigged to yield another version the liar. | |||
From: William D. Hart (The Evolution of Logic [2010], 4) | |||
A reaction: [He cites Hans Herzberger 1980-81] The machinery is Tarski's device of only talking about sentences of a language by using a 'metalanguage'. |
5 | p.144 | 13509 | We can establish truths about infinite numbers by means of induction |
Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof. | |||
From: William D. Hart (The Evolution of Logic [2010], 5) | |||
A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence. |
9 | p.236 | 13511 | Model theory is mostly confined to first-order theories |
Full Idea: There is no developed methematics of models for second-order theories, so for the most part, model theory is about models for first-order theories. | |||
From: William D. Hart (The Evolution of Logic [2010], 9) |
9 | p.238 | 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 |
Full Idea: The beginning of modern model theory was when Morley proved Los's Conjecture in 1962 - that a complete theory in a countable language categorical in one uncountable cardinal is categorical in all. | |||
From: William D. Hart (The Evolution of Logic [2010], 9) |
9 | p.238 | 13513 | Models are ways the world might be from a first-order point of view |
Full Idea: Models are ways the world might be from a first-order point of view. | |||
From: William D. Hart (The Evolution of Logic [2010], 9) |
p.350 | p.350 | 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions |
Full Idea: The conception of Frege is that self-evidence is an intrinsic property of the basic truths, rules, and thoughts expressed by definitions. | |||
From: William D. Hart (The Evolution of Logic [2010], p.350) | |||
A reaction: The problem is always that what appears to be self-evident may turn out to be wrong. Presumably the effort of arriving at a definition ought to clarify and support the self-evident ingredient. |