In the small-improvement argument, the incomparability as vagueness view might say that it is indeterminate whether banking is better or worse than philosophy, or precisely equally good. One taxonomic complication is distinguishing the view that incomparability is vagueness, combined with epistemicism about vagueness, from epistemicism about incomparability.
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One possibility is that this is all a mistake: that there is no genuine incomparability, and when it seems like none of the three trichotomous comparisons apply, in fact one of them does but we do not know which. This is where the small-improvement argument goes wrong: one of the trichotomous comparisons does apply between banking and philosophy. According to this view, apparent incomparability is merely ignorance. An advantage of this account is that the various puzzles surrounding incomparability dissolve rather quickly.
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A totally ordered set is exactly a partially ordered set in which every pair of elements is comparable. It follows immediately from the definitions of comparability and incomparability that both relations are symmetric, that is x is comparable to y if and only if y is comparable to x, and likewise for incomparability.
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Whereas Kuhn and Feyerabend's concepts of incommensurability do not imply complete incomparability of scientific concepts, this incommensurability of meaning does.
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An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.
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Every strict weak ordering < is also a semi-order. More particularly, transitivity of < and transitivity of incomparability with respect to < together imply the above axiom 2, while transitivity of incomparability alone implies axiom 3. The semiorder shown in the top image is not a strict weak ordering, since the rightmost vertice is incomparable to its two closest left neighbors, but they are comparable.
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Finally, a set of philosophers led by John Broome has argued that incomparability is vagueness. This theory says that it is vague or indeterminate which trichotomous comparison applies. The argument for this position is complex, and how 'incomparability as vagueness' is to be understood depends on one's theory of vagueness. But the main idea behind the theory is fairly simple.
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There are four main philosophical accounts of incommensurability/incomparability. Their task is to explain (or explain away) the phenomenon, and the small improvement argument. Some philosophers are pluralists about the phenomenon: they think that (for example) genuine incomparability might be the correct account in some cases, and parity in others. One way to understand the difference between the theories is to see how they respond to the small-improvement argument.
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Joseph Raz has argued that in cases of incomparability, no comparison applies. Neither option is better, and they are not equally good. On this view, the small-improvement argument is sound.
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The class of trapezoid graphs properly contains the union of interval and permutation graphs and is equivalent to the incomparability graphs of partially ordered sets having interval order dimension at most two. Permutation graphs can be seen as the special case of trapezoid graphs when every trapezoid has zero area. This occurs when both of the trapezoid’s points on the upper channel are in the same position and both points on the lower channel are in the same position. Like all incomparability graphs, trapezoid graphs are perfect.
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In this section the language of a trial returns with the demand for Israel to bear witness to YHWH's deeds, although they are blind and deaf (cf. Isaiah 42:18), to declare the incomparability of YHWH.
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Tanzih () is an Islamic religious concept meaning transcendence. In Islamic theology, two opposite terms are attributed to Allah: tanzih and tashbih. The latter means "nearness, closeness, accessibility". However, the fuller meaning of tanzih is 'declaring incomparability', i.e.
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Dilworth's theorem for infinite partially ordered sets states that a partially ordered set has finite width w if and only if it may be partitioned into w chains. For, suppose that an infinite partial order P has width w, meaning that there are at most a finite number w of elements in any antichain. For any subset S of P, a decomposition into w chains (if it exists) may be described as a coloring of the incomparability graph of S (a graph that has the elements of S as vertices, with an edge between every two incomparable elements) using w colors; every color class in a proper coloring of the incomparability graph must be a chain. By the assumption that P has width w, and by the finite version of Dilworth's theorem, every finite subset S of P has a w-colorable incomparability graph.
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This seems to show that one of our assumptions was incorrect. Defenders of incomparability will say it is most plausible that it is the assumption that banking and philosophy are equally good that is incorrect. So they conclude that this assumption as false, and thus that none of the trichotomous comparisons apply.
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Choice between incomparable options is no more than choice between options when we do not know which is better. The main objection to this kind of view is that it seems very implausible, for similar reasons to epistemicism about vagueness. In particular, it is hard to see how we could be ignorant of the kinds of facts involved in incomparability.
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Given a channel, a pair of two horizontal lines, a trapezoid between these lines is defined by two points on the top and two points on the bottom line. A graph is a trapezoid graph if there exists a set of trapezoids corresponding to the vertices of the graph such that two vertices are joined by an edge if and only if the corresponding trapezoids intersect. The interval order dimension of a partially ordered set, P=(X, <), is the minimum number d of interval orders P1 … Pd such that P = P1∩…∩Pd. The incomparability graph of a partially ordered set P=(X, <) is the undirected graph G=(X, E) where x is adjacent to y in G if and only if x and y are incomparable in P. An undirected graph is a trapezoid graph if and only if it is the incomparability graph of a partial order having interval order dimension at most 2.
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There are many counter-arguments which can be made to this. One of the simplest was made by Cubitt. His paper shows that the argument rests on some very strong assumptions and is tautological: to say that X acts as a money pump is no different from saying that X has intransitive preferences, and does not add anything to evidence for or against the existence of intransitive preferences. A second argument is more fundamental, and this rests on the possibility of incomparability.
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According to Vandenbroucke (2004) it was Kish who used the word "confounding" in the modern sense of the word, to mean "incomparability" of two or more groups (e.g., exposed and unexposed) in an observational study. Formal conditions defining what makes certain groups "comparable" and others "incomparable" were later developed in epidemiology by Greenland and Robins (1986) using the counterfactual language of Neyman (1935)Neyman, J., with cooperation of K. Iwaskiewics and St. Kolodziejczyk (1935). Statistical problems in agricultural experimentation (with discussion).
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This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have ai preceding bi, which would contradict the incomparability of ai and bi in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by Sn.
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A strict weak ordering can be defined on x by declaring two items to be incomparable when they have equal utilities, and otherwise using the numerical comparison, but this necessarily leads to a transitive incomparability relation. Instead, if one sets a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each other are declared incomparable, then a semiorder arises. Specifically, define a binary relation < from X and u by setting x < y whenever u(x) ≤ u(y) − 1\. Then (X,<) is a semiorder.
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Chang is the author of Making Comparisons Count, and the editor of the first volume on the topic of incommensurability of values in the Anglo-American world, Incommensurability, Incomparability, and Practical Reason, and has authored numerous articles and book chapters. Ruth Chang is also widely known for her work on 'hard choices' and decision-making, and her research has been the subject of radio, newspaper, and magazine articles around the world.See fn. 5. Her TED talk on the subject has had over 7 million views, and her ideas have been presented in many popular publications.
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In this final version, the first nine plagues form three triads, each of which God introduces by informing Moses of the main lesson it will teach. In the first triad, the Egyptians begin to experience the power of God; in the second, God demonstrates that he is directing events; and in the third, the incomparability of Yahweh is displayed. Overall, the plagues are "signs and marvels" given by the God of Israel to answer Pharaoh's taunt that he does not know Yahweh: "The Egyptians shall know that I am the ".
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Bitolerance graphs are incomparability graphs of a bitolerance order. An order is a bitolerance order if and only if there are intervals Ix and real numbers t1(x) and tr(x) assigned to each vertex x in such a way that x < y if and only if the overlap of Ix and Iy is less than both tr(x) and t1(y) and the center of Ix is less than the center of Iy.Kenneth P. Bogart, Garth Isaak. Proper and unit bitolerance orders and graphs. Discrete Mathematics 181(1–3): 37–51 (1998).
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The Financial Accounting Standards Board (FASB) issued Statement of Financial Accounting Standards No. 157: Fair Value Measurements ("FAS 157") in September 2006 to provide guidance about how entities should determine fair value estimations for financial reporting purposes. FAS 157 broadly applies to financial and nonfinancial assets and liabilities measured at fair value under other authoritative accounting pronouncements. However, application to nonfinancial assets and liabilities was deferred until 2009. Absence of one single consistent framework for applying fair value measurements and developing a reliable estimate of a fair value in the absence of quoted prices has created inconsistencies and incomparability.
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Therefore, by the De Bruijn–Erdős theorem, P itself also has a w-colorable incomparability graph, and thus has the desired partition into chains . However, the theorem does not extend so simply to partially ordered sets in which the width, and not just the cardinality of the set, is infinite. In this case the size of the largest antichain and the minimum number of chains needed to cover the partial order may be very different from each other. In particular, for every infinite cardinal number κ there is an infinite partially ordered set of width ℵ0 whose partition into the fewest chains has κ chains .
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The original motivation for introducing semiorders was to model human preferences without assuming (as strict weak orderings do) that incomparability is a transitive relation. For instance, if x, y, and z represent three quantities of the same material, and x and z differ by the smallest amount that is perceptible as a difference, while y is halfway between the two of them, then it is reasonable for a preference to exist between x and z but not between the other two pairs, violating transitivity., p. 179. Thus, suppose that X is a set of items, and u is a utility function that maps the members of X to real numbers.
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Let X be a set of items, and let < be a binary relation on X. Items x and y are said to be incomparable, written here as x ~ y, if neither x < y nor y < x is true. Then the pair (X,<) is a semiorder if it satisfies the following three axioms: describes an equivalent set of four axioms, the first two of which combine the definition of incomparability and the first axiom listed here. #For all x and y, it is not possible for both x < y and y < x to be true. That is, < must be an asymmetric relation #For all x, y, z, and w, if x < y, y ~ z, and z < w, then x < w.
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For example, if one is trying to decide on some nice afternoon whether they should stay in to do work or go for a walk, on this view of practical reason they will compare the merits of these two options. If going for a walk is the better or more reasonable course of action, they should put aside their books and go for a stroll. The topic of incommensurability—and the topic of incomparability in particular—is especially important to those who advocate this view of practical reason. For if one's options in certain circumstances are of incomparable value, he or she cannot settle the question of what to do by choosing the better option.
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Among noncommutative rings, PI-rings satisfy the Köthe conjecture. Affine PI-algebras over a field satisfy the Kurosh conjecture, the Nullstellensatz and the catenary property for prime ideals. If R is a PI-ring and K is a subring of its center such that R is integral over K then the going up and going down properties for prime ideals of R and K are satisfied. Also the lying over property (If p is a prime ideal of K then there is a prime ideal P of R such that p is minimal over P\cap K) and the incomparability property (If P and Q are prime ideals of R and P\subset Q then P\cap K\subset Q\cap K) are satisfied.
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A partition of a set S is a family of non-empty disjoint subsets of S that have S as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition. and by Theodore Motzkin a list of sets.. An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set {a, b} are :{a}, {b}, :{b}, {a}, and :{a, b}. In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition.
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The De Bruijn–Erdős theorem shows that, for this problem, there exists a finite unit distance graph with the same chromatic number as the whole plane, so if the chromatic number is greater than five then this fact can be proved by a finite calculation. The De Bruijn–Erdős theorem may also be used to extend Dilworth's theorem from finite to infinite partially ordered sets. Dilworth's theorem states that the width of a partial order (the maximum number of elements in a set of mutually incomparable elements) equals the minimum number of chains (totally ordered subsets) into which the partial order may be partitioned. A partition into chains may be interpreted as a coloring of the incomparability graph of the partial order.
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Person-affecting views can be characterized by the following two claims: first, the person-affecting restriction holds that doing something morally good or bad requires it to be good or bad for someone; and second, the incomparability of non-existence holds that existing and non-existing are incomparable, which implies that it cannot be good or bad for someone to come into existence. Taken together, these claims entail what Greaves describes as the neutrality principle: "Adding an extra person to the world, if it is done in such away as to leave the well-being levels of others unaffected, does not make a state of affairs either better or worse." However, person-affecting views generate many counterintuitive implications, leading Greaves to comment that "it turns out to be remarkably difficult to formulate any remotely acceptable axiology that captures this idea of neutrality".
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Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.. A strict weak order that is trichotomous is called a strict total order.. The total preorder which is the inverse of its complement is in this case a total order. For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤". The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in the total preorder corresponding to a strict weak order we get a \lesssim b and b \lesssim a, while in the partial order given by the reflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order.
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Any finite semiorder has order dimension at most three.. Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial orders that have the largest number of linear extensions are semiorders.. Semiorders are known to obey the 1/3–2/3 conjecture: in any finite semiorder that is not a total order, there exists a pair of elements x and y such that x appears earlier than y in between 1/3 and 2/3 of the linear extensions of the semiorder.. The set of semiorders on an n-element set is well-graded: if two semiorders on the same set differ from each other by the addition or removal of k order relations, then it is possible to find a path of k steps from the first semiorder to the second one, in such a way that each step of the path adds or removes a single order relation and each intermediate state in the path is itself a semiorder.. The incomparability graphs of semiorders are called indifference graphs, and are a special case of the interval graphs.
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