en>Crasshopper at 21:50, 19 October 2014

2014-10-19T21:50:49Z

en>Crasshopper: /* Properties of simple functions */

2014-01-20T06:13:10Z

Properties of simple functions

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	~~I’m Lily from Haimer studying Arts~~ and ~~Sciences~~. ~~I did my schooling~~, ~~secured 95%~~ and ~~hope~~ to ~~find someone~~ with same ~~interests~~ in ~~Slot Car Racing~~.<br><br>~~my web site~~; [http://~~Agapehd~~.~~com~~/~~item~~/37-~~luna~~-de-~~sangre~~-~~cumplimiento~~-~~profetico~~-~~o-simp Fifa 15 Coin Generator~~]		{{Use dmy dates\|date=October 2013}}
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			'''Ernst Mally''' ({{IPA-de\|ˈmali\|lang}}; 11 October 1879 – 8 March 1944) was an Austrian philosopher affiliated with the so-called [[Graz School]] of [[Phenomenology (philosophy)\|phenomenology]]. A pupil of [[Alexius Meinong]], he was one of the founders of [[deontic logic]] and is mainly known for his contributions in that field of research.

			== Life ==
			Mally was born in the town of [[Kranj]] ({{lang-de\|link=no\|Krainburg}}) in the [[Duchy of Carniola]], [[Austria-Hungary]] (now in [[Slovenia]]). His father was of [[Slovenes\|Slovene]] origin, but identified himself with [[Austrian German]] culture (he also [[Germanize]]d the orthography of his surname, originally spelled Mali, a common Slovene surname of [[Upper Carniola]]). After his death, the family moved to the [[Carniola]]n capital of [[Ljubljana]] ({{lang-de\|link=no\|Laibach}}). There, Ernst attended the prestigious [[Poljane Grammar School\|Ljubljana German language Gymnasium]]. Already at a young age, Mally became a fervent supporter of the [[Pan-German]] nationalist movement of [[Georg von Schönerer]]. In the same time, he developed an interest in philosophy.

			In 1898 he enrolled to the [[University of Graz]], where he studied philosophy under the supervision of [[Alexius Meinong]], as well as [[physics]] and mathematics, specializing in [[formal logic]]. He graduated in 1903 with a thesis entitled ''Untersuchungen zur Gegenstandstheorie des Messens'' (Investigations in the Object Theory of Measurement). In 1906 he started teaching at a high school in [[Graz]], at the same time working Meinong's assistant at the university. He also maintained close contacts with the Laboratory for Experimental Psychology, founded by Meinong. In 1912, he wrote his faculty-rank (''Habilitation'') thesis entitled ''Gegenstandstheoretische Grundlagen der Logik und Logistik'' (Object-theoretic Foundations for Logics and Logistics) with Meinong as supervisor.

			From 1915 to 1918 he served as an officer in the [[Austro-Hungarian Army]]. After the end of [[World War I]], Mally joined the [[Greater German People's Party]], which called the unification of [[German Austria]] with Germany. In the same period, he started teaching at the university and in 1925 he took over Meinong's chair. In 1938, he became a member of the National Socialist Teachers' Association and two months after the ''[[Anschluss]]'' he joined the [[NSDAP]]. He continued teaching during the [[Austria at the Time of National Socialism\|Nazi administration of Austria]] until 1942 when he retired. He died in 1944.

			==Work==

			===Mally's deontic logic===
			Mally was the first ever logician to attempt an axiomatisation of ethics. He used five axioms, which are given below. They form a [[first-order theory]] that quantifies over [[proposition]]s, and there are several predicates to understand first. !x means that x ought to be the case. Ux means that x is unconditionally obligatory, i.e. that !x is [[logical truth\|necessarily true]]. ∩x means that x is unconditionally forbidden, i.e. U(¬x). A f B is the [[binary relation]] A requires B, i.e. A materially implies !B. (All [[Logical consequence\|entailment]] in the axioms is [[material conditional]].) It is defined by axiom III, whereas all other terms are defined as a preliminary.

			<math>
			\begin{array}{rl}
			\mbox{I.} & ((A\; \operatorname{f}\; B) \And (B \to C)) \to (A\; \operatorname{f}\; C) \\
			\mbox{II.} & ((A\; \operatorname{f}\; B) \And (A\;\operatorname{f}\;C)) \to (A\; \operatorname{f}\; (B \And C)) \\
			\mbox{III.} & (A\; \operatorname{f}\; B) \leftrightarrow\; !(A \to B) \\
			\mbox{IV.} & \exists U\; !U \\
			\mbox{V.} & \neg (U\; \operatorname{f}\; \cap)
			\end{array}
			</math>
			<!-- How could we write an upside-down U instead of \cap? -->

			Note the implied [[universal quantifier]]s in the above axioms.

			The fourth axiom has confused some logicians because its formulation is not as they would have expected, since Mally gave each axiom a description in words also, and he said that axiom IV meant "the unconditionally obligatory is obligatory", i.e. (as many logicians have insisted) UA → !A. Meanwhile, axiom 5 lacks an object to which the predicates apply, a [[typo]]. However, it turns out these are the least of Mally's worries (see below).

			===Failure of Mally's deontic logic===

			Theorem: This axiomatisation of [[deontic logic]] implies that !x [[if and only if]] x is true, OR !x is unsatisfiable. (This makes it useless to deontic logicians.)
			Proof: Using axiom III, axiom I may be rewritten as (!(A → B) & (B → C)) → !(A → C). Since B → C holds whenever C holds, one immediate consequence is that (!(A → B) → (C → !(A → C))). In other words, if A requires B, it requires any true statement. In the special case where A is a [[tautology (logic)\|tautology]], the theorem has consequence (!B → (C → !C)). Thus, if at least one statement ought be true, every statement must [[Material conditional\|materially entail]] it ought be true, and so every true statement ought be true. As for the converse (i.e. if some statement ought be true then all statements that ought be true are true), consider the following logic: ((U → !A) & (A → ∩)) → (U → !∩) is a special case of axiom I, but its [[consequent]] contradicts axiom V, and so ¬((U → !A) & (A → ∩)). The result !A → A can be shown to follow from this, since !A implies that U → !A and ¬A implies that A → ∩; and, since these are not both true, we know that !A → A.

			Mally thought that axiom I was self-evident, but he likely confused it with an alternative in which the implication B → C is ''logical'', which would indeed make the axiom self-evident. The theorem above, however, would then not be demonstrable. The theorem was proven by [[Karl Menger]], the next deontic logician. Neither Mally's original axioms nor a modification that avoids this result remains popular today. (Menger did not suggest his own axioms.) See also [[deontic logic]] for more on the subsequent development of this subject. Mally's grandson, Thomas, is currently at [[Emory University]] attempting to prove the theories of his grandfather.

			== See also ==

			*[[Graz School]]

			==External links==
			*Entry ''[http://plato.stanford.edu/entries/mally/ Ernst Mally]'' at the [[Stanford Encyclopedia of Philosophy]]
			*Entry ''[http://plato.stanford.edu/entries/mally-deontic/ Mally's deontic logic]'' at the Stanford Encyclopedia of Philosophy

			{{Authority control\|VIAF=18014325}}

			{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
			\| NAME = Mally, Ernst
			\| ALTERNATIVE NAMES =
			\| SHORT DESCRIPTION = Austrian philosopher
			\| DATE OF BIRTH = 11 October 1879
			\| PLACE OF BIRTH =
			\| DATE OF DEATH = 8 March 1944
			\| PLACE OF DEATH =
			}}
			{{DEFAULTSORT:Mally, Ernst}}
			[[Category:1879 births]]
			[[Category:1944 deaths]]
			[[Category:People from Kranj]]
			[[Category:Greater German People's Party politicians]]
			[[Category:Austrian Nazis]]
			[[Category:Austrian philosophers]]
			[[Category:20th-century philosophers]]
			[[Category:Austrian logicians]]
			[[Category:Phenomenologists]]
			[[Category:Austrian people of Slovenian descent]]
			[[Category:University of Graz alumni]]

86.31.60.217: /* Relation to Lebesgue integration */ Non-neg func is --> R+

2012-02-06T21:31:55Z

Relation to Lebesgue integration: Non-neg func is --> R+

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Simple function - Revision history

en>Crasshopper at 21:50, 19 October 2014

en>Crasshopper: /* Properties of simple functions */

86.31.60.217: /* Relation to Lebesgue integration */ Non-neg func is --> R+