text:
User talk:Arj/Archive1

Hello, welcome to Wikipedia.

You might find these links helpful in creating new pages or helping with the above tasks: How to edit a page, How to write a great article, Naming conventions, Manual of Style. You should read our policies at some point too.

If you have any questions, see the help pages, add a question to the village pump or ask me on my talk page. If you made any edits before you got an account, you might be interested in assigning those to your username. I hope you enjoy editing here and being a Wikipedian!

 * If you made any edits before you got an account, you might be interested in assigning those to your username.

 * You can sign your name using three tildes, like this: . If you use four, you can add a datestamp too.

 * If you ever think a page or image should be deleted, please list it at the votes for deletion page. There is also a votes for undeletion page if you want to retrieve something that you think should not have been deleted.

UtherSRG 22:28, 2 Jan 2004 (UTC)

title: User talk:Arj/Archive1


text:
Euthyphro dilemma

The Euthyphro dilemma is found in Plato's dialogue Euthyphro, in which Socrates asks Euthyphro, "Is the pious (τὸ ὅσιον) loved by the gods because it is pious, or is it pious because it is loved by the gods?" (10a)

Although it was originally applied to the ancient Greek pantheon, the dilemma has implications for modern monotheistic religions. Gottfried Leibniz asked whether the good and just "is good and just because God wills it or whether God wills it because it is good and just". Ever since Plato's original discussion, this question has presented a problem for some theists, though others have thought it a false dilemma, and it continues to be an object of theological and philosophical discussion today.

The dilemma
Socrates and Euthyphro discuss the nature of piety in Plato's Euthyphro. Euthyphro proposes (6e) that the pious (τὸ ὅσιον) is the same thing as that which is loved by the gods (τὸ θεοφιλές), but Socrates finds a problem with this proposal: the gods may disagree among themselves (7e). Euthyphro then revises his definition, so that piety is only that which is loved by all of the gods unanimously (9e).

At this point the dilemma surfaces. Socrates asks whether the gods love the pious because it is the pious, or whether the pious is pious only because it is loved by the gods (10a). Socrates and Euthyphro both contemplate the first option: surely the gods love the pious because it is the pious. But this means, Socrates argues, that we are forced to reject the second option: the fact that the gods love something cannot explain why the pious is the pious (10d). Socrates points out that if both options were true, they together would yield a vicious circle, with the gods loving the pious because it is the pious, and the pious being the pious because the gods love it. And this in turn means, Socrates argues, that the pious is not the same as the god-beloved, for what makes the pious the pious is not what makes the god-beloved the god-beloved. After all, what makes the god-beloved the god-beloved is the fact that the gods love it, whereas what makes the pious the pious is something else (9d-11a). Thus Euthyphro's theory does not give us the very nature of the pious, but at most a quality of the pious (11ab).

In philosophical theism
The dilemma can be modified to apply to philosophical theism, where it is still the object of theological and philosophical discussion, largely within the Christian, Jewish, and Islamic traditions. As German philosopher and mathematician Gottfried Leibniz presented this version of the dilemma: "It is generally agreed that whatever God wills is good and just. But there remains the question whether it is good and just because God wills it or whether God wills it because it is good and just; in other words, whether justice and Goodness are arbitrary or whether they belong to the necessary and eternal truths about the nature of things."

Many philosophers and theologians have addressed the Euthyphro dilemma since the time of Plato, though not always with reference to the Platonic dialogue. According to scholar Terence Irwin, the issue and its connection with Plato was revived by Ralph Cudworth and Samuel Clarke in the 17th and 18th centuries. More recently, it has received a great deal of attention from contemporary philosophers working in metaethics and the philosophy of religion. Philosophers and theologians aiming to defend theism against the threat of the dilemma have developed a variety of responses.

Supporters
The first horn of the dilemma (i.e. that which is right is commanded by God because it is right) goes by a variety of names, including intellectualism, rationalism, realism, naturalism, and objectivism. Roughly, it is the view that there are independent moral standards: some actions are right or wrong in themselves, independent of God's commands. This is the view accepted by Socrates and Euthyphro in Plato's dialogue. The Mu'tazilah school of Islamic theology also defended the view (with, for example, Nazzam maintaining that God is powerless to engage in injustice or lying), as did the Islamic philosopher Averroes. Thomas Aquinas never explicitly addresses the Euthyphro dilemma, but Aquinas scholars often put him on this side of the issue. Aquinas draws a distinction between what is good or evil in itself and what is good or evil because of God's commands, with unchangeable moral standards forming the bulk of natural law. Thus he contends that not even God can change the Ten Commandments (adding, however, that God can change what individuals deserve in particular cases, in what might look like special dispensations to murder or stealing). Among later Scholastics, Gabriel Vásquez is particularly clear-cut about obligations existing prior to anyone's will, even God's. Modern natural law theory saw Grotius and Leibniz also putting morality prior to God's will, comparing moral truths to unchangeable mathematical truths, and engaging voluntarists like Pufendorf in philosophical controversy. Cambridge Platonists like Benjamin Whichcote and Ralph Cudworth mounted seminal attacks on voluntarist theories, paving the way for the later rationalist metaethics of Samuel Clarke and Richard Price; what emerged was a view on which eternal moral standards, though dependent on God in some way, exist independently of God's will and prior to God's commands. Contemporary philosophers of religion who embrace this horn of the Euthyphro dilemma include Richard Swinburne and T. J. Mawson (though see below for complications).

Criticisms

 * Sovereignty: If there are moral standards independent of God's will, then "[t]here is something over which God is not sovereign. God is bound by the laws of morality instead of being their establisher. Moreover, God depends for his goodness on the extent to which he conforms to an independent moral standard. Thus, God is not absolutely independent." 18th-century philosopher Richard Price, who takes the first horn and thus sees morality as "necessary and immutable", sets out the objection as follows: "It may seem that this is setting up something distinct from God, which is independent of him, and equally eternal and necessary."
 * Omnipotence: These moral standards would limit God's power: not even God could oppose them by commanding what is evil and thereby making it good. This point was influential in Islamic theology: "In relation to God, objective values appeared as a limiting factor to His power to do as He wills... Ash'ari got rid of the whole problem by denying the existence of objective values which might act as a standard for God's action." Similar concerns drove the medieval voluntarists Duns Scotus and William of Ockham. As contemporary philosopher Richard Swinburne puts the point, this horn "seems to place a restriction on God's power if he cannot make any action which he chooses obligatory... [and also] it seems to limit what God can command us to do. God, if he is to be God, cannot command us to do what, independently of his will, is wrong."
 * Freedom of the will: Moreover, these moral standards would limit God's freedom of will: God could not command anything opposed to them, and perhaps would have no choice but to command in accordance with them. As Mark Murphy puts the point, "if moral requirements existed prior to God's willing them, requirements that an impeccable God could not violate, God's liberty would be compromised."
 * Morality without God: If there are moral standards independent of God, then morality would retain its authority even if God did not exist. This conclusion was explicitly (and notoriously) drawn by early modern political theorist Hugo Grotius: "What we have been saying [about the natural law] would have a degree of validity even if we should concede that which cannot be conceded without the utmost wickedness, that there is no God, or that the affairs of men are of no concern to him" On such a view, God is no longer a "law-giver" but at most a "law-transmitter" who plays no vital role in the foundations of morality. Nontheists have capitalized on this point, largely as a way of disarming moral arguments for God's existence: if morality does not depend on God in the first place, such arguments stumble at the starting gate.

Responses to criticisms
Contemporary philosophers Joshua Hoffman and Gary S. Rosenkrantz take the first horn of the dilemma, branding divine command theory a "subjective theory of value" that makes morality arbitrary. They accept a theory of morality on which, "right and wrong, good and bad, are in a sense independent of what anyone believes, wants, or prefers." They do not address the aforementioned problems with the first horn, but do consider a related problem concerning God's omnipotence: namely, that it might be handicapped by his inability to bring about what is independently evil. To this they reply that God is omnipotent, even though there are states of affairs he cannot bring about: omnipotence is a matter of "maximal power", not an ability to bring about all possible states of affairs. And supposing that it is impossible for God not to exist, then since there cannot be more than one omnipotent being, it is therefore impossible for any being to have more power than God (e.g., a being who is omnipotent but not omnibenevolent). Thus God's omnipotence remains intact.

Richard Swinburne and T. J. Mawson have a slightly more complicated view. They both take the first horn of the dilemma when it comes to necessary moral truths. But divine commands are not totally irrelevant, for God and his will can still effect contingent moral truths. On the one hand, the most fundamental moral truths hold true regardless of whether God exists or what God has commanded: "Genocide and torturing children are wrong and would remain so whatever commands any person issued." This is because, according to Swinburne, such truths are true as a matter of logical necessity: like the laws of logic, one cannot deny them without contradiction. This parallel offers a solution to the aforementioned problems of God's sovereignty, omnipotence, and freedom: namely, that these necessary truths of morality pose no more of a threat than the laws of logic. On the other hand, there is still an important role for God's will. First, there are some divine commands that can directly create moral obligations: e.g., the command to worship on Sundays instead of on Tuesdays. Notably, not even these commands, for which Swinburne and Mawson take the second horn of the dilemma, have ultimate, underived authority. Rather, they create obligations only because of God's role as creator and sustainer and indeed owner of the universe, together with the necessary moral truth that we owe some limited consideration to benefactors and owners. Second, God can make an indirect moral difference by deciding what sort of universe to create. For example, whether a public policy is morally good might indirectly depend on God's creative acts: the policy's goodness or badness might depend on its effects, and those effects would in turn depend on the sort of universe God has decided to create.

Supporters
The second horn of the dilemma (i.e. that which is right is right because it is commanded by God) is sometimes known as divine command theory or voluntarism. Roughly, it is the view that there are no moral standards other than God's will: without God's commands, nothing would be right or wrong. This view was partially defended by Duns Scotus, who argued that not all Ten Commandments belong to the Natural Law in the strictest sense. Scotus held that while our duties to God (the first three commandments, traditionally thought of as the First Tablet) are self-evident, true by definition, and unchangeable even by God, our duties to others (found on the second tablet) were arbitrarily willed by God and are within his power to revoke and replace (although, the third commandment, to honour the Sabbath and keep it holy, has a little of both, as we are absolutely obliged to render worship to God, but there is no obligation in natural law to do it on this day or that). Scotus does note, however that the last seven commandments "are highly consonant with [the natural law], though they do not follow necessarily from first practical principles that are known in virtue of their terms and are necessarily known by any intellect [that understands their terms. And it is certain that all the precepts of the second table belong to the natural law in this second way, since their rectitude is highly consonant with first practical principles that are known necessarily". Scotus justifies this position with the example of a peaceful society, noting that the possession of private property is not necessary to have a peaceful society, but that "those of weak character" would be more easily made peaceful with private property than without.

William of Ockham went further, contending that (since there is no contradiction in it) God could command us not to love God and even to hate God. Later Scholastics like Pierre D'Ailly and his student Jean de Gerson explicitly confronted the Euthyphro dilemma, taking the voluntarist position that God does not "command good actions because they are good or prohibit evil ones because they are evil; but... these are therefore good because they are commanded and evil because prohibited." Protestant reformers Martin Luther and John Calvin both stressed the absolute sovereignty of God's will, with Luther writing that "for [God's] will there is no cause or reason that can be laid down as a rule or measure for it", and Calvin writing that "everything which [God] wills must be held to be righteous by the mere fact of his willing it." The voluntarist emphasis on God's absolute power was carried further by Descartes, who notoriously held that God had freely created the eternal truths of logic and mathematics, and that God was therefore capable of giving circles unequal radii, giving triangles other than 180 internal degrees, and even making contradictions true. Descartes explicitly seconded Ockham: "why should [God] not have been able to give this command [i.e., the command to hate God] to one of his creatures?" Thomas Hobbes notoriously reduced the justice of God to "irresistible power" (drawing the complaint of Bishop Bramhall that this "overturns... all law"). And William Paley held that all moral obligations bottom out in the self-interested "urge" to avoid Hell and enter Heaven by acting in accord with God's commands. Islam's Ash'arite theologians, al-Ghazali foremost among them, embraced voluntarism: scholar George Hourani writes that the view "was probably more prominent and widespread in Islam than in any other civilization." Wittgenstein said that of "the two interpretations of the Essence of the Good", that which holds that "the Good is good, in virtue of the fact that God wills it" is "the deeper", while that which holds that "God wills the good, because it is good" is "the shallow, rationalistic one, in that it behaves 'as though' that which is good could be given some further foundation". Today, divine command theory is defended by many philosophers of religion, though typically in a restricted form (see below).

Criticisms
This horn of the dilemma also faces several problems:

 * No reasons for morality: If there is no moral standard other than God's will, then God's commands are arbitrary (i.e., based on pure whimsy or caprice). This would mean that morality is ultimately not based on reasons: "if theological voluntarism is true, then God's commands/intentions must be arbitrary; [but] it cannot be that morality could wholly depend on something arbitrary... [for] when we say that some moral state of affairs obtains, we take it that there is a reason for that moral state of affairs obtaining rather than another." And as Michael J. Murray and Michael Rea put it, this would also "cas[t] doubt on the notion that morality is genuinely objective." An additional problem is that it is difficult to explain how true moral actions can exist if one acts only out of fear of God or in an attempt to be rewarded by him.
 * No reasons for God: This arbitrariness would also jeopardize God's status as a wise and rational being, one who always acts on good reasons. As Leibniz writes: "Where will be his justice and his wisdom if he has only a certain despotic power, if arbitrary will takes the place of reasonableness, and if in accord with the definition of tyrants, justice consists in that which is pleasing to the most powerful? Besides it seems that every act of willing supposes some reason for the willing and this reason, of course, must precede the act."
 * Anything goes: This arbitrariness would also mean that anything could become good, and anything could become bad, merely upon God's command. Thus if God commanded us "to gratuitously inflict pain on each other" or to engage in "cruelty for its own sake" or to hold an "annual sacrifice of randomly selected ten-year-olds in a particularly gruesome ritual that involves excruciating and prolonged suffering for its victims", then we would be morally obligated to do so. As 17th-century philosopher Ralph Cudworth put it: "nothing can be imagined so grossly wicked, or so foully unjust or dishonest, but if it were supposed to be commanded by this omnipotent Deity, must needs upon that hypothesis forthwith become holy, just, and righteous."
 * Moral contingency: If morality depends on the perfectly free will of God, morality would lose its necessity: "If nothing prevents God from loving things that are different from what God actually loves, then goodness can change from world to world or time to time. This is obviously objectionable to those who believe that claims about morality are, if true, necessarily true." In other words, no action is necessarily moral: any right action could have easily been wrong, if God had so decided, and an action which is right today could easily become wrong tomorrow, if God so decides. Indeed, some have argued that divine command theory is incompatible with ordinary conceptions of moral supervenience.
 * Why do God's commands obligate?: Mere commands do not create obligations unless the commander has some commanding authority. But this commanding authority cannot itself be based on those very commands (i.e., a command to obey commands), otherwise a vicious circle results. So, in order for God's commands to obligate us, he must derive commanding authority from some source other than his own will. As Cudworth put it: "For it was never heard of, that any one founded all his authority of commanding others, and others [sic] obligation or duty to obey his commands, in a law of his own making, that men should be required, obliged, or bound to obey him. Wherefore since the thing willed in all laws is not that men should be bound or obliged to obey; this thing cannot be the product of the meer  [sic] will of the commander, but it must proceed from something else; namely, the right or authority of the commander." To avoid the circle, one might say our obligation comes from gratitude to God for creating us. But this presupposes some sort of independent moral standard obligating us to be grateful to our benefactors. As 18th-century philosopher Francis Hutcheson writes: "Is the Reason exciting to concur with the Deity this, 'The Deity is our Benefactor?' Then what Reason excites to concur with Benefactors?" Or finally, one might resort to Hobbes's view: "The right of nature whereby God reigneth over men, and punisheth those that break his laws, is to be derived, not from his creating them (as if he required obedience, as of gratitude for his benefits), but from his irresistible power." In other words, might makes right.
 * God's goodness: If all goodness is a matter of God's will, then what shall become of God's goodness? Thus William P. Alston writes, "since the standards of moral goodness are set by divine commands, to say that God is morally good is just to say that he obeys his own commands... that God practises what he preaches, whatever that might be;" Hutcheson deems such a view "an insignificant tautology, amounting to no more than this, 'That God wills what he wills. Alternatively, as Leibniz puts it, divine command theorists "deprive God of the designation good: for what cause could one have to praise him for what he does, if in doing something quite different he would have done equally well?" A related point is raised by C. S. Lewis: "if good is to be defined as what God commands, then the goodness of God Himself is emptied of meaning and the commands of an omnipotent fiend would have the same claim on us as those of the 'righteous Lord. Or again Leibniz: "this opinion would hardly distinguish God from the devil." That is, since divine command theory trivializes God's goodness, it is incapable of explaining the difference between God and an all-powerful demon.
 * The is-ought problem and the naturalistic fallacy: According to David Hume, it is hard to see how moral propositions featuring the relation ought could ever be deduced from ordinary is propositions, such as "the being of a God." Divine command theory is thus guilty of deducing moral oughts from ordinary ises about God's commands. In a similar vein, G. E. Moore argued (with his open question argument) that the notion good is indefinable, and any attempts to analyze it in naturalistic or metaphysical terms are guilty of the so-called "naturalistic fallacy." This would block any theory which analyzes morality in terms of God's will: and indeed, in a later discussion of divine command theory, Moore concluded that "when we assert any action to be right or wrong, we are not merely making an assertion about the attitude of mind towards it of any being or set of beings whatever."
 * No morality without God: If all morality is a matter of God's will, then if God does not exist, there is no morality. This is the thought captured in the slogan (often attributed to Dostoevsky) "If God does not exist, everything is permitted." Divine command theorists disagree over whether this is a problem for their view or a virtue of their view. Many argue that morality does indeed require God's existence, and that this is in fact a problem for atheism. But divine command theorist Robert Merrihew Adams contends that this idea ("that no actions would be ethically wrong if there were not a loving God") is one that "will seem (at least initially) implausible to many", and that his theory must "dispel [an] air of paradox."

Restricted divine command theory
One common response to the Euthyphro dilemma centers on a distinction between value and obligation. Obligation, which concerns rightness and wrongness (or what is required, forbidden, or permissible), is given a voluntarist treatment. But value, which concerns goodness and badness, is treated as independent of divine commands. The result is a restricted divine command theory that applies only to a specific region of morality: the deontic region of obligation. This response is found in Francisco Suárez's discussion of natural law and voluntarism in De legibus and has been prominent in contemporary philosophy of religion, appearing in the work of Robert M. Adams, Philip L. Quinn, and William P. Alston.

A significant attraction of such a view is that, since it allows for a non-voluntarist treatment of goodness and badness, and therefore of God's own moral attributes, some of the aforementioned problems with voluntarism can perhaps be answered. God's commands are not arbitrary: there are reasons which guide his commands based ultimately on this goodness and badness. God could not issue horrible commands: God's own essential goodness or loving character would keep him from issuing any unsuitable commands. Our obligation to obey God's commands does not result in circular reasoning; it might instead be based on a gratitude whose appropriateness is itself independent of divine commands. These proposed solutions are controversial, and some steer the view back into problems associated with the first horn.

One problem remains for such views: if God's own essential goodness does not depend on divine commands, then on what does it depend? Something other than God? Here the restricted divine command theory is commonly combined with a view reminiscent of Plato: God is identical to the ultimate standard for goodness. Alston offers the analogy of the standard meter bar in France. Something is a meter long inasmuch as it is the same length as the standard meter bar, and likewise, something is good inasmuch as it approximates God. If one asks why God is identified as the ultimate standard for goodness, Alston replies that this is "the end of the line," with no further explanation available, but adds that this is no more arbitrary than a view that invokes a fundamental moral standard. On this view, then, even though goodness is independent of God's will, it still depends on God, and thus God's sovereignty remains intact.

This solution has been criticized by Wes Morriston. If we identify the ultimate standard for goodness with God's nature, then it seems we are identifying it with certain properties of God (e.g., being loving, being just). If so, then the dilemma resurfaces: is God good because he has those properties, or are those properties good because God has them? Nevertheless, Morriston concludes that the appeal to God's essential goodness is the divine-command theorist's best bet. To produce a satisfying result, however, it would have to give an account of God's goodness that does not trivialize it and does not make God subject to an independent standard of goodness.

Moral philosopher Peter Singer, disputing the perspective that "God is good" and could never advocate something like torture, states that those who propose this are "caught in a trap of their own making, for what can they possibly mean by the assertion that God is good? That God is approved of by God?"

False dilemma in classical theistic perspective
Augustine, Anselm, and Aquinas all wrote about the issues raised by the Euthyphro dilemma, although, like William James and Wittgenstein later, they did not mention it by name. As philosopher and Anselm scholar Katherin A. Rogers observes, many contemporary philosophers of religion suppose that there are true propositions which exist as platonic abstracta independently of God. Among these are propositions constituting a moral order, to which God must conform in order to be good. Classical Judaeo-Christian theism, however, rejects such a view as inconsistent with God's omnipotence, which requires that God and what he has made is all that there is. "The classical tradition," Rogers notes, "also steers clear of the other horn of the Euthyphro dilemma, divine command theory." From a classical theistic perspective, therefore, the Euthyphro dilemma is false. As Rogers puts it, "Anselm, like Augustine before him and Aquinas later, rejects both horns of the Euthyphro dilemma. God neither conforms to nor invents the moral order. Rather His very nature is the standard for value." Another criticism raised by Peter Geach is that the dilemma implies you must search for a definition that fits piety rather than work backwards by deciding pious acts (i.e. you must know what piety is before you can list acts which are pious). It also implies something can not be pious if it is only intended to serve the Gods without actually fulfilling any useful purpose.

Jewish thought
The basis of the false dilemma response—God's nature is the standard for value—predates the dilemma itself, appearing first in the thought of the eighth-century BC Hebrew prophets, Amos, Hosea, Micah and Isaiah. (Amos lived some three centuries before Socrates and two before Thales, traditionally regarded as the first Greek philosopher.) "Their message," writes British scholar Norman H. Snaith, "is recognized by all as marking a considerable advance on all previous ideas," not least in its "special consideration for the poor and down-trodden." As Snaith observes, tsedeq, the Hebrew word for righteousness, "actually stands for the establishment of God's will in the land." This includes justice, but goes beyond it, "because God's will is wider than justice. He has a particular regard for the helpless ones on earth." Tsedeq "is the norm by which all must be judged" and it "depends entirely upon the Nature of God."

Hebrew has few abstract nouns. What the Greeks thought of as ideas or abstractions, the Hebrews thought of as activities. In contrast to the Greek dikaiosune (justice) of the philosophers, tsedeq is not an idea abstracted from this world of affairs. As Snaith writes: "Tsedeq is something that happens here, and can be seen, and recognized, and known. It follows, therefore, that when the Hebrew thought of tsedeq (righteousness), he did not think of Righteousness in general, or of Righteousness as an Idea. On the contrary, he thought of a particular righteous act, an action, concrete, capable of exact description, fixed in time and space.... If the word had anything like a general meaning for him, then it was as it was represented by a whole series of events, the sum-total of a number of particular happenings."

The Hebrew stance on what came to be called the problem of universals, as on much else, was very different from that of Plato and precluded anything like the Euthyphro dilemma. This has not changed. In 2005, Jonathan Sacks wrote, "In Judaism, the Euthyphro dilemma does not exist." Jewish philosophers Avi Sagi and Daniel Statman criticized the Euthyphro dilemma as "misleading" because "it is not exhaustive": it leaves out a third option, namely that God "acts only out of His nature."

St. Thomas Aquinas
Like Aristotle, Aquinas rejected Platonism. In his view, to speak of abstractions not only as existent, but as more perfect exemplars than fully designated particulars, is to put a premium on generality and vagueness. On this analysis, the abstract "good" in the first horn of the Euthyphro dilemma is an unnecessary obfuscation. Aquinas frequently quoted with approval Aristotle's definition, "Good is what all desire." As he clarified, "When we say that good is what all desire, it is not to be understood that every kind of good thing is desired by all, but that whatever is desired has the nature of good." In other words, even those who desire evil desire it "only under the aspect of good," i.e., of what is desirable. The difference between desiring good and desiring evil is that in the former, will and reason are in harmony, whereas in the latter, they are in discord.

Aquinas's discussion of sin provides a good point of entry to his philosophical explanation of why the nature of God is the standard for value. "Every sin," he writes, "consists in the longing for a passing [i.e., ultimately unreal or false] good." Thus, "in a certain sense it is true what Socrates says, namely that no one sins with full knowledge." "No sin in the will happens without an ignorance of the understanding." God, however, has full knowledge (omniscience) and therefore by definition (that of Socrates, Plato, and Aristotle as well as Aquinas) can never will anything other than what is good. It has been claimed – for instance, by Nicolai Hartmann, who wrote: "There is no freedom for the good that would not be at the same time freedom for evil" – that this would limit God's freedom, and therefore his omnipotence. Josef Pieper, however, replies that such arguments rest upon an impermissibly anthropomorphic conception of God. In the case of humans, as Aquinas says, to be able to sin is indeed a consequence, or even a sign, of freedom (quodam libertatis signum). Humans, in other words, are not puppets manipulated by God so that they always do what is right. However, "it does not belong to the essence of the free will to be able to decide for evil." "To will evil is neither freedom nor a part of freedom." It is precisely humans' creatureliness – that is, their not being God and therefore omniscient – that makes them capable of sinning. Consequently, writes Pieper, "the inability to sin should be looked on as the very signature of a higher freedom – contrary to the usual way of conceiving the issue." Pieper concludes: "Only the will [i.e., God's] can be the right standard of its own willing and must will what is right necessarily, from within itself, and always. A deviation from the norm would not even be thinkable. And obviously only the absolute divine will is the right standard of its own act" – and consequently of all human acts. Thus the second horn of the Euthyphro dilemma, divine command theory, is also disposed of.

Thomist philosopher Edward Feser writes, "Divine simplicity [entails] that God's will just is God's goodness which just is His immutable and necessary existence. That means that what is objectively good and what God wills for us as morally obligatory are really the same thing considered under different descriptions, and that neither could have been other than they are. There can be no question then, either of God's having arbitrarily commanded something different for us (torturing babies for fun, or whatever) or of there being a standard of goodness apart from Him. Again, the Euthyphro dilemma is a false one; the third option that it fails to consider is that what is morally obligatory is what God commands in accordance with a non-arbitrary and unchanging standard of goodness that is not independent of Him... He is not under the moral law precisely because He is the moral law."

William James
William James, in his essay "The Moral Philosopher and the Moral Life", dismisses the first horn of the Euthyphro dilemma and stays clear of the second. He writes: "Our ordinary attitude of regarding ourselves as subject to an overarching system of moral relations, true 'in themselves,' is ... either an out-and-out superstition, or else it must be treated as a merely provisional abstraction from that real Thinker ... to whom the existence of the universe is due." Moral obligations are created by "personal demands," whether these demands come from the weakest creatures, from the most insignificant persons, or from God. It follows that "ethics have as genuine a foothold in a universe where the highest consciousness is human, as in a universe where there is a God as well." However, whether "the purely human system" works "as well as the other is a different question."

For James, the deepest practical difference in the moral life is between what he calls "the easy-going and the strenuous mood." In a purely human moral system, it is hard to rise above the easy-going mood, since the thinker's "various ideals, known to him to be mere preferences of his own, are too nearly of the same denominational value; he can play fast and loose with them at will. This too is why, in a merely human world without a God, the appeal to our moral energy falls short of its maximum stimulating power." Our attitude is "entirely different" in a world where there are none but "finite demanders" from that in a world where there is also "an infinite demander." This is because "the stable and systematic moral universe for which the ethical philosopher asks is fully possible only in a world where there is a divine thinker with all-enveloping demands", for in that case, "actualized in his thought already must be that ethical philosophy which we seek as the pattern which our own must evermore approach." Even though "exactly what the thought of this infinite thinker may be is hidden from us", our postulation of him serves "to let loose in us the strenuous mood" and confront us with an existential "challenge" in which "our total character and personal genius ... are on trial; and if we invoke any so-called philosophy, our choice and use of that also are but revelations of our personal aptitude or incapacity for moral life. From this unsparing practical ordeal no professor's lectures and no array of books can save us." In the words of Richard M. Gale, "God inspires us to lead the morally strenuous life in virtue of our conceiving of him as unsurpassably good. This supplies James with an adequate answer to the underlying question of the Euthyphro."

Atheistic resolutions
Atheism challenges the assumption of the dilemma that God exists (or in the original formulation, that the many gods in Greek religion existed). This eliminates the need to decide whether God is either non-omniscient or arbitrary, and also eliminates the possibility of God as the source of morality.

Secular humanism takes the positive stance that morality is not dependent on religion or theology, and that ethical rules should be developed based on reason, science, experience, debate, and democracy. Some secular humanists believe in ethical naturalism, that there are objective, discoverable laws of morality inherent to the human condition, of which humans may have imperfect knowledge. Others have adopted ethical subjectivism in the sense of meta-ethics – the idea that ethics are a social construct – but nonetheless by way of utilitarianism advocate imposing a set of universal ethics and laws that create the type of society in which they wish to live, where people are safe, prosperous, and happy. These competing resolutions represent different answers to a question similar to the original dilemma: "Is something inherently ethical or unethical, or is something ethical or unethical because a person or society says it is so?"

Rejection of universal morality
The other assumption of the dilemma is that there is a universal right and wrong, against which a god either creates or is defined by. Moral nihilism challenges that assumption by rejecting the concept of morality entirely. This conflicts with the teachings of most religions (and thus is usually accompanied by atheism) but is theoretically compatible with the notion of a powerful God or gods who have opinions about how people should behave.

Alexander Rosenberg uses a version of the Euthyphro dilemma to argue that objective morality cannot exist and hence an acceptance of moral nihilism is warranted. He asks, is objective morality correct because evolution discovered it or did evolution discover objective morality because it is correct? If the first horn of the dilemma is true then our current morality cannot be objectively correct by accident because if evolution had given us another type of morality then that would have been objectively correct. If the second horn of dilemma is true then one must account for how the random process of evolution managed to only select for objectively correct moral traits while ignoring the wrong moral traits. Given the knowledge that evolution has given us tendencies to be xenophobic and sexist it is mistaken to claim that evolution has only selected for objective morality as evidently it did not. Because both horns of the dilemma do not give an adequate account for how the evolutionary process instantiated objective morality in humans, a position of Moral nihilism is warranted.

Moral relativism accepts the idea of morality, but asserts that there are multiple potential arbiters of moral truth. This opens the possibility of disagreeing with God about the rules of ethics, and of creating multiple societies with different, equally valid sets of ethics (just as different countries have different sets of laws). "Normative moral relativism" asserts that behavior based on alternative systems of morality should be tolerated. In the context of religious pluralism, strong relativism it also opens the possibility that different gods and different belief systems produce different but equally valid moral systems, which may apply only to adherents of those faiths.

In popular culture
In the song "No Church in the Wild" from the album Watch the Throne, rapper Jay Z references the dilemma with the line, "Is pious pious 'cause God loves pious? Socrates asked whose bias do y'all seek."

In American legal thinking
Yale Law School Professor Myres S. McDougal, formerly a classicist, later a scholar of property law, posed the question, "Do we protect it because it's a property right, or is it a property right because we protect it?" The dilemma has also been restated in legal terms by Geoffrey Hodgson, who asked: "Does a state make a law because it is a customary rule, or does law become a customary rule because it is approved by the state?"

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Euthyphro dilemma


text:
Meša Selimović

Mehmed "Meša" Selimović (Мехмед „Меша” Селимовић; ; 26 April 1910 – 11 July 1982) was a Yugoslav writer, whose works are widely considered some of the most important in Bosnian and Serbian literature. Some of the main themes in his works are the relations between individuality and authority, life and death, and other existential problems.

Biography
Selimović was born to a prominent Bosnian Muslim family of Serbian descent on 26 April 1910 in Tuzla, Bosnia and Herzegovina, where he graduated from elementary school and high school.

In 1930, he enrolled to study the Serbian language and literature at the University of Belgrade Faculty of Philology and graduated in 1934. His lecturers included Bogdan Popović, Pavle Popović, Vladimir Ćorović, Veselin Čajkanović, Aleksandar Belić and Stjepan Kuljbakin. In 1936, he returned to Tuzla to teach in the gymnasium that today bears his name. At that time he participated in the Soko athletic organisation. He spent the first two years of the Second World War in Tuzla, until he was arrested for participation in the Partisan anti-fascist resistance movement in 1943. After his release, he moved to liberated territory, became a member of Communist Party of Yugoslavia and the political commissar of the Tuzla Detachment of the Partisans. During the war, Selimović's brother, also a communist, was executed by partisans' firing squad for alleged theft, without trial; Selimović's letter in defense of the brother was to no avail. That episode apparently affected Meša's later contemplative introduction to Death and the Dervish, where the main protagonist Ahmed Nurudin fails to rescue his imprisoned brother. After the war, he briefly resided in Belgrade, and in 1947 he moved to Sarajevo, where he was the professor of High School of Pedagogy and Faculty of Philology, art director of Bosna Film, chief of the drama section of the National Theater, and chief editor of the publishing house Svjetlost. Exasperated by a latent conflict with several local politicians and intellectuals, in 1971 he moved to Belgrade, where he lived until his death in 1982.

Identity
Selimović researched the roots of his family and found out that he originated from the Drobnjaci tribe. Most members of the tribe consider themselves to be Serbs, while some are Montenegrins. It is claimed that a part of the family converted to Islam in order to protect their Christian brethren. In his 1976 letter to the Serbian Academy of Science and Arts, Selimović stated for the historical record that he regarded himself as a Serb and belonging to the corpus of Serbian literature. Selimović was a full member of the Academy of Sciences and Arts of Bosnia and Herzegovina, and a member of Serbian Academy of Sciences and Arts.

In his autobiographical narration, Sjećanja, which Selimović complements with a memoir features thus transforming them into memoir prose, Selimović describes environment and milieu of his Bosnian Muslim origin. He is using discursive self-perception, and confronts and auto-reflect his identity as a complex and composite. Since perception of national belonging is distinctly subjective and simplistic, auto-perceptions are considered discursive creations, representamen, where memoirs overlap with socio-historical context. In doing so, and through lens of imagology, his autobiographical discourse becomes textual construct, or an imaginary discourse. Selimović's imaginarium turns his cultural self-reflection of his Bosnian Muslim identity into oddity, but he also describes it as a complex. His memories author then transpose on entire group, with a series of images.

Through the rhetoric of the image, Selimović confirms the cultural differences of Bosnian Muslims, and in that sense, his autobiographical narrative representamen confirms and strengthens the cultural and collective ethnic identity of Muslims. Selimović clearly define himself by stating, "I am a Muslim", and, "I am attached to my Bosnian and Muslim origins".

On the other hand, when Selimović brought forward the information about his Christian origin, some Bosnian Muslim critics attacked him, claiming that "they also knew about their origin", and asking him what is to be achieved with publicly expressing such information.

Critics consider this to be a rationalization of his choice to seek recognition as writer belonging to Serbian literary circle, by claiming that his paternal heritage was that of Orthodox Christian identity, alleging a conversion to Islam back in the 17th century for pragmatic reasons. The chapter Parents in his Sjećanje provoked reaction and criticism in his native country, and will be deemed a "constructed phantasm", or imaginary discourse. Critics contemplated about the reasons for this, as they called, "falsification of one's own family heritage", explaining it as a "vengeful act of defiance", and stating that Selimović's main conflicts trace back to his Muslim roots and his expression of disappointment in Bosnian environment and Bosnian Muslim milieu.

He was a communist and atheist.

Works
Selimović began writing fairly late in his life. His first short story (Pjesma u oluji / A song in the storm) was published in 1948, when he was thirty-six. His first book, a collection of short stories Prva četa (The First Company) was published in 1950 when he was forty. His subsequent work, Tišine (Silences) was published eleven years later in 1961. The following books Tuđa zemlja (Foreign land, 1962) and Magla i mjesečina (Mist and Moonlight, 1965) did not receive widespread recognition either.

However, his novel Death and the Dervish (Derviš i smrt, 1966) was widely received as a masterpiece. The plot of the novel takes place in 18th-century Sarajevo under Ottoman rule, and reflects Selimović's own torment of the execution of his brother; the story speaks of the futility of one man's resistance against a repressive system, and the change that takes place within that man after he becomes a part of that very system. Some critics have likened this novel to Kafka's The Trial. It has been translated into many languages, including English, Russian, German, French, Italian, Turkish and Arabic. Each chapter of the novel opens with a Quran citation, the first being: "In the name of God, the most compassionate, the most merciful."

The next novel, Tvrđava (The Fortress, 1970), placed still further in the past, is slightly more optimistic, and fulfilled with faith in love, unlike the lonely contemplations and fear in Death and the Dervish. The Fortress and Death and the Dervish, as well as the subsequent Ostrvo (The Island, 1974), featuring an elderly couple facing aging and eventual death on a Dalmatian island, are the only novels of Selimović that have thus far been translated into English. The posthumously published Krug (The Circle, 1983) has not been translated into English.

He also wrote a book about Vuk Karadžić's orthographic reforms Za i protiv Vuka (For and Against Vuk), as well as his autobiography, Sjećanja.

Poopak NikTalab introduces him as one of the three pioneers of children's and youth literature in the Balkans between 1950 and 1980 (along with Šukria Pandžu and Iskandar Klovnić), who played an important role in the development of Bosnian children's and youth literature.

Family
His brother’s granddaughter is Serbian actress Hana Selimović. Also, his cousin Amar Selimović is a Bosnian actor. Meša Selimović is the uncle of Bosnian politician Mirsad Đonlagić.

Translations into English

 * The Island, 1974, The Serbian Heritage Academy of Canada, ISBN 0-920069-00-2
 * Death and the Dervish, 1996, Northwestern University Press, ISBN 0-8101-1297-3
 * The Fortress, 1999, Northwestern University Press, ISBN 0-8101-1713-4

title: Meša Selimović


text:
Density of states

In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or states per unit energy range. The density of states is defined as $ D(E) = N(E) / V $, where $$N(E)\delta E$$ is the number of states in the system of volume $$V$$ whose energies lie in the range from $$E$$ to $$E+\delta E$$. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.

Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referred to as local densities of states (LDOSs).

Introduction
In quantum mechanical systems, waves, or wave-like particles, can occupy modes or states with wavelengths and propagation directions dictated by the system. For example, in some systems, the interatomic spacing and the atomic charge of a material might allow only electrons of certain wavelengths to exist. In other systems, the crystalline structure of a material might allow waves to propagate in one direction, while suppressing wave propagation in another direction. Often, only specific states are permitted. Thus, it can happen that many states are available for occupation at a specific energy level, while no states are available at other energy levels.

Looking at the density of states of electrons at the band edge between the valence and conduction bands in a semiconductor, for an electron in the conduction band, an increase of the electron energy makes more states available for occupation. Alternatively, the density of states is discontinuous for an interval of energy, which means that no states are available for electrons to occupy within the band gap of the material. This condition also means that an electron at the conduction band edge must lose at least the band gap energy of the material in order to transition to another state in the valence band.

This determines if the material is an insulator or a metal in the dimension of the propagation. The result of the number of states in a band is also useful for predicting the conduction properties. For example, in a one dimensional crystalline structure an odd number of electrons per atom results in a half-filled top band; there are free electrons at the Fermi level resulting in a metal. On the other hand, an even number of electrons exactly fills a whole number of bands, leaving the rest empty. If then the Fermi level lies in an occupied band gap between the highest occupied state and the lowest empty state, the material will be an insulator or semiconductor.

Depending on the quantum mechanical system, the density of states can be calculated for electrons, photons, or phonons, and can be given as a function of either energy or the wave vector $k$. To convert between the DOS as a function of the energy and the DOS as a function of the wave vector, the system-specific energy dispersion relation between $E$ and $k$ must be known.

In general, the topological properties of the system such as the band structure, have a major impact on the properties of the density of states. The most well-known systems, like neutron matter in neutron stars and free electron gases in metals (examples of degenerate matter and a Fermi gas), have a 3-dimensional Euclidean topology. Less familiar systems, like two-dimensional electron gases (2DEG) in graphite layers and the quantum Hall effect system in MOSFET type devices, have a 2-dimensional Euclidean topology. Even less familiar are carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. Systems with 1D and 2D topologies are likely to become more common, assuming developments in nanotechnology and materials science proceed.

Definition
The density of states related to volume $V$ and $N$ countable energy levels is defined as: $$ D(E) = \frac{1}{V} \, \sum _{i=1}^N \delta (E - E({\mathbf{k}}_i)). $$ Because the smallest allowed change of momentum $$k$$ for a particle in a box of dimension $$d$$ and length $$L$$ is $$ (\Delta k)^d = ({2\pi}/{L})^d $$, the volume-related density of states for continuous energy levels is obtained in the limit $$L \to \infty$$ as $$ D(E) := \int_{\mathbb{R}^d}{\frac{\mathrm{d}^d k}{(2\pi)^d}} \cdot \delta (E - E(\mathbf{k})),$$ Here, $$d$$ is the spatial dimension of the considered system and $$\mathbf{k}$$ the wave vector.

For isotropic one-dimensional systems with parabolic energy dispersion, the density of states is $D_{1D}(E) = \tfrac{1}{2\pi\hbar} (\tfrac{2m}{E})^{1/2}$. In two dimensions the density of states is a constant $D_{2D} = \tfrac{m}{2\pi\hbar^2}$, while in three dimensions it becomes $D_{3D}(E) = \tfrac{m}{2\pi^2\hbar^3} (2mE)^{1/2}$.

Equivalently, the density of states can also be understood as the derivative of the microcanonical partition function $$ Z_m (E)$$ (that is, the total number of states with energy less than $$E$$) with respect to the energy: $$ D(E) = \frac {1}{V} \cdot \frac{\mathrm{d} Z_m (E)}{\mathrm{d} E}.$$

The number of states with energy $$ E' $$ (degree of degeneracy) is given by: $$ g\left(E'\right) = \lim _{\Delta E\to 0} \int _{E'}^{E' + \Delta E} D(E) \, \mathrm{d} E = \lim _{\Delta E\to 0} D\left(E'\right) \Delta E,$$ where the last equality only applies when the mean value theorem for integrals is valid.

Symmetry
There is a large variety of systems and types of states for which DOS calculations can be done.

Some condensed matter systems possess a structural symmetry on the microscopic scale which can be exploited to simplify calculation of their densities of states. In spherically symmetric systems, the integrals of functions are one-dimensional because all variables in the calculation depend only on the radial parameter of the dispersion relation. Fluids, glasses and amorphous solids are examples of a symmetric system whose dispersion relations have a rotational symmetry.

Measurements on powders or polycrystalline samples require evaluation and calculation functions and integrals over the whole domain, most often a Brillouin zone, of the dispersion relations of the system of interest. Sometimes the symmetry of the system is high, which causes the shape of the functions describing the dispersion relations of the system to appear many times over the whole domain of the dispersion relation. In such cases the effort to calculate the DOS can be reduced by a great amount when the calculation is limited to a reduced zone or fundamental domain. The Brillouin zone of the face-centered cubic lattice (FCC) in the figure on the right has the 48-fold symmetry of the point group Oh with full octahedral symmetry. This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. As a crystal structure periodic table shows, there are many elements with a FCC crystal structure, like diamond, silicon and platinum and their Brillouin zones and dispersion relations have this 48-fold symmetry. Two other familiar crystal structures are the body-centered cubic lattice (BCC) and hexagonal closed packed structures (HCP) with cubic and hexagonal lattices, respectively. The BCC structure has the 24-fold pyritohedral symmetry of the point group Th. The HCP structure has the 12-fold prismatic dihedral symmetry of the point group D3h. A complete list of symmetry properties of a point group can be found in point group character tables.

In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. The DOS of dispersion relations with rotational symmetry can often be calculated analytically. This result is fortunate, since many materials of practical interest, such as steel and silicon, have high symmetry.

In anisotropic condensed matter systems such as a single crystal of a compound, the density of states could be different in one crystallographic direction than in another. These causes the anisotropic density of states to be more difficult to visualize, and might require methods such as calculating the DOS for particular points or directions only, or calculating the projected density of states (PDOS) to a particular crystal orientation.

k-space topologies

The density of states is dependent upon the dimensional limits of the object itself. In a system described by three orthogonal parameters (3 Dimension), the units of DOS is [Energy]−1[Volume]−1, in a two dimensional system, the units of DOS is [Energy]−1[Area]−1, in a one dimensional system, the units of DOS is [Energy]−1[Length]−1. The referenced volume is the volume of $k$-space; the space enclosed by the constant energy surface of the system derived through a dispersion relation that relates $E$ to $k$. An example of a 3-dimensional $k$-space is given in Fig. 1. It can be seen that the dimensionality of the system confines the momentum of particles inside the system.

Density of wave vector states (sphere)
The calculation for DOS starts by counting the $N$ allowed states at a certain $k$ that are contained within $[k, k + dk]$ inside the volume of the system. This procedure is done by differentiating the whole k-space volume $$\Omega_{n, k}$$ in n-dimensions at an arbitrary $k$, with respect to $k$. The volume, area or length in 3, 2 or 1-dimensional spherical $k$-spaces are expressed by $$\Omega_n(k) = c_n k^n$$

for a $n$-dimensional $k$-space with the topologically determined constants $$c_1 = 2,\ c_2 = \pi,\  c_3 = \frac{4 \pi}{3}$$ for linear, disk and spherical symmetrical shaped functions in 1, 2 and 3-dimensional Euclidean $k$-spaces respectively.

According to this scheme, the density of wave vector states $N$ is, through differentiating $$\Omega_{n,k}$$ with respect to $k$, expressed by $$N_n(k) = \frac{\mathrm d\Omega_n(k)}{\mathrm dk} = n\; c_n\; k^{n - 1}$$

The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as $$\begin{align} N_1(k) &= 2 \\ N_2(k) &= 2 \pi k \\ N_3(k) &= 4 \pi k^2 \end{align}$$

One state is large enough to contain particles having wavelength λ. The wavelength is related to $k$ through the relationship. $$k = \frac{2\pi}{\lambda}$$

In a quantum system the length of λ will depend on a characteristic spacing of the system L that is confining the particles. Finally the density of states N is multiplied by a factor $$s/V_k$$, where $s$ is a constant degeneracy factor that accounts for internal degrees of freedom due to such physical phenomena as spin or polarization. If no such phenomenon is present then $$s = 1$$. Vk is the volume in k-space whose wavevectors are smaller than the smallest possible wavevectors decided by the characteristic spacing of the system.

Density of energy states
To finish the calculation for DOS find the number of states per unit sample volume at an energy $$E$$ inside an interval $$[E, E + \mathrm dE]$$. The general form of DOS of a system is given as $$D_n\left(E\right) = \frac{\mathrm d\Omega_n(E)}{\mathrm dE}$$ The scheme sketched so far only applies to monotonically rising and spherically symmetric dispersion relations. In general the dispersion relation $$E(k)$$ is not spherically symmetric and in many cases it isn't continuously rising either. To express D as a function of E the inverse of the dispersion relation $$E(k)$$ has to be substituted into the expression of $$\Omega_n(k)$$ as a function of k to get the expression of $$\Omega_n(E)$$ as a function of the energy. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. More detailed derivations are available.

Dispersion relations
The dispersion relation for electrons in a solid is given by the electronic band structure. The kinetic energy of a particle depends on the magnitude and direction of the wave vector k, the properties of the particle and the environment in which the particle is moving. For example, the kinetic energy of an electron in a Fermi gas is given by $$E = E_0 + \frac{\left(\hbar k\right)^2}{2m} \ ,$$

where m is the electron mass. The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily.

For longitudinal phonons in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional k-space, as shown in Figure 2, is given by $$E = 2 \hbar \omega_0 \left|\sin\left(\frac{ka}{2}\right)\right| $$ where $\omega_0 = \sqrt{k_\text{F} / m}$ is the oscillator frequency, $$m$$ the mass of the atoms, $$k_\text{F}$$ the inter-atomic force constant and $$a$$ inter-atomic spacing. For small values of $$k \ll \pi / a $$ the dispersion relation is linear: $$E = \hbar \omega_0 ka $$

When $$k \approx \pi / a $$ the energy is $$E = 2 \hbar \omega_0 \left|\cos\left(\frac{\pi - ka}{2}\right)\right|$$

With the transformation $$q = k - \pi/a $$ and small $$q$$ this relation can be transformed to $$E = 2 \hbar \omega_0 \left[1 - \left(\frac{qa}{2}\right)^2\right] $$

Isotropic dispersion relations
The two examples mentioned here can be expressed like $$E = E_0 + c_k k^p$$

This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic because only the length and not the direction of the wave vector appears in the expression. The magnitude of the wave vector is related to the energy as: $$k = \left(\frac{E - E_0}{c_k}\right)^{{1}/{p}} ,$$

Accordingly, the volume of n-dimensional $k$-space containing wave vectors smaller than $k$ is: $$\Omega_n(k) = c_n k^n$$

Substitution of the isotropic energy relation gives the volume of occupied states $$\Omega_n(E) = \frac{c_n}{{c_k}^\frac{n}{p}} \left(E - E_0\right)^{{n}/{p}} ,$$

Differentiating this volume with respect to the energy gives an expression for the DOS of the isotropic dispersion relation $$D_n\left(E\right) = \frac {\mathrm d}{\mathrm dE}\Omega_n(E) = \frac{n c_n}{p {c_k}^\frac{n}{p}} \left(E - E_0\right)^{\frac{n}{p} - 1} $$

Parabolic dispersion
In the case of a parabolic dispersion relation (p = 2), such as applies to free electrons in a Fermi gas, the resulting density of states, $$D_n\left(E\right)$$, for electrons in a n-dimensional systems is

$$\begin{align} D_1\left(E\right) &= \frac{1}{2\sqrt{c_k \left(E - E_0\right)}} \\[1ex] D_2\left(E\right) &= \frac{\pi}{2c_k} \\[1ex] D_3\left(E\right) &= \pi \sqrt{\frac{E - E_0}{c_k^3}} \,. \end{align}$$

for $$E > E_0$$, with $$D(E) = 0$$ for $$E < E_0$$.

In 1-dimensional systems the DOS diverges at the bottom of the band as $$E$$ drops to $$E_0$$. In 2-dimensional systems the DOS turns out to be independent of $$E$$. Finally for 3-dimensional systems the DOS rises as the square root of the energy.

Including the prefactor $$s/V_k$$, the expression for the 3D DOS is $$N(E) = \frac {V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^\frac{3}{2}\sqrt{E - E_0} ,$$

where $$V$$ is the total volume, and $$N(E-E_0)$$ includes the 2-fold spin degeneracy.

Linear dispersion
In the case of a linear relation (p = 1), such as applies to photons, acoustic phonons, or to some special kinds of electronic bands in a solid, the DOS in 1, 2 and 3 dimensional systems is related to the energy as:

$$\begin{align} D_1\left(E\right) &= \frac{1}{c_k} \\[1ex] D_2\left(E\right) &= 2\pi\frac{E - E_0}{c_k^2} \\[1ex] D_3\left(E\right) &= 4 \pi \frac{\left(E - E_0\right)^2}{c_k^3} \end{align}$$

Distribution functions
The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter. The following are examples, using two common distribution functions, of how applying a distribution function to the density of states can give rise to physical properties.

Fermi–Dirac statistics: The Fermi–Dirac probability distribution function, Fig. 4, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium. Fermions are particles which obey the Pauli exclusion principle (e.g. electrons, protons, neutrons). The distribution function can be written as $$f_{\mathrm{FD}}(E) = \frac{1}{\exp\left(\frac{E - \mu}{k_\mathrm{B} T}\right) + 1}.$$

$$\mu$$ is the chemical potential (also denoted as EF and called the Fermi level when T=0), $$k_\mathrm{B}$$ is the Boltzmann constant, and $$T$$ is temperature. Fig. 4 illustrates how the product of the Fermi-Dirac distribution function and the three-dimensional density of states for a semiconductor can give insight to physical properties such as carrier concentration and Energy band gaps.

Bose–Einstein statistics: The Bose–Einstein probability distribution function is used to find the probability that a boson occupies a specific quantum state in a system at thermal equilibrium. Bosons are particles which do not obey the Pauli exclusion principle (e.g. phonons and photons). The distribution function can be written as $$f_{\mathrm{BE}}(E) = \frac{1}{\exp\left(\frac{E - \mu}{k_\text{B} T}\right) - 1}.$$

From these two distributions it is possible to calculate properties such as the internal energy per unit volume $$u$$, the number of particles $$N$$, specific heat capacity $$c$$, and thermal conductivity $$k$$. The relationships between these properties and the product of the density of states and the probability distribution, denoting the density of states by $$g(E)$$ instead of $$D(E)$$, are given by $$\begin{align} u &= \int E\, f(E)\, g(E)\,\mathrm dE \\[1ex] N &= V \int f(E)\, g(E)\,\mathrm dE \\[1ex] c &= \frac{\partial}{\partial T} \int E\, f(E)\, g(E) \,\mathrm dE \\[1ex] k &= \frac{1}{d}\frac{\partial}{\partial T} \int E f(E)\, g(E)\, \nu(E)\, \Lambda(E)\,\mathrm dE \end{align}$$

$$d$$ is dimensionality, $$\nu$$ is sound velocity and $$\Lambda$$ is mean free path.

Applications
The density of states appears in many areas of physics, and helps to explain a number of quantum mechanical phenomena.

Quantization
Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced. For quantum wires, the DOS for certain energies actually becomes higher than the DOS for bulk semiconductors, and for quantum dots the electrons become quantized to certain energies.

Photonic crystals
The photon density of states can be manipulated by using periodic structures with length scales on the order of the wavelength of light. Some structures can completely inhibit the propagation of light of certain colors (energies), creating a photonic band gap: the DOS is zero for those photon energies. Other structures can inhibit the propagation of light only in certain directions to create mirrors, waveguides, and cavities. Such periodic structures are known as photonic crystals. In nanostructured media the concept of local density of states (LDOS) is often more relevant than that of DOS, as the DOS varies considerably from point to point.

Computational calculation
Interesting systems are in general complex, for instance compounds, biomolecules, polymers, etc. Because of the complexity of these systems the analytical calculation of the density of states is in most of the cases impossible. Computer simulations offer a set of algorithms to evaluate the density of states with a high accuracy. One of these algorithms is called the Wang and Landau algorithm.

Within the Wang and Landau scheme any previous knowledge of the density of states is required. One proceeds as follows: the cost function (for example the energy) of the system is discretized. Each time the bin i is reached one updates a histogram for the density of states, $$g(i)$$, by $$ g(i) \rightarrow g(i) + f$$ where $f$ is called the modification factor. As soon as each bin in the histogram is visited a certain number of times (10-15), the modification factor is reduced by some criterion, for instance, $$ f_{n+1} \rightarrow \frac{1}{2} f_{n}$$ where $n$ denotes the $n$-th update step. The simulation finishes when the modification factor is less than a certain threshold, for instance $f_n < 10^{-8} $.

The Wang and Landau algorithm has some advantages over other common algorithms such as multicanonical simulations and parallel tempering. For example, the density of states is obtained as the main product of the simulation. Additionally, Wang and Landau simulations are completely independent of the temperature. This feature allows to compute the density of states of systems with very rough energy landscape such as proteins.

Mathematically the density of states is formulated in terms of a tower of covering maps.

Local density of states
An important feature of the definition of the DOS is that it can be extended to any system. One of its properties are the translationally invariability which means that the density of the states is homogeneous and it's the same at each point of the system. But this is just a particular case and the LDOS gives a wider description with a heterogeneous density of states through the system.

Concept
Local density of states (LDOS) describes a space-resolved density of states. In materials science, for example, this term is useful when interpreting the data from a scanning tunneling microscope (STM), since this method is capable of imaging electron densities of states with atomic resolution. According to crystal structure, this quantity can be predicted by computational methods, as for example with density functional theory.

A general definition
In a local density of states the contribution of each state is weighted by the density of its wave function at the point. $$N(E)$$ becomes $$n(E,x)$$

$$n(E,x)=\sum_j |\phi_j (x)|^2\delta(E-\varepsilon_j)$$

the factor of $$|\phi_j (x)|^2$$ means that each state contributes more in the regions where the density is high. An average over $$x$$ of this expression will restore the usual formula for a DOS. The LDOS is useful in inhomogeneous systems, where $$n(E,x)$$ contains more information than $$n(E)$$ alone.

For a one-dimensional system with a wall, the sine waves give

$$n_{1D}(E,x)=\frac{2}{\pi\hbar}\sqrt{\frac{2m}{E}}\sin^2{kx}$$

where $k = \sqrt{2mE} / \hbar$.

In a three-dimensional system with $$x > 0$$ the expression is

$$n_{3D}(E,x) = \left(1-\frac{\sin{2kx}}{2kx}\right)n_{3D}(E)$$

In fact, we can generalise the local density of states further to

$$n(E,x,x') = \sum_j \phi_j (x)\phi^*_j (x')\delta(E-\varepsilon_j)$$

this is called the spectral function and it's a function with each wave function separately in its own variable. In more advanced theory it is connected with the Green's functions and provides a compact representation of some results such as optical absorption.

Solid state devices
LDOS can be used to gain profit into a solid-state device. For example, the figure on the right illustrates LDOS of a transistor as it turns on and off in a ballistic simulation. The LDOS has clear boundary in the source and drain, that corresponds to the location of band edge. In the channel, the DOS is increasing as gate voltage increase and potential barrier goes down.

Optics and photonics
In optics and photonics, the concept of local density of states refers to the states that can be occupied by a photon. For light it is usually measured by fluorescence methods, near-field scanning methods or by cathodoluminescence techniques. Different photonic structures have different LDOS behaviors with different consequences for spontaneous emission. In photonic crystals, near-zero LDOS are expected, inhibiting spontaneous emission. Similar LDOS enhancement is also expected in plasmonic cavity. However, in disordered photonic nanostructures, the LDOS behave differently. They fluctuate spatially with their statistics, and are proportional to the scattering strength of the structures. In addition, the relationship with the mean free path of the scattering is trivial as the LDOS can be still strongly influenced by the short details of strong disorders in the form of a strong Purcell enhancement of the emission. and finally, for the plasmonic disorder, this effect is much stronger for LDOS fluctuations as it can be observed as a strong near-field localization.

title: Density of states


text:
Price elasticity of supply

The price elasticity of supply (PES or Es) is a measure used in economics to show the responsiveness, or elasticity, of the quantity supplied of a good or service to a change in its price. Price elasticity of supply, in application, is the percentage change of the quantity supplied resulting from a 1% change in price. Alternatively, PES is the percentage change in the quantity supplied divided by the percentage change in price.

When PES is less than one, the supply of the good can be described as inelastic. When price elasticity of supply is greater than one, the supply can be described as elastic. An elasticity of zero indicates that quantity supplied does not respond to a price change: the good is "fixed" in supply. Such goods often have no labor component or are not produced, limiting the short run prospects of expansion. If the elasticity is exactly one, the good is said to be unit-elastic. Differing from price elasticity of demand, price elasticities of supply are generally positive numbers because an increase in the price of a good motivates producers to produce more, as relative marginal revenue increases.

The quantity of goods supplied can, in the short term, be different from the amount produced, as manufacturers will have stocks which they can build up or run down.

Definition & Real-Life Example
The slope of a supply curve relates changes in price to changes in quantity supplied. A steeper curve means that price changes are correlated with relatively small quantity changes. Steep supply curves derive that the quantity supplied by producers are not particularly sensitive to price changes. Oppositely, flatter supply curves imply that price changes are associated with large quantity changes. Markets with flat supply curves will see large movements in quantity supplied as prices change.

The concept of elasticity expresses the responsiveness of a value to changes in another (particularly, responsiveness of quantities to prices). An elasticity is the ratio of the percentage change in one value to the percentage change in another. The concept of elasticity applies to demand and supply curves and agents like producers and consumers.

$$E^S =\left ( \frac{%\bigtriangleup Q^S}{%\bigtriangleup P} \right )$$

Suppose there is an increase in demand for apartments. There will be a shortage of apartments at the old level of apartment rents and pressure on rents (price) to increase. Ceteris paribus, the more responsive (elastic) the quantity of apartments supplied is to changes in monthly rents, the lower the increase in rent required to eliminate the shortage and to bring the market back to equilibrium. Conversely, if quantity supplied is less responsive (inelastic) to price changes, price will have to increase more to eliminate a shortage caused by an increase in demand.

Determinants

 * Availability of raw materials: For example, availability may cap the amount of gold that can be produced in a country regardless of price. Likewise, the price of Van Gogh paintings is unlikely to affect their supply.

 * Length and complexity of production: Much depends on the complexity of the production process. Textile production is relatively simple. The labour is largely unskilled and production facilities are little more than buildings – no special structures are needed. Thus the PES for textiles is elastic.

 * Mobility of factors: If the factors of production are easily available and if a producer producing one good can switch their resources and put it towards the creation of a product in demand, then it can be said that the PES is relatively elastic. The inverse applies to this, to make it relatively inelastic.

 * Time to respond: The more time a producer has to respond to price changes the more elastic the supply. Supply is normally more elastic in the long run than in the short run for produced goods, since it is generally assumed that in the long run all factors of production can be utilised to increase supply, whereas in the short run only labor can be increased, and even then, changes may be prohibitively costly. For example, a cotton farmer cannot immediately (i.e. in the short run) respond to an increase in the price of soybeans because of the time it would take to procure the necessary land.

 * Inventories: A producer who has a supply of goods or available storage capacity can quickly increase supply to market.

 * Spare or excess production capacity: A producer who has unused capacity can (and will) quickly respond to price changes in his market assuming that variable factors are readily available. The existence of spare capacity within a firm, would be indicative of more proportionate response in quantity supplied to changes in price (hence suggesting price elasticity). It indicates that the producer would be able to utilise spare factor markets (factors of production) at its disposal and hence respond to changes in demand to match with supply. The greater the extent of spare production capacity, the quicker suppliers can respond to price changes and hence the more price elastic the good/service would be.

Elasticity versus slope
The elasticity of supply will generally vary along the curve, even if supply is linear so the slope is constant. This is because the slope measures the absolute increase in quantity for an absolute increase in price, but the elasticity measures the percentage change. This also means that the slope depends on the units of measurement and will change if the units change (e.g., dollars per pound versus dollars per ounce) while the elasticity is a simple number, independent of the units (e.g., 1.2). This is a major advantage of elasticities.

The slope of the supply curve is dP/dQ, while the elasticity is (dQ/dP)(P/Q). Thus, a supply curve with steeper slope (bigger dP/dQ and thus smaller dQ/dP) is less elastic, for given   P   and   Q. Along a linear supply curve such as     Q = a + b P    the slope is constant (at  1/b)   but the elasticity is  b(P/Q),    so  the elasticity rises with  greater   P   both from the direct effect and the increase in  Q(P).

Another special feature of the  linear supply curve  arises because its elasticity can also be written as  bP/(a + bP), which is less than 1 if    a < 0   and greater than 1 if    a > 0. Linear supply curves which cut through the positive part of the price axis and have zero quantity supplied if the price is too low (P < -a/b) have a < 0   and hence they always have elastic supply. Curves which cut through the positive part of the quantity axis and have positive quantity supplied  (Q = a) even if the price is zero have    a > 0  and hence  always have inelastic supply. Curves which go through the origin have   a = 0   and hence have an elasticity of 1.

When looking at the price elasticity of supply, there are five types. The five types are perfectly inelastic supply, relatively inelastic supply, unit elastic supply, relatively elastic supply, and perfectly elastic supply. These five types help to show how different products supply quantity changes when faced with changed in price.

Perfectly inelastic supply: This is when the Es formula equals to zero, meaning that there is no change in the supply when there are price changes. This can be the case where there is a limited quantity of supply, for example, if there is only 200 of a certain product made and there will never be any more made, there will be no increase or decrease in the quantity of supply.

Relatively inelastic supply: This is when the Es formula gives a result between zero and one, meaning that when there is a change in price, the percentage change in supply is lower than the percentage change in price. For example, if a product costs $1 and then increases to $1.10 the increase in price is 10% and therefore the change in supply will be less than 10%.

Unit Elastic supply: This is when the Es formula equals to one, meaning that quantity supplied and price change by the same percentage. Using the previous example to show unit elasticity, when there is a 10% increase in price, there will also be a 10% increase in quantity supplied.

Relatively elastic supply: This is when the Es formula gives a result above one, meaning that when there is a change in price, the percentage change in supply is higher than the percentage change in price. Using the above example to show an elastic supply, when there is a 10% increase in price there will be more than a 10% increase in supply.

Perfectly elastic supply: This is when the Es formula actually gives an infinite result, meaning that the quantity that can be supplied is infinite, however, that is only at a specific price and if the price changes there will be no quantity supplied at all. For example, there may be an infinite supply of product at a price of $1 but if that price changes to $1.10 then the supply becomes zero.

Short-run and long-run
Since firms typically have a limited capacity for production, the elasticity of supply tends to be high at low levels of quantity supplied and low at high levels of quantity supplied. At low levels of quantity supplied, firms typically have substantial capacity available for use, so small increases in price make it profitable for firms to begin to use this idle capacity. Thus, the responsiveness of quantity supplied to changes in price is high in this region of the supply curve. However, as capacity becomes fully utilised, increasing production requires additional investment in capital (for example, plant and equipment). Since the price must rise substantially to cover this additional expense, supply becomes less elastic at high levels of output.

Selected supply elasticities

 * Heating Oil: 1.58 (Short run)
 * Gasoline: 1.61 (Short run)
 * Tobacco: 7.0 (Long run)
 * Housing: 1.6–3.7 (Long run)
 * Cotton
 * 0.3 (Short run)
 * 1.0 (Long run)
 * Steel: 1.2 (Long run, from Minimills)
 * Land: 0, except when land reclamation is taking place

title: Price elasticity of supply