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It refers to a hypothetical situation wherein an ass that is equally hungry and thirsty is placed precisely midway between a stack of hay and a pail of water. Since the paradox assumes the ass will always go to whichever is closer, it will die of both hunger and thirst since it cannot make any rational decision to choose one over the other. The paradox is named after the 14th century French philosopher Jean Buridan, whose philosophy of moral determinism it satirizes. A common variant of the paradox substitutes two identical piles of hay for the hay and water; the ass, unable to choose between the two, dies of hunger.
The paradox predates Buridan; it dates to antiquity, being found in Aristotle's On the Heavens. Aristotle, in ridiculing the Sophist idea that the Earth is stationary simply because it is circular and any forces on it must be equal in all directions, says that is as ridiculous as saying that
...a man, being just as hungry as thirsty, and placed in between food and drink, must necessarily remain where he is and starve to death.
— Aristotle, On the Heavens, ca.350 BCE
However, the Greeks only used this paradox as an analogy in the context of discussions of the equilibrium of physical forces.
The 12th century Persian Islamic scholar and philosopher Al-Ghazali discusses the application of this paradox to human decision making, asking whether it is possible to make a choice between equally good courses without grounds for preference. He takes the attitude that free will can break the stalemate.
Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them both. Surely he will take one of them, through a quality in him, the nature of which is to differentiate between two similar things.
Although Buridan nowhere discusses this specific problem, its relevance is that he did advocate a moral determinism whereby, save for ignorance or impediment, a human faced by alternative courses of action must always choose the greater good. In the face of equally good alternatives Buridan believed a rational choice could not be made
Should two courses be judged equal, then the will cannot break the deadlock, all it can do is to suspend judgement until the circumstances change, and the right course of action is clear.
— Jean Buridan, 1340
Later writers satirised this view in terms of an ass which, confronted by both food and water must necessarily die of both hunger and thirst while pondering a decision.
Some proponents of hard determinism have granted the unpleasantness of the scenario, but have denied that it illustrates a true paradox, since one does not contradict oneself in suggesting that a man might die between two equally plausible routes of action. For example, Baruch Spinoza in his Ethics, suggests that a person who sees two options as truly equally compelling cannot be fully rational:
[I]t may be objected, if man does not act from free will, what will happen if the incentives to action are equally balanced, as in the case of Buridan's ass? [In reply,] I am quite ready to admit, that a man placed in the equilibrium described (namely, as perceiving nothing but hunger and thirst, a certain food and a certain drink, each equally distant from him) would die of hunger and thirst. If I am asked, whether such a one should not rather be considered an ass than a man; I answer, that I do not know, neither do I know how a man should be considered, who hangs himself, or how we should consider children, fools, madmen, &c.
Other writers have opted to deny the validity of the illustration. A typical counter-argument is that rationality as described in the paradox is so limited as to be a straw man version of the real thing, which does allow the consideration of meta-arguments. In other words, it is entirely rational to recognize that both choices are equally good and arbitrarily (randomly) pick one instead of starving. This counter-argument is sometimes used as an attempted justification for faith or intuitivity (called by Aristotle noetic or noesis). The argument is that, like the starving ass, we must make a choice in order to avoid being frozen in endless doubt. Other counter-arguments exist.
Buridan's principle 
The situation of Buridan's ass was given a mathematical basis in a 1984 paper by American computer scientist Leslie Lamport, in which Lamport presents an argument that, given certain assumptions about continuity in a simple mathematical model of the Buridan's ass problem, there will always be some starting conditions under which the ass will starve to death, no matter what strategy it takes.
Lamport calls this result Buridan’s principle, and states it as:
- A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time.
Application to digital logic: metastability 
A version of Buridan's principle actually occurs in electrical engineering. Specifically, the input to a digital logic gate must convert a continuous voltage value into either a 0 or a 1 which is typically sampled and then processed. If the input is changing and at an intermediate value when sampled, the input stage acts like a comparator. The voltage value can then be likened to the position of the ass, and the values 0 and 1 represent the bales of hay. Like the situation of the starving ass, there exists an input on which the converter cannot make a proper decision, resulting in a metastable state. Having the converter make an arbitrary choice in ambiguous situations does not solve the problem, as the boundary between ambiguous values and unambiguous values introduces another binary decision with its own metastable state.
See also 
- Analysis paralysis
- Dining philosophers problem
- Hobson's choice
- Metastability in electronics
- Morton's fork
- Spontaneous symmetry breaking
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- Definition of term at wordsmith.org