The noun assumption refers to the act of laying claim to something, or a statement that is taken for granted.

The noun presumption refers to a belief that something is true even though it hasn't been proved, an attitude or belief that's been determined by probability, or the overstepping of proper bounds.

Also see the usage notes below.

Examples:

• "Your assumptions are your windows on the world. Scrub them off every once in a while, or the light won't come in."
  (attributed to Isaac Asimov)

• "We've all been to social functions where someone seems to know it all, and he makes the ridiculous assumption that people enjoy hearing him unload his vast stores of knowledge."
  (Jim Camp, Start with No: The Negotiating Tools that the Pros Don't Want You to Know. Crown, 2002)
• "[I]n the classical era (before the late eighteenth century) there was no presumption of innocence until guilt was proven. Even a mild degree of suspicion was taken to indicate a mild degree of guilt."
  (Frances E. Gill, The Moral Benefit of Punishment. Lexington, 2003)

Usage Notes:

• Assume and Presume; Assumption and Presumption
  "Assume means (1) to take upon oneself; to take over duties and responsibilities; or (2) to take for granted or without proof. Presume means (1) to take for granted or without proof; or (2) to undertake without permission. So one may assume a rank or office or authority by right, but another may presume to exercise the privilege of a rank, office, or authority despite a lack of right. Presume often carries a sense of arrogance, of going too far in acting without justification--hence the adjective presumptuous. . . .
  
  "The connotative distinction between [assumption and presumption] is that presumptions are more strongly inferential and more probably authoritative than mere assumptions, which are usually more hypothetical. Presumptions may lead to decisions, while assumptions typically don't."
  (Bryan Garner, Garner's Modern American Usage, 3rd ed. Oxford University Press, 2009)

• Assertions, Assumptions, and Presumptions
  "[P]resumption is a kind of speech act that is halfway between assertion and (mere) assumption. An assertion carries with it a burden of proof on the proponent. An assumption can go forward without proof. An assumption can even be contrary to known facts. And anyone in a dialogue is free at any time to reject an assumption without having to disprove it. Like an assumption, a presumption does not need to be backed up by evidence. But like an assertion, it does bear a relationship to evidence. However, that relationship is an oblique or negative one."
  (Douglas Walton, The Place of Emotion in Argument. Penn State Press, 1992)
• Lemony Snicket on Assumptions
  "Assumptions are dangerous things to make, and like all dangerous things to make--bombs, for instance, or strawberry shortcake--if you make even the tiniest mistake you can find yourself in terrible trouble. Making assumptions simply means believing things are a certain way with little or no evidence that shows you are correct, and you can see at once how this can lead to terrible trouble. For instance, one morning you might wake up and make the assumption that your bed was in the same place that it always was, even though you would have no real evidence that this was so. But when you got out of your bed, you might discover that it had floated out to sea, and now you would be in terrible trouble all because of the incorrect assumption that you'd made. You can see that it is better not to make too many assumptions, particularly in the morning."
  (Lemony Snicket, A Series of Unfortunate Events #5: The Austere Academy. HarperCollins, 2000)
• Presumption in Debates
  "'Presumption' is the assumption that the current state of events (status quo) is correct until proven otherwise. It is the ground or starting point of the debate that is taken for granted. Just as in law, where we are presumed innocent until proven guilty, a debate begins on the grounds that the current system (status quo) is given to be acceptable unless and until the affirmative proves otherwise by its case in support of the proposition."
  (Jon M. Ericson, James J. Murphy, and Raymond Bud Zeuschner, The Debater's Guide, 4th ed. Southern Illinois University Press, 2011)

Practice:

(a) "What an outrageous ______ it was that age must be infallible! Their idea of filial duty was that he should accept their authority, not because they were wise, but because they were old."
(William Somerset Maugham, The Hero, 1901)

(b) "With no apparent irony, Goldstein (the skeptical critic of clinical expertise) defended his _____ as scientific, although lacking in experimental evidence."
(Richard H. Gaskins, Burdens of Proof in Modern Discourse. Yale University Press, 1992)

Answers to Practice Exercises

Glossary of Usage: Index of Commonly Confused Words