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What is the name (or adjective) for an assumption made for the sake of contradiction?

To be clear, I'm in search of an expression in the form "a(n) $\underline{\quad \quad \quad \quad}$ assumption".

Edit: Why are there votes to close this question? This question at least fits the tag "mathematical-writing" and is pertinent to the writing of mathematical research papers, no doubt.

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    $\begingroup$ I once heard Robert Solovay refer to an assumption of this type as a "paranoid fantasy" ... but I don"t think that's an official term. $\endgroup$
    – Nik Weaver
    Commented Jun 12, 2020 at 4:34
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    $\begingroup$ Aside: as an undergraduate I discovered Scott's Group Theory with most of its terse proofs by contradiction beginning "Deny. Then ...", which I took as a model of a perfect proof in subsequent homework assignments. I was rapidly disabused of this, fortunately. $\endgroup$
    – J.J. Green
    Commented Jun 12, 2020 at 8:25
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    $\begingroup$ In the tradition of the trials against heretics I suggest "Praesumtio diaboli". $\endgroup$
    – user130903
    Commented Jun 12, 2020 at 8:38
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    $\begingroup$ You could call it “not” as in “Suppose not.” 🙂 $\endgroup$
    – RobPratt
    Commented Jun 12, 2020 at 12:56
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    $\begingroup$ counterfactual ... $\endgroup$
    – Michael Engelhardt
    Commented Jun 12, 2020 at 14:14

4 Answers 4

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Not the name, but a name: "contradictive assumption".

Google knows about 80 some odd uses of this in a mathematical context. I like it because the word "contradictive" has a dictionary definition that is more-or-less suited for the job, and the word immediately invokes the method of proof we are using.

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Especially in the philosophy of religion, the term reductio premise is sometimes used. A Google Scholar search for "reductio premise" (in quotation marks) turns up a few dozen references; one of the most highly cited is Robust vagueness and the forced-march sorites paradox, by Terence Horgan.

However, among mathematicians, I don't think there is any standard terminology.

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    $\begingroup$ I was reminded today that I once used the term "reductio hypothesis" in one of my own papers. $\endgroup$
    – Timothy Chow
    Commented Jul 31, 2020 at 18:05
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Straw Man Proposal

A straw-man (or straw-dog) proposal is a brainstormed simple draft proposal intended to generate discussion of its disadvantages and to provoke the generation of new and better proposals.

https://en.wikipedia.org/wiki/Straw_man_proposal

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I liked the suggestion "the marked assumption", found in the comments. Unlike Willie Wong's suggestion the "contradictive assumption", the "marked assumption" only partially suggests the nature of the expression. This is hard to fix unless the expression is made standard.

An advantage of the "marked assumption" is that it can serve double-duty: Even in cases where the truth of the assumption is unknown, the expression is still relevant. This is particularly notable in the study of the Riemann Hypothesis where the Riemann Hypothesis is occasionally assumed and consequences derived therefrom. If the marked assumption holds, then all we have done is discover consistent structure. If the marked assumption fails, then our presumptive work can just be seen as a search for a contradiction.

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    $\begingroup$ In this case, why not just label your assumption "Assumption A" at the beginning of the proof, and refer to it as "Assumption A" throughout the course of the argument? If you are going to mark it, might as well actually mark it. (People already do this for "positive" assumptions that they need to use repeatedly in the same paper.) $\endgroup$
    – Willie Wong
    Commented Jun 19, 2020 at 14:48
  • $\begingroup$ @WillieWong Of course this is a valid tactic. That said, the whole point of this question is to pindown an acceptable adjective in such cases where such an adjective is desirable. $\endgroup$
    – Benji
    Commented Jun 19, 2020 at 22:28
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    $\begingroup$ I thought the whole point of the question was to find a name for an assumption made for the sake of contradiction. $\endgroup$
    – Nik Weaver
    Commented Jul 16, 2020 at 11:41
  • $\begingroup$ @NikWeaver Imagine trying to motivate a theory or definition, or even just proceeding through mathematical research. How often is the truth state of an assumption known? $\endgroup$
    – Benji
    Commented Jul 16, 2020 at 23:34

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