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In the crime novels by Sir Arthur Conan-Doyle, central character, detective, Sherlock Holmes described his approach to evidence-analysis as the discarding of the impossible; then, whatever remains, however unlikely, must be the truth.

How to express this in Latin?

Sequentially:

an ablative absolute: "impossibilibus missis" = "with the impossible (things having been) discarded";

"quodcumque manet (originally, quodcumque relinquit)" = "whatever remains"; (An interesting debate [Comments] on the passive/ active forms of "relinquo" unfolded. A case can be made for either.)

"quantalibet incredulitas" = "no matter how great the disbelief";

a neuter, impersonal gerundive: "deinde est credendum" = "then it-ought-to-be-believed"/ "then it must be believed"/ (using English impersonal pronoun "one") "then one must believe (it).

Alternatively, result clause: "ut id crederetur" = "so it is to be believed" --this does not provide the element of obligation.

Giving:

"impossibilibus missis, quodcumque manet, quantalibet incredulitas, deinde est credendum" =

"With the impossible discarded, whatever remains, no matter how great the disbelief, then it must be believed."

Is this correct?

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    I think you want manet rather than relinquit: relinquo means "leave" in the sense of "abandon", not "be remaining". – Colin Fine May 8 at 22:40
  • @Colin Fine: Thanks. I've changed it. – tony May 9 at 8:52
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    @tony Or you could say: quod relinquitur (canonical Caesar example: relinquebatur una via), or, even simpler: relictum. – Sebastian Koppehel May 9 at 9:59
  • Yes, @SebastianKoppehel, a passive form of relinquo is better than manet. – Colin Fine May 9 at 10:33
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    @tony That is one meaning. It has many other nuances that imply no unwantedness (such as reliquit filiam adulescentulam = “he [died and] left behind a very young daughter”). Note the dictionary entry explicitly mentions the passive sense “to remain,” but as usual in L&S this crucial information is the needle in the haystack of untranslated examples. Colin rightly pointed out that the use in the active voice does not fit here. – Sebastian Koppehel May 9 at 18:04

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