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Simplify is very gullible: if it gets a False assumption, it'll believe anything is True:

Simplify[x > y, False]
(* Simplify::fas -- Simplify: Warning: one or more assumptions evaluated to False. *)
(* True *)

Simplify[1 > 2, False]
(* Simplify::fas -- Simplify: Warning: one or more assumptions evaluated to False. *)
(* True *)

This also holds for $Assumptions that turn out to be False (which is how I ran into this problem):

$Assumptions = {x > 0};
Simplify[1 > 2]
(* False *)

x = -1;
Simplify[1 > 2]
(* Simplify::fas -- Simplify: Warning: one or more assumptions evaluated to False. *)
(* True *)

I'm sure there's a reason why this happens (and I'd be curious to hear it), but I find this behavior potentially misleading. Is there a good way to work around this behavior, so that the warning message shows but False assumptions are ignored?

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    $\begingroup$ Most likely this is based on logic fundamentals like de Morgan's laws. E.g. Implies[False, a] yields True and so the issue with Simplify turns out no longer to be misleading. $\endgroup$
    – Artes
    Commented Jun 28, 2022 at 0:59
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    $\begingroup$ You get a warning message telling you this is GIGO. I don't see what more Simplify could do here. Wash your dishes? $\endgroup$
    – Daniel Lichtblau
    Commented Jun 28, 2022 at 14:57
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    $\begingroup$ The rule $(F \Rightarrow e) \mapsto T$, as inferred, is applied in Simplify[e, False] only if the expression $e$ (or its parts) is a traditional boolean-valued expression. Yes in Simplify[x + y && x + y, False] but not in Simplify[x + y && x + y // Simplify, False] (because the e here becomes simply x + y). An odd example: Simplify[False + y, False] becomes True + y, and two wrongs Simplify[False + False, False] make a double right (too true!). But these inputs seem nonsensical and of not much real practical importance. $\endgroup$
    – Michael E2
    Commented Jun 28, 2022 at 14:57
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    $\begingroup$ It is not possible to disregard false assumptions. If your assumptions are {a<b,a>b}, which do you discard? $\endgroup$
    – Daniel Lichtblau
    Commented Jun 28, 2022 at 15:19
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    $\begingroup$ @DanielLichtblau Hmm, I guess a general solution is trickier than I thought. Luckily in my context, the problem will be single expressions that are False on their own, which are easily screened out. $\endgroup$
    – Chris K
    Commented Jun 28, 2022 at 15:56

2 Answers 2

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This is the expected behaviour. Think Simplify[q,p] as $p⇒q$. You are evaluating the truth of "$q$ under the premise $p$".

From a false premise, you can get either True or False implications. So $\left(False⇒q\right)$ is a True statement regardless of the value of $q$.

We can build a truth table

Column[
   Row[
      {
        "(",#2,"\[Implies]",#1, ") is ", 
        Simplify[#1,#2]
      }
   ]& @@@ Tuples[{True,False},2]
]

enter image description here

You get the General::fas warning.

"This message is generated when the symbol False is encountered in a position where valid assumptions are expected."

You are been warned that you are starting from a false premise, but you get the expected truth table.

As discussed in the chat, you can see this on Wikipedia under Truth Table on column 11 (Credit to @CarlLange).

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Thanks to @Artes for the comment explaining the reason. I haven't tested these thoroughly, but as for a workaround:

Simplify[1 > 2, Assumptions -> DeleteCases[$Assumptions, False]]
(* False *)

seems to do the trick in the case of False $Assumptions.

SetOptions[Simplify, Assumptions :> DeleteCases[$Assumptions, False]]

could be a blanket solution, and

If[Cases[$Assumptions, False] != {}, Message[Simplify::fas]];

could provide the warning if desired.

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