cognitive psychology. PhD 
Are there any programming languages with *intentionally* unsound static type systems?
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Wouldn't explicit UB in any ISO language (C, C++, etc) count?
It does, although I've never been 100% clear on how intentional it is. I know of claims that it's necessary for compiler optimizations, but I haven't seen any arguments as to why.
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Chandler Carruth has a somewhat interesting talk on the subject, but even having watched it, I'm still not sure that unsoundness is strictly *necessary* even for systems languages.
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It's absolutely not _necessary_, but it is a deliberate choice by language designers.
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A trivial example is that casting a function pointer (in C) to return a different type is undefined behaviour, so different return types can be returned in different ways (general-purpose register vs floating-point register).
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Not sure if this counts, but undefined signed overflow is useful for several kids of optimizations:https://kristerw.blogspot.com/2016/02/how-undefined-signed-overflow-enables.html?m=1 …
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