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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeSep 2nd 2012
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   Author: TobyBartels
   Format: MarkdownItexI put a bunch of stuff [[binary function|there]] that might be of interest to the logicians and foundationalists among us, although it's still pretty trivial.

   I put a bunch of stuff there that might be of interest to the logicians and foundationalists among us, although it’s still pretty trivial.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeSep 2nd 2012
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   Author: Mike Shulman
   Format: MarkdownItexI think in type theory the "curried" definition of a binary function as a function whose codomain is a function type is at least as prevalent as having the domain a cartesian product. It's certainly more convenient in some formalizations, like Coq (it avoids the need to "destruct" the ordered pair explicitly). One could even consider a judgment like $$ (x:A),\, (y:B) \;\vdash\; (f(x,y):C) $$ to define a binary function without reference to function types *or* product types. True, when type theory is used as a foundation for mathematics, one generally does include both product and function types, but arguably neither of them is necessary for a definition of "binary function".

   I think in type theory the “curried” definition of a binary function as a function whose codomain is a function type is at least as prevalent as having the domain a cartesian product. It’s certainly more convenient in some formalizations, like Coq (it avoids the need to “destruct” the ordered pair explicitly). One could even consider a judgment like

   $$(x:A),\, (y:B) \;\vdash\; (f(x,y):C)$$

   to define a binary function without reference to function types or product types. True, when type theory is used as a foundation for mathematics, one generally does include both product and function types, but arguably neither of them is necessary for a definition of “binary function”.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeSep 2nd 2012
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   Author: TobyBartels
   Format: MarkdownItexYeah, I had started to write some stuff about type theory but took it out unfinished. All that's left is the vague statement about axiomatising the concept instead of defining it. (That a judgement like your example is meaningful is due to a common principle that axiomatises $n$-ary functions for all $n$ at once.)

   Yeah, I had started to write some stuff about type theory but took it out unfinished. All that’s left is the vague statement about axiomatising the concept instead of defining it. (That a judgement like your example is meaningful is due to a common principle that axiomatises $n$-ary functions for all $n$ at once.)