Who translated "deductive reasoning" into 演绎推理? The word 演绎 had been used by Chinese literati long before it acquired the precise modern meaning. 

I wonder exactly who made the link between 演绎 and deduction?

The definition on baidu makes you want to puke. It gives ancient use cases followed by modern definitions. Did ancients understand 演绎 as modern Chinese do? What was the ancients' definition of 演绎?
For Deduction


  Definition of Deduction: (From Webster's Dict)
  
  
  a :  an act of taking away
  b :  something that is or may be subtracted 
  a :  the deriving of a conclusion by reasoning
  b :  a conclusion reached by logical deduction
  


Let check the 演绎


  Definition of 演绎:(I don't have official Chinese dictionary on hand so I used my common sense)
  
  
  verb. Acting, Show
  noun. the process of getting a conclusion by logical reasoning.(as poopsed to 'sum up')
  


Let be aware of the history of word'演绎'. For the English-Chinese translation experts(around the 1900s), using the '演绎' was the reasonable choice at that time. 

And since it's like more than 100 years. I believe nobody would remember who firstly use this word. It's naturally used because its meaning of so close(or even the same) to "deduction". 

The same logic could be applied to "米国"(Japanese) and "USA"(English). No one ever remembers who created "米国"

There's a detailed blog regarding ancient usage of 演绎 and演译.