If he's trained to be a super-soldier, then anyone with any kind of covert minded thinking on the training team would know that when a person of pure logic acts in a purely logical way, it comes off unsettling. It's not enough to look like you belong, but also act like it. So saving a person might be logical because not saving them would blow cover, which would mean an unsuccessful operation. So if he's working in Los Angelos, he might have to refer to the nearest interstate as "The 405" instead of "I-405" because while most of the English speaking world says "I-405", Angelenos' particular dialect calls it "the 405".
It might be that, when dealing with the public on the street, an emotional trainwreck if ever there was one, acting like he is emotional is the best course of action. Give him the ability to slip into different personalities so convincingly that to those who know the truth, it's rather jarring.
Another avenue is that you can not fight human emotions. Even famed logical minded adherant Spock was capable of emotions but was trying to not let those emotions affect his judgement. He even once explained that when the sitution is governed by illogical rules, the only logical tactic is to act illogically. Meanwhile the character Data was unable to think emotionally, but was able to understand emotional responses and even adjust when corrected. The episode "Measure of a Man" demonstrates that Data still has important self-qualities and treasured mementos because he recognized the significance of the meaning behind the trinkets. He explains the value of personal achievements, personal friendships, even his own value in a way that does not betray an emotional connection to any of these, but ones of logic. When pressed on the matter of a prior intimate relationship, he declines to specify the reasoning for keeping the memento, as it is logical not to discuss the matter openly, until it's pointed out that given the sitution, the person who gave him the memento would likely encourage the breech of socilal protical.
I'd also point to Terminator 2 where we see two emotionless killing machines (The terminator and the nemisis T-1000) show emotional displays. In the former's case this is a learned behavior and even then, he is capable of flexibility (when ordered not to kill anyone, the Terminator acknowledges the order, and immediately shoots a guard... while it looks like he is killing the guard, he then points out to the horrified kid he complied with the order. He kneecapped the guard... who is howling in pain, but did not kill him. Sure, he hurt the guy and violated the spirit of the law... but the exact word of the law said nothing about pain. Meanwhile, the T-1000 is emotionless the entire time, but can act, and his initial dialog frames him as a friendly police officer just looking for a kid to talk with him (at this point, the film is shot so a first time viewer believes Arnold's character is a villain, like in the first film, and the T-1000 is the hero... pay close attention to how Arnold enters and behaves when he and John Connor firsts meets him.).
This is another important aspect of logic. As it's said in the computer world, "Garbage In, Garbage Out." That is, a computer will literally follow any instructions given. If it does not perform the intended response, the problem was with the instructions you gave it, not the computer. Thus, any unexpected behavior is expected from the completely logical computer's eyes... if you wanted something different, you would have said so. To put this in a less technical term, The law that can never be broken, can be bent... into a pretzel if necessary.
Mathmatically, Logic often relies on logical opperators AND, OR, NOT (!) and NOR. From here the orders parsed in an algebraic formla where variables represent statements (For example X = "Do you like chocolate ice cream?" and Y = "Do you like Vinella ice cream?") and they are either truthful or false.
If I am asked (X AND Y), then my response is no: X may be true, but Y is false... therefore I cannot like chocolate and vilnella Ice Cream (to make this a true statement, you would have to do (X AND !Y) with the exclamation point turning my response to Y into a negative of the response. If asked !(X AND Y) then my response is also yes as it asks if the statement (X AND Y) is a false statement... which it is because only one condition is true.
However, if you ask me (X OR Y) then my response is yes: I like one of those flavors (or both) so yes is an adequate logical response (that does give you the information you wanted to know but does answer your statement truthfully). It's only false if both of those statements are false. A fun one of these is "are you crazy or just stupid" answered with Yes. (I can be crazy, but my answer shows logical understanding of the question asked... or I can be stupid but perfectly sane and logical. Or I'm a crazy idiot.).
Make sure his answers fit appropriate responses to questions with And and Or present such that they follow only if the two conditions are correctly met to make the whole statement logical.