Ideally logic is a set of rules. People can change those rules, but if you ask me it ceases to be simply logic at that point. They can implement logic to change those rules like "I'm not going to drive because I'm taking the bus with a friend."
Context is everything, so says cultural constructivism. Not everyone absorbs or looks at the same things even if they are sitting right next to each other. If your concern is that they come to correct/accurate conclusions, then I would have to say yes, though in order for it to be true logic, bias would have to be removed from it.
I wouldn't say this is bias in the sense that it's an unsubstantiated claim. If I observe a ball rolling down a hill a million times (This is the sum total amount of balls I've seen rolled down a hill)and infer through logical induction that every ball roll down hills, and you observed independently of I that there was a moment in which it didn't, then we'd have two subjective logical processes. We use informal logic relating to the use of language, since language is not a atomic while observations and expressions are infinite there is no perfectly reducible form. You could define logic as that of logical/mathematical operators expressing fundamental operators (Conjunction, Disjunction, Element, Set, Implies, Equals, Not-Equal, All/Exists etc) but I'd put that in the scope of mathematical logic. Since mathematics is pure abstraction it's exact, there can be no subjectivity pertaining to it. Logic has many sub-fields, I think the use of language in particular and the nature of experience combined with logical induction being a valid way to call a conclusion "logical" makes subjective logic frequent.
It goes to say what we mean by subjective logic when we think of the non-contradiction principle: It is either logical or illogical. The person employing his own subjective logic should be the logic that exists in the world, right? Though, since we don't understanding causality and causality is often a posteori in which we can't understand all variables that cause a particular outcome (And we couldn't even simulate the variable output if we the initial conditions, i.e chaos theory), then subjective logic is a consequence of our lack of omniscience and taking reductionist approach to multi-variate solutions.
Sure, ideally we want logic to be objective. Society doesn't rely on subjective logic. However, the individual does rely on subjective logic when he had made an observation about one time that the ball did not roll down the hill. Likewise, the one that hadn't observed this (me), employs the observations he's used to logically induce the conclusion of a ball rolling down the hill. The state of logic in its idyllic form, separate from the machinations of man is one we try to approach but never manifest in words. You could parallel truth to the theory of forms, in which logic shines the light towards truth but inevitably we are still aware of the darkness.
For me, reality is probabilistic and so is logic to that degree. No entity (in my belief) can confirm for any of us why a certain theory behaves the way it does. We had Newtonian physics which was superseded by Relativity, sure newton's law's of motions are easier than working them out using relativity and the approximation is virtually exact. Who's to say QM or relativity is superseded by another idea? String theory, Bohmian mechanics? We supersede ideas of this nature because they explain more phenomena in an understandable fashion, yet we're always plagued by expanding our knowledge from axiomatic truths to logical theories. Dark energy is an inference from the universe expanding and that light cannot optically observe this. Phantom energy is an inference due to the fact that galaxies are separating at an exponential rate, suggesting that the repulsive force of dark energy is not enough to explain the increase in speed of the expansion.
Lastly, I think most of our logical processes are not inferred using objective measurements. If you wish to call that illogical you may do so. When I discuss an idea with you, I cannot present to you a situation precisely illustrating the empirical evidence I've acquired which infer a different logical solution to your own. The logic you've accrued from your experience is different from my own, yet in the context of the individual it is logical unless they gather more evidence. Sometimes this is impossible to do.
