Every single person is convinced by logic and evidence, but each person is convinced by one more than the other. Here is the basic question: which is more important for discerning truth or fiction: logic or evidence?
I'm big on the Myers-Briggs personality typology. It is tremendously helped me understand the people around me. One of the personality functions is the Perceiving function, of which there are two kinds: Intuitive and Sensing. The Perceiving function has to do with the way your personality takes in new information and concepts and then stores it.
Sensing types take in information through their senses, i.e. touch, smell, sight, hearing, and taste. Because this is the primary way they take in information, the kind of information that they generally take in is concrete. They tend to have a better head for details, and have a photographic memory. They also tend to have a static view of the world. They are more concerned about what is, then what could be, or even will be.
Intuitive types take in information by reasoning. They remember things based on its importance or purpose. Often they have a conceptual framework in their mind, through which they decipher the usefulness of data, and only take that bit in. This means that the kind of information they take in is abstract. They are more likely to remember things like stories, concepts, and relationships (such as a cause and effect relationship). They tend to have a very dynamic view of the world. Viewing things in terms of purpose and relationship, they are quick to figure out how to use various things in new ways, or even whether a structure has real value.
My point in bringing this up is that Sensing types tend to be more convinced by evidence, which Intuitive types tend to be more convinced by logic. If you think about it, this makes sense. A sensing type has a very concrete way of looking at things. Evidence represents what is tangible, and thus, to them, reliable. An intuitive type is going to be more trusting of logic, because it places things in perspective, and stresses the importance of something over its mere presence.
But Which Is Best
I think it is clear that I am going to say both. This is because both evidence and logic reflect the reality of something, but in different ways. Also, the use of one without the balance of the other can easily be manipulated.
The flaw of using logic without evidence is obvious. Evidence is, after all, derived from the physical world. There is a clear sense of reality that evidence brings with it. No matter how pretty and powerfully a point is argued, it doesn't mean anything if it contradicts the tangible reality around us. I can make a brilliant logic as to why the sky must be red, but it only take a casual observation to point out that it isn't.
This is one reason why postmodern and liberal ideas do not have a lot to stand on. Because we have no objective demonstrable access to God, the only means of discerning information about Him is through revelation. When God's revelation of Himself is discredited, or treated as merely a perspective, we loose the only source of evidence there is, and then anything can be said about God.
However, the use of evidence without logic is equally flawed. Logic is what gives something importance or relevance. Without a well crafted, and reasoned logic, there isn't any guarantee that the evidence provided actually has anything to do with the discussion, let alone that it actually makes the other position's point.
This is why many are not convinced by proof-texting. Often people quote a Bible verse, expecting the other person to merely relent because the Scripture has been spoken, and are shocked when the other person asks for silly things like context, and exegetical interpretation. This is because evidence without good logic is meaningless.
A classic example is what is known as cum hoc ergo propter hoc or the logical fallacy of correlation implying causation. For instance, it was noted that ice cream sales correlated with heat stroke. Therefore it was suggested ice cream causes heat stroke. While one can quote statistics to back this up, simple logic shows that they both have a common cause, rather than that one causes the other.
Another is what is known as "argument by verbosity" where one quotes a myriad of facts, without linking those facts together, or aptly demonstrating the facts as having to do with the subject. For more see machine gun hermeneutic.
My point in all this is think about what you say, and make sure your arguments are backed by both logic and evidence. Think, don't just regurgitate what someone once told you.