Another misconception about philosophy is that we just argue with logic to prove a point. This is nominally true but does not capture the enterprise of philosophy. To use the tools of argument and the rules of logic purely for the purpose of dogmatically defending a point belongs to the realm of sophistry, the origins of modern lawyers (In ancient Greece “sophist” was a derogatory term used against those who were employed by the rich to use their skills in argumentation to argue for the rich and powerful). Lawyers are not concerned with truth but with winning their arguments. Conversely, philosophers are concerned with truth and use argument to test the strength of hypotheses. If a hypothesis doesn't withstand argument we modify it or reject it. Once again if you study the history of any philosopher you will see this is what happens. This is a formal practice in both philosophy and science. You publish a paper. Your peers criticize it in written form, you try to reply to the criticisms. If you can't reply then you modify or reject the hypothesis. This happens all the time.
Regarding your comment about the motives of others and their concern for truth, Hume would agree with you, and so would many psychological studies. It even has its own term in psychology: motivated reasoning. Most people begin with a position and when they are subjected to countervailing evidence, rather than modifying their position as we would expect, they reject the evidence and further entrench themselves in their view. This is because most people are emotionally attached to their position, not the quest for truth. Their beliefs define who they are and releasing a central belief is painful and is an admission of fault. Philosophers aren't emotionally committed to any particular beliefs. We are committed to a method and we are committed to approaching truth as best as can be expected from fallible humans.
Regarding the everyday person's interest in philosophy, I am going to disagree with your assessment that most people aren't interested. Anytime you see a long string of comments on someones facebook page, blog, or hear a passionate discussion, you can bet that it involves a philosophical issue. Just because the issue wasn't raised in a classroom does not preclude it from being philosophy. Now, of course not everyone is interested in every area of philosophy but everyone has some philosophical curiosity and inclination, otherwise they wouldn't be human. Most people are concerned with the original question of classical philosophy, which is "what does it mean to live a good live"? (i.e. what are the necessary conditions).
So, to summarize good philosophy (again, as with science) is not a competition about who's right and who's wrong; it is about a commitment to a method of inquiry and a commitment to seeking truth. You might disagree with the deliberate methods of logic and hypothesis testing for choosing one set of beliefs over another but seeing who can scream louder has yet to yield better results.
Finally, to repeat, I will always be grateful to you if, when I am unclear in my explanations, that you let me know where and (if possible) why, and I'll do my best to improve my explanation...
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