Before I start explaining my ideas concerning logic, let me clarify something. I adressed this in my first post, but I might as well elaborate upon it, because often this is the biggest problem many people have with my ideas. Particularly in debates with objectivists, (<3 @ Kuh, TGS, MiB, et al.), I am often accused of trying to use logic to "disprove logic". This is absurd. I doubt there's anyone out there who believes that logic is totally useless. Most philosophers who are accused of being "against" logic simply believe that logic and rationality have limits - that they are tools, and like any other tool, they have things they're suited for and things they're not suited for. Reason alone cannot be used to lift a rock, for example. So it's rather obvious reason has limits of some kind. The question is what exactly these limits are. The difference between Ayn Rand and Friedrich Nietzsche isn't that one "likes" rationality and the other "doesn't". Rather, it's that one believes that logic and rationality can work by itself, and can be applied to almost every facet of one's life, and the other believes that it's simply impossible for logic to do this without the support of other things. But I'm getting ahead of myself.
What is logic? Logic is simply the way of relating ideas together to produce new ideas which are useful. Depending upon the context, the usefulness can be expressed in terms of (among other things) correspondence to reality, prediction of future outcome, or discovery of the best method to achieve a desired outcome.
Logic relies upon assumptions. There's no getting around this. Trying to use logic without having assumptions to operate upon is like trying to build a building without any material. The initial assumptions that logic uses cannot be logically proven, because a logical proof of these assumptions must either be based upon those same assumptions, or based upon other assumptions, in which case those assumptions must be proven. Thus, any attempt to ground the assumptions themselves in logic must involve either circular reasoning or infinite regression. Logical assumptions are "alogical", i.e. outside the provability of logic (as opposed to being "illogical", in which case they would be directly contradictory to the what logic would prove).
What this means is that if logic operates upon two different sets of assumptions, it can achieve two different, and perhaps even contradictory, results, and both results would be equally "logical".
In the context of descriptive judgments (i.e. judgments concerning what things exist and how those things exist), our logical assumptions are basically taken from our intuitions. True, we have other assumptions. For example, in elementary-school science we were all taught about atoms; however, at the time we had to basically accept the existence of atoms without really understanding (except in a very vague way) why it makes sense to believe this. However, in these cases we can find out why it makes sense to believe in the existence of atoms. So, in the case of the individual, they're assumptions, but in the case of the entire logical system by which society in general (and the institution of science in particular) operates, they are not. The only assumptions on this scale are, as I've said, human intuitions.
So what are these intuitions? It would be very hard to really catalogue and adequately describe all of them, and even if one could, they would probably seem to be tautologies. For example, one of our intuitions is that we can trust our senses. Not trusting our senses would not be illogical, as some people might believe. There are two reasons one would have for believing distrust of the senses to be illogical: 1) As a direct contradiction to one of our fundamental assumptions, it would seem "wrong". 2) Assuming distrust of the senses would lead to a logical system with conclusions drastically different than the conclusions of the system that is based upon our assumptions.
Any attempt to prove that trusting the senses is more "logical" than not trusting the senses would either a) assume trust of the senses in the proof(*1), or b) Involve deeper assumptions whose denial would cause the exact same problem. I doubt option b is even possible, as I believe that trust of the senses is one of the fundamental assumptions. However, neither case would really solve the problem.
As for applying logic to ethics, things get a bit dicier. Ethics is based upon prescription (what one should do) rather than description (what is). However, there is no way to derive prescription from description, and therefore in the case of ethics our assumptions are composed of not only the intuitions but also the desires. One desire cannot be more logical than another; however, one method of achieving a particular set of desires can certainly be more logical than another. (For a more in-depth explanation of this please consult
this thread.)
How do logical descriptions and predictions relate to reality? We cannot know. The problem is, one of our inuitions/assumptions is that our logical desription of the universe is directly correspondent to the state of the universe itself. In fact, the division between objective and subjective reality itself is an intuition/assumption. Any attempt to "logically"(*2) determine whether logic is "objectively" embedded in the universe would itself involve the assumption that this is the case. So the only reason we would have for believing that the universe operates via logical laws is that one of our fundamental assumptions is that it does. Believing otherwise would not
necessarily be illogical, if said belief was part of a logical system that did not involve the aforementioned assumption.
(*1) People who argue that not trusting the senses leads to trouble are missing the fact that such "trouble" is only trouble if we're already under the assumption that the senses are accurate. Trusting that the picture of reality presented us by the senses is "correct" is only more practical than not doing so if we base our judgments of practicality on the results of actions which we are observing through our senses.
(*2) "Logically" here meaning using the commonly-accepted logical system - i.e. the one which uses out intuitions as its assumptions.