First, I shall ask the question "what is logic?" And I shall answer it. In the context of the present poll, "logic" means those methods of reasoning that are guaranteed to produce, from true premises, only true conclusions. And the poll is asking whether classical logic is it.
Particular formalisms used to model particular things are not, in this sense, logic, although they may be expressed in logic. For example, number theory is not logic. Neither is geometry, or physics, or probability theory. Neither, I claim, is fuzzy logic, despite the word "logic" in its name. You can say, "here is a set of functions (which I shall call fuzzy logic truth tables), and here are some theorems about how they behave (which I shall call fuzzy reasoning), and here are some physical systems whose description uses these functions." That does not mean that those functions are actually a form of logic, as I just defined it. Bang-bang controllers like the room thermostat were invented (in 1883) long before fuzzy control theory (about which I've heard anecdotally that the term was invented only to avoid someone's patent claims).
The closest anyone has come to promulgating an alternative system is intuitionistic logic, which is a pessimistic version of classical logic, in which the axiom of the excluded middle is dropped. In intuitionistic logic, you cannot infer P from not-not-P, or carry out proof by contradiction. However, I think intuitionism is simply a mistake, a historical accident which would never have happened if there had not been a half century between the codification of mathematical logic and the invention of the computer. Everything that is useful in intuitionism is given by computability theory and classical logic.
In the context of the present poll, "logic" means those methods of reasoning that are guaranteed to produce, from true premises, only true conclusions. And the poll is asking whether classical logic is it.
I think you're begging the question. I think you've given a definition of "classical logic" rather than "logic".
Despite being (IMO) a philosophy blog, many Less Wrongers tend to disparage mainstream philosophy and emphasize the divergence between our beliefs and theirs. But, how different are we really? My intention with this post is to quantify this difference.
The questions I will post as comments to this article are from the 2009 PhilPapers Survey. If you answer "other" on any of the questions, then please reply to that comment in order to elaborate your answer. Later, I'll post another article comparing the answers I obtain from Less Wrongers with those given by the professional philosophers. This should give us some indication about the differences in belief between Less Wrong and mainstream philosophy.
Glossary
analytic-synthetic distinction, A-theory and B-theory, atheism, compatibilism, consequentialism, contextualism, correspondence theory of truth, deontology, egalitarianism, empiricism, Humeanism, libertarianism, mental content externalism, moral realism, moral motivation internalism and externalism, naturalism, nominalism, Newcomb's problem, physicalism, Platonism, rationalism, relativism, scientific realism, trolley problem, theism, virtue ethics
Note
Thanks pragmatist, for attaching short (mostly accurate) descriptions of the philosophical positions under the poll comments.
Post Script
The polls stopped rendering correctly after the migration to LW 2.0, but the raw data can be found in this repo.