I quote from the first part of the first tutorial:
Consistency and Validity
The subject matter of Logic
Logic is not, however, concerned much with the actual truth-values of beliefs and sentences, but rather with such questions as:
Is such and such a set of beliefs or sentences consistent?
Is such and such an argument valid?
Too often, people confuse these things: logic is in essence a language, with a particular very strict syntax. Logic may or may not produce results that correlate with a given aspect of reality- it all depends on how one chooses one's axioms, and whether the particular aspect of reality under consideration is, in fact, logical.
Is all of "reality" logical? Heck no, in the sense that it can be encapsulated without paradox or "completed." We can say "reality has some undecidable propositions." Now sometimes somebody (usually a religious "apologist" of dubious intent) will size on this aspect of existence, and say "X doesn't hold to the 'Law of the Excluded Middle'" with the implication that those who adhere to X are raving lunatics for not seeing that, ahem, they're just not being rational. In fact, there are times when the "Law of the Excluded Middle" simply doesn't apply: Suppose
Ω = ÈkSk ; Sk~=Sj for j~=k.If an Sk can't be broken down into a smaller set, then there is no excluded middle of Sk and appeal to the Law of the Excluded Middle is simply an error of logic, not a virtue.
The, um, truth is though, that there are things logic and reason cannot encompass (it is possible to demonstrate this logically). The language of logic has limitations which logic itself freely admits. My question is: why doesn't the apologist admit this?
(As for references, see also the famous GEB; if you haven't read it by now, shame on you. Or, if you want something less technical - you can read The Illusion of Technique. If you want something more technical, I can recommend any number of books on logic and switching theory and automata as well...)
We know that.
So to summarize:
1. Logic is a language.
2. It can be very useful for resolving simple propositions.
3. Logic may or may not correspond to reality.
4. Reality has certain "undecidable propositions"
5. The law of the excluded middle "works" depending on the axioms/assumptions.