Philosophers have the embarrassing habit of apologizing for formal logic. Mathematicians don’t bother because they don’t care -- they’re just interested in the pretty pretty symbols and waste no part of their lives checking to see if their activities actually mean anything. But philosophers worry about everything, and the more obvious a thing or its explanation might be, the more worrisome it becomes to them. And since a particularly large part of philosophy in the last 140 years has specifically centered itself around the importance of formal logic -- which is “obviously” important -- this becomes especially problematic.
Formal -- which is to say, deductive (I’ll use the terms interchangeably) -- logic is “obviously” important because it is “obviously” the key to good reasoning. The trouble is, the more one pursues this thesis (as it has been expressed!) the harder it becomes to actually justify its claims with evidence or, gods help us, logic. Jaakko Hintikka -- a logician of singular ability -- examined these claims with no small measure of skepticism, and could scarcely find a single text on the subject of formal logic published in the past 60 years that was worthy of the bother.
Part of the traditional explanation for this situation is that deductive logic does not provide you with anything you did not already have. Which is to say, all the data that you might ultimately find in your deductive conclusion was already present in your premises. All you achieved with your formal shuffling of symbols was just and only that: a formal shuffling that merely rearranged what you already had in hand.
The error in this traditional view is imagining that such rearrangement is a matter of small or trivial consequence. Even the authors of the vast majority of logic texts Hintikka examined and referenced suffered from this belief; hence the apologetic nature of their approach to the subject of whose importance they were so desperately trying to persuade their readers.
You see, formal logic IS important because the content in the conclusion is not all present in the premises until the conclusion itself is formed! Just because the symbols are there does not mean the information is equally present. (In this regard, the “information” that is being discussed is not a mathematical structure per se, but a consequence and result of inquiry.) Only when the premises are ordered in a manner that makes evident their implicative relations is the information genuinely present.
Let me offer you an example. This example is not explicitly scientific in character and comes, rather, from fundamental issues relating to the Constitution of the United States and a particularly vexed notion of legal right. Nevertheless, I find it a particularly useful example in that it exemplifies the above in a very tangible way.
The claim is occasionally made in American legal and political circles that there is no such thing as a right to privacy. The reason this asserted is because there is no mention of such a right anywhere in the Constitution. (The technical term for this falls under the heading of “no non-enumerated rights.” Since the right to privacy is not enumerated in the Constitution, it does not exist.) The issue is then variously wrangled back and forth by appealing to other texts and legal precedents, but a core problem with the above is typically ignored: namely, the claim as made does not form a formally valid argument. But one might easily miss this fact if one first neglects to formalize the argument and second fails to understand such formalizations in the first place. So let us actually formalize the above.
For purposes of historical connection, let us have the letter “E” stand for the phrase “There is no …” and “A” stand for “All.” Conveniently -- again for historical purposes -- we can use “M” to stand for “Mention in the Constitution” and its various cognate phrases, and “P” for “A right to Privacy” again in various equivalent phrasings. Finally, while it is not entirely idiomatic, it is useful to let “R” stand for “A Right that exists.” So the above claim, “There is no mention in the constitution of a right to privacy, therefore there is no right that exists that is a right privacy” can be schematized as:
E M P
E R P
And here is where the problem becomes evident. The above is a syllogism, but it is missing a premise. If the missing premise could be filled in as one of the valid syllogistic forms, then the above would be what is known as an “enthymeme.” Moreover, if we understand the forms of valid syllogisms, then we can indeed fill in the missing premise, and we can fill it in without even knowing what the various terms stand for:
E M P
A R M<-- The missing premise!
E R P
This is now a valid "Celerant" syllogism . Note how the “M” -- which is known as the “middle” term -- stands in a diagonal between the first and second premise; this is part of the formal structure that assures us of the syllogism's validity. So far, we've filled in the blanks without even knowing the interpretation of the premise that we put in the missing space. Now, recalling that “A” stands for “All” we can read off what the second premise says from our previous interpretations: “All Rights that exist are Mentioned in the Constitution.” This is now a logically valid argument, and if both premises are true, then the conclusion follows by necessity. There is just one problem; the second premise is actually false.
The 9th Amendment of the Constitution states:
“The enumeration in the Constitution of certain rights shall not be construed to deny or disparage others retained by the people.”
This quite clearly denies the second premise, demonstrating that while the argument is valid, the conclusion does not follow from the argument as formed. (One might conceivably still argue for the conclusion from other premises, based on claims outside the Constitution that there are no non-enumerated rights. While not an expert in the field, I confess that I’ve never seen such an argument even attempted, much less imagined one that was remotely plausible.)
From the perspective of logic as inquiry, the above makes perfect sense. Formalizing the relationships enabled us to gain a better understanding of the implicative structure of the whole, and even made an implicit -- and previously unexamined -- assumption explicitly a part of that implicative structure. Formalizing the argument enabled us to ask better questions, which is to say, it made it possible to engage in more effective inquiry. This is the power of formal logic, as an instrument of general logic it enhances our ability to organize the information we have in hand, recognize what information we might need, and thereby press our inquiries forward in a well-organized and unambiguous manner.
 Full disclosure, I’ve met Professor Hintikka. Quite aside from the fact that he’s one of the best logicians to have ploughed the field in the last half-century and more, he is also an exceptionally decent fellow who is particularly generous with his time and attention. And he likes cats …
 "Is Logic the Key to All Good Reasoning?" Argumentation 15: 35–57, 2001. The one exception that Hintikka identified was Introduction to Logic, Patrick Suppes, Van Nostrand, Princeton, 1957. Republished by Dover Publications, Mineola, 1999. It is, indeed, a good book, except that there are no published solutions to the exercises so you either have to already know the material or know someone who does in order to usefully navigate the book.
 See http://www.science20.com/inquiry_inquiry/logic_inquiry for a more extensive discussion of this position.
 Indeed, this is really the only example I’ve ever found of an Aristotelian syllogism that is at once useful and interesting from a practical perspective, and relatively transparent when the terms are interpreted. Generally speaking, one has to seriously pretzel one’s terms in order to fit them into the Procrustean bed of syllogistic forms. So it amuses me to find a syllogism that actually works. Writing the above argument up in modern FOPL (First Order Predicate Logic) is scarcely any kind of exercise at all, the contemporary instrument is so much more powerful and flexible than its syllogistic ancestor.