This is a very common and quite understandable question especially given the difficulty of the subject and the abstract nature of much of formal logic. What possible purpose could be served in learning logic? When will you ever use logic? Let's look at some reasons and attempt to make the case for the importance of learning logic.
I think there are several distinct arguments that can be made.
1. The relevance argument
2. The argument from the nature of logic
3. The mental exercise argument
4. The general knowledge argument
A common question concerning logic is how it could possibly be relevant to the major of the student or their career. This is a stunning point if one thinks about it in light of the nature of logic which we'll address below. However, one consideration is offered by the economist Thomas Sowell. Relevance is something you can only assess after you've learned a subject. You can't tell until then whether something is relevant to your life or not. Closely related to this point is the fact that none of us knows for sure what will happen next in their life and so we can never be absolutely sure that a subject, any subject, will not be relevant. Think about it. As a 20 year old you might say, I know I'm never going to use this in my career or life. Assuming (which is a safe assumption) that you'll live for 50-70 more years, how can you say this? How do you know for that length of time what will be relevant to your life and what won't? I took logic as a sophomore majoring in telecommunications and could have easily believed at the time that I might never use logic again. Who knew I'd end up teaching it!?
Logic is concerned with training the mind to think clearly. Given this, let's ask the question again: When will I ever need logic? OK, so you're really asking: When will I ever need clear thinking? Now, the importance of learning logic should be crystal clear. There isn't a single area of life where clear thinking wouldn't be beneficial to some degree. The real issue here is not whether logic is useful, but how can logic and what we do in a formal logic class help improve our thinking. The real question is not whether clear thinking is needed but how can learning about the categorical syllogism, truth tables, and natural deduction improve our ability to think clearly. I think I have some answers to those questions.
One of the reasons such questions come up in the first place is the apparent strangeness of formal logic. It looks so different than our ordinary use of language because it is a formal symbolic system abstracted from ordinary language. In fact, what we are attempting to analyze in logic is the underlying nature of inferential thinking and to do so we must inspect the form of our reasoning by separating it from its content. Doing so makes it look irrelevant for the same reason that looking at a car engine out of context looks irrelevant to the working of the car. Think about it. If you looked at the engine of a car without ever addressing the purpose or context you would never connect that mechanism with driving your car. If you never bothered to look under the hood you might not even know there is such a thing as an engine! For all you know there's nothing under the hood. So, if you look at the engine for the first time it looks strange. So too does the inner workings of thinking and reasoning. It's only after studying formal logic that you begin to see that the principles of logic are connected with ordinary thinking. So by studying the underlying depth of the subject you can get a greater appreciation of the ordinary application and in the process become better at that application.
Let's look at what formal logic (or symbolic logic) is forcing us to do not from the standpoint of using the principles of formal logic directly, but from the standpoint of the underlying skills these principles are drawing on. In categorical logic you have to read statements carefully to make distinctions between terms that on the surface sound similar. You have to recognize general rules by looking at specific cases. In propositional logic you have to recognize the general rules underlying the use of certain words and recognize these principles in different situations. Finally, in natural deduction you have to take a set of rules and apply it in an orderly method to solve a problem. OK, look at the skills being used here: rule recognition, abstraction, planning, problem solving, making distinctions. It is these skills that logic is training you to improve and it is these skills that represent the real practical benefits of learning logic.
Perhaps an analogy will help. Some people go to the gym to workout. They lift weights, do stair climbing, walking machines, etc. Do they do these things to improve their ability to lift weights, climb stairs, and walk? No, they do them for some other benefit which these exercises indirectly lead to. It's the same with the mind. We need some form of exercise for the mind and that's what logic is. The benefits to logic are indirect. That is, what we do in logic is improving our ability to do something else.
So, why don't we just practice the thing itself instead of practicing the skills indirectly by learning logic? Well, the answer is that we do this as well but logic represents a more rigorous form of exercise. Look at it like this. When you walk to your car or to class you are getting some exercise benefit. But, you may also go to the gym. When you carry groceries from the store to your car you are getting some exercise benefit. But, you may also lift weights. Why? Because the more causal form of the exercise isn't really vigorous enough to get the true benefit of doing the exercise. It's the same with logic. When you read a book or magazine you are getting some benefit to your mental exercise. But, you need to train your mind in a more rigorous fashion forcing it to do more involved thinking. Like all exercise your muscles are sore at first until you get used to the exertion. In time you find you are better able to handle the exercise and your general thinking skills improve as a result.
Learning is a funny thing in how it works. The more you know, the more you can learn and the more you know the easier it is to learn something new. Learning, then, is about making connections. The more general knowledge you have the easier it is to make connections. So, in this vein logic adds one more subject of knowledge to your mind allowing you to make more connections thus making the learning of anything else easier. Not only that, since the subject matter of logic is based on inferential reasoning the very skill of making connections is being learned as you go through your training in logic.
If you know something about psychology, understanding philosophy is easier because they're connected. That's an easy one to see. But there are so many ways that one field of knowledge is connected to another that we don't see immediately. Learning about logic makes it easier to learn about computers. How so? Computer programs are nothing more than logic commands. OK, but what if you're not programming a computer, only using it? Still, there is a benefit to understanding the method of following rules, thinking in an orderly fashion, recognizing that one step follows from another. All skills that logic teaches. There are other connections to consider as well but too numerous to list here. However, you might consider that others have thought about these issues long before you got into the classroom. It is not to punish you that they have decided you should take such classes as logic. Their experience in life (sometimes 20, 30, 40 more years of experience than you have) has shown them that there are benefits to learning about subjects such as logic. You can benefit from their experience. Of course, the alternative is to learn about the benefits the hard way. That is, by forgoing these skills now and discovering that you need them at the worst possible moment.
Given that you don't know what might be useful later in life, and the obvious utility of logic in training the mind, and the connections between different subjects which make learning easier, it only makes sense to obtain as much knowledge as you can while you can get it. There's no better time than right now to add to your knowledge. Even knowledge which has no direct benefits now might be useful to you later on in a direct way, and is certainly useful to you now in indirect ways. Some students come to logic class (and perhaps many others) with the attitude of just wanting to get the grade. That's unfortunate. First, youre paying for something you're not taking delivery on which doesn't make a lot of sense. Like going to the store and purchasing a new big screen plasma TV and paying for it but not picking it up! Students have told me that once they leave the class they'll forget everything they've learned. That's unfortunate too. Again, you're paying for something and not taking delivery. But also you're missing out on something that WILL benefit you in the near future and MAY benefit you later in life as well. Are you really willing to discard useful information so quickly? Think twice before doing that. Not only with logic but with every other subject you encounter. Besides, it costs you nothing to hang on to knowledge. The brain is not a sieve leaking out old information to make room for new or a small container, which must be cleared out to make room for new information. The brain is a complex organ capable of making connections between different knowledge sets, and the more connections you make the more knowledgeable you become. Imagine how much better at information processing and using you'd be if you learned the basis of making connections, inferences, arguments, and deductions. Guess what, Logic teaches all of those skills!
In academia we've known for years that people with limited knowledge are easy targets for scam artists, tricksters, politicians, and demagogues. What you need to know is that the scam artists, tricksters, politicians, and demagogues know this too. Francis Bacon was right; knowledge is power. And protection. You spend money to protect your computer against viruses, your cars against theft, your health against illness. You're also spending money to protect your mind against harm. The product you purchase to do this is NOT a grade. It is NOT a piece of paper called a degree. It's the knowledge behind them that will help you. You paid for it. Take it!