Defining logic as "the study of information encoded in the form of logical sentences" is like defining the sentence as "a group of words combined in the form of a sentence."
But Stanford Professor Genesereth's definition of logic (offered in the first segment of an online Coursera course on logic) is worse than circular. Logic is not (fundamentally) about propositions, just as carpentry is not (fundamentally) about hammers or nails. Carpentry is about how to build or repair, and then building or repairing; say, a house. Logic, too, is about method: how to think, and then thinking, perhaps by building an argument or assessing one. Logic is the science and art of reasoning in such a way that one's thought is consistently tied to reality, and therefore capable of yielding knowledge. Translating statements into logical forms (like "All S are P") and understanding the implications of such sentences are a means of doing that. It is a critical part of the method. But logic as science of method is about more than the study of propositions or other "logical sentences."
American Heritage's definition of logic is not bad: "The study of principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning." But if we read past the comma, this definition seems to give short shrift to inductive logic. The procedure of induction is to draw inferences from premises already adopted as true--to excavate what is implicit in those premises. Induction involves forming by observation and experiment the generalizations that may then serve as premises for subsequent deductions. Once we know that there are aardvarks, and what makes aardvarks like other aardvarks and different from all the non-aardvarks, we can feel logically justified in deducing, "Here's another damn aardvark." Once we know something about the laws of gravity by inductive inference (thanks Galileo, Newton, et al.), we can deduce where the asteroid is headed.
Dictionary.com defines logic as "the science that investigates the principles governing correct or reliable inference." "Inference" we can take to be not quite a synonym of "reasoning." They're both about how to get from starting points to conclusions (although you could say the same thing about the running of foot races). American Heritage says that to infer is to "conclude from evidence or by reasoning." To reason is "to use the faculty of reason; think logically." Dictionary.com says to reason is "to form conclusions, judgments, or inferences from facts or premises" and that to infer is "to derive by reasoning; conclude or judge from premises or evidence."
In his introduction to his logic text The Art of Thinking, David Kelley says that his book is about "how to think." Thinking is a "cognitive process we use in the attempt to gain knowledge or to understand something, as distinct from our emotional responses to things." The subject matter of logic is "the rules and strategies of thinking, certain standards that tell us when we have achieved a clear understanding of some subject or succeeded in proving a case." Kelley distinguishes thinking as a directed mental activity from the "thinking" that has to do with the flotsam and jetsam that floats through one's mind even when one is not attempting to direct thought. I'm going to go out on a limb (a fairly stout limb given the title of Kelley's book) and declare that "thinking" in the narrower sense Kelley spells out is synonymous with "reasoning." Or we can say that reasoning is a kind of thinking: the directed-toward-finding-out-things kind of thinking.
Reasoning and being logical are closely allied. But reasoning has more broadly to do with the exercise as such of the human conceptual faculty to find out about the world. Logic is the method we follow (not always formally and explicitly) to keep ourselves on track in that process. Formal logic is explicit and precise in its vocabulary and directives--perhaps precisely wrong, if and when logicians go off the rails with symbolic arcana or non-fundamental definitions. If we mean by logic only formalized and explicitly applied rules of logic, i.e., logic as a developed science, then its overlap with reasoning as such is not as great as the overlap of any even implicit inferential process that attends to the identity of things.
The principle at the root of logic is the law of identity: the fact that things are what they are. All injunctions of logic are forms of recognizing, accommodating, applying the law of identity and ensuring that it is not violated in thought as we go about identifying facts and drawing further conclusions, i.e., making further identifications on the basis of past identifications. Following the law of noncontradiction is the negative aspect of the means by which we abide by the law of identity: we're taking care to avoid the bad thing in order to continue adhering to the good thing. Whenever we see that we have stated or implied that A is non-A, we know we've made a mistake.
How many different ways can we say that A is non-A without doing so willfully? Lots of ways, some of them very subtle. Hence the need for a science of logic, for a textbook as opposed to a 3x5 card saying "remember that things are what they are." Ayn Rand stressed that identifying the nature of things must follow non-contradictory procedures when she offered a boiled-down definition of logic as "the art of non-contradictory identification." Identification of what? Not, ultimately, identification of the logical character of sentences.
Formal rules of implication help us recognize fallacies and flaws in reasoning, but also to positively assess the strength of our evidence and the level of support for our conclusions. We want "logical sentences" and their relationships to be valid. But we don't want to safeguard from contradiction only "logical sentences." The concern is broader. We also want to know about relationships within and between arguments as well as between propositions in one sequence of an argument; and most important, the relationship between our argument and reality. We want to know what constitutes supportive but inadequate evidence for a generalization. We want what we're saying about reality to be true, or as true as our means and effort make possible.
Classification and definition are among the aspects of reasoning with which logic is properly concerned. You don't convey what logic as the method of reasoning is basically about if your definition of logic--what you write on the blackboard as worth encoding--is "the study of information encoded in the form of logical sentences." Concepts of method have something to do with the purposes the method is designed to achieve. The purpose regulates the method. Logic is fundamentally about guiding our reasoning. Logic is not, fundamentally, about the study of information encoded in units of logical form (although it is also and importantly that). Logic is the study of any principles and standards designed to help ensure that our identifications identify. If we lose sight of that and merely manipulate symbols according to stipulated rules without regarding to whether these rules as practiced can help us to achieve and sustain a consistent knowledge of reality, what we are doing is not logic.
