“Logic” originally came from the meaning, “to look (up)”, and originally assumed the inclusion of what is now called, ‘science’. But this depended upon apiriori beliefs. That is, statements of fact assumed to be universally true by its meaning in all conditions. An example of such is, “This sentense is a sentence.”
When many argued that the input statements of an argument cannot all bbe treated as ‘apriori’, which means a fact that needs no prior argument, a separation between the deductive aspects of argument from the inductive ones based upon one’s senses was sought.
Thus, “science” was added to the jargon of intellectual analysis that meant to specify most particularly the process of using one’s senses to infer to initiating statements of an argument. The root of the term literally means, “to see” or “to sense” with an old spelling. Then the ‘formal’ logic referred to the process of VALIDATING input statements used to draw conclusions. It is best to think of logic as the symbolic means of determining functioning conclusions with as much certainty as the input premises are when assumed ‘true’.
So both play a role. Think of logic as a machine that has one or more inputs and one specific output. (More complex machines can have many outputs based upon these. They call mechanisms that give one specific conclusion, “functions”, whereas you CAN have in general more than one conclusion. This is just conventional and one can technically define a system of reasoning initially based upon multiple outputs/conclusions. Some set theories do this. These mechanisms of argument are then referred to as ‘relations’, to which a ‘function’ is kind of relation(ship).
I think that many falsely take the stance that science now includes logic, but I still prefer treating science as a subset of logic that deals specifically with determining the initial inputs. Because sensed reality is relatively subjective, science acts as the collective subjective senses (often referred to as ‘objective’ without care to differentiate the meaning distinct from a gods-eye-view of reality.) As such, ‘science’ is a convention of people’s shared subjective declarations about what they sense along with any inclusive interpretations upon them. I think it has to remain separate from the act of validating conclusions but should remain under a common heading. However, I’m not sure if it should be considered ‘science’ and that where science uses logic, such as in theorizing, it should be treated as a hybrid. Science can be thought of as the “logic of inferring conclusions based upon the senses inductively” whereas formal logic is sometimes interpreted as “the science of deductive inference.” As long as one understands that they both are needed this should not matter. But those who tend to favor the former tend to think in terms of stereotypical forms of thinking given they prioritize sensation while trivializing logic as unreal but just a tool (as math is).
I understand your reference to fallacies. But note that even those are mostly inductive and lack universal inclusion of all cases beyond the deductive fallacies. They can be political and ignorant of how interpretation is often perceived through culture of the times. [‘paradigm’]
Fallacies are just the set of common errors (declared) and most have exceptions. Nevertheless, it is a good introduction to arguing and eventually necessary for understanding as one advances in any intellectual analysis.
Note that I happen to prefer interpreting reality as made up of mechanisms to which logic is just a symbolic or ‘abstract’ type of machine. To me, the ideal goal we need to seek intellectually is to bring the model that the abstract logic refers to as close as possible to the reality that science seeks to determine. Science can then be thought of as trying to guess, by patterns of our collective senses, what the mechanisms of Nature are or might be, and logic as its validator. That is, upon guessing what the possible function of some observed phenomena is scientifically, we then test this mechanism for the logic we assumed. If it predicts successfully any novel inputs we place into it, then this ‘validates’ the system or particular logical nature of the guessed mechanism.
Science then works backwards from some given indeterminate ‘object’, where logic works forwards in specifically determining conclusions.
