But note that some logics -- e.g., calculus, Newtonian mechanics, special relativity theory, etc. -- work even though they are far removed from most people's intuitive understandings.
Cf. Tim van Gelder, Rationale - Why Does It Work? - Semi-Formality (May 2, 2007):
There is also the point that it might legitimately be said that an exotic or strange-looking but valid logic is a logic in search of a domain in which it applies. (Human beings have the peculiar and astonishing ability to invent or discover logics before they can figure out how to apply them.)
Mathematics, formal logic, programming languages, artificial intelligence, and many games such as chess are all formal. Natural language, conversation, politics, and humour on the other hand are informal, even if they exhibit a few of the features of formality to some extent.
Now human reasoning and argumentation are standardly informal. There are primitives (words, or propositions), meanings, and principles or norms, but these are not defined with the kind of precision, rigour or reliability one finds in formal modes such as mathematics or chess. In this respect, human reasoning is often reflecting the nature of the domains or issues about which the reasoning is being conducted. For example, politics is an inherently informal domain, and this informality is reflected in the nature of our default intellectual tools for thinking about and debating over political topics.
Noting on one hand the problems associated with ordinary human reasoning and argumentation, and on the other the often impressive achievements of their formal counterparts, the temptation has often been to recommend shifting human reasoning into a formal gear. Thus introductory logic textbooks are usually dominated by elementary formal logic (Aristotelian syllogisms, propositional logic, predicate calculus), making the assumption that people would reason more effectively if they replaced their instinctually informal thinking habits with logical formulae and proofs. This tendency culminates in the aspiration of mainstream artificial intelligence to recreate human-grade intelligence in the formal medium of digital computation.
Unfortunately this generally doesn’t work. AI research has discovered that it is extremely difficult, and perhaps impossible, to engineer formal systems capable of reliably making the mundane inferences underlying everyday conversation or humour, let alone engaging complex argumentation of the kind found, for example, in legal practice. Conversely, when people struggle with everyday reasoning and argumentation, they are not helped by attempting to translate their premises into some logical calculus and draw inferences by reference to its official rules. Indeed for innumerable commonplace reasoning challenges, formal techniques are so hopelessly impractical that recommending their use seems like a kind of sophistical sadism, another sad manifestation of Bacon’s Idols of the Theater.
There are of course notable exceptions to the claim that human reasoning is not improved through the adoption of formal techniques. Clearly, for example, the proper use of statistical methods can help us draw better conclusions about subtle correlations and causal relationships. Formal modes of thought certainly have broad and important areas of application. The point being made here is that in general our practices of informal reasoning and argumentation cannot be enhanced by transposing into a formal key.
It may be, however, that some degree of formality can be helpful even where full-blown formal modes such as mathematics, formal logic and computation get no useful traction. The distinction between formality and informality is not a binary opposition. Rather, there is a spectrum of cases depending on which of the characteristics of formality are adopted and how those characteristics are manifested.
In the case of reasoning and argumentation, there have been many contexts in which these practices have been made more formal than they would normally be. Consider for example the medieval theory of disputation, which presented a sophisticated framework of rules governing moves in argumentation. Or consider contemporary high-school debating or “forensics;” etc. These sorts of elaborations of our ordinary practices are aimed, at least in part, at improving those practices; i.e., at enabling us to reason and argue more effectively. Implicitly they are proposing that the optimal mode for human reasoning is not the informal or the formal, but rather something intermediate, a “best of both worlds” scenario.
The conjecture, then, is that for general reasoning and argumentation there is a “sweet spot” somewhere on the spectrum between ordinary informal practices at one end, with their sloppiness and disorder, and purely formal techniques at the other, with their rigidity and limited range of application.
The principle of abstraction [and] its ilk depend fundamentally for their successful application on intuitive human judgments based on experience and practice. To the best of our knowledge it is impossible to cash them out as formal rules capable of mechanistic implementation. Thus, argument mapping has irreducibly informal dimensions even as it makes reasoning activities more formal than they would normally be.