Logic 2: formal and informal systems, applied ideas

In the previous post, I sketched out some reasons why it would be useful for all of us to know something about the fundamental concepts in formal logic. My claim is that because we use a vast array of models, theories and systems in our lives, we need to understand the concepts that describe how models of reality work consistently (or otherwise); they provide real information in application and can be improved. I know that many philosophers and logicians will be cringing at the simplifications and assumptions I have made to jump to that conclusion, but I think the gaps can be filled in (and have been) by others in more academic forums.

In this post, I want to think a bit more about the link between formal systems and the informal ones that we mainly use every day. Everything I have to say owes much to the work of others across books and articles too numerous to mention here (I’ll put together a bibliography at some point), although by no means do I claim that I accurately represent others without error. But I do want to show how important it can be to understand the way formal systems work, and how the very idea of a formal system affects how we think about proof, meaning and truth elsewhere.

Formal systems

Formal systems, studied by logicians, mathematicians, philosophers and others, are  (usually) characterised by a specified and restricted set of symbols, rules for making statements from those symbols, plus rules for generating new statements. So from these basic rules and instructions, statements can be generated that are regarded as products of the system; they are called theorems. In this form, the symbols are just marks on the page or screen and mean nothing at all (in the sense that we understand the meaning of ordinary words, signs, or gestures). They have no interpretation. An interpretation is a way of mapping the statements onto facts or truths in a world (it doesn’t have to be the real one) so the theorems do not contradict each other. Theorems so interpreted are said to be consistent. A system may use a symbol, ‘+’ say, but its ‘meaning’ is determined purely by how it is to be used in the rules, and this need not be the meaning we understand by the concept of addition in mathematics. A proof within a formal system is just a setting out of the sequence of rules that have been applied to get from one statement to another: you prove that a statement can be produced by applying the rules, nothing more.

What this means in practice is that you can create a real (or theoretical) device that follows the instructions of symbol manipulation without having to build in any ‘understanding’ as an extra power for the machine. So when we ask ‘Yes, but how does the computer know to move the cursor when I move the mouse?’ the answer is ‘It doesn’t. It just follows a set of symbol manipulation rules physically coded into electrical circuits that switch parts of the circuits on or off in complex patterns. It is we who see this as the cursor moving to match a mapped movement of the mouse in a different space.’

All kinds of interesting ideas flow from grasping what a formal system is. In the 1930s Kurt Gödel (1906-78) discovered that once they reach the level of sophistication to be self-referential when they are interpreted (that is, the statements can be about statements in the system itself), they automatically generate ineliminable complexity that produces consistent theorems that cannot be proved or disproved within the system. [This is a difficult idea that does take a time to appreciate – it has taken me a while! – and I recommend the first part of Douglas Hofstadter’s Gödel, Escher, Bach as a good place to start, if this is of interest.]

Deep questions and applications

Now, here are the deep questions: what does this extra ‘understanding’ of meaning that we have consist in? When we add an interpretation to a formal system and start to talk about a statement having a meaning that makes it true or false, what else has been added to the simple rules of symbol manipulation?

Perhaps surprisingly, we are increasingly discovering that maybe the answer is that there is nothing extra at all beyond ultra-complexity. The sophistication of our brains, the deeply embedded systems of self-reference in language and thought, and the infinitely beautiful patterns of our ongoing dialogue of concepts as a species, generate these ‘extras’ without the need for anything more. Inside these questions is a vast part of the research in artificial intelligence (AI), cognitive science, the philosophy of mind and a whole host of sciences and theories that fundamentally shape the modern world. The spin-offs from this research are ways of thinking and modelling people, their behaviour, and the world. The ideas are driving forward society in the fields of cybernetics, cyber security, automation, expert systems, engineering, medicine, finance, big data analysis, economics, social research, psychology, education and training … the list is increasing daily as we use more and more technology, and, importantly, formal modelling to understand ourselves and our world.

Once it would have been commonplace to suppose that studying logic and formal systems was an interesting but esoteric activity, of use to nerdy computer scientists, mathematicians and some ivory-tower philosophers. However, we are coming to see that we cannot understand ourselves and our world without the logic, mathematics and philosophy that flow from these fields. I deliberately include philosophy because conceptually there is much to explore, and we would be mistaken if we did not recognise that we interact with each other, and fundamentally understand each other as persons, not machines (I will have more to say on this in a later post); this means there is still much to do to explore what such thinking about formal systems really means in practice. However, we would be equally mistaken if we thought that leaving the thinking about such things entirely to others was acceptable. Understanding and exploring logic and formal systems should be as accepted and embedded in our thinking as the theory of evolution through natural selection – incidentally, a process that can be captured and modelled by computers.

In a business context, we use and reuse ideas about customer behaviour derived from big data analysis all the time. They are often informally applied and are integrated into other theories about culture, economics and choice. But precisely how these models work, how they produce apparently consistent patterns and predictions may not be well understood; we may blur the mathematics and logic into the interpretation of the patterns that result. If we understand how such models work conceptually, even without having to grasp the intricacies of the detailed statistics, we will be much better informed about where errors may lie, or where misapplication can be avoided.

Getting logic right, understanding its fundamental place in our modern world, a world replete with models, theories and informal thinking which assume logical systems and are built out of them, is one of the great adventures of our age. Everyone who uses models or theories, computers or mathematics in any shape or form in research, business, policy, leisure or communication, can learn something from taking the time to reflect on logic, formal systems and their place in our lives.

To find out more about learning aspects of formal logic and systems for professional and business purposes, please do get in touch; I am always happy to help.