Tag Archives: Non-Monotonic Reasoning

Logic

Logodynamics

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What is logic for?

Is logic about thinking? I used to think so, believing that logic was something like the ‘doctrine of thinking’, or even the ‘doctrine of correct thinking’. A closer look, however, reveals that what we call logic, and the field of study that goes by this name, is about proving rather than thinking. Classical logic is in fact the science of the proof.

But there’s a lot more to thinking than proving. If you want to proof something, first you have to find the proofs. Then you have to assess these proofs in context – a context that can change. And what do you do about contradictions? I believe it is the job of logic to investigate the question of how we think in a more general sense. It should be more than just a science of proof. But how do we arrive at such an extended version of logic?

The decisive step for me was the realisation that there are two types of logic: one static and one dynamic. Only when we dare to leave the safe garden of static logic can we begin to examine real thinking.

Classical logic = logostatics

Classical logic shaped Western intellectual life for more than two millennia – from the syllogisms of Aristotle to the scholasticism of the Middle Ages including the teachings of Thomas Aquinas, to the first order logic (FOL) of mathematicians, which represents the widely accepted state of the art today. These systems of logic are truly static. Every statement within them has a generally valid, absolute truth value; the statement is either true or false – and that must not change. In other words: the logical building is static. Mathematicians call such logic monotonic.

Logodynamics

Although contradictions cannot be tolerated in a classical system of logic, in a dynamic one they make up crucial elements in the network of statements. It’s the same in our own minds, where contradictions are nothing more than starting points for our thinking. Finally, contradictions, e.g. observations that are incompatible with one another, force us to take a closer look. If statements are contradictory, it makes us want to reflect on where the truth lies. Contradictions, forbidden in classical logic, are actually the starting point for thinking in dynamic logic. Just as in physics, where an electric voltage supplies the energy that allows current to flow, in logic a contradiction provides the tension that drives us to carry on thinking.

But continuing to think also means always being open to completely new statements. This is another way that logodynamics differs from classical logic. The classical system first defines its ‘world’, i.e. all the elements that may be used subsequently, or indeed at all. The system must be closed. Classical logic requires a clear demarcation (definition) of the world of a system of statements (both true and false) before any conclusions can be drawn in this closed world of statements. By contrast, our thinking is by no means closed. We can always include new objects, test new differentiations for known objects, find new reasons and re-evaluate existing ones. In other words: we can learn. Therefore, a system of logic that approximates the way people think must always be open.

In a classical system of logic, time does not exist. Everything that is true is always true. The situation is very different in a logodynamic system. What is considered true today may be recognised as an error tomorrow. Without this possibility there is no learning. The logodynamic system recognises time as a necessary and internal element.  This fundamentally changes the logical mechanism, the ‘basic switch’ of logic, namely the IF-THEN. The IF-THEN of dynamic logic always has a time element to it – the IF always comes before the THEN. A static system could, at most, recognise time as an object for consideration, along the lines of one of its variables, but not as something that plays a role in its own functioning.

Thus, a logodynamic system has the following three properties that differentiate it from a logostatic one:

  1. Non-monotony: contradictions in the system are allowed.
  2. Openness: new elements can appear in the system at any time.
  3. System-internal time: time passes between IF and THEN.

(Translation: R. Waddington)

Logic

Non-Monotonic Reasoning (NMR)

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Concept Molecules and NMR

In the article Two types of coding 1, I described the challenge of getting computers to ‘understand’ the incredibly diverse range of medical diagnoses that may crop up in a text. To meet this challenge, the computer has to convert the various diagnostic formulations encountered into a consistent format that represents all the semantic details in an easily retrievable form.

With concept molecules we have succeeded in doing this. We were aided here by two properties of the concept molecules method:
a) the consistently composite representation of semantics, and
b) a non-monotonic reasoner.
At the time, the use of a non-monotonic reasoner was very much out of vogue. Most research groups in the field of medical computational linguistics were in the process of switching from First Order Logic (FOL) to Description Logic (DL), believing that DL is the best way to get computers to interpret complex semantics. As it turned out, however, it was us – a small private research company without state support – that was successful. Instead of the accepted doctrine of FOL and DL based upon a monotonic approach, we used a non-monotonic method.

What is monotonic logic?

In logic, monotony means that the truth of statements does not change even if new contradictory information subsequently crops up. Thus, what has been recognised within the system as true remains true, and what has been recognised as false remains false. Under non-monotony, on the other hand, conclusions drawn by the system can be called into question on the basis of additional information.

So, what’s the problem with non-monotony?

It is clear that proof is only possible in a monotonic system. In a non-monotonic system, on the other hand, there is always the possibility of another argument cropping up that leads to completely different conclusions. Since proof is essential in mathematics, it is obvious that mathematical logic relies on monotony.

Computational linguistics, however, is not about proof, but about the correct assignment of words to concepts. Thus, the advantage of being able to supply proof – as important as it clearly is for mathematics – is irrelevant to our task.

And the problem with monotony?

A system that cannot change its conclusions is not able to learn in any real sense. The human brain, for example, is in no way monotonic.

Moreover, a monotonic system must also be closed, whereas in practice scientific ontologies are not closed, but grow as knowledge progresses. Progress of this kind is also evident in the development of an interpretation program with its complex algorithms:  here, too, there is continuous improvement and expansion that poses problems for monotonic systems.

In addition, monotonic systems are not particularly efficient when it comes to dealing with exceptions. It is well known that there are exceptions to every rule, and a non-monotonic system can handle these in a much more effective and straightforward way.

Non-monotony in practice

If we compare rules-based systems, I believe that non-monotonic systems are clearly preferable to monotonic ones for our purposes. Non-monotony is by no means the easy option and has a few pitfalls and knotty issues of its own, but the ease with which even detailed and complex fields can be modelled decides the issue in its favour.