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Logic

Static and dynamic IF-THEN, Part 2

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(This blog post continues the introduction to the dynamic IF-THEN.)

Several IF-THENs next to each other

Let’s have a look at the following situation:

IF A, THEN B
IF A, THEN C

If a conclusion B and, at the same time, a conclusion C can be drawn from a premise A, then which conclusion is drawn first?

Static and dynamic logic

In terms of classical logic, this does not matter since A, B and C always exist simultaneously in a static system and do not change their truthfulness. Therefore it does not matter whether one or the other conclusion is drawn first.

This is completely different in dynamic logic – i.e. in a real situation. If I opt for B, it may be that I “lose sight” of option C. After all, the statement B is usually related to further other statements, and these options may continue to occupy my processor, which means that the processor does not have any time at all for statement C.

Dealing with contradictions

There is an additional factor: further conclusions drawn from statements B and C lead to further statements D, E, F, etc.  In a static system, all the statements resulting from further valid conclusions must be compatible with each other. This absolute certainty does not exist in the case of real statements. Therefore it cannot be ruled out that, say, statements D and E contradict each other. And in this situation, it does matter whether we reach D or E first.

A dynamic system must be able to deal with this situation. It must be able to actively hold the contradictory statements D and E and “weigh them up against each other”, i.e. analyse their relevance and plausibility while taking their individual contexts into consideration – if you like, this is the normal way of thinking.

In this process, it matters whether I “weigh up” B or C first. Depending on the option I choose, I will end up in a totally different “field” of statements. It is certain that occasionally, statements from the two fields contradict each other. For static logic, this would be tantamount to the collapse of the system. For dynamic logic, however, this is perfectly normal – indeed, a contradiction is the reason for having a closer look at the system of statements from this position. It is the tension that drives the system and keeps the thought process active – until the contradictions are resolved.

Describing this dynamism of thinking is the objective of logodynamics.

Truth – a search process

The fact that the truthfulness of the statements is not determined from the start may be regarded as a weakness of the dynamic system. Then again, this is exactly our own human situation: we do NOT know from the start what is true and what isn’t, and we first have to develop our system of statements. Static logic is unable to tell us how this development works – it is precisely for this that we need a dynamic logic.

Thinking and time

In real thinking, time plays a part. It matters which conclusion is drawn first. Admittedly, this makes things in dynamic logic slightly more difficult. However, if we want to track down the processes in a real situation, we have to accept that real processes always take place in time. We cannot remove time from thinking – nor can we remove it from our logic.

Yet static logic does so. This is why it is only capable of describing one result of thinking, the final point of a process in time. What happens during thinking is the subject of logodynamics.

Determinism – a cherished habit

If I can conclude both B and C from A and if, depending on which conclusion I draw first, the thought process evolves in a different direction, then I will have to face another disagreeable fact: namely that I am unable to derive from the initial situation (i.e. the set of statements A that I accept as being truthful) what direction I will pursue. In other words: my thought process is not determined – at any rate not by the set of what I have already recognised.

On the one hand, this is regrettable, for I can never be quite sure whether I draw the right conclusions since I simply have too many options. On the other hand, this also provides me with freedom. At the moment when I must decide to follow the path through B or C first – and that without already seeing through the system as a whole, i.e. in my real situation – at this moment I also gain the freedom to make the decision myself.

Freedom – there is no certainty

Logodynamics thus explores the thought process for systems which still have to find truth. These systems, too, are not in a position to examine an unlimited number of conclusions at the same time. This is the real situation. This means that these systems have a certain arbitrariness in that they are capable of making decisions at their own discretion. The thought structures that emerge in the process, the advantages and disadvantages they have and what may be taken for granted, is explored by logodynamics.

It is clear that derivability cannot be taken for granted. This is regrettable, and we would prefer to be on the safe side. Yet it is only this uncertainty that enables us to think freely.

The dynamic IF-THEN is necessary

In terms of practical thinking, the point is that static logic is not equal to the process of finding truth. Static logic merely describes the found consistent system. The preceding discussion of whether the system resolves the contradictions and how it does so, will only become apparent through a logodynamic description.

In other words: static logic is incomplete. To examine the real thought process, the somewhat trickier dynamic logic is indispensable. It deprives us of certainty but provides us with a more realistic tool.

This is a blog post about dynamic logic. The preceding post made a distinction between dynamic and static logic IF-THEN.

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)

Information, Logic

Is ‘IF-THEN’ static or dynamic?

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IF-THEN and Time

It’s a commonly held belief that there’s nothing complicated about the idea of IF-THEN from the field of logic. However, I believe this overlooks the fact that there are actually two variants of IF-THEN that differ depending on whether the IF-THEN in question possesses an internal time element.

Dynamic (real) IF-THEN

For many of us, it’s self-evident that the IF-THEN is dynamic and has a significant time element. Before we can get to our conclusion – the THEN – we closely examine the IF – the condition that permits the conclusion. In other words, the condition is considered FIRST, and only THEN is the conclusion reached.

This is the case not only in human thinking, but also in computer programs. Computers allow lengthy and complex conditions (IFs) to be checked. These must be read from the computer’s memory by its processor. It may be necessary to perform even smaller calculations contained in the IF statements and then compare the results of the calculations with the set IF conditions. These queries naturally take time. Even though the computer may be very fast and the time needed to check the IF minimal, it is still measurable. Only AFTER checking can the conclusion formulated in the computer language – the THEN – be executed.

In human thinking, as in the execution of a computer program, the IF and the THEN are clearly separated in time. This should come as no surprise, because both the sequence of the computer program and human thinking are real processes that take place in the real, physical world, and all real-world processes take time.

Static (ideal) IF-THEN

It may, however, surprise you to learn that in classic mathematical logic the IF-THEN takes no time at all. The IF and the THEN exist simultaneously. If the IF is true, the THEN is automatically and immediately also true. Actually, even speaking of a before and an after is incorrect, since statements in classical mathematical logic always take place outside of time. If a statement is true, it is always true, and if it is false, it is always false (= monotony, see previous posts).

The mathematical IF-THEN is often explained using Venn diagrams (set diagrams). In these visualisations, the IF may, for example, be represented by a set that is a subset of the THEN set. For mathematicians, IF-THEN is a relation that can be derived entirely from set theory. It’s a question of the (unchangeable) states of true or false rather than of processes, such as thinking in a human brain or the execution of a computer program.

Thus, we can distinguish between
  • Static IF-THEN:
    In ideal situations, i.e. in mathematics and in classical mathematical logic.
  • Dynamic IF-THEN:
    In real situations, i.e. in real computer programs and in the human brain.
Dynamic logic uses the dynamic IF-THEN          

If we are looking for a logic that corresponds to human thinking, we must not limit ourselves to the ideal, i.e. static, IF-THEN. The dynamic IF-THEN is a better match for the normal thought process. This dynamic logic that I am arguing for takes account of time and needs the natural – i.e. the real and dynamic – IF-THEN.

If time is a factor and the world may be a slightly different place after the first conclusion has been drawn, it matters which conclusion is drawn first. Unless you allow two processes to run simultaneously, you cannot draw both conclusions at the same time. And even if you do, the two parallel processes can influence each other, complicating the matter still further. For this reason along with many others, dynamic logic is much more complex than the static variant. This increases our need for a clear formalism to help us deal with this complexity.

Static and dynamic IF-THEN side by side

The two types of IF-THEN are not mutually exclusive; they complement each other and can coexist. The classic, static IF-THEN describes logical states that are self-contained, whereas the dynamic variant describes logical processes that lead from one logical state to another.

This interaction between statics and dynamics is comparable with the situation in physics, where we find statics and dynamics in mechanics, and electrostatics and electrodynamics in the study of electricity. In these fields, too, the static part describes the states (without time) and the dynamic part the change of states (with time).

This is a blog post about dynamic logic. The next post specifies the topic of the dynamic IF-THEN.