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[ΛΟΓΟΣ] De Principiis Decernendi

PostPosted: Sat Jul 15, 2017 1:20 am
by Gaius Florius Lupus
Salvete, amici!

After having seen the rules of logic it is now time to put them to use in our daily life. One of the questions mentioned before, was whether we should carry an umbrella or not. This is a quite common question that we face almost every morning before leaving the house.

Let us see where propositional logic leads us.

We define the following propositions:
r: It is raining.
u: I carry an umbrella.
w: I get wet.

We can write the problem as a simple Modus Tollens:
[(r ∧ ¬u) ⇒ w] ∧ ¬w ⇒ ¬(r ∧ ¬u)
In words: IF it rains AND I do NOT carry an umbrella, THEN I get wet AND I should NOT get wet. THEREFORE it should NOT be the case that it rains AND I do NOT carry an umbrella.

Using the 1st De Morgan Theorem on the right side of the formula, the side that tells us the logical conclusion, we get:
¬(r ∧ ¬u) ≡ ¬r ∨ ¬(¬u) ≡ ¬r ∨ u
This tells us that we can either carry an umbrella and be always on the safe side or hope for ¬r (that it does NOT rain).

Now we do not want an umbrella, if it is not absolutely necessary, therefore we add the condition ¬(¬r ∧ u)
Applying the 1st De Morgan Theorem we can write it as r ∨ ¬u
Adding this condition we get:

(¬r ∨ u) ∧ (r ∨ ¬u)

We remember the definition of an implication (p ⇒ q) ≡ ¬p ∨ q

Therefore we can write this as two implications:
(r ⇒ u) ∧ (¬r ⇒ ¬u)

In natural language this means:
Given the premises that
1. when it rains and we do not carry an umbrella, we will get wet and
2. that we should not get wet as well as
3. that we should not carry an umbrella, if it does not rain
we conclude that
if it rains we should carry an umbrella and if it does not rain, we should not carry an umbrella

This is quite easy to understand, and we would not have needed symbolic logic to find that out.
However it does not give us a clear answer, because the decision depends on the unknown variable r. Depending on whether r or ¬r is true (whether it rains or not), logic demands either u or ¬u (to carry an umbrella or not to carry one).
We now have to make a decision under uncertainty.
Unfortunately this is a quite common situation in our daily life. In fact most decisions in life have to be made without full knowledge of all relevant factors.
But logic has a way to deal with this problem. This is by using a decision matrix.

Decision Matrix

The mutually exclusive alternatives are named A1 and A2.
The future state of a particular variable is uncertain and either S1 or S2.
The payoff is indicated by a numeric value in each cell depending on the decision and the unknown future state. Important: This numerical value must be proportional to the total benefit for the decision maker, and the options to consider must be broken down into mutually exclusive alternatives if necessary.

matrix.jpg (6.59 KiB) Viewed 1328 times

Different decision strategies have been suggested:

Maximax Gain
Choose the alternative that allows the largest maximum possible gain. This is alternative A1 in this case, hoping for the payoff of +100.

Maximin Gain
Choose the alternative that allows the largest minimum possible gain.
A2, because the minimum possible gain is -200.

Minimin Loss
Choose the alternative that allows the smallest minimum possible loss.
A2, because the loss of -200 is less than the loss of -250.

Minimax Loss
Choose the alternative that allows the smallest maximum possible loss.
A2, because the worst that can happen is a loss of -200.

Hurwicz' Rule
Choose the alternative that has the maximum optimism-weighted value.
If you are 60% sure of an optimistic outcome:
A1 = 0.6 × (+100) + 0.4 × (-250) = -40
A2 = 0.6 × (+50) + 0.4 × (-200) = -50
So choose A1.

None of these rules are reasonable. (So best you forget them right away!) They all take the personal attitude (optimistic or pessimistic) of the decision maker into account, even when there is no objective reason to assume a correlation between the decision and the probability of the uncertain variable. The probabilities of S1 and S2 are independent from the personal attitude of the decision maker. And there is no rational justification for taking a disproportionate risk hoping for a favorable outcome or being overcautious fearing an unfavorable outcome.
The first four rules could only be justified, if there is a secondary non-numerical criterion involved that actually supersedes the importance of the numerical values, for example a certain limit that needs to be surpassed, while all numerical values above this limit or all below equally satisfy or do not satisfy the secondary criterion. But in this case an inappropriate payoff matrix would have been used, since the numbers would not reflect the total payoff because they would not be proportional to the secondary criterion.

The rational decision strategy is described by the following two rules:

Laplace Utility Rule
Choose the alternative that has the maximum Laplace utility.
Consider each outcome is equally likely.
A1 = (+100-250) / 2 = -75
A2 = (-200+50) / 2 = -75
It is a tie. There is no optimal decision.

Expected Utility Rule
Choose the alternative that has the maximum expected utility.
If S1 will occur with a probability of 60%:
A1 = 0.6 × (+100) + 0.4 × (-250) = -40.
A2 = 0.6 × (-200) + 0.4 × (+50) = -100
So chose A1.

The Laplace Utility Rule is a special case of the Expected Utility Rule, when the probabilities of S1 and S2 are assumed to be equal. This assumption is reasonable, when no further information is available that would give a higher probability to either outcome (principle of insufficient reason).
An analysis of the situation according to the Laplace Utiliity Rule should always be the first step before attempting to estimate probabilities of S1 and S2, which might often be imprecise or speculative.

If there are justified reasons to assume that the probabilities of S1 and S2 are not equal, then the Expected Utility Rule will provide the optimal decision in a situation.
If the decision has a feedback effect on the probability of S1 and S2, different probabilities have to be used in either row of the payoff matrix. This means for A1 the probability of S1 may be higher or lower than for A2, if the decision has an effect on the uncertain variable.

Let us now use this method to answer the question about the umbrella!
We have first to quantify the benefit of each possible outcome.
Let us assume the burden of carrying the weight of an umbrella to be -10 and the discomfort of getting wet be -100.
Then we get the following decision matrix:


If we have no reason to assume a higher probability for rain than for good weather (principle of insufficient reason), then we have to assume equal chances for each option (Laplace Utility Rule).
For u we get an average utility of (-10-10)/2 = -10
For ¬u we get an average utility of -100/2 = -50
In this case the decision is clear. The Laplace Utility Rule tells us to carry an umbrella.

Now let us assume we have good reasons to assume a low probability for rain. We can use Google to find this out:
Enter the search terms "weather Rome" we get the following result:

weather rome.jpg
weather rome.jpg (24.06 KiB) Viewed 1328 times

We can see a general probability for rain of 2% for today, but during the next hour an elevated probability of 5%.

Now let us use the Expected Utility Rule and enter the probabilities found in Google:
u: 0.05x-10 + 0.95x-10 = -10
¬u: 0.05x-100 + 0.95x0 = -5
This tells us that based on the low probability for rain according to Google the optimal decision is not carrying an umbrella.

As we can see, logic provides always one optimal decision even in everyday situations under uncertainty.
There is no justification for "gut feelings" or "opinions". Logic tells us exactly what to do. And its methods are universally valid and agreeable and reproducible for all rational beings.
There is no need for disagreement and quarrels, if we simply follow logic.

Keywords: uncertainty; decision matrix; Laplace Utility Rule; Expected Utility Rule; principle of insufficient reason


Re: [ΛΟΓΟΣ] De Principiis Decernendi

PostPosted: Sat Jul 29, 2017 6:32 pm
by Gaius Florius Lupus
Salvete, amici!

I hope I was able to show that there are precise and clear rules how to make in every situation always the optimal logical decision. However we also know from our daily experience that human beings often make irrational decisions and even refuse to listen to logical arguments.
How is this possible?

It is clear that we can not draw a decision matrix every time we have to make a decision in life.
Higher organisms like animals permanently have to make innumerable decisions in life: when to breathe in and when to breathe out, how to move the legs in order to keep the balance, how to move the hands and arms or the mouth in order to grab an object.
Most decisions in life are apparently made subconsciously (e.g. walking or other simple motoric functions). But controversial decisions are made consciously. It is only the latter type which is relevant here, because all the other decisions are taken care of by our instincts and conditioning.

So how do conscious decisions work?
The decision-making process begins with the occurrence of a situation that requires a conscious choice among several feasible options. It is a situation, which is too complex for a simple subconscious reaction based on instinct or conditioning of the nervous system. The conscious mind is therefore alerted and made aware that its intervention is required.
Ideally the conscious mind would now begin a rational analysis of the situation. It would take all available data into consideration without any bias. It would calculate the probabilities of the resulting scenarios after every possible choice. Then it would evaluate the benefit of these possible outcomes.
Finally the option with the most beneficial outcome and a reasonable probability would be selected.
The individual would then act according to the decision that has been consciously made.
This would be the ideal process of decision-making that leads to an optimal decision.

Ideal decision-making process:
  1. Situation requiring a conscious choice among several options
  2. Rational analysis of all options
  3. Selecting the option with the most beneficial outcome and a reasonable probability
  4. Acting according to the decision

However the decision-making process in human beings does not follow this ideal pattern, since humans are subject to emotional bias. It is therefore commonly assumed that this emotional bias can play a significant role when making conscious decisions and in some cases can lead to a non-optimal decision. This means the result of the decision is either less beneficial or even harmful (often in the long term) or the desired beneficial outcome is simply unlikely and based on wishful thinking.
We are all aware that a high level of emotional stress can for example lead to aggressive behavior that can result in serious harm for oneself. Although a violent reaction can bring immediate relief from an emotional stress situation like a perceived threat to the social status, in most cases it endangers the physical integrity of the person reacting violently. And even if this is not the case, there is a high risk of later reprisal or social consequences like conflict with laws or the institutions that enforce them.
Considering these obvious influences of emotional and other irrational motivations on the human decision-making process, it is therefore commonly assumed that in real life the decision-making process in human beings follows a slightly different pattern.

We start again with a situation that requires a conscious choice among several apparently viable options of reacting.
Similar to the ideal pattern of decision-making it is assumed that the human mind makes a rational assessment of the feasible options and the most beneficial result with a reasonable probability based on the available data. Additionally to this rational analysis it is believed that emotional bias and personal inclinations are taken into consideration. This is believed to be the cause of sub-optimal decisions. Emotional reasons or instinctive impulses are given priority over rational analysis.
The resulting decision can therefore be based either on the application of reason and logic or on emotional bias. The human mind is believed to be given the freedom to choose between these two options, either acting according to reason or acting according to his feelings.
Later when the human is questioned about his decision, he is expected to be able to explain the reasons for his decisions, either by justifying them by reproducing the logical analysis or by admitting his emotional inclinations. In cases that the latter ones might be embarrassing for him, he might attempt to cover them up and provide false rational explanations. However the human is considered to be mostly aware of the real motives for his actions and the false nature of his explanation.
In some cases however it is believed that it might be difficult for the human being to identify the real emotional motives behind his actions. This is thought to cause a certain degree of confusion and emotional discomfort due to the inconsistencies of his personality. In a mentally healthy person this is not believed to be a major problem. But if this happens to a major extent, so that somebody is not able to understand his own decisions and actions in the aftermath, it is believed to be an indicator of a psychotic or neurotic personality or some other kind of mental disorder.

Commonly assumed human decision-making process:
  1. Situation requiring a conscious choice among several options
  2. Taking into consideration rational analysis and emotional bias or instinct
  3. Decision is made either based on reason or on emotions or instinct
  4. Acting according to the decision
  5. If questioned afterwards, the rational or emotional motives for the decisions are explained.

However this model is still too optimistic about the true nature of the human decision-making process. In fact rational analysis plays no role at all when humans make a decision. Reason and logic only get involved AFTER the actual decision has already been made. And the decision has always been made based on totally different motives that have nothing to do with a rational analysis of the situation.

The decision-making process starts with the situation that requires a conscious choice among several viable options.
The human mind then makes a decision ENTIRELY based on emotional bias, instinct or conditioning.
Only AFTER the decision is already made, the human mind uses reason and logic to make up arguments to justify his decision in case he will be questioned later.
Then he acts according to his instinctive or emotional decision.
The order of the last two steps can be reversed. The human might only make up rational arguments for his decision after he had already acted or he does not make any attempt to produce a justification for his decision at all. Reason and logic are unnecessary for the whole process.

Real human decision-making process:
  1. Situation requiring a conscious choice among several options
  2. Decision is made based on instinct or emotional bias
  3. Fabricating rational arguments to justify the decision in case of being questioned later
  4. Acting according to instinctive or emotional decision

It is interesting to compare this decision-making process in humans with the analog process in other animals.
In this case the animal faces a situation that requires a conscious choice among several possible options.
The animal is unable to conduct any advanced rational analysis of the situation and therefore makes its decision solely based on its instinct or its emotions.
It then acts according to its instinctive decision.
It is later unable to justify its decision and soon forgets about it.

Decision-making process in animals:
  1. Situation requiring a conscious choice among several options
  2. Decision is made based on instinct or emotional bias
  3. Acting according to instinctive decision
  4. Unable to justify the decision

There is not much of a difference in the decision-making process of humans and animals. It is essentially the same, because there is no fundamental difference between humans and other animals. Humans and animals are both biological systems and have developed according to the same principles of biological evolution. They function in the same way. And their decision-making process follows the same pattern. They share the same biological matrix, which is responsible for the way their system functions. Their organisms use hormones and other molecules that serve as messengers between cells and stimulate or dampen the neuronal activity. These hormones are in most cases responsible for the emotions and inclinations that are the determining factor of conscious decisions. They are the biochemical background of the models that we have developed.
The capability of humans for advanced rational analysis is not used in determining the actions of human beings. It only has a subordinate function that is not involved in human behavior but only in the process of justification, either towards other humans or towards oneself.
The only instance where reason and logic takes part in the human decision making process is after the decision has already been made in order to justify it fabricating reasons that have never had anything to do with the actual decision. The decision-making process itself is identical in humans and animals.

There is one exception in this pattern though. Human beings show a high ability to act logically and reasonable in their assigned profession. The highly evolved level of technology and civilization would not be possible without this ability. This can only be explained by a distinctly different process of decision-making depending on whether a human being is in a professional environment or making decisions on a personal level.
The difference is his emotional involvement with the decision. When making decisions on a personal level, he is emotionally affected by the decision. The outcome of the decision matters for him personally. He is not indifferent towards the decision. In a professional environment this is not the case. Many professions are highly specialized, so that the worker is totally alienated from his work. He is performing his job mechanically without any emotional engagement. It does not matter for him what particular decision he makes, since any outcome is not important for him on a personal level. This is just as true for simple mechanical work as for complex calculations of an engineer. Whatever might be the result of an engineer's calculation, it does not matter for him personally. This is why he is able to have his professional decisions based solely on reason and logic.

The decision-making process in a professional environment is therefore as following:
We have a situation that requires a conscious choice among several viable options.
Emotional bias and instinct are unable to develop any preference for a particular decision.
By rational calculation a human is able to provide logical justifications for a particular decision while not so for others. At the same time he is aware that not being able to justify his decisions could have negative consequences for him.
He therefore decides in favor of the rational solution, because it makes the justification of the decision easier.
Then he acts according to his decision, which has a fully rational justification.

Decision-making process in the state of emotional equanimity:
  1. Situation requiring a choice between several options
  2. Decision cannot be made based on instinct or emotional bias due to total indifference regarding the subject of the decision
  3. Decision is made based on rational analysis in order to make its justification easier
  4. Acting according to the decision

The human decision-making process cannot be changed. It is part of the way the human brain operates. But if step 2 is skipped due to indifference, the brain is forced to make the decision according to reason and logic in step 3.

For rational decisions, emotional indifference is indispensable.
In order to make optimal rational decisions in daily life, we have to train emotional indifference.
This is what the Stoics called apatheia, a state of mind without emotions and passions. It is a requirement for rational thinking and acting.

The Logos as Method of Persuasion

Since humans that are not conditioned for emotional indifference make their decisions and arrive at their opinions based on their emotional bias and their instincts and totally unaffected by logic and reason, it is pointless to attempt to change their opinion by logical reasoning (logos). Even if we succeed in pointing out the fallacies of their reasoning, this would have no effect on their preformed opinions and would only require them to adjust their reasoning in order to arrive at the same decision or opinion as before.
The effort done by attempting to discuss a certain subject rationally with irrational persons creates the false illusion of a logical analysis of the problem. It is delusional to think that different rational arguments that appear contradicting were seriously considered in an attempt to eliminate false or weak arguments and to achieve a better solution, which would be accepted by everybody involved in the discussion. It is a waste of time and effort, has no possibility to contribute anything useful to the rational analysis of a problem and should therefore be avoided. It can even destabilize the emotional indifference towards the subject by mistaking the discussion as a personal challenge or mistaking it as an indication of a fallacy in our own logical reasoning and causing the urge to verify and reaffirm it. Although verifying and reaffirming own conclusions are no mistake per se, doing so driven by the urge to defend our personal prestige is.

If it is necessary to change the decision or opinions of irrational persons, it is usually more efficient to do this by changing their emotional state or invoking different instincts (pathos as method of persuasion). Logical arguments will have no effect. It can be achieved by creating either positive or negative emotional motivation (incentives or threats - argumentum ad baculum) or appealing to fear or sympathy.

Another method of persuasion is ethos. This is done by seeking moral high-ground and pointing out the own virtues, while at the same time attacking the opponent personally (argumentum ad hominem) to undermine his reputation. Nothing of this invalidates his arguments, but the audience is less likely to listen to them.

A rational discussion of a subject is only useful in a professional environment, where the participants of the discussion are indifferent towards the outcome, or where the participants have learned to suppress their emotional bias in order to reach an indifferent state of mind.

If we are trained in logic and the identification of fallacies, we can notice when these strategies are used on us. We should then immediately discard the fallacious argument brought forth and only take the valid logical arguments into consideration.

The most important issue is equanimity. If we want to be able to act rational, then we have to train ourselves to be able to maintain equanimity just as the Stoics did. This is the essential and most difficult part.
All that has been said in earlier posts, all the logical inference rules and formulae were easy. We can take our time to use them properly and we can get help from others when we make formal mistakes.
What is difficult is achieving the state of equanimity, because it goes against our biological nature.
But as Homines sapientes we are supposed to be "wise men", not driven by instincts and biology like animals. This is what the Stoics assumed to be the human nature - being rational. And living according to our nature as humans means always being rational in our decisions and never being influenced by emotional bias.

This is what we should focus on. And this is what all future threads about the topic of logic will focus on.
We have talked about the tools of logic. Now we have to learn how to use them without distraction. We will then be able to cooperate efficiently with others, we will have fewer conflicts and we will always make optimal decisions.

Optime valete!
C. Florius Lupus