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.

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:

.................r.............¬r

_____________________

u.............-10..........-10

¬u..........-100...........0

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:

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

Valete!