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IFN521: Foundations of Decision Science
The purpose of this document is to explain some essential points of event-propositions in the context of
Shannon’s and Dretske’s theories of information. It is intended to provide a complete description of all
relevant aspects. The document should be used in conjunction with the other learning materials. The
essential points will be introduced based on the following example scenario.
Example:
“No Swimming” sign at a popular swimming spot. The sign appears to be quite old, and some of the
paint is missing from it. The swimming spot is in an area known by the locals to occasionally have
crocodiles at certain times of the year. There are other people swimming in the creek. There’s an
odd rustling sound coming from the banks of the creek.
Person A: a tourist
Person B: a local person
Goal: Decide whether to swim, or not.
What is an event-proposition?
Please keep in mind the following important characteristics for event-propositions:
• It is a proposition. This means that it is a statement that is either true or false, e.g., “It is safe
to swim” is a statement that is either true or false, but Person A or B might not be able to
establish the truth of the event-proposition. Therefore, they assign subjective probabilities
to its truth. Shannon and Dretske’s theories use such probabilities in their respective
definitions of information.
• It is specific. For example, “It is safe” is a true/false statement but is not specific, because it
does not express what the safety relates to. The scenario is about swimming, therefore “It is
safe to swim” is a more specific event-proposition which is more relevant to the goal of the
scenario.
• It is atomic. The event-proposition is concise, which means there are no unnecessary extra
details, for example, “It is safe to swim, because other people are swimming”. The relative
clause “because other people are swimming” is an extra unnecessary detail that does not
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need to be included in the event-proposition (but it can be included in your justifications.
See below).
• It is relevant. It expresses an important, essential aspect. For example, “There is water in the
creek” is an event-proposition as it is a true/false statement that is atomic and specific.
However, it expresses an unimportant aspect of the scenario that has little relevance to the
goal of the scenario.
How to extract an event-proposition from a scenario.
It is helpful to follow some steps.
The first step is to list facts. Facts are true statements that are given in the scenario description, for
example,
• There is an odd rustling sound.
• There are people swimming.
• The sign is partially readable.
The second step is to consider the goal.
Both Person A and Person B must decide whether to swim or not (the goal). Imagine that you are
either of these people. What do I need to know in order to make the decision? Imagine that I am
the tourist (Person A). It is hot and I want to go for a swim. I can’t read the English instructions on
the sign and the surroundings are very different to my home country. Therefore, I need to know
whether it is safe to swim.
Person B is familiar with the surroundings and knows about the danger of crocodiles. Therefore,
if I am such a person, I need to know if there are crocodiles around.
The third step is to specify the event-propositions:
• “It is safe to swim”
• “There are crocodiles around”
How to assign probabilities to event-propositions.
Subjective probabilities are assigned to event propositions. How probabilities are assigned depends on
the characteristics of the person and the facts. The way the probabilities are written depends on the
theory (Shannon or Dretske). Here are some examples.
Shannon: Person A
P(“It is safe to swim” | “there are people swimming”) = 0.9
Justification: Person A is a tourist and doesn’t know the surroundings and can’t read the sign. Because
other people are swimming, he/she thinks that it is probably safe to swim as well. For this reason, the
probability of the event-proposition “It is safe to swim” is high.
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The preceding example uses a conditional probability. The statement after the “|” is the evidence.
Evidence is provided by facts. Evidence influences the probability of the event-proposition.
P(“There are crocodiles around” | “there is an odd rustling sound”) = 0.01
Justification: Person A is a tourist and can’t read English. He/she sees a crocodile icon on the sign but can’t
be informed about the danger. Therefore, he/she can’t connect the rustling sound to the possibility of a
crocodile. For this reason, the probability of the event-proposition “There are crocodiles around” is very
low.
N.B. It is not necessary to use conditional probabilities, which use facts as evidence. For example, it is also
possible to do the following:
P(“It is safe to swim”) = 0.9
Justification: Person A is a tourist and doesn’t know the surroundings and can’t read the sign. Because
other people are swimming, he/she thinks that it is probably safe to swim as well. For this reason, the
probability of the event-proposition “It is safe to swim” is high.
Note how the fact “there are people swimming” is used in the justification for the high probability.
Shannon: Person B
P(“It is safe to swim” | “there are people swimming”) = 0.3
Justification: Person B is a local person and knows about the danger of crocodiles. Even though he/she
sees other people are swimming, he/she knows that this is not necessarily a sign that it is safe to swim,
e.g., tourists often swim without knowing the danger. For this reason, the probability of the eventproposition “It is safe to swim” is quite low.
P(“There are crocodiles around” | “there is an odd rustling sound”) = 1
Justification: Person B is a local and is familiar with the behaviour of crocodiles and knows that they can
quietly try to sneak up on their prey. Therefore, the rustling sound is taken as a sign of an approaching
crocodile. Therefore, they judge probability of the event-proposition “There are crocodiles around” is
certain because there is a lot of fear involved.
Dretske’s theory is based on a signal r and background knowledge k. The signal r is provided by the
situation. The knowledge k differs depending on the person: kA is used to denote the knowledge of
Person A and kB is used to denote the knowledge of Person B. The way probabilities are written is
different than used by Shannon.
All facts are digitized because they are true, and therefore have a probability equal to 1 for both Persons.
For example, Person A and B digitize the information content that people are swimming. This is added to
his or her knowledge (denoted kA* or kb*)
P(“There are people swimming” | r, kA) = 1
P(“There are people swimming” | r, kB) = 1
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Dretske: Person A
P(“It is safe to swim” | r,kA*) = 0.9
Justification: Because he/she knows other people are swimming, he/she thinks that it is probably safe to
swim as well. For this reason, the probability of the event-proposition “It is safe to swim” is high. The
probability is less than one, so the event-proposition is not digitized as information content, so person A
does not know if it is safe to swim.
As an exercise you can try is to describe the cases using Drestke’s theory and supply appropriate
justifications.
After you have extracted the relevant event-propositions and applied the probabilities
(with justification), you can then calculate Shannon’s information value, and describe the
digitisation of the information based on Dretske’s theory.