Module A Supplement

Here are some general comments of mine on the exercises you handed in on Friday.  Review how I discuss these exercises and try to take advantage of it in your self-evaluation.

Unit A3 Exercise 2 (Drug Trafficking)

In this exercise, you were asked to represent a hypothetical drug trafficking scenario as a mathematical structure as a combination of sets, relations, and functions.  Thus, you should be

A very simple hypothetical scenario might be as follows: Drug producers in some country sell drug to traffickers. The traffickers around the world sell drug to dealers in certain target countries.  The dealers sell drug to users.
Note:  It was very good that you came up with all sorts of realistic and/or revealing scenarios.  More complex scenario will end up with more complex representations.  Here, I use a simple one to illustrate how you can represent it in a way we discussed in class.
In this scenario, the key players include producers, traffickers, dealers, and users.  They are all people.  So, we can consider a set of people.  The essential connection between these people is the act of selling.  So, we can consider a relation "sells", i.e., "___ sells (drug) to ___".

Then, we have a mathematical structure, say, DrugTrafficking, consisting of the following components:
A relation connects two objects, in this case two people. As shown above, if one individual is involved in two different people, you will need to show both instances.  You will also need to analyze whether this relation is a function (you must be able to analyze this by observing that producer1 is involved with two different people).

Structure DrugTrafficking is different from several structures discussed in class.  For example, it is different from String, PrimitiveCounting, and Max because the structure patterns are different (the latter structures involve a function, not a relation).  It is different from BarbieDoll because the latter involves multiple relations and also a function.  However, it is similar to a few structures, e.g., ObjectsInRoom, NorthPole, Professional, Map.  All of these involve a set and a relation (which cannot be in general a function).  In fact, this kind of structure is called Graph, as we discussed in class.  Although I did not clarify in class, DrugTrafficking is in general not a Tree (check the properties shown on slide) because it is quite possible that there are disconnected groups of people (e.g., a drug trafficking links from South America to the US and another entirely separate one locally taking place in a remote community near the Arctic).

Unit A4 Summary Exercise (about why logic can convey precise meaning)

While we can discuss a variety of aspects, I will focus on a few essential ones.  First, primitive statements (property and equality) can convey precise meaning because there is a precise way of interpreting whether an individual has a property (by checking set membership) and whether an equality holds (by checking the identities of the referent).  Next, complex statements can convey precise meaning because there are several ways of combining component statements (primitive or complex), which are syntactically precisely defined. Each of building a complex statement has a precise rule to interpret the complex statement out of its component statements.  For example, the truth value of a complex statement that involves "and" depends on the truth values of both component statements.

In certain cases, the truth values of a statement depends on the context.  In practice, it is very important that we be able to analyze a variety of contexts to interpret logical statements.  For this reason, we tackle a somewhat challenging exercise (Module A Comprehensive Exercise 2).  When we discuss two standard versions of logic (in Module B), the semantics of a statement must be interpreted independent of the context.

This summary exercise appears to be more challenging.  Although we discussed all the points described above, not many students were able to point out the essential points.  In this course, we try to prepare ourselves for the future when we will surely face all sorts of uncertainty.  So, we will do exercises that are not repeating what are shown or said earlier.  We need to be able to analyze (into principles) and synthesize (from principles) depending on the situation.