Thread: A Brief Introduction to Formal Logic and Set Theory

I have spent some time writing responses to posts in here and I think the most cordial thing I can do is describe the type of logic I use.

Definition 1 A preposition is a statement that identifies an instance as a member of a class. For example, Sam is a violinist is a preposition that identifies or claims Sam to belong to the class of violinists. In set theoretic terms a preposition is of the form "A is an element of B" or it may also be saying that "A is a subset of B"

The amazing thing about a preposition is that it must be either true or false and so can form the basis of a complete logical discussion with only a little more added machinery. If someone sets forth a preposition, "dogs are black". We can disprove the satement by noting only one dog that is not black. Or we can prove the statement by examining every dog and varifying that they are all black. The first is the easier approach and is much preferrred as a method of debunking.

Definition 2- A qualifying statement is a statement or adjective that determines a subset. For instance "black dogs" uses the term black to qualify the term dogs, we then know to think of dogs that are black. So that the statement "Black dogs are mean." is meant to imply that the subset of all dogs, namely black dogs all share the property of being mean.

In examining more closely the statement "Black dogs are mean." it may become important to split hairs and ask ourselves what do we mean when we say the words, black, dog, and mean. So far in our discussion words like is and are are given a formal meaning, these words infer set inclusion and allow us to construct prepositions. In any given argument the precise nature of definitions of words may become important. For instance mean may be defined as one menacing growl toward a human by one researcher and as a pattern of distemper towards humans by another. So inorder to determine whether an argument is sound we need not only to examine the logical structure but the consistency of definitions.

The next step is to start combining prepostions into phrases. One very useful phrase is the implication.

Definition 3 an implication is a combination of two prepositions P1 and P2 with the words If and Then to form a true statement of the form

IF P1 Then P2 or more graphically P1->P2

Examples:

If John is over six feet then he will not enjoy driving in a Yugo

The truth of an implication is determined by the four possibilities of truth that come from its propositions.

For example if both are true the stament is true, that is if John is over six feet tall and people over six feet tall universally do not enjoy riding in Yugos then the whole statement is true. If however not very person over six feet dislikes the tight space of a yugo then John may enjoy it and we cannot verify the statement as true. Suppose John is not over six feet tall Then the statement remains true because it is predicated by if. So here is the truth table for implications based upon the truth or falsehood of the component prepositions.

1) If True Then True - True
2) If False Then True - True
3) If True Then False - False
4) If False Then False - True*

Each of these has it sublties but I have starred the last one for a reason. How can a statement constructed from two non-truths be true? This is a logical convention a choice that is made that is useful for consistency at higher levels in fact the statement is generally undecidable. Numbers 1-3 are the important cases when examining the logical consistency of someones argument, most sober people will not string together two false statements and claim that it reveals truth.

There is much more to the study of formal logic. Anyone interested can get started at the following link.

http://www.math.csusb.edu/notes/logic/logn...gnot/node1.html

Of course the best way to form a solid argument is using number one
above. The stringing together of true statements is called a direct argument and essentially is a chain of truth leading from A to B, A implies B implies C implies D. Since we have confirmed that A, B, C and D are true it must be that B implies D is true. Knowledge of the consequences of B implies D help millions of people make wise decisions every day.

There are other forms of argument that are useful, one of them is contradiction. A solid contradiction argument tries to put together a series of implications that lead to a contradiction thus implying that at least one of the implications in the argument was false because valid arguments cannot lead to false conclusions.

Proof by contrapositive tries to show A implies B by showing that not B implies not A. This relies upon the fact that not B will be a subset of not A if and only if A is a subset of B. A Venn diagram illustrates this well.

More generally one can use a whole set of logical operators and phrases to construct a variety of valid arguments. The close analysis of the logical operators and phrases used in an argument can also reveal flaws.

Basically there are only two logical arguments that people commonly use: direct proof and contradiction. The errors in logic people make usually come from false prepositions (we assume too much). Although arguments that are built upon assumptions are often set forth as true they are not valid if any of the assumptions prove to be wrong.

A method of pseudo-logical investigation is to string together assumptions until one arrives at a statement that is obviously true. This discovery of truth lends credence to the assumptions made along the way but does not confirm them. Lawyers often argue this way and have convinced many jurors of guilt and innocence using this technique although it is really more like an argument about the chance that something will occur. I will talk about this next.

An extension of prepositional logic is fuzzy logic in which the truth of prepostions is based upon probability. A fuzzy argument works like this.

There is a 90% chance that John is over six feet. There is also a 60% chance that a person over six feet will not enjoy driving a Yugo. The probability that the statement, If John is over six feet Then he will not enjoy driving Yugo applies to John is .90*.60 =.54 or 54%. It is based upon the familiar multiplication rule for probabilities of independent events. So that persons familiar with the statistics related to the component prepositions can make an estimate of the likelihood of truth of an implication. A savvy saleperson would likely decide not to try to sell John a Yugo if they believe there is only a 46% chance that he will be happy with it. As a sales strategy if 90% of your customers are over six feet tall then pushing Yugos will probably not lead to record growth. this highlights the old phrase "Know your market."

In a sort of ad hoc way fuzzy logic is quite close to how most humans navigate through their daily lives and make plans for the future. We estimate the chances of certain things happening and then in some crude way make a decision upon how we feel the scales are balanced in line with our desires. If I want to go swimming at the beach because it is 98 degrees, I may risk swimming at a beach that has had E-coli problems in the past. How far in the past may affect my decision, how much I know about E-coli may affect my decision. Regardless of the factors I am comparing risk and potential benefits to arrive at a conclusion. If I do so using valid logic I sell more Yugos or I switch to selling Mustangs which have a much larger market share.

All feedback is desired and necessary. Remember to check out the link above if you want to learn more.

Enjoy!

EJ