How definitions can help you in your daily life

Back in college I took a class called Abstract Mathematics where i learned things that would stick with me for years to come. One of these things were my professor’s emphasis on definitions. He said before we can start understanding how to perform a mathematical proof, we really had to get a sense of what everything was.

For example, understanding a definition of something can start by playing around with conrete examples. This is because implications that can be made about concrete example may help in understanding the general structure of a definition. These fact elicitation methods used on definitions are commonly known as deduction and induction. Deduction states that “Because a structure is defined to works like this, it should imply this other behavior is true”. Or in the case of induction you can ask “because this concrete example works like this, we can generalize some of these properties to other cases of this definition”.

This process of developing proofs is great for evaluating the logical consistency of arguments. Whenever I have an argument with someone of a certain belief I always check for logical consistency. Note that this is different from questioning whether any standalone subjective view I don’t like is logical, but rather it’s the assessment of whether one’s view is logically consistent with the other views in their belief system.

The crazy thing about logic is that you can form an argument about general statements as long as you have logical assertions that relate them together. For example, the logical operator for implication (usually denoted as =>) can be used to say that the statement A being true will always imply that the statement B is true (A => B). The ‘~’ symbol means the negation of something. So if A is true and we write ‘~A’, that means the expression evaluates to false.

The thing to note is that logical consistency isn’t really checking the validity of these statements themselves, but rather if the proper usage of logical connectors are employed. Whatever statement I choose for A, and whatever statement I choose for B are irrelevant. Everyone can start with their own base assumptions, but I shouldn’t be able to contradict myself if I am to be logically consistent.

Another example can be made using these statements: Statement A is said to be true, A being true implies statement B is true (A => B), B being true implies statement C is true (B => C), and C being true implies statement A is not true (C => ~A), then I have a logical inconsistency.

If I were to write all the statements in mathematical notation it would look like:

A
A => B
B => C
C => ~A

When dealing with an argument we require all the expressions to evaluate to true to classify it as logically consistent. In this case I said A is true which implies B to be true. Because B being true implies C is true, I can write:

A => B => C

and actually just drop the middle part by using a mathematical concept called transitivity:

A => C

Now if we couple all our statements together I can show that the argument is logically inconsistent:

A
A => C
C => ~A

A is true, and A implies C is true. C unfortunately implies A is not true which causes us to reach a contradiction because it was already stated that A IS true. You can put in any sentence you want for A, B, and C but based on your logical connections of your facts, the argument will always be logically inconsistent.

Let’s take a concrete example of what was just written:

A = “Austin is a man”
B = “Austin likes women”
C = “Austin wants to date women”

A
A => B
B => C
C => ~A

In logic terms I’m saying “Austin is a man” is true, I’m asserting that because “Austin is a man” is true “Austin likes women” is true, as well as “Austin likes women” being true means “Austin wants to date woman”, but the last statement says that Austin wanting to date women somehow means he’s not a man. Now you could argue about whether my actual statements or implications are true in the first place (like whether A=>B) but the structure of this argument will always breed logical inconsistencies for whatever statements are put in for A, B, and C.

Leave a comment

Your email address will not be published. Required fields are marked *