Systems Of Logic/Mathematical Logic

Logic is the strict following of several simple rules. These rules have to do with persistence and relationships. In logic, a statement can only be true or false - there is nothing in between. Following this, a statement can only be formulated so that everyone can state whether it is true or false independently of your relationship with the statement. Lets illustrate this with some examples:
 * Statement: 1 + 1 = 2. Value: True.
 * Statement: 1 + 1 = 5. Value: False.
 * Statement: Jonny is a nice fellow. Value: ''There is no logical truth value as it depends on your relationship with Jonny. It is also not clear who Jonny is and what you mean by "nice" (friendly, honest, cooperative or a completely different meaning), so in logic you cannot give a truth value to this statement.

In logic, the truth value of a statement remains unchanged. If a statement is true, it remains true. The truth value of a statement can lead to the truth value of another statement, which in the same way remains unchanged.

Let's take a look at the relationship between two statements named p and q. The relationship is: If p is true then q is true too. (p = True => q = True) Can q be false? Not as long as p is true. What about when p is not true? (p = False) Since q is only dependent on p when p is true then q is unsolved as long as p is false.

Relationship 1:
 * p = True => q = True
 * p = False => q = unsolved

If we do not know the result of p but know the result of q, then can we tell if p is true or not? If p is true then q has to be true, but if p is not true then q is unsolved (can be true or not). Following this we can say that if q is not true then p can also not be true as a true p leads to a true q (p = True => q = True). If q is true (q = True) p can be true but it can also be not true as q is independent of p when p is False. We have now used relationship 1 to set up a new relationship.

Relationship 2:
 * q = False => p = False
 * q = True => p = unsolved

How about the persistence of p  or  q?

p is  p.  We know this much. And we know that q is q. Do we really? Well can't they change? We must define these statements as persistent. If they change, then they are defined as new statements.

Examples
What does this mean? Let's look at some real world statements. Here are two:

The sky is blue. The water is blue.

Now if the sky is blue, then is the water blue? Well in many natural areas this is the case, however in many natural areas, the water is muddy and brown or glacial and green and even these colors are a matter of perspective. Some might say that the water is clear and any color in the "water" comes from something that is not H2O at all. There are too many assumptions and conditions to state any useful logical relationship about these two statements except that they do not necessarily "follow" one another logically.

Here are two different statements:

The sky is blue. There are no clouds in the sky.

Does one of these statements follow from the other? I'll suggest that in order to be certain of something we must assume that we are in the day without special circumstances such as a dust storm, volcanic ash haze, or a solar eclipse. Then I can safely say that:

If there are no clouds in the sky then the sky is blue. If p  then  q.

What about the other way around?

If q  then  p. If the sky is blue, then there are no clouds in the sky.

I don't think that works. Have you ever seen a brilliant blue sky and one cloud in the sky at the same time? Sure! So p  doesn't follow  q.  But  q  does indeed follow  p  (under our special assumptions about volcanoes and such).

What about if q  is false? Is p?

If the sky is not blue, then there are clouds in the sky.

Yes! This seems to ring true in all cases. (Again under our special assumptions.) Logicians call this statement the contrapositive of the first statement.

Statement: If p  then  q.

Contrapositive: If not q  then not  p.

Statements
There are four types of statements in logic. A bar over a letter negates the statement. For example, "My name is Tristan" would become "My name is not Tristan". The four types of statements are the:


 * Statement: p implies q
 * $$p \to q$$


 * Contrapositive: not q implies not p
 * $$\bar{q} \to \bar{p}$$


 * Converse: q implies p
 * $$q \to p$$


 * Inverse: not p implies not q
 * $$\bar{p} \to \bar{q}$$

The statement and contrapositive are said to be "logically equivalent". The converse and inverse are also logically equivalent. What this means is that if the statement is true, the contrapositive is also true. The converse and inverse are not necessarily true, but can be. Here are some basic statements to illustrate this:

p = Today is New Year's Day. q = Today is a holiday.

The statement, which is $$p \to q$$, would read, "If Today is New Year's Day, then Today is a holiday".

The contrapositive, $$\bar{q} \to \bar{p}$$ is also true, "If Today is not a holiday, then Today is not New Year's Day".

In this case, the converse, $$q \to p$$, is not true. It would read, "If Today is a holiday, then Today is New Year's Day.

Since the converse and the inverse are logically equivolent, the inverse, $$\bar{p} \to \bar{q}$$ is not true, either. It reads, "If Today is not New Year's Day, then Today is not a holiday".