Logic & Truth Tables

27 April 2011

What is the whole idea about the Laws of Logic we study in Mathematics in our lives?

The laws of logic studied in Mathematics applies in real life, and is all about
developing a valid reasoning to determine what is true, false or indeterminable.

What is Logic?
Logic is our method of determining the validity of a statement. When we solve problems, we use logic and determine what are the possible ways to solve it. Using logic in our daily problems can automatically refute things that CANNOT be done, but not things that can be done.

For example, a question like this, "What is in this box I'm holding now?"

The possible answers are infinite, as long as you have a good imagination. However, we know what CANNOT be in the box, using logic alone to refute them. We know that The Grand Canyon is not in the box. We know that big things such as elephants, trees, cars, humans and so on cannot be in the box unless it's a miniature toy figure.

Logic requires:
- Linear reasoning, not circular.
- Valid.
- Based on facts, not opinions.
- Not subjective, not personal.
- True, false or indeterminate.
- A good mastery of language (English)

Truth Tables
In Mathematics, logic is proved or justified using truth tables. We only deal with propositions in Mathematics, where a statement can only be TRUE or FALSE, but not INDETERMINATE. Now, what exactly does the truth table show?

Let p = "I study hard."
and q = "I pass my exams."

Given the premise, IF p THEN q (p → q) the truth table is shown below.

pqp → q

How did we fill the truth table? Well, we put in all possible truth values for both statements p and q. Then we can determine whether the premise "if p then q" is true or false. So it has all combinations of true and false for p and q.

The If-Then statement specifies that IF the condition is true, THEN the following will be so and so. The IF clause is checked first, before we can proceed to the THEN clause. Now that we know what does IF-THEN mean through our basic English revision, we can now correctly fill in the truth table.

a) First Row
p = True, q = True
I study hard, I passed my exams. IF I study hard, I pass my exams (p → q).

Therefore, p → q is TRUE. Why? Because if we say, "IF I study hard THEN I pass" it means that you will pass your exams IF you study hard. You studied hard, and you passed, so the If-Then statement remains true.

b) Second Row
p = True, q = False
I study hard, I did not pass my exams. IF I study hard, I pass my exams.

Therefore p → q is FALSE. Why? Because "IF I study hard, I pass my exams" specifies exactly that if you did study hard, THEN you WILL pass your exams. If you studied hard, and still failed your exams, then p → q cannot be true and is a false statement, or a lie.

c) Third Row
p = False, q = True
I didn't study hard, I passed my exams. IF I study hard, I pass my exams.

Therefore p → q is still TRUE. Why? Because the If-Then statement only specifies that when you study hard, you will pass the exams. But did it say that if you don't study hard, you will fail? No. You may have passed the exams by cheating or some other method, who knows? The main thing is, the premise does not specify that you will fail, if you do not study. It only specifies that you will pass, if you study.

This fallacy is called Denying the Antecedent which goes like this:
If p then q.
Not p.
Therefore not q.

A better example:
If I am Eric Clapton then I own more than 5 guitars at home.
I am not Eric Clapton.
Therefore, I don't own more than 5 guitars at home.

This is an invalid argument, a fallacy. This fallacy is quite often seen in many daily arguments, just that people tend not to realize it.

d) Fourth Row
p = False, q = False
I did not study hard. I did not pass my exams. IF I study hard, I pass my exams.

Therefore, p → q is TRUE. Why? Because it has no influence over the premise. The premise is only concerned that you DID study hard, THEN so and so will happen. If you did not, it doesn't specify what happens. So the statement p → q remains true.

The Truth Table & Reality
But wait, how come p → q is True even when it has a possibility of being False?

In the truth table, a true statement tells you that it is a VALID statement but not necessarily factual. A valid statement is a statement constructed with the correct form of logic. In short, whatever that is regarded as TRUE in the truth table basically means there is a possibility of it being true or false, but it is not DEMONSTRABLY false.

The false values in the truth tables are DEMONSTRABLY false, which means it is invalid, it has been PROVEN to be false, using logic alone. It CANNOT be true. Remember the box example earlier? We can name what's NOT in the box using logic alone. In other words, we can say what is FALSE using logic alone, with full basis and confidence.

In reality, you might face statements that you cannot determine whether it is true or false. Some of these are CLAIMS that are not mere opinions. What do we do with claims? Claims need to be proven, by the people who make that particular claim. This is called the burden of proof. The burden of proof rests on the person who makes the claim, not on others. If a claim does not have proof or evidence, the default position would be to not believe in it, as it is INDETERMINATE. We have no reason to believe in anything that we don't know whether it's true or false.

This is the logical position to be in for every claim without proof. If we choose to BELIEVE that a claim is true because we can, or just because we want to do so because it feels better, then a belief remains a belief. The claim is still INDETERMINATE, and our beliefs CANNOT influence reality, nor does it deserve any special respect, because there is no reason to believe, apart from our willingness to believe.

Real Life Logic
In Mathematics, we can assume that the premises are relevant to each other, and are correct. When we say, p → q and p is true, we don't have to review whether p is actually really true as we speak or not. Therefore, we can clearly come to the conclusion that q MUST be true.

But in real life, we need to look out and see whether there is relevance in the premises and conclusions or not. Otherwise, we will only have a bunch of logical fallacies. We also need to look out whether the arguments are following the correct logic or not, which is hard to detect in everyday speech. We'll need a good mastery of English to understand what exactly a statement means.

Here are 3 common fallacies in daily life:

a) Non-Sequitur
- A conclusion that does not follow its premise.
This apple is rotten.
Rotten apples can cause stomach pains.
Therefore, this apple is red in colour.

The conclusion is about the colour of the apple. It has no relation to whether it causes stomach pains or not. Even if the apple is red, there is no way to draw that conclusion from the premises. Any conclusion that does not follow the premise is invalid, unless there are other premises supporting that conclusion. The correct conclusion should be, "Therefore, this apple can cause stomach pains."

b) Circular Reasoning
- Reasoning that goes around in a circle, coming back to its original statement.
You can get what you want.
By having lots of will power.
By wanting to get what you want.
By having lots of will power.
... so on.

This goes in a circle, a loop that when you ask questions, it comes back to the same answer. In the end, it basically means, "This is so because this is so." Logic is linear, not circular. It either reaches an end answer, or it doesn't, in which the answer would be "I don't know" and not any other claims or opinions.

c) Appeal to Emotion
- The truth is influenced by emotions, in other words, biased reasoning.
I feel good about X.
I feel bad about Y.
Therefore, X is correct and Y is wrong.

Even if you say that X is PROBABLY correct, and Y is PROBABLY wrong, it is still a fallacy to determine the probability of it being correct or wrong based on your emotions. Your emotions do not change reality, and does not determine probability. Also, you need to know all the parameters, variables and data of an observable event before you can actually determine probability, unless you're generating the numbers out of nowhere.

This is one of the most common fallacies in almost anywhere. Such as saying that being gay is wrong, because of the feeling that it's not right/natural or because you find it disgusting. Being disgusted about something does not influence anything but yourself. Other emotions such as FEAR is also one of the most common for this fallacy.

We need to know how to use English well in order to describe or state something as it is - brevity. It helps if you apply the laws of logic in life to analyze different situations and refute what is not correct with logic alone.

We must learn the different types of logical fallacies so that we do not become victims of it ourselves. Such as, getting 3 heads in a row by flipping a coin does not make it more probable to get a tails in the next flip (Gambler's Fallacy). How many of us disregard logic and use faith as reasoning instead? Stop being biased and illogical. The only method of determining what is true and false is LOGIC, and that is the strong point of human beings, otherwise we'll be useless pieces of shit but still arrogant enough to claim that we're superior to other animals.