Principles of Logic

[Graham]

Principles of Logic​

I would like to take this fantastic opportunity to introduce Sythe to the great topic of mathematical logical reasoning. No this is not too simple of a subject, but I will be showing you the parameters of the subject which could allow you to grasp a sense of the topic for proofs. Once you have a fundamental understanding of logic, you can see that it pertains to the English language and shall allow us to establish the truth of statements, premises and their conclusions.

- To create the truth tables which are standard we will use the letters "p" and "q". These are the accepted letters which have been used for many years. We are going to establish every possible truth value for p and q. T will mean true and F will mean false, for those who need clarification

p q
T T
T F
F T
F F

- These values are know throughout logic and to gain an understanding of the math you must understand these standards. To further explain the above chart; when p is T, we have q as true - case 1. When p is T, we have q as false - case 2. When p is F, we have q as T - case 3. When p is F, we have q as F - case 4. This will give us every possible truth value for p and q.

Example- Refer to line 1 of the chart where p is true and q is true.
p = Randy is very athletic
q = Randy is strong

1.Randy is very athletic, randy is strong.
2. Randy is athletic, randy is not strong.

Here we gather from human experience and the above standards that according to line 1 of the truth table, statement number 2 is incorrect.

Now you will take hat we have learned above and you will carry it over to the next stage of learning.

Propositional Logic Symbols​

First we will speak of the 5 fundamental symbols used in propositional logic. Using these 5 symbols we can establish the full truth table which can be used for multiple topics in logic.

1. Conjunction- Whenever you see the symbol "." that is the universal symbol for conjunction which means "and".

• Symbol= . (basically a multiplication symbol)

• p q / (p . q)
• T T / T (when p is T and q is T than (p . q) is T)
• T F / F (when p is T and q is F than (p . q) is F)
• F T / F (when p is F and q is T than (p . q) is F)
• F F / F (when p is F and q is F than (p . q) is F)

________________________________________________​

2. Disjunction- Whenever you see the symbol "v" you can translate this to the word "or".

• Symbol = v (basically a lower case "v")

• p q / (p v q)
• T T / T (when p is T and q is T than (p v q) is T)
• T F / T (when p is T and q is F than (p v q) is T)
• F T / T (when p is F and q is T than (p v q) is T)
• F F / F (when p is F and q is F than (p v q) is F)

________________________________________________​

3. Negation- When we see the symbol "~" this will mean the negation of a statement. We can translate by say this is not, or this is not the case. For example we can negate T and make it F.
Example-

Statement- It is true that my computer will last 3 hours on battery life.
Negation- Is is not true that my computer will last 3 hours on battery life.

• p q / (~p) / (~q)
• T T / F / F (when p is T than p is F) (when q is T than q is F)
• T F / F / T
• F T / T / F
• F F / T / F

________________________________________________​

4. Conditional- When we see the symbol of a "u" tilted 45 degrees to the left. For our purposes we will use "u". This means if than. So if we have this, than we have that.
p q / (p u q)

• T T / T (when p is T and q is T than (p u q) is T)
• T F / F (when p is T and q is F than (p u q) is F)
• F T / T (when p is F and q is T than (p u q) is T)
• F F / T (when p is F and q is F than (p u q) is T)

________________________________________________​

5. Tribar- When we see the symbol of three lines stacked on top of each other that is the symbol for tribar. For our purposes we will use the "=" symbol. This means that both must be the same for the statement to be true.

• p q / (p = q)
• T T / T (when p is T and q is T than (p = q) is T)
• T F / F (when p is T and q is F than (p = q) is F)
• F T / F (when p is F and q is T than (p = q) is F)
• F F / T (when p is F and q is F than (p = q) is F)

 

[Necromancer24]

Ah Mathematical Reasoning My favourite. I don't have any problems with this topic, but this is a good guide.

I suggest formatting it to make it look better. Use different font

 

[Graham]

Thanks for the feedback,it's much appreciated. I guess I just have this type of style when I write my guides :laugh:

 

[b0b3rt]

Your conjunction and disjunction symbols are highly non-standard. Where did you learn this?

 

[Graham]

The standard for disjunction is "v". There are multiple options you can use and it's how you feel comfortable. The symbol for conjunction can either be "^" or ".". Like I said, it's not wrong; it's a preference and this is the way it's taught in my college.

 

[b0b3rt]

I apologize - there seems to be some sort of distinction between the symbols of set theory and propositional logic that I don't understand. When I first learned it ~5 years ago, I learned a heavy mixture of both with the symbols U and ∩.

 

[Graham]

I apologize - there seems to be some sort of distinction between the symbols of set theory and propositional logic that I don't understand. When I first learned it ~5 years ago, I learned a heavy mixture of both with the symbols U and ∩.

Yes, there is a difference between propositional theory and the symbols of set point. My symbols may also look odd because I don't have the proper tools to create the correct image, but the presented ones are pretty similar to what I'm getting at.