How Logic and Argumentation Works

Although the definition of logic can be somewhat of a controversial issue, it can be defined as the study of principles behind correct reasoning. 

Logic is the pinnacle of human reasoning, and is what everyone ought to adhere to when assessing one's own tenets, beliefs, and arguments (something not as easy as one might think).

It can be separated into two categories:

• Formal Logic - Study of logic using purely formal content that can be used as an abstract rule. Aristotelian logic held the early genesis of this branch of logic. Symbolic and Mathematical logic can be put into this branch as well. Computers use this system in order to function. All in all, it's a purely deductive-based system.  
• Informal Logic - Study of natural language arguments, or the type of arguments that you hear in everyday life. A grand majority of it relies on induction to make one's argument strong. The study of informal fallacies is a supremely important part of this branch of logic. 

The Laws of Logic
-There are three laws of logic that are integral to the structural integrity of the study.

1. The Law of Identity: A is A. (Example: A rock is a rock. A rock cannot not be a rock. That would be in violation of the law of identity.)  
2. The Law of Non-Contradiction: A is not non-A (Example: It cannot be raining and not raining at the same time in the same place. Or another way of saying it is that the statement "It is raining" cannot be both true and false in the same sense. That would be in violation of the law of non-contradiction.) 
3. The Law of the Excluded Middle: Either A or non-A (Example: For any proposition, the proposition is either true or false. "I was born in the U.S." is either a true or false statement, and cannot be neither true or false. This law only applies to statements which imply a truth value.)  

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An argument is composed of premises and a conclusion, where the premises are supposed to give support to the conclusion. An example of this type of structure can be seen as so: 

P1. If it is Sunday, I will take off work. 
P2. It is Sunday.
—————————————————————
C. I will take off work.

-There must always be at least two premises that supports the conclusion in order to be considered a logical structure. 

-Not all sentences or statements are considered premises. For example, a command is not a premise that supports any conclusion. A premise must be a truth statement. "Go do your work." is not a truth statement. However, something like "All dogs are brown." is a truth statement, and can be evaluated on the grounds of it either being true or false. 

Now of course, not all arguments in reality are going to be as neatly formatted as the one stipulated above, but getting into the habit of deconstructing one's own or others' arguments will be supremely helpful in illustrating the strength and weaknesses of one's arguments. 

All in all, arguments that rely on logical systems can be then separated into two categories: 

• Deductive logical arguments - This type of argument is truth-preserving. In other words, There will always be a 100% guarantee of a conclusion that logically follows from its premises, as opposed to a chance or probability that it may follow. This kind of logic is concerned with the following, respectively: 
• Validity - This is concerning only the logical structure of the argument, and not the truth of it. For example:  

     P1. If all bunny rabbits are white, I can fly.
     P2. All bunny rabbits are white. 
     —————————————————————
     C. I can fly. 

-When assessing validity, we must assume that the premises are true in order to understand if the logical structure of the argument stands. The premises are obviously untrue, but the structure makes sense if we assume the premises to be true, so therefore, the argument is considered valid.
-Arguments deemed invalid are also called by the Latin phrase as Non sequitur, which is formally translated to "does not follow."

• Soundness - This is concerning the truth of the argument. An argument must first be considered valid, and its premises must be true in order for the argument to be considered sound. For example: 

     P1. Socrates is a man.

     P2. All men are mortal. 

     —————————————————————
     C. Socrates is mortal. 

-Since the argument is both valid, and the premises of the argument are true, the argument can now be considered sound.

• Inductive logical arguments - reasoning in which the premises intend to supply strong evidence for (but not absolute proof of) the truth of the conclusion. This kind of logic is concerned with the following, respectively: 
• Strength - Whether the argument is strong or weak depends on whether the premises provide strong probable support for the conclusion. For example: 

     P1. Every swan I've seen in the state of New Jersey is white.
     P2. I've seen 3,000 swans. 
     —————————————————————
     C. All swans are probably white. 

-The argument is strong if there are only close to 3,000 swans in the world. The argument is weak if there are severely much more than 3,000 swans in the world, like 3,000,000. In other words, a large part of the strength of inductive arguments relies on good sample sizes that can be representative of the whole. 

• Cogency - An inductively strong argument with true premises. An argument must be strong before it can be cogent.

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Common types of deductively valid reasoning:

-Modus Ponens (Affirming the Antecedent) 

If p, then q. 

p.

Therefore, q. 

-Modus Tonens (Denying the Consequent) 

If p, then q. 

Not q.

Therefore, not p. 

-Hypothetical Syllogism

If p, then q.

If q, then r.

Therefore, if p, then r. 

-Disjunctive Syllogism

Either p or q. 

Not p. 

Therefore, q. 

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Common types of deductively invalid reasoning: (Common errors in reasoning seen in everyday life) 

-Affirming the Consequent 

If p, then q. 

q. 

Therefore, p. 

-Denying the Antecedent

If p, then q.

Not p. 

Therefore, not q. 

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In reality, much argumentation relies on some deduction and a lot of induction to provide strength to one's arguments. However, in that process, it's more often than not the case that one commits fallacious reasoning through informal fallacies, simply because it's very easy to commit them, especially if one isn't trained to spot and avoid them in their own arguments. A comprehensive list of the informal fallacies can be found here.