a. Proportional Logic
This one is more dazing because it is a lot similar to math and everybody hates math, amirite?
First, the most basic form o Proportional Logic is when you break up a proposition in variables that are given the symbols of p, q, r, etc. and execute a logic operation (that is, the connection between variables/propositions).
There are 6 basic logic operations:
- Negation, symbolized by ~ (e.g Naruto is not a Senju)
- Conjunction, symbolized by ^ (e.g Naruto knows Rasengan and Kage Bunshin no Jutsu)
- Inclusive disjunction, symbolized by V (e.g Some nins are male and/or black)
- Exclusive disjunction, symbolized by W (e.g Or a nin is from Kumo or is from Konoha)
- Conditional, symbolized by => (e.g If one knows Sage Mode, then he can learn Frog Katas)
- Biconditional, symbolized by <=> (e.g One can be a Chuunin if and only if s/he passes the Chuunin exams)
Now there are some rules concerning the veracity of the proposition considering the logic operations:
- If p is true, than ~p is false and vice-versa
- A proposition that comes from a conjunction is only true if its variables (p^q or p^~q) are both true. It is false in any other way.
- A proposition that comes from an inclusive disjunction is only false if its variables (pVq or pV~q) are both false. It is true in any other way.
- A proposition that comes from an exclusive disjunction is false if its variables (pWq or pW~q) are both true or both false. It is true in any other way.
- A proposition that comes from a conditional is only false if antecedent (p) is true and the consequence (q or ~q) is false. It is true in any other way.
- Finally, a proposition that comes from a biconditional is true if its variables (p<=>q or p<=>~q) are both true or both false. It is false in any other way.
What’s a tautology? It’s a composed proposition that, no matter how many variables are false, is always true.
For example, the tautology [(p=>q)^~q]=>~p (e.g, If Tobi is Obito, then he survived the falling rocks and if he didn’t survive the rocks, then Tobi can’t be Obito) is always true. Why? We know that for a conditional to be false, the consequence (q in the first conditional, ~p in the second) must be false but the antecedent must be true (p in the first, (p=>q)^~q in the second). We also know that a conjunction is only true if both variables are true and that if p and/or q are true, then ~p and/or ~q are false.
Let’s solve this damn equation:
- If p and q are both true, ~p and ~q are false and p=>q is true. If that’s so, the conjunction (p=>q)^~q is false because ~q is false. But since both variables of the second conditional are false, than the conditional must be true, therefore the proposition is true.
- If p is true and q is false, ~p is false and ~q is true and p=>q is false. If that’s so, the the conjunction (p=>q)^~q is false because the conditional is false. But since both variables of the second conditional are false, than the conditional must be true, therefore the proposition is true.
- If p is false and q is true, ~p is true and ~q is false and p=>q is true. If that’s so, the conjunction (p=>q)^~q is false because ~q is false. But since the consequence of the second conditional is true, than the conditional must be true, therefore the proposition is true.
- If p and q are both false, ~p and ~q are true and p=>q is true. If that’s so, the conjunction (p=>q)^~q is true because both variables are true. But since both variables of the second conditional are true, than the conditional must be true, therefore the proposition is true.
See? Needs a little bit of work to understand, but it pays off.
Contradictions are simply the opposite of tautologies, so their false no matter how many true variables it has. The simplest form of contradiction is the negation of a tautology.
d. Formal fallacies of the Proportional Logic
There are two forms of fallacies in Proportional Logic: affirming the consequence and denying the antecedent.
i. Affirming the consequence
Here’s an example of this.
“If Tobi=Obito, then Tobi can’t be Madara and if Tobi isn’t Madara, then Tobi=Obito”
Equating this, this is [(p=>~q)^~q]=>p. Pretty similar to the tautology above, right? WRONG!!! If p (Tobi=Obito) is false and ~q is true (Tobi isn’t Madara), then the second conditional is false since the consequence is false and the antecedent true, therefore the proposition can be proven wrong, so it becomes fallacious.
ii. Denying the antecedent
“If Tobi=Madara, then Tobi can’t be Obito and if Tobi isn’t Madara, then Tobi=Obito”
The equation for this, [(p=>~q)^~p]=>q, is also very similar to the tautology shown, but yet again it is not what it looks. If p (Tobi=Madara) is false and ~q (Tobi can’t be Obito) is true, then the second conditional is false since the consequence is false and the antecedent true, therefore the proposition can be proven wrong, so it becomes fallacious.
Like in everything, some of it is good, some of it is bad, so what makes a good argument?
The premises the conclusion is made of must be acceptable to whom the argument is made for. It doesn’t matter how true the conclusion is if the line of thought is heavily invalid.
All premises must be relevant to the conclusion. Showing how you know how to use Wikipedia doesn’t have anything to do with how Buddhist concepts fit in the manga.
The conclusion must be strongly supported by its premises, therefore the premises must be enough and strong to hold the theory. If this doesn’t happen, it all falls down like a house of cards.
d. Resilience against Refutability
The more complicated it is to refute your theory while the refuter remains valid, the best. That means everything above was applied correctly and the conclusion is very valid. Now if something as shallow as point out that the flowers you thought were the same aren’t the same, that means you’re doing a crappy job at it.
3) Informal Fallacies
It’s the Fallacy Flying Circusssssssssssssssssss!!! (I’ll only refer to fallacies relevant to literature and the discussion of it)
a. General Informal Fallacies
i. Argument from ignorance
Occurs when someone appeals to the unknown nature of the matter being argued instead of providing propositions about it.
Example: “Naruto can still be Senju because we don’t who were Minato’s and Kushina’s parents”
ii. Argument from repetition / ad nauseam
Occurs when someone doesn’t seem to give up on a flawed argument and keeps repeating it.
Example: Making dozens of posts about your Yin/Yang theory when it’s been already refuted again and again
iii. Argument from silence
Occurs when someone assumes the silence of others means the argument s/he made is irrefutable. Most of times, the other person just doesn’t know how to overcome the sheer stupidity of the argument being made.
iv. Proof by Verbosity
Occurs when someone uses verbose arguments that have little to no relevance to the argument or are even wrongly applied/spelled, just to make the author of the argument look smart and overwhelm others with information
Example: “Shinbuitsu shugo [it’s spelled Shinbutsu Shuugou] is based on the theory of the merger between Buddhism and Shinto practices [that is never witnessed in the manga].
And shibutsu Bunri [it’s spelled Shinbutsu Bunri] is based on the separation of those practices passed latter on [never realizing that the Shinbutsu Bunri wasn’t successful in its purpose, putting Shintoism above Buddhism]
v. Shifting the burden of proof
Occurs when someone, instead of making a proposition that defends the argument, asks others to make a proposition that refutes the argument. The burden of proof should always be on the defending side.
Example: If Tobi isn’t Obito, then who do you think he is?
vi. Circular reasoning
Occurs when someone uses or assumes the conclusion in the premises.
Example: Naruto is Senju because Minato’s mother is Tsunade or else Naruto wouldn’t be Senju.