Since it seems we’re on theory season in the Naruto Manga section and seeing that some theories are… less than stellar, to put it nicely, I thought that this section needs a thread that can serve as reference for both those who create theories and those who refute them.

**1) Deductive thinking**** a. Proposition**

When we state something, be it affirmative or negative, we’re making a proposition that is composed by subject and predicate. The subject is the target of the statement and predicate is the attribute given/denied to the subject.

For example, in the proposition “This manga’s name is Naruto”, the manga is the subject and Naruto is the predicate. But why am I talking about this?

**b. Syllogism**
Because of that big word. Syllogism is the form of deductive thinking with 3 propositions where 2 of them become premises and the other becomes the conclusion. The conclusion is a proposition with new information that is based on the premises.

For example:

- Naruto is a Genin
- All Genins graduated from Academy
- Therefore, Naruto graduated from Academy

You may noticed there are 3 terms that are shared among the propositions. They are the major term (T), the minor term (t) and the medium term (M). The minor term serves as subject of the conclusion, the major term is its predicate and the medium term is the link between the premises that doesn’t appear in the conclusion. Also, the premise with the minor term is the minor premise, while the premise with the major term is the major premise. Taking the previous example:

- Naruto (t) is a Genin (M) (minor premise)
- All Genin (M) didn’t graduate in the Chuunin Exams (T) (major premise)
- Therefore, Naruto (t) didn’t graduate in the Chuunin Exams (T) (conclusion)

Simple, right? But notice the wording, I wrote “all Genin” instead of just Genin. That’s because the wording has a role to play in the validity of the conclusion. There are four kinds of propositions (each with its symbolic vowel):

- Universal and affirmative (A)
- Universal and negative (E)
- Particular and affirmative (I)
- Particular and negative (O)

Keep in mind that the predicate in affirmative propositions is non-distributed (that is, either some or all of the subject is some of the predicate) while the predicate in negative propositions is distributed (that is, either some or all of the subject is none of the predicate).

Also, how the terms display themselves in the premises alter the nature of the conclusion. There are four figures, going from more spontaneous to more technical:

- First Figure: Middle term is subject of the major premise and predicate of the minor
- Second Figure: Middle term is predicate of both premises
- Third Figure: Middle term is subject of both premises
- Fourth Figure: Middle term is predicate of the major premise and subject of the minor

Taking again the example, the syllogism belongs to the First Figure. Now these are the principles behind those figures

- First Figure: the major premise is always universal (A/E), while the minor is always affirmative (A/I); the conclusion is universal (A/E) if the minor premise is also universal (A) and it’s negative (E/O) if the major premise is negative (E). These are the aspects of this figure (major-minor-conclusion): AAA, EAE, AII (the aspect of the example given) and EIO.
- Second Figure: the major premise is always universal (A/E), while one of the premises must be negative (E/O), the conclusion is always negative (E/O); the conclusion is universal (E) if the minor premise is also universal (A/E). The aspects are EAE, AEE, EIO and AOO.
- Third Figure: the minor premise is always affirmative (A/I) and the conclusion is always particular (I/O). The aspects are AAI, EAI, IAI, AII, OAO and EIO
- Fourth Figure: When the major premise is affirmative (A/I), the minor is universal (A/E); if the conclusion is negative (E/O), the major premise is universal (E) and when the minor is affirmative (A/I), the conclusion is particular (I/O). The aspects are AAI, AEE, IAI, EAO and EIO.

Now, if you’re not bored out of your skulls like when I first learned this, you’ll notice that are some aspects that never appear, like two negative premises or two particular premises. That’s because there are rules when forming a syllogism.

- There are 3 terms and ONLY 3 terms
- The medium term cannot exist in the conclusion
- The medium term must be distributed equally between premises
- No term can be more meaningful in the conclusion than in the premises
- We can’t conclude anything from 2 negative premises
- We can’t conclude negatively if both premises are affirmative
- If one of the premises is negative and/or particular, the conclusion must be negative and/or particular
- We can’t conclude anything from 2 particular premises.

**c. Formal fallacies of syllogisms**

Yes, all this blabber was only to reach to these pearls of (lack of) logic that people in this forum seem to love. Since we made a list of rules, let’s see what fallacies are connected to them by order.

** i. Fallacy of Four Terms**

One would think people would be wiser to avoid such a silly fallacy, but some just seem to love it. Here’s an example:

- A Rinnegan user was dubbed the 7th Path
- The 7th Realm is a step to reach Buddhahood
- Therefore, Rinnegan users can reach Buddhahood

This is fallacious because there are clearly four terms: the 7th Path is not the 7th Realm in Buddhism, therefore this syllogism lacks a middle term, making the conclusion invalid.

**ii. Fallacy of Intrusive Middle Term**

Again, another fallacy that should be easy to avoid but somehow it isn’t.

- All Senju descend from the Second Son.
- The Uzumaki are related to the Senju
- Therefore, all Uzumaki are Senju because they descend from the Second Son.

This is fallacious because the middle term (Senju) is present in the conclusion, giving the proposition 2 predicates when it can only have one (in this case, being descendant of the Second Son). The correct conclusion is “All Uzumaki descend from the Second Son”.

**iii. Fallacy of Undistributed Middle Term**

Seemingly similar to the previous fallacy, but there’s a significant difference.

- All Senju are descended from the Second Son
- All Uzumaki are descended from the Second Son
- Therefore, all Uzumaki are Senju.

This is fallacious because, even though the middle term is not in the conclusion, it still makes the conclusion invalid. Why? Think of it for a bit: the middle term is being the predicate for both affirmative premises, so it’s never distributed in any of them. That’s why the Second Figure has a negative premise and a negative conclusion, to give distribution to the middle term.

**iv. Illicit minor/major**

Another fallacy that’s connected to the distribution of terms

- All Edo Tensei summons are undead
- Some Uchiha can’t be Edo Tensei summons
- Therefore, some Uchiha can’t be undead

This is fallacious because the major term (being undead) is augmented in the conclusion. It was non-distributed in the premise but somehow distributed in the conclusion.

** v. Fallacy of Exclusive Terms**

And this is why you don’t have two negative premises.

- Tobi isn’t Tobirama
- Tobi isn’t Naruto
- Therefore, Tobirama isn’t Naruto

This is a blatant fallacy, since the conclusion is useless (we didn’t need the syllogism to know that Tobirama isn’t Naruto) and the middle term is irrelevant (who says Tobi can say any character in the context of the syllogism)

** vi. Illicit affirmative/negative**

More nonsense!

- Tobi uses Sharingan
- The Uchiha use Sharingan
- Therefore, Tobi isn’t an Uchiha.

Another blatant fallacy, because, like in math, when you add two affirmatives you can’t have a negative as result.

** vii. Universal/Affirmative conclusion when one of the premises is Particular/Negative**
One more silly fallacy:

- Some redheads aren’t Uzumaki
- Karin is a redhead
- Therefore, Karin is an Uzumaki

This is fallacious, because the major term is negative and it’s the predicate of the conclusion, therefore its negative property also has to be present in the conclusion.

** viii. Fallacy of Particular Terms**

And taking the above example, it is also fallacious because Karin is a particular term that also happens to be the minor, but since we’re taking two particular terms, the link between premises is too weak to validate the conclusion.