Knowledge, to my understanding, is the mental grasp of entities in reality, as opposite to those in hallucination, or dreams. Surprisingly enough, there's actually a set of rules of these entities and their attributes and relations. More surprisingly, these rules can be observed and investigated by human. Some people stepped into this field much farther than others, and they wrote down what they found and abstracted. So today we lucky descendants have the fortune of learning their great discoveries. Most fundamental one, is what we call “logic”. As I'm more intrigued by classical logic, the aspect dealing with the real world, than those abstract branches, I also write down what I have learned so far.   

 

 

1. Two Postulates   

   

First of all, there is the objective reality, and it is independent from any consciousness. It is unchangeable, through different ages and spaces, unaffected by whatever impact there could be. For example, the phenomena or attributes that indicates the principle of conservation of energy. In this regard, a noteworthy confusion is that, the distinction between objective reality and other objects in reality. In other words, not all things that exist are objective. Those things created by human, thus changeable by human, like societies, traditions, or kingdoms, are fine examples of things that are not objective.    

   

Another postulate is that, the objective reality is perceivable. Specifically, that perceived by human consciousness is what interests me the most. That includes an object's existence, attributes, metaphysics, relations, developments, and all other aspects. In Bertrand Russell's definition, everything in reality is "sense-data". (Link )

 

 

2. Law of Identity    

   

Its basic idea is "an object is identical to itself", or in Gottfried Leibniz's words, "Everything is what it is." A reflexive relation between an object and itself is therefore demonstrated.   

   

A derivation is instantly known, that there cannot be two objects identical to each other. If two objects share one identity, they are in fact one object. So, this is to say, that identity is the object itself.   

   

While it seems like a redundant claim, it does reveal some defects of our daily language. An especially troublesome issue is that, our definition of an object could be incomplete due to our limited knowledge, or inertial thinking, thus could not capture the identity of an object, which could lead to some confusing questions, like the Ship of Theseus, and Heraclitus's River. However, the object itself, or the identity, is not affected by our definition. So if I completely capture the objects in question, for example in four-dimensional space like Perdurantism did, or if I understand that the time behavior is one of the object's properties, not enough to change the identity, the issue would not be so troublesome.    

   

The key here, as I think, is to try to understand an object as its own, independent of the language I use, or the knowledge I have, or anything that could possibly limit my perspective, like social surroundings or traditions. An identity is not an object's property, it's the object itself. An object (or the identity) is defined by all the properties it has, not the ones we recognized at a time.   

 

 

3. Law of Non-contradiction   

   

It says "two contradictory statements cannot both be true at the same time." Or simply put, "contradictions don't exist." For the sake of rigor, the "at the same time" part is usually necessary. It's not as obvious as the first law, but it is also how objective reality is, thus should be how I recognize the world.   

   

A particularly useful application of this law is the elenctic method, or Socratic Method, where I assume a statement I need to investigate is true, and then derive a sequence of conclusions using it as a premise. If I get a contradictory, to the original statement, or to a previously proved conclusion, I know that the original statement is irrefutably false, as contradictions do not exist.   

   

There's an interesting story, "deductive explosion", that points out that any statement, no matter how ridiculous it is, can be proven if the absence of the law of non-contradiction. Say we have two contradictory statements, "All fish can swim" and "Some fish cannot swim", both true, and an obviously ridiculous one, "censorship is virtue." Since it is true that "All fish can swim", the or-statement is true that "All fish can swim or censorship is virtue". However, it is also true that "Some fish cannot swim", i.e. the first part of that or-statement is false, in order to accomplish that or-statement, the second part, "censorship is virtue", must be true.   

 

 

4. Law of Excluded Middle   

   

"For every proposition, either it is true or its negation is." Or "there cannot be an intermediate between contradictories." But whether this proposition is true, or its negation is, has nothing to do with this law. A derivation is that "Ambiguity cannot exist in the facts, only in names", a confirmation that the definition by a language can be inaccurate.   

   

There's a useful derivation. If a statement p is sufficient for another statement q, then the negation of statement q is sufficient for the negation of statement p. In symbolic representation, it is written as (p → q) → (~q → ~p), here → means "be sufficient for", ~ means "the negation of".     

   

The proof of this derivation:   

Using elenctic method, I assume that, on the basis of p → q, the negation of q is not sufficient for the negation of p. Based on the Law of Excluded Middle, it means that the negation of q is sufficient for p, in symbols ~q → p. And because of the starting assumption, that p → q, I can get that ~q → p → q. That is a contradiction, which does not exist, thus false. That means my assumption is false in the first place, therefore I get (p → q) → (~q → ~p).     

  

There's something called "false dilemma", which is pretty much a fake appeal to the law of excluded middle. When I see two options that are opposite and I need to choose one, but the two options are in fact not negations to each other. There is an intermediate between them, a third option. This is not where the law of excluded middle should be used.   

 

 

5. Law of Sufficient Reason   

   

According to Leibniz, "There can be no fact real or existing, no statement true, unless there is a sufficient reason." This principle may be an explanation of those "non-objective" objects I mentioned before, that there are sufficient reasons for such entities, like decisions made by people, or actions taken by people, or things affected by human will. It's just that these reasons are not objective.    

   

Some statements about this principle that is helpful to me:   

"When a reason is explicitly or implicitly given, then there must exist a consequent; and vice versa."   

"Where there is no reason there can be no consequent; and vice versa, where there is no consequent there can be no reason."   

"We all make choices, but in the end, our choices make us."    

 

 

6. Logical Reasoning    

   

This includes Deductive Reasoning, Inductive Reasoning, and other methods. But I assume those other methods can all be found derived from the first two using the laws above, so the two are my preference.   

   

Deductive Reasoning is in fact an instance of the relation of superset and subset of Set Theory. In daily language it says "if A, then B. Now A is true, therefore B is true." In set theory it's written as "A ⊆ B, ∀x∈A; ∴x∈B". Deductive reasoning guarantees the truth of its conclusion, given the adequate conditions. But the "adequate conditions" technically come from abstract models. That's why some people like to call it "priori truth", because the ability to understand this kind of truth is born in people's mind. Mathematics, in particular, is a system of deductive reasoning, nothing else. That's why it deals with abstract issues, and not counted as natural science.    

   

There are several detailed forms of deductive reasoning, such as modus ponens, which is exactly the example above to demonstrate deductive reasoning itself, and modus tollens, which is modus ponens plus that derivation of the law of excluded middle I mentioned before. And the one that says "if A then B, if B then C, therefore if A then C." And many others, which I also assume are all derivations, and don't wanna think too much about.   

   

On the contrary, there's Inductive Reasoning, where the "adequate conditions" come from the real world, or our observation of it. That's to say, if I observed an adequate amount of examples, and found that all of them are followed by the same result, I may get the conclusion that all such things as the examples will lead to that consequence. Seems like a good idea but when somebody exams it closely he'll find not so much. My observation cannot cover the whole, therefore my conclusion is not guaranteed for all such instances. In other words, what happened before is not guaranteed to happen in future. Therefore, when faced with the real world, at some point sooner or later, I'll have to get off the track of guaranteed truth.   

   

That's why David Hume and his skepticism, questioning the uniformity of nature, the vital point of inductive reasoning. He said that we have to use experience to assist the conclusion of induction, so it's not priori, thus not provable by deductive reasoning, which can guarantee its truth. Also, it's not provable by inductive reasoning because that would be circular argument, as it is exactly inductive reasoning itself we need to prove.   

   

So, at this point, things go back to the two postulates at the beginning of this article, the second one in particular. I have to assume the truthiness of uniformity of nature, or that the truth lying in objective reality can really be perceived. In Immanuel Kant's words, uniformity of nature is priori truth. Only after that postulate can I really study every other piece of knowledge.   

   

A noteworthy point is the difference between Induction Reasoning and Mathematical Induction. The latter guarantees the truth of its conclusion because it has a terminate condition that comes from deductive reasoning, or is a piece of mathematical fact.