Should We Trust Logic? A Brief Defense of the Laws of Logic

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So here is a basic question that philosophers have struggled with for a while now: “why should we trust reason?” We could appeal to the laws of logic, but that simply takes the question back one step further to where we must ask, “Why should we trust the laws of logic?” The funny thing about these questions is that they are so simple yet incredibly loaded with assumptions. When we ask why we should trust reason, the question is really, “Are we thinking and processing the world around us well?” “Can we trust the fundamental aspects of our logic?” I think it is helpful to substitute the word “reason” with “good thinking.” How do we know that we are thinking well?

This article’s title originally said, “Should We Trust Reason?…” I think that with the definition of “reason” being “good thinking” we can safely agree that we all can trust reason by merit of its definition. If someone does not want to think well, then they will be left to nonsense by virtue of their decision to abandon “good thinking” or “reason.”

If we can agree that we all want to think well (which I sincerely doubt anybody would disagree with), then we can begin the real fun which is establishing what it actually means to “think well” or to “be reasonable.” It is the opinion of this author that the three traditional laws of logic are the three pillars which uphold good thinking. These laws have come into question by some who think that believing these laws requires too many unfounded assumptions. I, however, think that with careful consideration, the laws of logic can be proven to be true. I am probably a little too bold to take on such a topic in a blog post, but here goes nothing.

There are three laws of logic:

1. The law of identity: Something is equal to itself.
2. The law of non-contradiction: Two contradicting statements cannot both be true (We are going to change this definition, but this is the traditional definition even though I do not like it).
3. The law of the excluded middle: Either something is true or it is false.

In order to justify all three laws, I think it is important to start with the law of non-contradiction. Some people think it is necessary to prove this law; however, this is not the case thanks to the definition of “contradiction” alone. A contradiction is a set of premises that cannot all be true. When you state the traditionally defined law of non-contradiction, you are really only establishing the definition of a contradiction. This is why people fail when they try to prove this law. They must assume that there are contradictions to prove that contradictory things cannot contradict, and they reason in circles.

However, given the very definition of a contradiction, it is entirely unnecessary to prove the law of non-contradiction true. Instead, we must only prove that there are indeed examples of contradictions to which the law (which is really just a restatement of the definition) applies. The traditional definition of the law of non-contradiction is that two contradictory statements cannot both be true. I personally define the law of non-contradiction simply by saying contradictions exist in the sense that there is at least one proposition which necessitates the falsity of at least one other proposition. If contradictions exist, then by definition of “contradiction” both contradictory statements cannot be true (so restating that fact would be superfluous). If this law of logic is false, then there must not be any contradictions – that is there is no statement which necessitates the falsity of another statement. Since the opposite of the law of non-contradiction is that there can be no contradictions, then all I have to do is prove that there is at least one contradiction to prove the law of logic true.

We will be proving the law of non-contradiction via argumentum ad absurdum. In other words, we will take the opposite of the law and prove the opposite of the law false to prove that the law must be true.

Consider the following premises:

1. There are no contradictions. (The opposite of the statement, “There are contradictions.” Also note that this premise could be phrased “There is no statement which, if true, disqualifies the truth of at least one other statement.”)
2. There are some contradictions. (Given. Since there are no contradictions, it must be true. If any statement is false, then there must be a statement which is true which contradicts it. There are no contradictions, so there can be no false statements.)
3. Premise 2 is only true if some contradictions exist.
4. If some contradictions exist then premise 1 is false.
5. Premise 2 is the opposite of premise 1.
6. If premise 1 is true then premise 2 is also true. (By its own definition)
7. If premise 2 is true then Premise 1 is false. (Premise 4)
8. Premise 1 is true (Given)
9. Premise 2 is true (Premise 6)
10. Premise 1 is therefore false (Premises 4 and 7)

Even if we define premise 1 as true, we can still deduce that it is false. We can hardly allow a statement to be both true and false (we would allowing acknowledging the existence of a contradiction which violates premise 1 as well as asserting that two contradictory statements are both true), therefore premise 1 must be false.

In other words, there is a contradiction for the statement “There are no contradictions” therefore there are contradictions, therefore the law of non contradiction is proven true.

We have established that the law of non-contradiction is true in that there are some statements which necessitate the falsity of others\, so we can now apply it to the other laws of logic and our job just got a whole lot easier.

Let’s use the law of non-contradiction to prove the law of identity.
The law of identity would say that something is itself. If the law of identity is not true, then something is not itself. Something obviously has to be something, otherwise it would be nothing. Something cannot be nothing, because that would be a contradiction per the definitions of the two terms and we have already shown the law of non-contradiction to be true. Something must then be something, and we must now talk about something in particular that we will call P. P must be itself. If P is not P then somebody simply got confused and wrongly labeled something P (perhaps a lower-case q). This would simply be an error in judgment, and it has no bearing on the law of identity. So P is P, but if the law of identity is false then P is P and not P at the same time. This would be a contradiction, and since we have already demonstrated the truth of the law of non-contradiction we know that two contradicting statements cannot both be true. Therefore, P is either rightly labeled P and is itself, or P never was P and is not P but is still itself. If it was not itself, then it would simply be something else that was itself. If it was not anything, then it would simply be nothing at all. Since something cannot be itself and not itself at the same time, the law of identity is true given the falsity of the opposite.

This brings us to the final law of logic which is the law of the excluded middle. This law is very similar to the law of non-contradiction in that it simply says that P is either P or not P and it cannot be both. In other words, I am either reading this sentence, or I am not. I cannot be doing both at the same time (I dare the reader to try). The law of the excluded middle is not hard to prove after we have proven non-contradiction. If I said that you are both reading and not-reading this sentence at the same time, then I would be contradicting myself. Since I am contradicting myself, both statements cannot be true, and there is no middle ground (except in the possible sense that you are reading this sentence and not paying attention, but that is not the kind of middle ground we are talking about).

Since there is no middle ground in contradictory statements in that a statement is either true or false (A statement cannot be true along with its contradictory statement), the law of the excluded middle is proven.

The keystone of this entire argument is the law of non-contradiction. If the law of non-contradiction is proven false then the other two laws fall with it. However, the law of non-contradiction can be proven true, and along with it the other two laws. Therefore, we do not have to assume the laws of logic are true. We simply have to think about them a little bit. Because the laws of logic can be justified, we can continue to trust our reason as long as it is properly applied. The proper application of reason, rather than the merit of reason itself, is where the fun is really at.

Kyle Huitt
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Kyle Huitt

Part of the multitude that has lost their faith, but part of the few that has returned to it. This blog is my attempt to describe why I returned to the faith, and to maybe prevent somebody else from leaving it in the first place. Studying philosophy and history at Hillsdale College. Member of Delta Tau Delta fraternity.
Kyle Huitt
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