The Foundations of Logic

People have been participating in debates since the beginning of time. It is a wonderful tool that humans have developed over the millennia to communicate ideas in hopes of reaching an amicable and constructive conclusion. However, in my experience, individuals who partake in these discourses often fail to use logic. You will sometimes even encounter the saying “it's a matter of opinion.” Well, actually, it's not. You either have evidence to support your premise(s) (i.e., demonstrate the truth of your premise) or you don't. You can either structure a good argument following the rules of logic or your argument is bad and should be rejected. It's that simple.

But, how do we know that logic is right? Why should we trust it? Simply, it works. Time and time again, logic has lead to correct answers. However, let's delve into the details of how it is that we can consistently arrive at correct conclusions when we apply the rules of logic.

The Axiom

An axiom is anything that we take as a self-evident truth that requires no proof or a universally accepted principle or rule. Essentially, the axiom is going to be the foundation of your theory from which everything else will follow. For example, the most commonly used foundational system for mathematics is called set theory. Within set theory, the axioms (sometimes referred to as postulates) from which all mathematics is built upon are known as the Zermelo-Fraenkel Axioms [1]:

1) Axiom of Extensionality: If X and Y have the same elements, then X = Y.


Explanation: This is basically saying that if X and Y have the same parts, then they must be the same. For example, if X is composed of the natural numbers [1,2,3] and Y is also composed of the same natural numbers [1,2,3], then clearly X = Y. Note, the natural numbers are composed of all the positive integers (i.e., 1,2,3,4,.....).


There are a total of nine axioms that compose the logical foundations for mathematics. However, they become increasingly more complex to the untrained eye, so we will stop here.


As another example, let's assay the branch of mathematics concerned with questions of shape, size, properties of space, etc. Geometry, as with set theory, is also an axiomatic system (i.e., all theorems or true statements are derived from a small number of simple axioms). In particular, the type of geometry that you were first introduced to is what's known as Euclidean geometry. This particular type of geometry has the following axioms [2]:

  1. A straight line segment can be drawn joining any two points.

  2. Any straight line segment can be extended indefinitely in a straight line.

  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

  4. All right angles (i.e., a 90 degree angle) are equal to one another.

  5. If a line segment intersects two straight lines forming two interior angles on the same side that sum to less that two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.


Explanation: The first four axioms/postulates are self-explanatory. However, while the wording on the fifth is a bit confusing, in reality, it is something that you are already familiar with. This axiom is known as the parallel postulate, which is equivalent to what's known as the triangle postulate. This second postulate is the one that you are already familiar with as you were taught, early on in school, that the sum of all the interior angles of a triangle must add up to 180 degrees. Interestingly, this axiom/postulate means that in non-Euclidean geometries (e.g., hyperbolic, elliptic, or even the Riemann geometry used in General Relativity) the sum of the interior angles of a triangle doesn't necessarily add to 180 degrees anymore.


Once more, axioms are the foundations from which everything else is built. As in mathematics, logic also has a set of axioms from which all else follows.

Axioms of Logic

The axioms/rules for logic or “laws of thought” are [3]:

  1. The Law of Identity: Something is what it is and isn't what it is not. Something that exists has a specific nature.

  2.  Law of Non-Contradiction: Something cannot be both true and false at the same time in the same sense.

  3. The Law of Excluded Middle: A statement is either true or false, without a middle ground.

  4. The Law of Transitive Properties: The properties of one premise must carry over to the other premises.


These can also be represented mathematically as:

  1. A = A.

  2. A != !A

  3. Everything is either A or !A.

  4. If A = B and B = C, then A = C.


Note, within the mathematical descriptions, the precedent “!” negates the object that follows. Thus, != reads “not equal to” and !A reads “not A.” Further, the fourth axiom isn't technically apart of the traditional laws of thought; only axioms 1-3 are. Nonetheless, I felt it was important enough to be included here.

Examples

Expounding upon the rules of logic described above, we have:

1) I am me. I am not you. I am not my dog. My dog is a dog. You are not my dog. You, my dog and I are all separate as we are each composed of our own unique set of characteristics.

2) A cat is a cat. It has fur, two eyes, whiskers, claws and is alive. A rock is a rock. It is hard, generally roundish in nature, and is not alive. We cannot have the situation where a cat is also a rock (i.e., a cat-rock); it's verboten. Once we have assigned a label to a given object, then it cannot simultaneously assume the label of something else. If this were to happen, then we would arrive at the situation where something could simultaneously be both alive and not alive. Clearly, this makes no sense. What is more, this isn’t a matter of opinion; it’s fact.

A square is an object that is composed of four line segments and four right angles; which, when you add all of the interior angles add to 360 degrees. A triangle, as discussed before, is composed of three line segments and has interior angles that add to 180 degrees. Therefore, a square-triangle is something that cannot exist as you cannot have an object which simultaneously has interior angles that add up to both 180 and 360 degrees. It doesn't make any sense; it has to be one or the other. Again, this isn’t a matter of opinion. If we adhere to the axioms of geometry, this is fact.

3) You are either dead or alive; there is no middle ground in this case. Note, in everyday discourse (i.e., informal logic), this axiom can fail unless strict definitions are agreed upon by both parties. It is generally reserved for use in formal logic (i.e., symbolic logic).

4) Joe has less money than Ashley, Ashley has less money than Heather, and Heather has less money than Rob. Does Rob have more money than Joe? Now, before we answer this question, we can rearrange the premises into a syllogism:

  • Joe has less money than Ashley.

  • Ashley has less money than Heather.

  • Heather has less money than Rob

  • Therefore, Joe has less money than Rob.


As you can see, the property of money transitioned to each premise, which eventually allowed us to reach the conclusion that Rob has more money than Joe. Note, in order to structure a good argument using this axiom, it is imperative that the premises are true. For example, if the premise that “Joe has less money than Ashley” was not true, than the conclusion that “Joe has less money than Rob” cannot be guaranteed. This results in a bad argument, which must then be rejected.

You apply logic in everyday life without even thinking about it. For example, consider the scenario where you notice that your cell phone has low battery. You then choose to plug it in. Without giving it any serious thought, you went through the following syllogism in your head:

  • The battery percentage display on my cellphone is designed to tell me how much battery is left.

  • I know that the battery display works.

  • My cell phone is displaying that I am low on battery.

  • Therefore, I must be low on battery.


This is a simple example of deductive logic in everyday life (I'm charging my phone right now and applied the same exact syllogism demonstrated here before I did so). If I were to dismiss this as a matter of opinion or simply choose to ignore it, my phone would eventually run out of battery and become useless. As you can clearly see here, the rules of logic are important and, when properly applied, give us the best possible chance of arriving at a correct answer.

Conclusion

As demonstrated above, logic, like everything else, has a starting point or foundation from which the rest is built. These axioms direct us towards crafting a cogent thought process, which allow us to reach true conclusions. Implicitly, this is something that all of use on a daily basis to make important decisions. However, more often than not, I have observed people (myself included) failing to use logic when engaging in a discourse, which will result in an incorrect conclusion. Moreover, we are all entitled to have opinions, there's absolutely nothing wrong with that; but, this doesn't mean that your opinion is tantamount to facts or the immutable rules of logic. Hence, if you find yourself presenting your opinion when confronted with evidence, you are most certainly wrong and must re-evaluate your position. As a Critical Thinker, it is paramount that you have a thorough understanding of these rules and implement them regularly to ensure that the course of your life is being directed by good arguments.

References

[1] Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource.

[2] Weisstein, Eric W. “Euclid's Postulates." From MathWorld--A Wolfram Web Resource.

[3] "Laws of thought". Cambridge Dictionary of Philosophy. Robert Audi, Editor, Cambridge: Cambridge UP. p. 489.