Truth Reconciled

In a previous article, I wrote about what I considered to be the ultimate question: Why does something exist rather than nothing? Why does anything exist at all? Before we can answer this question, or any question about existence, it is necessary to think about what the word “exist” means. This crucial step seems to have been overlooked or underestimated by many who have tried to tackle the questions of existence. If existence is not defined, then no progress can be made, because a question that is undefined cannot have a definite answer. Continuing without a definition will only lead to confusion.

Confusion

To give an example of the kind of confusion that I’m referring to, let’s consider a question that everyone should immediately know the answer to: Does Santa Claus exist? I’m sure you would say that he does not, and yet, if I ask you to tell me about Santa Claus, you will begin to talk about his red suit, his sleigh, his reindeer, and so on. If a child asks you who that man in a red suit is, what will you say? Will you say that he is Santa Claus? But how can that man be Santa Claus if Santa Claus doesn’t exist? You might say that the man is not the real Santa Claus, but is just representing Santa Claus. And then I will ask you, how can he represent something that doesn’t exist? What is the real Santa Claus? How can something nonexistent have all these attributes that you speak of? Why am I wrong if I say that this nonexistent person wears a blue suit? Isn’t my blue-suited Santa Claus just as real as your red-suited one? If not, then you must think that Santa really does exist in some way, as some kind of commonly accepted idea or concept. What do you mean then, when you say that he does not exist?

Another example is the existence of mathematical objects. Very often in mathematical proofs, it is said that abstract mathematical objects “exist.” But do they really? What do we mean by that? For example, if a mathematician is presented with the equation , he can say, “the solution exists.” The solution is 3, but does the abstract number 3 actually exist physically somewhere, somehow? If so, where is it? What about the function , does it exist? What about Euler’s relation , which describes a deep underlying relation between a few common and seemingly unrelated constants? Does it exist? If so, where is it? If not, why is it useful? These mathematical concepts are certainly not objects that we see in our physical world. If they exist at all, it could only be in some sort of ethereal realm of logic. But is that really what we mean by existence?

Definition

The main source of confusion in these examples is the lack of a clear definition of existence, so let’s think about what existence means and write down a good definition. In mathematics, the property of existence is usually equivalent to the property of being included in a set or collection of objects. In other words, an object exists if it is one of the “elements” or members of a given set. For example, when we say the solution to doesn’t exist, what we are really saying is that, within the set of complex numbers, we cannot find any element x that satisfies that equation. So when I say Santa Claus doesn’t exist, I’m saying that, within the set of living organisms on Earth, there is not one that satisfies all the characteristics of Santa Claus. However, there is a Santa Claus that does exist in the set of fictional cultural icons of our modern society.

A set is a collection of objects that satisfy certain predefined conditions. In order for an entity to exist within a given set, it needs only to be consistent with the conditions of the set. For example, consider this set of even numbers:

{2,4,6,8,10,12,…}.

The number 2 exists in this set, as does the number 4. But the number 3 does not exist in this set because it doesn’t satisfy the condition of evenness. However, if an entity does not exist in one set, it may exist in another. The number 3, for example, exists in the set of odd numbers.

In order to generalize the definition of existence, we can simply remove the conditions imposed by a set. The only condition that cannot be removed is that an entity be consistent with itself. One example of an entity that is not consistent with itself is a number n that is both odd and even. The two properties of oddness and evenness contradict one another, and this makes it impossible for this paradoxical number n to exist anywhere, in general. Therefore, we can say that an entity exists “generally” if it is self-consistent, or in other words, if it is not a contradiction. A generally existing entity can further exist “locally” in a set if it satisfies the conditions necessary to belong to the set. We can now present the formal definitions:

General existence: An entity exists generally if it is self-consistent.

Local existence: An entity exists locally in a set if it exists generally and satisfies the conditions of the set.

At first sight, it may seem that the definition of general existence is not well-defined. For example, suppose entity a exists, and another entity b is self-consistent but contradicts a. Then the compound entity a AND b is self-contradictory and cannot exist, according to the definition. Apparently, we have just created a case in which b cannot exist even though it is self-consistent, which disproves the definition.

However, it turns out that there is never a case where a self-consistent entity contradicts another self-consistent entity. In other words, truth never contradicts truth. To prove this, let A and B represent the statements “a exists” and “b exists,” respectively. It can be shown that whenever B contradicts A, then B contradicts itself. We can write the statement as 

 (A ∧ (B ⇒ ¬A)) ⇒ (B ⇒ ¬B)

and build a truth table, following the method described in my article titled Foundations of Logic and Mathematics. This statement is a tautology (it is always true). We can rest assured that our definition of existence is well-defined.

This implies that all self-consistent entities must necessarily be consistent with all other self-consistent entities. This can be proven by analyzing the following statement:

 (¬(A ⇒ ¬A) ∧ ¬(B ⇒ ¬B)) ⇒ ¬(A ⇒ ¬B).

In words: “If A doesn’t contradict itself and B doesn’t contradict itself, then A doesn’t contradict B.” Upon constructing a truth table for this statement, you will find that it is also a tautology.

Conclusion

Questions of existence have historically been notoriously difficult to answer. The reason for this difficulty is a poor understanding of the meaning of the word “existence.” In this article, we have solidified a working definition, and distinguished between different types of existence. The most general and abstract definition involves nothing more than checking whether or not a proposed entity is consistent with itself. Further existence in a more localized setting requires that an entity pass the requirements necessary to occupy that spot.

As a corollary, this definition of existence formally solidifies many of the axioms of mathematics. Many axioms involve assuming the existence of certain mathematical objects. Any axioms of this sort can now be proven by simply applying the definition of existence. If the object in question is consistent with itself and the conditions which it is claimed to have, then it necessarily exists.

With a proper definition of existence, we can now begin to answer the question of why there is something rather than nothing. The definition of general existence seems to automatically answer this question. Something exists because something is consistent with itself. But this answer is hardly satisfying, because the existence we have defined so far is purely abstract, while the concepts of something and nothing that come to our mind when we ask the question are not abstract, but physical in nature. Why does the physical universe exist? I’ll tackle that question in my next post.