What, exactly, is logic? How does it exist? How is it true? How do we know that it is real, and that it applies universally to all being?
I would define logic (at least for our current purposes) thus: A is A and is not non-A. That is, things are what they are and they are not what they are not. This is also called the law of identity or the law of non-contradiction.
Some people are content just to note that without logic, propositional claims cannot have meaning and knowledge is impossible. This is true (as I have argued here), but it is an insufficient argument for the truth and necessity of logic. All this proves is that we must assume logic if we want to have knowledge and we want our statements to be meaningful. But wanting something does not make it so. Perhaps the universe and all claims about it are simply meaningless and no knowledge is possible. We can't reject that possibility simply on the grounds that we don't like it; we must prove it wrong (or accept it if it is right).
There are a number of philosophers today arguing that contradictions can sometimes be true and meaningful, and that this is compatible with true knowledge and meaningful propositions. This position is called dialetheism. I will not here go into an analysis and refutation of dialetheism (though I plan on attacking one of its main arguments in my next post), but its existence helps to point out the importance of exploring the ontological foundations of logic--that is, what logic really is, how it fits into reality, and how we can know that it is true and universal.
So what exactly is logic, and how does it fit into reality? Logic is an essential characteristic of the category of being. To understand this, we have to realize that all our knowledge is based ultimately on observations of reality. And reality turns to have lots of differentiated stuff in it. There are desks, and kangaroos, and electrons, and people, and grass, and so on. In order to maintain a grasp on the world, we create categories rooted in language in order to sort out the diversity of reality. We talk about desks as a distinct category from kangaroos which is a distinct category from electrons, for example. Each of these categories has definite substance and meaning, and there are characteristics of each category that are essential to it. There are essential characteristics that are involved in the idea of a desk, and there are a different set of essential characteristics involved in the idea of a kangaroo. The differences between these sets of essential characteristics is what creates the distinction of categories.
In order to have meaning in our statements, we must make sure that when we use certain words or put forth certain ideas, like the idea of a kangaroo, we stay within the set of essential characteristics of that idea. So when we talk about kangaroos, we'd better have in mind an animal that is a marsupial, that lives wild in Australia, etc., rather than, say, a vegetable that grows wild in the jungles of Brazil. If we jump free of the essentials that define an idea, our words lose their meanings. Of course, we can redefine our terms if we want to. There is no inherent loss of meaning if I simply wish to use the word kangaroo from now on to refer to what most people call pumpkins. But if I do so, I'd better be clear about it in my own mind and in conversation with others. What I can't do, however, is lose sight of the essential meaning of a word while thinking to myself or articulating to others that I am not redefining it in some clear way. This is to slip from meaningfulness into meaningless gibberish, and my words and ideas thus lose all meaning and content. They become equivalent to meaningless grunts or streams of nonsense words like "Gorfuffle op to scorialobotomo hulp."
This process of categorizing the elements of experience allows us to examine our categories to figure out what their essential components are, and this helps us to get a clearer understanding of what reality is like and the sorts of things that are in it. So I can consider the category of kangaroo, for example, and determine certain things from it, such as that "all kangaroos are marsupials." This allows me to make universal claims that apply in all of reality. No matter where one goes in reality--whether past, present or future, whether in this galaxy or another, whether in this universe or another--we can be certain that it is always true that "all kangaroos are marsupials," so long as the words have not been redefined.
I talk about all of this in more detail in two previous posts found here and here.
One of our categories is the category of being. It is a very broad category. It is basically synonymous with categories such as existence, things that exist, reality, real things, etc. It is a category that has been drawn, like the category of kangaroo, from our experience. We can examine the concept of being, then, to find out its essential characteristics.
So what are the essential characteristics of being? For one thing, to be a being, a thing has to exist rather than not-existing. Being excludes non-being. We use the words in an exclusive way. If "X exists," we include in this idea the rejection of the other idea that "X does not exist." If we prove the first, we believe we have disproved the second, and vice versa. Being and non-being have different and conflicting sets of essential characteristics, and so to be the one involves not being the other, just as the categories of kangaroo and pumpkin have different and conflicting sets of essential characteristics. To be a pumpkin requires that one lack some of the essential characteristics of kangaroos, and vice versa.
Another characteristic of being is that all beings must have particular characteristics, and these particular characteristics will exclude other characteristics. One cannot be a being in general; one must be some particular being. There are multiple characteristics to choose from--orange, blue, 2 meters tall, vegetable, organic, fluffy, etc.--and any being must have particular characteristics. To have no particular characteristics (other than the particular characteristic of having no other particular characteristics) is to be a non-being and not a being. Since these particular characteristics are different from each other, to choose some is to reject others. So we cannot speak of beings in the abstract. We must think of specific beings, such as pumpkin-beings. Pumpkin-beings are different from kangaroo-beings because they have a different set of essential characteristics.
This characteristic that characteristics have of excluding other characteristics--such as kangaroo-ness excluding pumpkin-ness or being excluding non-being--is interesting. The basis of this is simply the fact of difference. For example, to look at two simple characteristics, red is different from blue. The two characteristics are simply and fundamentally different from each other. They are irreducible to each other. This we can see from direct observation of them. It is evident from such observation. And it is this fact of difference that means that red always excludes blue. To the extent that an object is red, it cannot in the exact same way, at the exact same time, etc., be blue. Difference creates, and really just is, exclusion.
This principle of difference, and thus exclusion, is of course manifest in all distinctions of categories. And what is this principle of difference? What name can we give to it? Of course, it is nothing other than logic, or the law of identity, or the law of non-contradiction. Therefore, since the category of being essentially excludes the different category of non-being, and since all the particular characteristics that beings might have--such as redness, pumpkin-characteristics, etc.--exclude all that which is not themselves and which is therefore different from themselves--such as non-redness, blueness, kangaroo-characteristics, etc.--we can say that logic is an essential universal characteristic of all being. Here we have the metaphysical, ontological foundation of logic. Logic is an essential characteristic of the concept of being. If we try to remove logic from being, we find that we have lost the essential substance of being and have slipped into meaningless gibberish, just as if we tried to remove non-blueness from the concept of red or non-pumpkinness from the concept of kangaroo.
So it turns out that logic is a characteristic of reality and the things in it in the same way that being marsupial is a characteristic of kangaroos, and the method of figuring this out is the same in both cases. We start with observation, we make categories to organize our observations, and we analyze our categories/concepts and express the implications of what we have found. All kangaroos are marsupials. All being is logical. Logic is not something irrationally mystical or strange. It is not an independent substance, or some Platonic Form. It is simply an essential characteristic of the world we live in, and we know this is true the same way we know anything else--by observation and analysis of our observations.