It is commonly held that deductive logic is properly basic, that is, that it does not need to be grounded in anything else. Thus, a fine response to "why does logic work" is held to be, "because it does, are you crazy?" I do not quite disagree with this position, but, while our belief in the propositions of basic deductive logic may be properly basic, our belief that our belief in those propositions is properly basic is not, itself, properly basic.
I am going after two questions which come out as one. Why does logic work? And, What makes something properly basic?
Logic may be considered as nothing more than a bunch of symbols and their rules of use, together defined to as to be tautologous. Thus, in symbolic logic, I am conjoining symbols which stand for possible truth values by means of operators which are defined in terms of what outputs they give for each set of input truth values. In this case, however, logic still rests on the principle of non-contradiction. In this way, it is, thus far, circular.
If I have a pair of propositions, p and q, and know their truth values--I know both p and q to be true, for example--then I also know the truth value of the conjunction p & q. The system is defined so that the conjunction is given by the truth-values of the input propositions p and q. In the system, it is a matter of definition that if p is true and q is true, then p & q is true. Likewise for any of the other symbols and their outputs.
Thus far, however, there is nothing grounding the correspondence of the system to the world. If the principle of non-contradiction does not hold in the world, then logic is useless. On what basis can we claim that the principle of non-contradiction, and thus logic in general, actually holds of the (messy) world?
If we lack logic, then most of this paper will turn out to be self-refuting, and we will be left with no reason not to hold that the principle of non-contradiction holds. A refutation of logic will have to rely on some kind of logic. However, a proof of logic, while it will involve logic, need not be useless. The grounding of logic will show why logic works for us, and thus may allow us to see what may be included in logic which has not, thus far, been included, or been included more warily, etc.
Essentially, the grounding of logic will be the grounding of a properly basic belief, viz., that the principle of non-contradiction applies to reality. It will thus, if successful, form a case-study for what makes something properly basic.
Suppose non-contradiction does not apply. If that is the case, then, besides the fact that we have as yet no reason not to hold that the principle also does apply, we also are left with no confidence on which to rest our actions. For any belief we might act on, we have no reason not to also hold of the belief that it is false. In order to go on with our lives, in order to live, we must act according to the belief that some proposition, or some category of propositions, is true rather than false. We must hold, that is, that we have reason to act in this way, and that it is not the case that we have no reason to act in this way.
This is not a logical argument per se. Although it does rely on a logical form. What the argument draws out, is, however, the basis proper: our inability to go on without this principle grounds our use of this principle. This is an experienced fact which does not rely on logical form. We are compelled to adopt this principle by nature, that is, by our very make-up, by our being agents who act for reasons. Our very need to act rather than merely happen to behave gives us a need for reasons which may be relied on. Thus, we need the principle of non-contradiction in order to be as humans.
Thus far, however, this basis lacks grounds itself. Before going on to ground it, however, I want to point out what this grounding is not: it is not a pragmatic grounding. I am not arguing that, because we cannot go on without this principle, we ought to treat the principle as true, but that, because we cannot go on without this principle, the principle is, in fact, true. This shows how much in need of grounding this basis, our inability to go on without the principle, needs to be grounded qua basis.
Essentially, the principle now in need of grounding is the movement from a human need for something to its existence. This is a far more disputed principle. We have gone from the more certain to the less certain, and are asking the less certain to ground the more certain. This would be a bad thing if we were trying to prove to each other that logic works, but the dispute in our actual case is over how we can know that logic works. I am trying to show that this is a solid basis for our trust in logic, and thus a place from which to appeal for beliefs being properly basic, rather than that for these reasons you should trust logic to work.
So, then, why should we hold that because humans need a certain belief in order to go on, that belief is true? I noted that this is not a pragmatic move. However, it is close, and we will move through the pragmatic point of view to arrive at the non-pragmatic acceptance.
First, because we need this principle in order to go on, we cannot but hold this principle. To dispute this principle in action is to give up agency. Our choice is between agency and giving up the principle of non-contradiction. We cannot give up agency, as that would be an act of agency, and thus self-refuting. We would be affirming by our actions that agency both exists and cannot exist. Insofar as agency relies on non-contradiction, and giving up agency violates non-contradiction, agency and non-contradiction rise and fall together.
We are stuck as agents, however. The only way to get rid of all our agency is to rid ourselves of life. Even in our most capricious moments, we exercise some degree of agency. Thus, giving up non-contradiction consigns us, or at least prompts us, to a final act of irrational agency into death--suicide.
Suicide is a negating of self and world in one act. It is a denial of the world's fit to me, and to accept that the world finally fails to fit me grounds such an act as, if not rational, at least no less rational than continued attempts to conform to reality. To deny that the world fits me, which is what is required by denying non-contradiction, is to give up the possibility of a rational existence in the world.
The choice, then, is between my basic fit to the world or death: to hold that I can live, or that I cannot live (though this degree of binarity is exclusive only from the side of basic fit). Thus, either non-contradiction holds, or humanity lives in a world which rejects it. In this latter case, we are no part of the world, but foreign to it. To hold that non-contradiction does not really hold, that it is a useful fiction, even, is to deny a fit between myself and the world which, if true, undermines my ability to make my way through the world. A pragmatic affirmation of non-contradiction does not leave me solidly enough fit to the world, but leaves the possibility of a final lack of fit somewhere down the line on precisely this point. It leaves life uncertain of itself.
To get to the solid truth of the principle of non-contradiction, then, it is necessary to exclude our own death from our choices. Essentially, what we have said is, however, precisely what is noted by most who hold that some beliefs are properly basic: you must be crazy, out of touch with the world, in order to deny these beliefs. Our options have been shown to be between absurdity or this properly basic belief.
There is a degree of circularity to my argument around the edges. I assume that neither death nor absurdity are live options for most of us, and thus that we must choose the principle of non-contradiction. Some will accept absurdity, however, particularly at the edges of their lives. I am arguing that the degree of absurdity here is one which reaches deeply into our lives, that to reject these beliefs in words can never square with our lives so long as we live. Thus, the degree of absurdity moves beyond words and into daily actions, and thus presses us toward death if we deny these basic beliefs.
I am confident that any who agree with my argument will not disagree with my choice in favor of non-contradiction and other properly basic beliefs, since, were they to choose otherwise, they would not remain long for this world. Is this not merely pragmatic? Perhaps it would be, except that I hold that the world must basically fit us. One may arrive at this from any number of directions, whether evolution or theism. It is another principle which is embedded in how we live.
Here I have not grounded the principle that what we need in order to go on exists on some further principle, but have simply laid out our choices as I see them, and thereby, likely, helped show what I mean by "go on." This is a basic principle in all my philosophy: that it is (always for everyone) possible to live, to go on, or, put another way, that suicide is, in every case, irrational given reality. I am not sure it is possible to fully ground it further, but it is the point at which epistemology and ethics become one in life as lived, and thus why I sometimes speak of my philosophy as existential.
Thus, my argument is:
1. There must be some way to tell what beliefs are properly basic,
2. Here is a basis for a properly basic belief (that non-contradiction holds).
3. This basis is satisfactory.
4. Thus, this basis may be appealed to in other cases.
If I am correct, then a certain kind of human need for a belief in order to go on grounds that belief. This does not mean that there can be no other bases for properly basic beliefs. If it is the only satisfactory basis, however, then beliefs will be properly basic if and only if humans have a need for them in this way.
I suspect, but will not show here (as the argument itself has been put forth by C. S. Lewis and others elsewhere), that one can reach a belief in something quite like heaven from this point. There is also only a slight stretch from here to reach the need of a salvation which is not by works but by grace. In brief, since: if it is by works, then we may reach a point of no return of failure--a final failure that is irremediable, and thus disfits us to the world. This latter relies on the "always for everyone" in my principle, and someone else may prefer a more restrained principle, but I see no reason to privilege some positions with respect to this principle over others. I do not doubt that the principle may be used wrongly, to reach false conclusions, wherein we may find us using the arguments in both directions: some as modus tollens, others as modus ponens. If you hold you need x to go on, and I hold that x is not available to you, then I will also hold that you do not, truly and deeply, need x in order to go on.