A Formal Justification of Agnosticism
by Bill Schultz
There has long been a trend among some strong atheists to savagely attack any who declare themselves to be agnostics. The usual mode of attack is to bifurcate the decision, claiming that if you fail to assert that God exists (and to thereby become a theist), the only other option available is to declare that you lack a belief in any God or gods, and that this is one of the accepted definitions for atheism. I have long opposed such people by asserting this argument to be a fallacious form of bifurcation.  Now, another carefully reasoned argument for the fallaciousness of this same argument has surfaced. This new argument is the subject of this brief essay. It argues that the rules of inference from Classical logic (meaning the logic as defined by Principia Mathematica by Whitehead and Russell) cannot be used in discourse about unexperienced truth values. As a direct consequence, the attempt at bifurcation is formally invalid.
The usual argument used by atheists to attempt to convince a nonatheist and nontheist to convert to atheism is that there are only two alternatives, either you believe that God exists or you do not believe that God exists. If you believe that God exists, you are a theist, and in the alternative, if you do not believe that God exists, you are an atheist. The atheist assumes that those are the only two options, and will sometimes badger an agnostic until and unless they assert their willingness to convert to atheism.
In the formal language of logic, the proposition that the atheist (described above) is advocating is this one, wherein G stands for “God exists”  :
P1: G v ¬G (Either God exists or God does not exist.)
The above is just based upon a restatement of the principle from Classical logic known as the Principle of the Excluded Middle. In Classical logic, this principle is seen as a tautology, which cannot be false. Accordingly, assertions based upon that principle have great force, at least for so long as Classical logic is seen as absolutely valid. But, as Floy Andrews has explained, in his essay The Principle of Excluded Middle Then And Now: Aristotle And Principia Mathematica:
It is clear that those who subscribe to the Classical position on Bivalence and Excluded Middle, do express some form of ‘realism’ [Dummett’s characterization]: there is for them a “world” which is what it is whether we know it or not; and true propositions are simply those that express what is the case with the world. …
To trace the history of this (largely) twentieth century ‘realism’ would be long, and out of place here. Whatever its origin, in Frege’s rejection of what he calls “psychologism” or elsewhere, this much can be said, that the logic which he invented was so radical and so remarkably compelling and productive that it effected its own revolution. Frege succeeded in reducing all of classical mathematics to a single formal system. But that would have had only esoteric interest. His revolution in logic lies in his doctrine that singular sentences are atoms, and more complex propositions are simply truth-functions of these atoms. Once a mechanical calculus of truth-functions was provided, then Classical logic became an extraordinarily powerful technique for manipulating these atoms, ordering them in ways that serve the prevailing interest in the logic of contingencies, the concatenation of `facts’ for practical and theoretical purposes. … In its wonderful simplicity, power and decidability it is said to have advanced in a very short time beyond everything achieved in logic in its previous 2000 year history. In this case, it is the logic which carries the metaphysics with it: the `realism’ of our times is consequent on the unquestioned success and authority of Classical logic.
If anything else is to be gleaned from the matters analyzed here, perhaps it is this, that Classical logic is a particularly inept instrument to analyze those philosophies which stand opposed to the ‘realism’ it demands. Yet we see such analyses everywhere in philosophical literature, presented as though they were objective assessments of positions they can at best dogmatically oppose.
The mention of Dummett and his definition of “realism” as the metaphysical underpinnings of Classical logic, as described above, serves to introduce my main thesis herein: that the argument advanced by atheists, above, is formally invalid, for the reasons explained by Dummett and his followers. This essay of mine is based upon a book length attack on “realism” and Classical logic: Language, Logic and Experience: The Case for Anti-Realism, by Michael Luntley, who is a proponant of Dummett’s “antirealism” school of philosophy. I shall now summarize Luntley’s argument from the referenced book.
A realist believes that P1 (the Principle of Excluded Middle, above) is valid and applies in all cases. In fact, the entire structure of Classical logic is based upon some version of that formulation, because (as Andrews suggests) “the three principles of Aristotelian Logic are clearly only interdefinitions in Classical logic.” Thus, the Principle of the Excluded Middle, the Principle of Contradiction (that a thing may not be and not be at the same time), and the Principle of Identity (that a thing has an immutable nature and is thus always identical with itself) are all just themes and variations of the same underlying principle. The realist believes in the universal applicability of those principles. However, is the realist justified in that belief? The key insight of Luntley’s book, above, is the idea that a failure to have any experience of some given element’s truth value leaves that element logically undefined, and this undefined state cannot properly be treated in the way that classical logic treats it, as represented by P1, above. Andrews, in his essay, makes similar observations about contradictory propositions where some common element (the subject of both propositions) is logically false. An example of this would be the propositions that “God is good” and “God is not good” if the subject (God) does not exist. In that case, both propositions fail, and the Principle of the Excluded Middle falls with them.
Luntley extends his argument to invalidate the entire foundation of Classical logic, and along the way, P1 falls as well. I will now quickly run through Luntley’s arguments. But before I do, I should first stop and explain a couple of terms that are used in the argument. The first term is “content.” By “content” it is meant to refer generally to information, as opposed to “vehicle,” which refers generally to the method by which information is stored or transmitted. This is based upon a definition from the philosophy of mind:
mental content – As distinguished from vehicle, mental content is that aspect of mentality which, ideally, refers to an object, property or relation and specifies some properties of that item.
If I have a mental concept of a book (said concept being the “content” in this example), then my brain is the vehicle for containing that concept. There are strong arguments over whether or not “content” of this sort is objective or subjective in nature. The full range of this dispute is beyond the scope of this essay. However, given the recent discoveries in brain imaging which illuminate the human thought processes, I believe it is fair to say that for sure, the realist school accepts that a mental concept (“content”) is objective in nature. In fact, when the content refers to a clearly perceivable (experienced) external object, then the objective status of the mental concept is difficult to deny. The question, then, lies within the distinctions drawn for content which is unperceived.
Luntley (on behalf of the antirealist school) accepts the idea that "content" is objective in nature. He represents this idea this way  :
Objectivity of content: The characterization of content requires the subject’s possession of a conception of a world beyond that which is experienced.
However, Luntley claims that the fact that content can be objective without experiencing it does not flow naturally into the idea that truth can be objective without experiencing it. The realist asserts the opposite, this way  :
Thesis R: The contents we grasp are contents that have a determinate truth value independent of our knowledge of that value.
Luntley expresses the distinction between merely accepting the objectivity of content and the objectivity of the truth value in this way:
The objectivity of content for some thought P commits one to the idea that if there is a fact of the matter, it is of such and such kind; the realist objectivity-of-truth thesis commits one to the idea that there is a determinate fact of the matter.  .
The concept of “a determinate fact of the matter” extends beyond what we might grasp with gradual change between one condition (such as “hirstute”) to another condition (such as “bald”). What the realist worldview commits to is that no matter what are the descriptive or prescriptive conditions which comprise the particular content in question (such as, “John has exactly 10,384 hairs on his head), that content will always be precisely either true or false in the world external to the observer. The content of “John is bald” is falsified by finding at least one hair on John’s head. The difficulty is in handling situations where the observation has not yet been made which would determine the truth or falsity of the content. This is what Andrews identifies as a problem which Aristotle himself recognized, as discussed in part B of the Andrews essay.
So, is it really true, as the realist believes, that there is always a determinate fact of the matter? There are at least some well proven examples where there is not. One of these might well be Gödel’s Incompleteness Theorum, which asserts “that any consistent formal system powerful enough to contain arithmetic must contain at least one proposition whose truth or falsity cannot be proven within the system.” In other words, no matter how you attempt to formalize a mathematical or logical proof, there will always be at least one “undecidable” proposition contained within said proof. However, the most obvious of all or those examples would appear to arise due to the Heisenberg Uncertainty Principle, which holds that both the position and the momentum of a subatomic particle cannot be determined precisely at the same time. You can make one into a determinate measure only at the cost of making the other into an indeterminate measure. Thus, this proves at least one notable (and broadly accepted) exception "to the idea that there is a determinate fact of the matter." And where an exception clearly exists, an allegedly universal law is clearly falsified. It is obvious that where the fact of the matter is unknown (by reason of its having not been experienced), then the fact of the matter might also be indeterminate, and thus the realist "idea that there is a determinate fact of the matter" cannot always be true. Any indeterminancy is incompatible with the required universal determinancy of the realist position. This circumstance falsifies a key foundation of Classical logic.
The implication of the above is that where we are dealing with empirical (real world) facts, Classical logic is invalid, because the foundation upon which it rests is destroyed by indeterminancy. Instead, in the empirical world, we must actually experience a truth before we can logically assert it. Luntley uses this concept to develop what he calls the manifestation constraint:
Constraint C: For any content P if it makes sense to say that subject S knows that P, P must be an experiencable state of affairs and this entails that S’s knowing that P makes a detectable difference to experience. 
As Luntley himself notes, “the reference here to ‘experience’ is vague.” For instance, what should we make of knowledge claims that come from others? For the claim to possess validity, there must be some person who has experienced the required “detectable difference,” and then the knowledge claim (“content”) may be relayed by any number of different means (“vehicles”) until it is part of the knowledge of many other people. And frequently, the knowledge claim itself is really a hypothesis about a situation which has never been actually observed (such as claims about the “Big Bang” origin of the universe). But still, such claims are based upon the experiences of various scientific observers who analyze their observed facts in light of the currently popular theories to produce the relevant knowledge claims. These claims have a probability of truth that is based upon numerous factors which ought to be obvious to the serious student of the methodologies of science and information transfer between people. But there is nothing in this whole business that denies that these truth claims are an attempt to describe an observed (past tense) objective reality.
This is not really a new concept by any means. In fact, it can easily be argued that Thomas Henry Huxley’s definition of agnosticsm anticipated Constraint C when Huxley defined agnosticism in this way:
Agnosticism is not a creed but a method, the essence of which lies in the vigorous application of a single principle. Positively, the principle may be expressed as in matters of intellect, follow your reason as far as it can take you without other considerations. And negatively, in matters of the intellect, do not pretend that matters are certain that are not demonstrated or demonstrable.
— “Agnosticism”, 1889
Accordingly, the antirealist does not deny the existence of objective knowledge. However, only knowledge that satisfies constraint C (in some unspecified way) can properly qualify as an objective truth. And the agnosticism of Huxley, requiring “demonstrated or demonstrable” facts really requires substantially the same thing as does Constraint C. So then, Luntley develops the following:
A-R Thesis: An area of discourse is objective if and only if the propositions of the area of discourse are subject to standards of correctness and incorrectness of assertion encoded in a system of evaluation in such a way that through attendance to the system of evaluation the area of discourse is able to generate knowledge claims which satisfy constraint C.
Having run through Luntley’s proof rather quickly (with some additions and rearrangements of my own), it should now be obvious that a proposition in the form of P1 is invalid when we are discussing matters of speculative (unexperienced) knowledge. The potential of false common terms and indeterminate results kicks the supports out from under the foundations of Classical logic and invalidates at least the three principals mentioned above. The end result of Luntley’s investigation is to prove that these classical rules of inference cannot be used for philosophical discourse when the underlying facts are unobserved. This is a far reaching claim that few people will readily grasp. However, I would assert that a proof far stronger than what either Luntley or I have discussed can be constructed by a thoughtful philosopher because the groundwork has clearly been laid for just such a proof.
Professor Ted Drange developed a system of definitions  of what it means to be a theist, an atheist, an agnostic, and a noncognitivst. While I don’t entirely agree with the system he presents, he does seemingly agree that P1, above, should not be used to force agnostics into the atheist camp. This is what certain advocates of the definition of atheism formulated by George Smith and Michael Martin have attempted, largely by relying upon P1.
However, as Luntley demonstrates, in the world of hypothetical (unexperienced) data, P1 is not supportable by valid rules of inference. Instead, if no knowledge of what is true or not true can reach the senses of the observer, then the observer cannot draw any inference at all. This is a proof of the need for agnosticism whenever the truth or falsity of any fact necessary to the conclusion cannot actually be determined because the data is lacking (nothing has been sensed). The atheists who attempt to apply P1 to force agnostics to declare themselves to be atheists are relying upon false logic that cannot be validly employed in the context for which it is invoked.
Accordingly, agnosticism is not only a valid choice for those of us who see insufficient evidence to decide the matter as to the truth or falsity of some particular God concept. Agnosticism is, in fact, the only valid choice in such circumstances under the rules of inference applicable to discourse based upon the evaluation of hypothetical (unexperienced) data.
I would like to thank Dean Stretton for his lengthy and detailed critique of an earlier version of this essay. His comments led to a considerable expansion and improvement from that earlier draft.
 See Language, Logic and Experience: The Case for Anti-Realism, by Michael Luntley, pp. 56-62, where this same proposition is treated generally.
 Ibid, pp. 3-4.
 Ibid, p. 4.
 Ibid, p. 29.
 Ibid, p. 46 et. seq.
The text of this essay is Copyright © 2001, by William A. Schultz. All Rights Reserved. Used by permission of the author.