Have you ever tried to argue about science or politics with a postmodernist or a creationist? It is next to impossible, because the person subscribes to radically different metaphysics, epistemology and methods for finding knowledge. People who refuse to let go of their belief that demons cause disease will never be convinced that we should treat sick people with medication. Someone who believes that a supernatural power will punish them with eternal damnation if they use condoms will probably not use condoms, no matter how many studies you provide that they are generally safe and effective against unwanted pregnancy and many sexually transmitted infections. In order to resolve those conflicts, one has to examine the underlying assumptions and beliefs made further down in their worldview. For people who share many aspects of their worldview, it may be sufficient to retreat to discussing morality in order to resolve political disputes. For people with extremely divergent worldviews, it may require discussing what exists, what truth is and how knowledge about the world is gained.
However, people do not want give up on their cherished beliefs, so this approach involves a tremendous intellectual struggle on the part of those who defend a rational and evidence-based worldview. In many cases, they will refuse to answer questions, make assertions without argument or evidence or even dismiss the notion that knowledge is possible or champion the idea that all axioms are arbitrary. In other words, it is a profoundly waste of time. However, it might be interesting to flesh out some of the absurd consequences that follows from the rejection of the existence of non-arbitrary axioms.
For those that believe that there are no non-arbitrary axioms, three disastrous implications follow: the statement is self-referentially incoherent, knowledge cannot exist, and any proposition P and its negation ~P becomes true.
Self-referential incoherence
Consider the proposition “all propositions are false”. Can this proposition ever, in principle, be true? Well, let us assume, for the sake of argument, that it is true. That means that all propositions are false. But “all propositions are false” is itself a proposition, and thus must be false. But if you can show that P is false by assuming P is true, you have stumbled onto a contradiction, so the initial statement must be logically false. More specifically, it is self-referentially incoherent. There are many self-refuting ideas like this one (such as relativism, solipsism, philosophical skepticism etc.), but the key feature is that they are either directly contradictory and/or create a context that implies that it is false or in principle can never be true.
Consider the proposition that “there are no non-arbitrary axioms”. If there are no such non-arbitrary axioms, then whatever axioms this statement is built on have to be arbitrary. If they are arbitrary, then the proposition is not true in a general sense, but rather follows from an arbitrary assumption. But there is no obligation to accept claims that are based on arbitrary axioms. What is worse is that such a proposition can never be rationally justified. So the notion that no non-arbitrary axioms exists is self-refuting and therefore false.
Münchhausen trilemma
If knowledge is not built on non-arbitrary axioms, what can it be built on? It turns out that, according to Münchhausen trilemma, there are only three options: knowledge claims are either based on arbitrary axioms, are circular or it is based on an infinite regress of justifications. But none of these three can justify knowledge. Basing knowledge on arbitrary assumptions cannot work, because someone else can infer the exact opposite knowledge claim from other arbitrary assumptions. Circular arguments are pathetically weak as they merely assume that their position is true, without any argument or evidence, and then uses it to prove itself. It would be like saying that planets are mammals because planets are mammals. In a similar way, anyone advocating the opposite knowledge claim can be just as an “impressive” circular argument: planets are not mammals because planets are not mammals. Clearly, circular arguments do not produce knowledge. Finally, infinite regress cannot justify knowledge either, because you can never tell if a regress is truly infinite or will end up being based on arbitrary assumptions or circularity. Finally, you never ever get down to the foundation, so you can never tell if it is reasonable.
Thus, in the absence of non-arbitrary axioms, knowledge is not possible. This means that people who believe that all axioms are arbitrary have to give up science, math and logic. Most people are not willing to surrender these areas, and so they must reject their position that there are no non-arbitrary axioms.
Principle of explosion
As was argued earlier, it is possible to infer both a proposition P and its falsehood ~P from a set of arbitrary axioms. But if you can do that, then all statements are both true and false simultaneously. This follows from an argument known as the principle of explosion.
Assume that a specific proposition P is true. This can be the existence of Pluto, the notion that all cows are mammals or that cars are made of foam. It does not matter. If P is true, then we can construct the statement “P or S is true” where S is an arbitrary proposition. As long as one of P or S is true, the statement “P or S is true” is true.
If we allow contradictions (that is that ~P can also be true) then we can infer S because ~P implies that P is false, and “P or S is true” cannot be true if both P and S are false. Since we assumed that ~P is true, the P part in “P or S is true” is false, and so S must be true. But since S was an arbitrary proposition, we have now shown that contradictions imply that everything and its negation is true. Thus, it is not possible to distinguish truth from falsehood anymore.
Coherentism: the last-ditch case against non-arbitrary axioms
People who reject the existence of non-arbitrary axioms often appeal to a philosophical system of truth and justification called coherentism. It states that a belief B is true and justified if it is a part of, and mutually reinforced by, a coherent system of beliefs. But there are many fatal problems with coherentism. It is not clear what it means for a set of belief to cohere. Does it require logical consistency? If so, then coherentism is based on the non-arbitrary axiom of non-contradiction and coherentism is false. Any justification reduces to “consistency”, but consistency cannot be sufficient for knowledge. The easiest way to realize this is to see if you can convince someone of an idea simply by saying that it is consistent with your personal, arbitrary assumptions. It is possible to create two different systems that are coherent, but contradict each other. So coherentism lacks a method of distinguishing which one is true and which one is not. If they accept that both can be true, then they fall to contradiction. If they respect non-contradiction, they have assumed a non-arbitrary axiom. Thus, coherentists shamelessly borrow from foundationalists in order to avoid refutation.
The rejection of non-arbitrary axioms cannot be sustained. It is self-refuting. It eliminates the possibility of knowledge and it makes everything both true and false at the same time. Appealing to coherentism does not work because consistency is not sufficient for knowledge and there can be no model selection between two equally internally coherent, but mutually contradicting, systems.