Amy would say that nothing disconfirms that two plus two equals four. It makes sense to her, is consistent with her own beliefs and her experience, and she sees no way it could be false. Though “two plus two equals four” is not dependent on physical matters, its truth can be at least demonstrated with examples from the physical world. Amy could hold two crayons in one hand and two crayons in the other hand, and then put all the crayons on a table. When she counted them, she would arrive at the conclusion that there are four crayons, thus showing that in this case, two plus two does indeed equal four. Amy could go around for the rest of her life showing examples of “two plus two equals four” in the physical world, but she would also say that she does not need to, because the physical examples she is giving us are only representations of “two” and “four.” Even if there were no crayons, or indeed any objects to demonstrate with, Amy would say that two plus two is still four, because the numbers themselves are concepts. Let’s say we give Amy a physical example where two plus two seems to equal five. We hold two crayons in each hand, and we throw them down on the table so that one crayon breaks. When we count the pieces, we get five. Amy would tell us that in that case, we did not really add two and two; we simply broke one of the twos so that we were adding two and three, getting five. Maybe then, we show her another example. With no broken crayons and two in each hand, we put them down on the table, and then count. We arrive at five. Amy, after seeing that there was no way that either of the twos became a three, would simply say that we counted wrong. If she were to count our end product and arrive at five, she would insist that we had had three in one hand before. She would always believe that if the answer was five, there was some other explanation other than “two plus two equals five.” Being inconsistent with her logic, she would dismiss all contradictory claims, saying that her knowledge that two plus two is four is a priori and can’t be violated.

       Now we move to Basil. Similar to Amy, he believes that nothing can contradict that God exists, and he will continue to believe God exists no matter what the evidence. “God exists” is consistent with Basil’s experience and beliefs; he believes he sees evidence of God’s existence in his daily life, and he cannot comprehend a universe being created without a God. So, like Amy, he has reasons to believe that “God exists” is an a priori truth. If we give Basil a physical example that might point towards God not existing, he will be able to dismiss it. For example, if we asked God to give us a streak of lightning if he exists, and it didn’t happen, Basil would say that God is not obeying our commands for his own reasons, and that we don’t know what those might be. The fact that God didn’t listen to us does not prove that he doesn’t exist. Either he didn’t hear us, or he didn’t want to do our bidding. In my short story “Dear God,” a little boy writes in his letter to God: “Maybe you can help us but you won’t, right? Maybe you CAN do anything and you just choose not to.” Like this example, Basil says that God does not have to manifest himself to us at all in order for him to exist.

       Say we ask Basil why he believes in God. He’s likely to give concrete answers relating to the physical world; most theists do. They will say life must have begun somewhere, and thus God must exist to create it. But the existence of life does not prove the existence of God, it only proves the existence of life. Perhaps they will turn to the Bible, which obviously does not prove that God exists, only that the Bible says he does, and that the Bible exists. The fact is, there is no physical proof that God exists; everything that theists like Basil will point to can either be explained or assigned no specific explanation. (It is not necessarily true that just because we don’t know why something is, it means God did it.) There being no “proof” of God’s existence, Basil is then restricted to faith.

       Basil does not have any qualms about saying his belief in God is due to faith. Many say that proof denies faith. But what is his faith based on, if there are no physical proofs of God’s existence? If God does not intrude in the realm of the physical at all, he may still exist, but what does Basil have to show us that? He does not have the physical examples that Amy does with her mathematical belief, but not only does he not have that, his concept doesn’t necessarily make sense. It is not necessarily true that God exists. Amy’s claim that two and two are four is true because of the definitions of two and four, yet Basil doesn’t have that when defining God. God is defined as the one supreme being, the creator and ruler of the universe. But where in that definition does it say he exists? The answer is nowhere. Greek Gods such as Thor and mythical creatures such as unicorns can also be “defined” just like God can, but it does not necessarily guarantee that these things exist. Yet “two” does not have to physically exist in order to team with itself and get “four.”

       We go finally to Clara, who can’t be persuaded that an elephant is not in her apartment. For whatever reason, the idea that an elephant is in her living quarters makes sense to her, and she sees no way it could be false. We can give Clara a lot of reasons why there couldn’t physically be an elephant in her apartment, but she will refute them with equally ridiculous claims. If we claim that we can’t see any elephant, she might say he is invisible. If we claim that an elephant can’t fit in her apartment, she might say he is a small elephant. Here it might be wise to point out that “an elephant is in Clara’s apartment” is not a concept; it is a claim. “God” and “two plus two is four” are concepts, but “an elephant is in my apartment” is purely physical. Take away the elephant or the apartment and the claim loses its meaning. There is no abstract way to define “elephant in Clara’s apartment” as there is with “God” and “two plus two is four.” We can take both A (Amy’s proposition) and B (Basil’s proposition) out of the physical world and still have grounds for them; C (Clara’s proposition) holds no water outside her apartment. Despite this, we will examine it.

       Perhaps we shall ask Clara why in the world she believes there is this elephant in her apartment. “Well,” she replies, “I found its droppings in my kitchen! That’s how I know.” Using logic, we jump to the conclusion that somehow elephant dung got into Clara’s kitchen, but we don’t assume it was put there directly by an elephant. Perhaps it was the nasty boy next door who has a vendetta against her. Clara might insist there are also other reasons why the elephant must be there; maybe she hears it moving in the night. This could also be explained with possibilities much less outlandish than ones involving invisible elephants. In fact, it seems any physical proof of this elephant can be refuted. Say we get to the point where Clara has no more evidence for the elephant in her apartment, but she still insists it’s there because she “just knows it.” This is going to put her belief in the elephant into the same category as Basil’s belief in God, which we have already discussed. It goes into this category because without physical evidence, it goes on faith.

       How is A different from B and C? I have shown that B and C must turn to faith since their physical examples can be refuted, but what is different about “believing in” A? How can we say that we aren’t taking A on faith? Though A can be demonstrated with physical examples or absolute faith like the other two, it is also logically sound. To most of us, C seems ridiculous. Most of us would label Clara crazy for believing that the physical evidence available points to an elephant in her apartment, or we might label her crazy for believing without evidence or in the face of contradictory evidence. In Basil’s case, his belief in God can also be traced to faith, but most people find a belief in God far less outlandish than a belief in an elephant housemate. But is it less outlandish? Not really, given that they are based on the same principles. Though there are physical examples which could point to the existence of God, they could also, as I’ve said, be explained other ways; saying “God did it” is simply answering a mystery with a bigger mystery, exactly the same as the elephant. If Clara hears something moving in the night and then begins finding elephant droppings on her floor, attributing them to a phantom elephant is exactly the same as creating a God concept.

       So what is so different about Amy? We would say that “two plus two is four” is an a priori truth. She has physical examples whose results other than four can be traced back to bad counting or to replication between the time of counting groups of two and the time of counting the total. Logically, there has to be an explanation for where that fifth crayon came from if we started with two and two. We cannot simply answer it with the bigger mystery: “two and two is usually four, but once in a while it equals five.” This would mean we have no clue whether when we add two and two we are going to end up with four or five. It seems that if we look hard enough for the reason for the fifth object’s creation (assuming it isn’t obvious), we will either find it or give up but not give in. The explanation for the fifth crayon might even be unavailable to us. But we might leave it at that: two and two are still four, and even though we can’t find why it seems two and two are five this time, two and two are still four. “Two plus two is five” doesn’t have any evidence behind it, physical or otherwise, because the concept of two does not double into five. It is a matter of definition. Even if there were no such thing as amounts of things, these concepts have these definitions without application. Both God and the elephant have to exist somewhere in order for B and C to be valid; “two” and “four” do not have to exist anywhere for A to work.

       “Two plus two is four” doesn’t really explain anything. It is true of its own sake, and though more advanced mathematics can be based on it or found in the same way, it was not proposed as an explanation for a problem. Amy did not one day notice that there were four crayons lying around and thus assume that someone must have at one time put two and two together; well, she might have done that, but it is not her reason for deciding that two plus two is four. She believes it because of the definitive nature of the proposition, not because it is a good explanation for things she has seen. Most likely, Clara and Basil are using their claims to explain things; Clara using the elephant to explain the messes in her kitchen and the noises at night, and Basil using God to explain the existence of the universe. But suppose Basil and Clara just decided they had no evidence and no claims attached to their propositions. Suppose Basil is only claiming that God exists, not that he created the universe, and suppose Clara is only claiming that an elephant is in her apartment, not that it is leaving any evidence. Should we say that their belief is enough for truth, we would be condoning the giving of truth value to any harebrained statement. Therefore, we need evidence. Physical evidence isn’t enough, as it seems to be easily refutable. What is irrefutable, then? We can’t quite say logic is, since Amy believes two and two are four but Basil and Clara also believe in their statements. Just because “two plus two is four” makes sense to Amy, that does not make it true. God’s existence and the apartment elephant make perfect sense to Basil and Clara, respectively, for whatever reason. So what is it about “two plus two is four” that makes it irrefutable while B and C are not?

       Again, we come back to definitions. Why does Basil believe in God? And is that belief based on necessary truths? When we examine the definitions in Basil’s belief, we find flaws. God, as I said before, is defined as the one supreme being and the creator of the universe. But the definition does not include his existence. Going to the definition of “universe,” we find that it also, by definition, needs no creator. The universe is all known and unknown objects and the phenomena they engage in, all existing things. But it is not necessarily “created by God”; actually, there is a bit of a contradiction there. God is the creator of the universe, yet the universe includes all existing things. If God created the universe, he must have pre-dated it . . . yet if all existing things are in the universe, how could he exist if at one time he wasn’t in the universe? Digressions aside, we realize that the definition of God does not necessarily imply existence. We can take definitions to Clara’s elephant too. What’s an elephant? As far as I know, they aren’t tiny or invisible. In order to believe there is an elephant in there, Clara would have to rearrange the definition of elephant to mean “an invisible, imperceptible something that lives in my apartment.” This is hardly “elephant” anymore, but say it is now “Clara-elephant.” If Clara-elephants are by nature invisible and imperceptible, there is no way to know that they exist (since we can’t perceive them), so why believe? Clara has no better reason to believe in her elephant roommate than she does in Martians that invade her bathtub. And of course, as I said before, we can’t take the elephant (or the Clara-elephant) out of the context of elephant-ness or Clara’s apartment without losing the validity of the claim, so it has to remain in the physical.

       Amy’s definitions of “two” and “four” are the only ones that hold up. Amy’s claim makes the other two claims look backwards. When faced with physical evidence against her claim, Amy could simply say there is no explanation for why two and two seemed to come out as five that time. Basil and Clara, on the other hand, would have to look for evidence to support their claims in order to preserve them. The difference is, Amy’s is true until proven wrong, while Basil’s and Clara’s are wrong until proven true. And because of the nature of Amy’s claim, it can’t be proven wrong because there always might be another explanation other than “two plus two equals five.” With Basil’s and Clara’s, there is always another explanation other than “God exists” or “there is an elephant in Clara’s apartment.” Both Basil and Clara can be refuted, or, at least, offered possibilities that can’t be entirely ruled out. If they refuse to accept that their beliefs aren’t 100% impervious to error, they are simply clinging to their beliefs petulantly. Amy is another matter; she might actually be clinging to “two plus two is four” in the face of conflicting evidence, but if she is offered that conflicting evidence, she can show that what is actually conflicting is the definitions. If someone asserts that two and two are five, she can reply that they are not talking about the same thing anymore. Since the concept of “two” is defined as “one plus one,” and “one” is defined as “a single unit,” we can see that “one plus one” plus “one plus one” is indeed four. In order to get otherwise, we would have to change a definition in there somewhere. And if we did, we would again be talking about something other than one, two, or four. As we have seen, Clara can’t manipulate “elephant” to mean something in her apartment that is invisible and imperceptible and still make sense, and Basil can’t redefine God as necessarily existing. Given these arguments, it seems that in order to be a priori knowledge, something must be irrefutable in the context of definition.