Proving Negatives

This notion contradicts our standard epistemological principles. In the strictest sense, proof is only found in mathematics. Typically, when we refer to "proof," we mean "evidence-based support." In mathematics, we can demonstrate nonexistence either directly or via proof by contradiction. In the empirical realm, a similar approach is taken through statistical methods known as hypothesis testing.

To grasp hypothesis testing, we must first comprehend proof by contradiction. This involves assuming the truth of the statement we wish to refute (let’s call it p) and then using established knowledge to derive a conclusion that we know to be false. Since our logical system must remain consistent, we conclude that p must be false.

However, the assumption that p holds true is only significant within the context of our proof. If we do not reach a contradiction, we cannot make any claims about p. This means that if we derive logical inferences from our assumption p without encountering a contradiction, we cannot determine whether p is genuinely true or false; we must simply acknowledge this uncertainty.

In the empirical world, the situation is not as straightforward, but that is acceptable. Instead of assuming p, we start with a related hypothesis we wish to disprove, termed H0. We aim to demonstrate that, assuming H0 is true, the probability of our observations being accurate is low. Since our observations are indeed true, we are likely to encounter a contradiction. Essentially, hypothesis testing is a statistical adaptation of proof by contradiction.

Given that our null hypothesis is akin to our assumed statement p, the truth of the assumption is relevant solely within the framework of our hypothesis test. Just like in proof by contradiction, if we do not identify a contradiction, we cannot make any statements about our null hypothesis.

To summarize, we set our null hypothesis to whatever we seek to disprove. We then attempt to demonstrate that our observations are improbable under our assumed hypothesis. If we succeed, we can reject the null hypothesis; if we fail, we cannot make any claims regarding it. Moreover, there is no "null hypothesis in life"; it only holds meaning within the confines of hypothesis testing.

Applying this to the concept of proving negatives: If we start with the assumption of existence for a particular entity and possess a list of expected observations under that assumption, we can conduct a series of observations to demonstrate that such observations are unlikely if existence were true. This does not absolutely disprove existence, but if our observations are improbable given the assumption of existence, we can be reasonably confident in asserting nonexistence.

For a more in-depth discussion on proving negatives, consider reading Thinking Tools: You Can Prove a Negative by Steven D. Hales, Ph.D., or explore his various works on logic.

Burden of Proof

The burden of proof is another significant topic. Many believe that only existential claims carry this burden. Indeed, I would argue it is a burden in itself. An argument often put forth is that negative claims cannot be proven, which I have already addressed. If we assume that negative claims lack a burden of proof, we find that the statement "no claim suffers from a burden of proof" is, in itself, a negative claim. Consequently, it would be up to the opposing party to demonstrate that any claim bears a burden.

Moreover, we can typically relate claims of existence and nonexistence to each other. For example, the statement "there is a god" implies that "a universe without a god does not exist." While this may seem convoluted, it holds true. If a negative claim implies the existence of a god, how can the assertion "there is a god" be held to a burden of proof while the negative claim is exempt? This reasoning is illogical; we would perpetually seek equivalent nonexistence claims to escape our burden.

Ordinary vs. Extraordinary Claims

A common approach to discern the burden of proof revolves around whether a claim is ordinary or extraordinary. This perspective is somewhat valid; commonly accepted positions are generally easier to embrace. Similarly, claims aligned with foundational axioms are typically more readily accepted.

However, challenges arise. What constitutes ordinary or extraordinary varies by context. Different groups will hold diverse opinions regarding what claims are considered ordinary or extraordinary. Additionally, while this strategy is frequently employed by naturalists to evade burdens associated with naturalistic claims, the majority view leans toward the belief in some form of supernatural entity or deity. Thus, if we classify extraordinary claims as those less frequently asserted, naturalism itself could be considered extraordinary.

The Elephant Analogy

Consider the scenario where I aim to prove my neighbor does not have an elephant in their backyard. I start by recognizing that elephants are large, noisy, and leave noticeable footprints, particularly in wet conditions. Is this an absolute truth? Not necessarily; it is possible that an elephant could exist without these characteristics. However, based on our observations, we infer these traits.

To reinforce my argument, I ask my neighbor if they agree with these statements. If they do, I can proceed, establishing a solid foundation.

With this foundation in place, I can observe my neighbor's yard. I find no large animals visible over the fence, hear no sounds from their yard, and see no footprints, even after recent rain. These observations serve as a secondary layer, leading to the conclusion that my neighbor likely does not have an elephant.

Backward Chaining

Humans typically find it straightforward to draw conclusions from a set of assertions by working forward. In computer science, however, it can be advantageous to work backward, beginning with the conclusion and using various initial assertions to determine its truthfulness. This method is referred to as backward chaining and is a valid approach.

Unanchored Arguments

The intriguing aspect arises in the context of religious claims. Many arguments begin with a conclusion asserted as true, then attempt to validate that conclusion without a solid foundation. A claim is made, which leads to the conclusion's validation, but often lacks substantiation. When pressed, the individual searches for another claim to support the previous assertion, creating a chain without a solid anchor. I refer to this as unanchored backward chaining.

Validity of Use

This does not imply that unanchored backward chaining is inherently "wrong." It can serve as a useful investigative tool to uncover anchor links and assess the reasonableness of claims. However, it should not be employed within formal arguments, as it is too easily undermined. Once an anchor is established, an opponent must break a link in the chain to challenge the conclusion. In contrast, with an unanchored chain, the opponent only needs to reject the latest link, leading to potentially endless searches for supporting evidence.

Absence of Evidence

The absence of evidence does not equate to evidence of absence. Just because we fail to find evidence for something's existence does not mean it does not exist. To demonstrate absence of evidence, we must first define what would constitute evidence of existence, then conduct observations to show those observations contradict the existence assumption. This brings us back to hypothesis testing.

For example, I cannot claim that the wind likely does not exist merely because I cannot see it. There is no expectation for me to see the wind, even if it is present. Similarly, I cannot assert that the lack of moving leaves indicates wind absence if no trees are around. Failing to establish reasonable expectations for existence and conduct appropriate observations means we cannot make claims about nonexistence.

Occam's Razor

Occam's razor serves as a philosophical tool designed to simplify arguments and positions. Typically, razors are probability statements: a claim is deemed less likely to be true if certain conditions are met. However, Occam's razor does not adequately address "truth." William of Occam initially sought a tool to support his arguments for god's existence. Although various formalizations of Occam's razor exist, none definitively clarify the probabilities of competing theories. It serves merely as a tool to guide predictions.

To apply Occam's razor effectively, we compare two competing models consistent with a given dataset. The model with fewer assumptions is preferred, as additional assumptions must be valid for the model to be sound. However, we cannot ascertain the probability of any assumption's validity, leaving us uncertain about the likelihood of either model's correctness.

Dr. Yaser S. Abu-Mostafa has authored several books on machine learning that delve into this topic, including Learning from Data.

Probability and Unlikelihood

The terms "probably" and "unlikely" are frequently employed to evade the burden of proof. Statements like "There probably is a god" or "gods are unlikely" still represent claims. All claims carry a burden of proof, and one cannot disprove a claim simply due to a lack of evidence. Absence of evidence does not make a claim unlikely. We cannot assert that someone "probably beats their wife" or "probably lacks an education" without absurdity. The same principle applies here. A statement of probability necessitates information about the probability space, which can only be derived from constructs or evidence.

Proof by Obviousness

This methodology appears in various contexts. Theists may assert that god's existence is self-evident due to the universe's existence. Sometimes this "obviousness" is implied rather than explicitly stated, such as when there is a direct leap from "the universe exists" to "there is a god."