Arguments from Silence

For any given domain of inquiry, some argument structures and patterns tend to be more central than others. For example, statistical sciences tend to emphasize arguments about causality and correlation. Arguments from precedent tend to frequently arise in legal domains, especially within the context of legal interpretation. Arguments using analogies tend to underpin many situations where we need to evaluate some situation in terms of criteria used to evaluate related situations. You can get a sense of an arguments prominence by googling it and observing what is returned. For example, argument from precedent seems to exclusively return results from legal domains. Searching for "correlational arguments" will return a plethora of results related to data analysis. But strangely, "Argument from Silence" returns quite a number of articles from apologetics websites. This is not obvious at first glance, especially if you are unfamiliar with the apologetic-industrial complex. Afterall, search engines use algorithms such as PageRank to return relevant search results from their index. Ranking algorithms use a number of heuristics when deciding what to return, such as:

• Relevance: How closely the content matches the query terms.
• Quality of Content: How valuable or trustworthy the content is, often inferred from links, authority, and user engagement.
• Page Authority: Based on the number of other reputable sites linking to the page (backlinks).
• Freshness: Some searches prioritize recent information, like news, while others don’t require the latest updates.

So the question is: why apologetics, as opposed to something more directly relevant like fields where the argument structure is commonly used? Why are so many sites linked to these apologetics pages and not academic pages? Why are apologetics websites mentioning it so much, and why are people linking to these sites so much? I think there are clear answers to these questions. As I will expand upon further, what we are observing is an example of the Fallacy Fallacy, along with a peculiar strand of motivated reasoning. Before I talk about that, I need to explain what this "argument from silence" even is, and when we inferences from this structure are considered plausible. 

As alluded to above, others are interested in this structure of argument. It arises very frequently within the historical analysis of classical texts. Lets take a look at the argument from an informal perspective and then something more rigorous. 

• P1: If some event occurred, or some hypothesis about the historical record is true, then we would expect to observe evidence indicating its occurrence
• P2: We do not observe any evidence
• C: Therefore, the hypothesis is false, or the event did not occur.

In The Argument from Silence, John Lange cites three major components of these arguments:

1. An extant document D in which no reference to an event E appears.
2. It is known that the intention of the author of document D was to provide an exhaustive list of all the events in the class of events to which E belongs
3. Event E is assumed to be a type of event which the author of D would not have overlooked, had the event taken place.

So we can see here that the argument is attempting to show Evidence of Absence; an author is expected to have generated content about the event under discussion, but did not. Presumably, this lack of evidence is explainable by the event not occurring. According to UMass historical methods, evidential silence refers to:

"Silence" means that the thing in question (call it X) is not mentioned in the available documents. If it were mentioned, then with the usual qualifications it would be proved to exist. Since X is not mentioned, X cannot be proved to exist. A natural further inference from this evidence is that X did not exist. The basic point is that if X did not in fact exist, then the only trace which that fact could leave, in the evidence, is the silence of the evidence as to X. At the same time, any such conclusion must be provisional. If documents are later found that do mention X, then X is after all proved to exist. A single positive may overturn any number of negatives. A single sound refutes all silences.

The possibility of such a future positive can never be ruled out. But until it occurs, the non-existence of X is the best inference from the absence of X in the evidence. The strength of that inference in a given case will depend on (1) how many documents there are, or in statistical terms how large the sample is, and, in literary terms, (2) how likely the thing is to have been mentioned in documents of that type in the first place. We might explore these concepts just a little 

... the argument from silence, like all historical arguments, is always conjectural. But it is not, as some claim, a fallacy. It is the correct default inference from silence. That inference can be strengthened by relevant evidence of a positive kind, or by the continued silence of further evidence.

More generally, the argument from silence (argumentum ex silentio) is a reasoning technique based on the absence of statements or evidence in a particular context. It suggests that if a source that should logically mention a particular fact, event, or entity does not, then that fact, event, or entity may not have existed or occurred. Here's the basic structure of the argument from silence:

• Context Establishment :Identify a source or set of sources (such as a text, record, historical account, or authority) that would reasonably be expected to mention a certain fact, event, or detail if it were true or relevant.
• Expectation of Mention: Argue that, under normal circumstances, if the fact or event in question had occurred, then this source (or sources) would likely have included it.
• Observation of Silence: Point out that the source is silent or does not mention the fact, event, or detail being discussed.
• Inference from Silence: Conclude that because the source does not mention it, the fact, event, or detail likely did not occur, or the entity did not exist.
• Limitations and Counterarguments: The information may have been omitted due to irrelevance to the source’s purpose. The source may not have had access to the information. The fact may have been considered too well-known to mention explicitly.

The argument from silence is generally considered weak if there is no strong expectation that the source should contain the information or if there are plausible alternative explanations for the "silence". 

This type of inference relies upon something like the Closed-World assumption; what is not currently know to be true is inferred to be false. It uses a presumptive argumentative structure relying on default logic to arrive at a conclusion. In other words, the inference is non-monotonic and therefore defeasible. Strictly speaking, from the perspective of classical logic, the argument is invalid. Practically speaking, it follows a common form of plausibilistic reasoning most people use every day. For example, if there is no evidence that my wife is cheating on me, I infer from that absent evidence that she is not cheating on me. This argument is usually mentioned in relation to historical arguments but the general pattern is observed in many scenarios characterized by default reasoning.

This form of reasoning is closely connected to the idea of the Burden of Proof. The idea is pretty straightforward; if you have not established evidence for some hypothesis then it's unreasonable to accept it, unless there are extraordinary extenuating circumstances that shift the presumption in favor of the hypothesis lacking evidence. It is unreasonable to accept a conclusion lacking evidence, unless by default we find it more probable than the alternatives which have evidence in their favor. I am describing prior likelihood and Bayes factors, which I'll elaborate on later. In the context of this argument, we have two scenarios: H is true or H is not true, where H can be some hypothesis about some historical event. If H did not occur, then we obviously would not expect to see evidence implying H. But if H did occur, we would expect to discover evidence confirming H. If we have not seen any evidence, then we infer not H.  This can be interpreted as the default rule. The proposition H has the burden of production. If nothing is produced, we infer that evidence is not producible, because not H is true. This argument becomes stronger if H is expected, but does not manifest. In other words, there are scenarios where H can be true, but E is not produced for extenuating circumstances such as suppression of evidence. If those circumstances are ruled out and it's highly expected that E would manifest under the assumption of H, this serves as a strengthening condition that not H is true. 

In Burden of Proof, Presumption, and Argumentation, Walton describes two senses of "Burden of Proof", as described by Wigmore:

Wigmore ( 1940 , 270) drew a distinction between these two meanings of  burden of proof. The first one he called the risk of nonpersuasion. Wigmore offered the following example (271) from “practical affairs.” Suppose A has a  property  and  he  wants  to  persuade  M  to  invest  money  in  it,  while  B  is opposed to M’s investing money in it. A will have the burden of persuasion  because unless he persuades M “up to the point of action,” he will fail and  B will win. Wigmore went on to show how the burden of persuasion works  in litigation, in a way similar to that of practical affairs, except that the prerequisites are determined by law (273), and the law divides the procedure  into stages (274). The second meaning is called the burden of production.  It refers to the quantity of evidence that the judge is satisfied with to be considered by the jury as a reasonable basis for making the verdict in favor of  one side (279). If this is not fulilled, the party in default loses the trial (279).  According  to  Wigmore  (284),  the  practical  distinction  between  these  two   meanings  of  burden  of  proof  is  this:  “The  risk  of  nonpersuasion  operates   when the case has come into the hands of the jury, while the duty of producing evidence implies a liability to a ruling by the judge disposing of the issue  without  leaving  the  question  open  to  the  jury’s  deliberations.”

I think historical reasoning resembles legal reasoning to a large extent. In the situation expressed by Wigmore, if you have not produced evidence, or if the evidence is somehow insufficient, the judge wouldn't even present it to a jury. In the context of historical reasoning, if you have some hypothesis but have not established evidence for it, historians are probably not going to consider it. I also bring this up in relation to Arguments from Silence because there is a crucial point that needs to be made. In the context of a legal trial, a defendant might fail the burden of persuasion, because their argument from silence does not consider a wider scope of possible evidence. A defendant relying on silence must convince others that the absence of evidence is itself meaningful and not merely coincidental or due to missing records. If the silence does not meet the burden of persuasion, the argument fails. For instance, arguing that someone didn’t commit a crime because no eyewitness mentioned them being at the scene might fail if other types of evidence (e.g., physical evidence) exist. I'll elaborate on this more later because there is a paper I want to criticize. The idea is that if you are merely asserting the non-expectation of evidence while ignoring alternative sources of evidence, you are reasoning fallaciously. 

In summary, the argument often relies on presumptive reasoning, which itself interacts with the burden of proof and proof standards. There are a few things to consider for an argument from silence to be compelling. A claim based on silence presumes that if something had occurred, evidence of it would exist. The opposing side can challenge the significance of the silence, showing alternative reasons for the lack of evidence (e.g., poor record-keeping, loss of sources over time). For example, if a historian argues that a figure’s silence on a major war indicates disinterest could be countered with the rebuttal "the silence could result from lost correspondence, fear of censorship, or irrelevant context", which itself would require evidence to justify. The argument’s weight depends on how reasonable it is to expect evidence. Silence is more compelling if Evidence should exist given the circumstances (e.g., silence on an event in a well documented era is more meaningful than in a poorly recorded one) and the absence is anomalous or surprising (e.g., official records omit a highly significant event).

Before I move on to a formal analysis of this argument, I want to first consider what "Evidence" even is. This usually seems to be taken for granted, but I think is crucial for understanding evidential arguments. In "The Evidential Foundations of Probabilistic Reasoning", David Schum argues that something must become evidence through the argumentative process of establishing it as such. Contrasted with the mere probability account of evidence, that is, evidence is merely that which raises the probability of some hypothesis, it seems like, silent evidence depends on something being recognized as evidence. For example, someone might say that E is evidence for H, where E is some historical document. But I could easily counter by saying that E is unreliable as evidence for H, or that its irrelevant. This is not trivial. Have you ever stopped to consider why something like hearsay is inadmissible in a court? What exactly constitutes hearsay and why does that classification render it irrelevant to a claim? Many people tend to be moved by hearsay! David Schum's argument establishes a critical distinction between evidence as a product of argumentation and the probabilistic account of evidence. This contrast has significant implications for how we understand silent evidence and the conditions under which something is deemed "evidence." Schum contends that something becomes evidence only through a process of reasoning and argumentation that establishes its relevance and reliability. This view emphasizes the social, interpretative, and justificatory nature of evidence. Evidence is not inherently self-evident; it requires interpretation, contextualization, and debate to be recognized as evidence. For example, a historical document might be relevant to a hypothesis (H), but it needs to be scrutinized for authenticity, contextual accuracy, and how it supports or refutes H. The argumentative process allows for counterclaims; E might be irrelevant, unreliable, or even misleading. For instance, a diary entry stating, "I saw John at the meeting" could be argued against as unreliable (the author might be lying or mistaken) or irrelevant (if the meeting isn't tied to H). Silent evidence—things that are missing, unrecorded, or overlooked—depends on their recognition as missing or significant through argumentation. In contrast, the probability account defines evidence as anything that raises the probability of a hypothesis being true, regardless of how it is interpreted or contextualized. In this view, evidence (E) is linked to a hypothesis (H) by a probabilistic rule: P(H∣E)>P(H). If observing E increases the likelihood of H, then E is considered evidence for H. It is purely based on conditional probabilities. Unlike Schum's approach, the probability account doesn’t require that E be argued as reliable or relevant. If it raises $P(H)$, it counts as evidence. In other words, the probability account identifies something as evidence based on conditional probabilities, but Schum's approach considers whether some piece of evidence should even be introduced into the dataset. I bring this up because P(~E|H) crucially depends on how we characterize something not being evidence. An interlocutor might claim that evidence is not absent, because of some observation E. However, if we reject E as irrelevant, and there are no alternative sources of evidence, the situation reduces to ~E. I think this is a limitation of the basic probability approach to evidence; it does not explicitly account for how something becomes evidence in the first place. This implies that our probability assessments are dependent on a dialectical process that establishes an observation as credible and useful in a probability calculation. Silent evidence illustrates the strength of Schum's approach in contrast to the probability account. For silent evidence to matter, someone must recognize that something is missing and argue for its relevance. In historical reasoning, the absence of records of a supposed major event might be evidence that the event did not occur. This depends on establishing that such records would normally exist if the event happened, which occurs through argumentation. In addition, Silent evidence often challenges the apparent probative value of present evidence. For example, "E is evidence for H" might be countered by arguing that silent evidence (the lack of corroborating documents) undermines E’s reliability. In Schum’s framework, this argumentative process determines whether E survives as evidence. Silent evidence requires an interpretative step to explain why the absence of data is meaningful. This is important, because as we will see later, many people misunderstand the argument from silent evidence by making ridiculous assertions that "there is no evidence for many common people existing in history, but we know they existed." Yes, and we know this through the evidence of persistent and reoccurring human reproduction.

Lets now move on to the formal Bayesian analysis. To provide a Bayesian analysis of the argument from silence, we analyze the odds ratio using the relevant probabilities. The goal is to evaluate how the absence of evidence (−E) affects the likelihood of a hypothesis ($H$).

Definitions

• $H$: The hypothesis that the event or entity in question exists/occurred.
• $-H$: The negation of $H$ (i.e., the event/entity does not exist/occur).
• $E$: The evidence we expect to observe if $H$ is true (e.g., a record or mention in a source).
• $-E$: The absence of that evidence.

Bayesian reasoning assesses the posterior odds as:

$$\text{Posterior Odds of } H = \text{Prior Odds of } H \times \text{Likelihood Ratio},$$

where the likelihood ratio is:

$$\text{Likelihood Ratio} = \frac{P(-E \mid -H)}{P(-E \mid H)}.$$

Here’s how these terms apply to the argument from silence:

Step 1: Assign Probabilities

1. $P(-E \mid H)$ The probability of silence (absence of evidence) given that $H$ is true.

   • This depends on how likely the evidence would be recorded or observed if $H$ were true. If silence is unlikely (evidence is expected), $P(-E\mid H)\; will\; be\; low.$

2. $P(-E\mid -H):\; The\; probability\; of\; silence\; given\; that$$H$ is false.

   • This depends on the context. If no evidence would arise regardless of $H$’s truth, $P(-E\mid -H)\; will\; be\; high.$

3. Prior Odds $P(H) / P(-H)$ Our initial belief in the likelihood of $H$ vs. $-H$.

Step 2: Likelihood Ratio

The argument from silence is strongest when:

$$P(-E \mid -H) > P(-E \mid H).$$

• If evidence is expected when $H$ is true but not when $H$ is false, the absence of evidence strongly supports $-H$.
• If $P(-E \mid H) \approx P(-E \mid -H)$, silence is not informative.

Step 3: Odds Update

Using Bayes' theorem, we update the odds:

$$\text{Posterior Odds of } H = \frac{P(H)}{P(-H)} \times \frac{P(-E \mid -H)}{P(-E \mid H)}.$$

$$$$

Here are some general implications of the argument:

1. Argument Strength: The argument from silence is stronger when $P(-E \mid H)$ is low (evidence is highly expected if $H$ is true).
2. Limitations: If $P(-E \mid -H)$ and $P(-E \mid H)$ are similar, the absence of evidence is weakly informative.
3. Uncertainty: Prior beliefs ($P(H)$) play a critical role in how strongly silence updates our confidence in $H$.

Here is a fictitious example:

• $P(H)$ Prior probability of $H$ (e.g., the event occurred) = 0.5.
• $P(-H)$ Prior probability of $-H$ = 0.5.
• $P(-E \mid H)$: Probability of silence if $H$ is true = 0.2.
• $P(-E\mid -H):\; Probability\; of\; silence\; if$$H$ is false = 0.8.

Likelihood Ratio:

$$\text{Likelihood Ratio} = \frac{P(-E \mid -H)}{P(-E \mid H)} = \frac{0.8}{0.2} = 4.$$

Posterior Odds:

$$\text{Posterior Odds of } H = \frac{0.5}{0.5} \times 4 = 4.$$

Thus, the posterior probability of $H$ given $-E$ decreases significantly because the silence is more likely if $-H$ is true. By comparing $P(-E\mid H)$ and $P(-E\mid -H)$, this Bayesian framework provides a systematic way to evaluate arguments from silence.

This formalism makes it clear what we are trying to do when arguing from silence. We are essentially claiming that the likelihood ratio is significantly greater than 1. I wrote this down because I found an article that I think is deeply problematic, highlighting probability trickeries that can be smuggled in if you're not aware. The Bayesian framework helps us make explicit our reasoning process, but can be misleading if you're unfamiliar with this epistemology.

In "The Argument from Silence" by Dr. timothy McGrew, describes the structure of the Argument from Silence using the Bayesian formalism. The first 10 pages or so are used to establish the argument structure. On page 13 the author establishes criteria for grading the argument:

Third, the strength of an argument from silence can be measured in terms of the ratio of these likelihoods, P(~E|~H)/P(~E|H). This mathematical fact has three consequences: (a) there is no upper limit on the strength of arguments from silence, since that ratio approaches infinity with a positive numerator as the denominator shrinks toward zero; (b) when the two likelihoods are equal—that is to say, when we expect ~E equally strongly whether or not H is true—the argument is completely forceless; and (c) when there is not a very high expectation of the evidence on the assumption that the event had occurred, that is, when P(E|H) is rather small, say less than 0.5, the denominator of the likelihood ratio, which is equal by definition to 1 – P(E|H), will be rather large, in this case greater than 0.5; and as the numerator can be no greater than 1, the argument from silence will have very little force.

I want to focus on (c), because while mathematically accurate, establishing the magnitude of P(E|H) is not trivial. This can be artificially inflated or deflated, depending on the objectives of the reasoner. This is where the author establishes criteria for assessing that probability:

1. If H (the event or fact in question) were true, how probable is it that the author in question would have noticed it (N)?
2. If H were true and the author had noticed it, how probable is it that he would record it (R)?
3. If H were true, and the author had both noticed and recorded it, how probable is it that this record would have survived and that contemporary historians would be aware of it (S)?

In more formal terms, these three questions amount to a request for three numbers: 

P(N|H), P(R|H & N), and P(S|H & N & R). Since (N & R & S) entails E, we can approximate the critical value P(E|H) by the product 

P(N|H) × P(R|H & N) × P(S|H & N & R) 

noting that this is equivalent to P(N & R & S|H), which in turn must be less than or equal to P(E|H). 

This is established by the author on page 13. Again, this is mathematically accurate if 1-3 are exhaustive and independent. By the rules of probability calculus, this joint conditional can be broken down to a product using the chain rule. And this is where the problems begin to emerge. The author correctly states that if any conditional probability is low, then the total product will be low. This will have implications for Bayes factor, since P(E|H) being low means P(~E|H) will be high, which means the ratio is closer to one, and hence the argument from silence has no force. The author continues on page 15 arguing why they think there are reasons a well informed author would omit information in a historical account. But why should we think that 1-3 are independent? Should we think that there are alternative conditions not listed that would inflate P(E|H)? Doesn't seem a bit narrow to consider a single author? Doesn't the third condition seem a tad bit artificial?

The author somewhat acknowledges this when he states that P(N & R & S|H) is less than or equal to P(E|H). After all, if the conditions they listed are exhaustive, then by definition P(N & R & S|H) = P(E|H). That is, the conditions are equivalent to the evidence. However, as I've alluded to above, I don't think there is good reason to assume that P(N & R & S|H) exhausts the set of all evidence relevant to the hypothesis. I think that, P(E|H) = "A bunch of other stuff" + P(N & R & S|H), and that the conditions implied by P(N & R & S|H) are not independent. I also think that by stating P(N & R & S|H), the author is arbitrarily stipulating events that should be aggregated together, and is therefore arbitrarily deflating P(E|H) to reduce the credibility of arguments from silence. By narrowing the analysis to a single author/source and placing somewhat restrictive conditions, the author of this article is artificially deflating P(E|H) and therefore inflating P(~E|H). 

There a tendency for people to underestimate the probability of something by arbitrarily adding many conditions, assigning one of them with low value, so that the joint probability is low. This is the deliberate use of the Conjunction Fallacy. When people arbitrarily add conditions to an event (even if some are unnecessary or unlikely), the joint probability of all conditions being true becomes smaller because:

$$P(A \text{ and } B) = P(A) \cdot P(B | A)$$

or, if the events are independent:

$$P(A \text{ and } B) = P(A) \cdot P(B)$$

Every additional condition multiplies the probability by a factor, so the more conditions you add, the smaller the joint probability becomes. For example:

• Event $A$: It rains tomorrow ($P(A) = 0.5$).
• Event $B$: A bird lands on your window ($P(B) = 0.1$).
• $P(A \text{ and } B) = 0.5 \cdot 0.1 = 0.05$.

If you keep adding conditions, such as "and I win the lottery," the joint probability quickly becomes negligible.

People may have the tendency to underestimating probabilities via low-value assignments in the subjective Bayesian framework,. When people assign a low probability to one of the conditions (even arbitrarily or incorrectly), it disproportionately impacts the perceived joint probability, making the overall probability seem implausibly small. This happens for a variety of reasons. People tend to focus on the most unlikely condition in the chain, leading them to ignore the base rates of the primary event. The representativeness heuristic also plays a role: people imagine detailed scenarios but fail to realize that adding conditions reduces likelihood. For example, Suppose the hypothesis is: "Someone notices a rare cosmic event." People might think:

• "They would need a telescope (low chance)."
• "They would need to be awake at that time (low chance)."
• "They would need to report it (low chance)."

Even if the base rate of someone noticing a rare event is $P(H) = 0.2$, adding arbitrary low-probability conditions can lead to $P(H)$ being perceived as near zero. The conjunction fallacy arises when people believe the probability of a detailed event is higher than that of a simpler one. In reality, adding conditions always reduces the probability. The classic example being:

• Scenario A: "Linda is a bank teller."
• Scenario B: "Linda is a bank teller and a feminist."

People often perceive Scenario B as more likely because it feels more "representative," but mathematically:

$$P(\text{Bank Teller and Feminist})\le P(\text{Bank Teller})$$

This happens because people over-focus on the added details and fail to recognize the constraints they impose on probability. They also often treat dependent events as independent or vice versa. Or they undervalue the base probability of the primary event. If we assume P(N) is very low, because we have implicitly defined P(N) to be the joint condition of extremely specific low probability events, then P(E|H) will end up being very low. For example, if I am implicitly assuming that P(N) is the joint condition "They were awake", "They had eyes", "They had a pen", "they had paper", and "They knew how to write", I could be radically deflating P(N). This is partially why philosophers use of Bayes Theorem drives me crazy; it seems rather arbitrary how we can measure P(N) in a reasonable way without base rates. 

Arbitrarily adding conditions and assigning one of them a low probability often leads to underestimating the overall probability. This bias is a common trap in probabilistic reasoning and decision-making, which I think manifests in the article, and has implications for how people reason about silent evidence. In the above scenario, the author is arguing that P(-E|H) is very high, on account of P(N,R,A|H)=P(E|H) being low, but they are arbitrarily selecting these extra conditions. This argument is a manifestation of probability dilution, where adding unnecessary or overly specific conditions artificially reduces the likelihood of an event. More specifically, It just seems that N is redundant. It also seems like S depends on a plethora of other conditions. Lets analyze this step-by-step using the Bayesian framework:

1. N (Awareness) Redundant

If the ultimate goal is to analyze $P(E|H)$ or $P(-E|H)$—the probability of observing or not observing evidence given the hypothesis—then $N$ (awareness) might indeed be unnecessary.

Why $N$ Could Be Redundant:

• Evidence Is Ultimately Tied to Writing ($R$): Whether someone was aware of a historical fact ($N$) only matters if they wrote it down ($R$). If no one wrote it down, $N$ has no impact on whether we observe evidence $E$.
• If the chain of reasoning always requires $R$, then: $$P(E|H) = P(R, A | H) = P(R | H) \cdot P(A | R, H)$$ $N$ can be omitted because awareness alone doesn’t produce observable evidence.
• $N$ would only matter if there’s a direct pathway where awareness alone could create evidence—for example, if awareness led to oral traditions that later became written records. But if you're focusing strictly on $R$ (writing) and $A$ (preservation), $N$ may add unnecessary complexity.

2. S (Survival of Writing) Depends on Many External Factors

$A$ (the survival of writing through history) is influenced by numerous independent and external factors. These factors could make the probability $P(A|R, H)$ seem arbitrarily low if the dependencies aren’t accounted for properly.

Why This Matters:

1. Complex Dependencies Inflate Uncertainty:

   • Historical survival depends on many unrelated events: wars, natural disasters, decay, or random loss of records. Modeling all these factors directly is almost impossible, so $P(A|R, H)$ can feel arbitrarily small.
   • This small value for $P(A|R, H)$ might unfairly dominate the overall $P(E|H)$, even if $R$ (writing it down) was highly probable.

2. The Fallacy of Over-Specification:

   • Treating $A$ as a single, unified condition hides the fact that $P(A|R, H)$ is actually a conjunction of many events, each of which reduces the total probability: $$P(A|R, H) = P(A_1 \cap A_2 \cap A_3 \dots | R, H)$$ where $A_1, A_2, A_3, \dots$ represent conditions like physical preservation, political continuity, and access to historical archives.

3. Alternative Pathways for Evidence:

   • If evidence can survive through indirect means (e.g., copies, translations, or secondary references), then modeling $A$ solely as the survival of the original writing may be overly restrictive.

Rather than modeling all these conditions explicitly, we might use a broader probability for $P(A|R, H)$ that reflects survival through any reasonable pathway (not just the original artifact).

3. Simplify the Representation

If $N$ is redundant, the probability of observing evidence $E$ depends on two key components:

1. $R$: The fact is written down.
2. $\mathrm{S:\; The\; writing\; survives\; through\; history.}$

$$P(E|H) = P(R, A|H) = P(R|H) \cdot P(A|R, H)$$

Probability of No Evidence:

$$P(-E|H) = 1 - P(E|H) = 1 - \left(P(R|H) \cdot P(A|R, H)\right)$$

4. Dealing with the Complexity of S:

Rather than modeling $A$ as a conjunction of highly specific and improbable events, treat it as a broader, aggregated probability:

• $P(A|R, H)$: Represents the likelihood of written evidence surviving through any pathway, not just direct preservation.

For example:

• $P(A|R, H)$ could account for:
  • Copies being made.
  • Translation into other languages.
  • Indirect mentions in other documents.

By broadening $A$, you avoid making $P(A|R, H)$ arbitrarily small due to overly specific assumptions. If someone is arguing that 

$P(-E|H)$ is high because $P(E|H) = P(R, A|H)$ is low:

• I would argue that $A$ (survival) depends on overly complicated and arbitrary assumptions. Suggest using a broader, more reasonable $P(A|R, H)$.
•  S (survival) is heavily dependent on external, independent factors and should not be modeled in an overly restrictive way. Simplifying it as a broader, aggregated probability avoids artificially deflating P(E∣H).

The analysis is overly restrictive if it focuses exclusively on one individual's written account as the sole determinant of $P(E\mathrm{\mid }H)\; or$$P(-E|H)$. By ignoring alternative sources of evidence—such as physical artifacts, other written records, or indirect inference from context—it artificially limits the pathways through which evidence $E$ could arise. Let’s expand the reasoning to account for these additional sources of evidence.

1. Evidence is Often Multimodal

Historical evidence rarely relies on a single source or pathway. The absence of evidence ($\sim E$) cannot reasonably hinge on just one author's potential recording of an event. Historical evidence often emerges from multiple pathways, so $P(\sim E|H)$ must account for all these sources of potential evidence. P(∼E∣H) Is Not Determined Solely by One Author. Instead, it typically arises from a combination of:

1. Direct Written Records:
   • The specific author’s mention (e.g., $R$ and $\mathrm{S\; in\; the\; current\; model).}$
2. Other Independent Written Accounts:
   • Records by other authors or cultures, often reinforcing or corroborating the event.
3. Physical or Archaeological Evidence:
   • Artifacts, ruins, or environmental traces that indirectly suggest the event happened.
4. Contextual Evidence:
   • Broader societal patterns, oral traditions, or logical implications derived from other known facts.

By focusing solely on the chain $N \to R \to A$, the current model excludes all these additional potential sources, which would generally increase $P(E|H)$ and reduce $P(-E|H)$.

2. Expanding P$<b>(</b>$$E$$\mathrm{\mid }$$H$$)$ to Include Multiple Pathways

To account for these additional sources, $P(E|H)$ should reflect a disjunction of pathways through which evidence could arise. Instead of just one pathway (the specific author), we sum the probabilities of all independent sources of evidence:

$$P(E|H) = P(\text{Author's Writing Survives}|H) + P(\text{Other Written Records Survive}|H) + P(\text{Physical Data Exists}|H) - \text{Overlap Terms (to avoid double-counting)}.$$

General Formula:

Let:

• $P(E_1|H)$: Probability of evidence from the original author's writing ($R$ and $A$).
• $P(E_2|H)$: Probability of evidence from other independent written accounts.
• $P(E_3|H)$: Probability of evidence from physical data.
• Overlap terms account for situations where multiple sources produce overlapping evidence.

The expanded $P(E|H)$ is:

$$P(E|H) = P(E_1|H) + P(E_2|H) + P(E_3|H) - P(E_1 \cap E_2|H) - P(E_1 \cap E_3|H) - P(E_2 \cap E_3|H) + P(E_1 \cap E_2 \cap E_3|H).$$

If the sources are approximately independent (a simplifying assumption), this reduces to:

$$P(E|H) \approx P(E_1|H) + P(E_2|H) + P(E_3|H).$$

3. Reassessing P(−E∣H)

Once $P(E|H)$ is expanded to include these alternative pathways, $P(-E|H)$—the probability of no evidence given $H$—is correspondingly reduced:

$$P(-E|H) = 1 - P(E|H).$$

If $P(E|H)$ becomes significantly larger because of multiple pathways, $P(-E|H)$ becomes much smaller.

4. Addressing the Redundancy of Focus on One Source

By restricting $P(E|H)$ to a single author's survival ($R$ and $A$), the analysis implicitly assumes:

• The author’s record is the only possible source of evidence for $H$.
• If that specific record is lost, no other data could provide evidence.

This is rarely true in practice, especially for historical or archaeological hypotheses. Other potential sources, such as:

• Parallel accounts by different observers or cultures.
• Physical remnants that confirm the hypothesis indirectly.
• Secondary writings that reference the original work (even if it’s lost).

All these sources contribute to $P(E|H)$, even if $R$ or $A$ fails.

5. Alternative Interpretation of Indirect Evidence

Even if no direct records exist (e.g., no author writes about the event), indirect evidence can still support $H$. For instance:

1. Archaeological digs might uncover artifacts consistent with $H$.
2. Geographical evidence (e.g., drought, volcanic ash) might align with $H$.
3. Social or cultural patterns (e.g., sudden abandonment of a city) might indirectly imply $H$.

These indirect forms of evidence are not captured by $P(\sim E|H)$ if the model focuses solely on written records.

Example:

• Hypothesis ($H$): A major volcanic eruption destroyed a historical city.
• Evidence ($E$): Archaeological ruins with volcanic ash layers, corroborating environmental data, and shifts in trade routes.

Even if no written records survive ($\sim E_1$), the physical data $E_2$ and contextual evidence $E_3$ could still strongly support $H$. Ignoring these sources severely underestimates $P(E|H)$ and inflates $P(\sim E|H)$.

Example:

If $H$ is "a major battle occurred in a particular region," evidence could include:

• Written accounts from multiple observers (not just one author).
• Archaeological findings, such as weapons or fortifications.
• Cultural artifacts that reflect the aftermath (e.g., monuments, traditions).

By ignoring these, the restrictive model vastly underestimates $P(E|H)$.

6. Accounting for Uncertainty in Alternative Pathways

While it's hard to precisely model all pathways, you can approximate their contributions:

1. Assign Probabilities to Additional Pathways:

   • Estimate $P(\text{Other Written Records}|H)$, $P(\text{Physical Data}|H)$, etc., based on the likelihood of independent documentation or physical traces.
   • Example: If the hypothesis concerns a widely known historical event, $P(\text{Other Written Records}|H)$ might be high.

2. Incorporate Conditional Independence:

   • Treat the different sources as conditionally independent given $H$, unless strong dependencies are known.

3. Adjust for Historical Context:

   • Consider factors like the time period, geography, and cultural context, which influence the likelihood of independent evidence pathways.

Let's now reconsider the formalism:

$$\text{Likelihood Ratio} = \frac{P(−E|H)}{P(−E|−H)}.$$

Expanding $P(E|H)$ to include multiple pathways strengthens the numerator, as it becomes harder for $P(E|H)$ to be arbitrarily small. Conversely, $P(E|-H)$—the probability of evidence arising without $H$—often remains low, as most sources of evidence are causally tied to $H$. By restricting P(E∣H) to one individual’s writing, the argument severely underestimates the probability of evidence. A more realistic model would:

1. Recognize that evidence $E$ can arise through multiple independent pathways, not just $R$ and $A$.
2. Expand $P(E|H)$ to include contributions from physical data, other writings, and indirect sources.
3. Adjust $P(-E|H) = 1 - P(E|H)$ accordingly, making $P(-E|H)$ less likely to be arbitrarily high.

This broader perspective ensures a more accurate and balanced assessment of $P(E|H)$ and the overall plausibility of $H$. This makes the argument from silence a far more realistic. It also shows that evidence does not exist in a vacuum. Reinterpretation of source material can transform old materials perceived to be irrelevant into something that could be added into P(E|H).  Focusing solely on one individual's record (and whether or not it survives) as the determinant of $\sim E$ (the absence of evidence) under $H$ is extremely restrictive and incomplete. Historical events often leave traces across multiple independent sources—including indirect evidence such as physical artifacts, secondary references, or even inconsistencies in unrelated records that can be analyzed probabilistically. This over-restrictive approach fails to account for the diversity of ways evidence can support $H$, which ultimately inflates $P(\sim E|H)$ to an unreasonable degree. Essentially, we can challenge the restrictiveness of $P(\sim E|H)$ by addressing the complementarity of evidence and presenting a broader framework for evaluating $P(E|H)$. Even if one pathway fails to produce evidence, others can compensate, thereby reducing $P(\sim E|H)$. This broader view ensures that the absence of one type of evidence (e.g., written records) doesn’t overly inflate $P(\sim E|H)$. Below is a summary of what we have discussed thus far:

1. Written Records Are Not the Sole Source

The argument overly restricts $P(E|H)$ by assuming:

• $E$ depends solely on the existence, recording, and survival of a single written account.
• $\sim E$ arises whenever this specific pathway fails.

However, history and archaeology show that events often leave redundant evidence across multiple domains. For example:

• Physical artifacts might confirm an event even if all contemporary writings are lost.
• References in secondary or unrelated records can fill gaps left by primary sources.

2. Physical and Contextual Evidence Are Complementary

Written records provide direct documentation, but physical and contextual evidence can serve as indirect confirmation:

• Physical Evidence: Archaeological artifacts (e.g., ruins, tools, weapons, environmental markers) can independently corroborate $H$. For example:
  • A battle might leave behind fortifications, graves, or weapon fragments.
  • A volcanic eruption might leave geological markers like ash layers.
• Contextual Evidence: Broader societal patterns or consequences indirectly support $H$. For example:
  • Economic disruption visible in trade routes might indicate a major event like a war or natural disaster.
  • Cultural shifts, such as the sudden adoption of specific religious practices, might hint at a historical turning point.

Even in the absence of direct written accounts, these complementary sources can significantly boost $P(E|H)$.

3. Redundancy of Evidence Reduces $P(\sim E|H)$

If evidence arises from independent and complementary pathways, the probability of complete absence of evidence becomes much smaller. Mathematically:

$$P(\sim E|H) = P(\sim E_1 \cap \sim E_2 \cap \sim E_3 \cap \dots | H)$$

Where:

• $\sim E_1$: No evidence from written records.
• $\sim E_2$: No evidence from physical data.
• $\sim E_3$: No evidence from contextual clues.

If these pathways are independent, the probability of joint failure decreases exponentially:

$$P(\sim E|H) \approx P(\sim E_1 | H) \cdot P(\sim E_2 | H) \cdot P(\sim E_3 | H)$$

For example:

• If $P(\sim E_1|H) = 0.8$, $P(\sim E_2|H) = 0.5$, and $P(\sim E_3|H) = 0.6$, then: $$P(\sim E|H) = 0.8 \cdot 0.5 \cdot 0.6 = 0.24$$

This is far smaller than the restrictive $P(\sim E|H) = 0.8$ derived from focusing on one pathway. Instead of focusing solely on one source, redefine $P(E|H)$ to include all possible pathways:

$$P(E|H) = P(E_1|H) + P(E_2|H) + P(E_3|H) - \text{Overlap Terms}$$

Where:

• $P(E_1|H)$: Evidence from written records.
• $P(E_2|H)$: Evidence from physical artifacts.
• $P(E_3|H)$: Evidence from contextual patterns.

Practical Simplifications:

If the pathways are roughly independent:

$$P(E|H) \approx P(E_1|H) + P(E_2|H) + P(E_3|H)$$

This makes $P(E|H)$ far larger than when relying solely on $P(E_1|H)$, reducing $P(\sim E|H) = 1 - P(E|H)$.

4. Likelihood Ratio and the Absence of Evidence

The real test for $\sim E$ is not whether $P(\sim E|H)$ is high, but how it compares to $P(\sim E|\sim H)$, the probability of no evidence under the alternative hypothesis. Expanding $P(E|H)$ makes $P(\sim E|H)$ smaller and shifts the likelihood ratio:

$$\text{Likelihood Ratio} = \frac{P(−E|H)}{P(−E|−H)}.$$

Example:

• If $P(\sim E|H) = 0.2$ and $P(\sim E|\sim H) = 0.9$, the absence of evidence is more consistent with $\sim H$, weakening the argument against $H$.

By including complementary pathways, you show that $P(\sim E|H)$ is not overwhelmingly high. This broader, multimodal approach reflects how evidence for historical events is typically evaluated and avoids artificially inflating $P(\sim E|H)$.

One of my main critiques is that the author is not considering total evidence. Arguments from silence are rarely ever presented with respect to one historical source. When they are presented as such, it's normally under the assumption that alternative pathways of evidence are also silent. One could argue that this "total evidence" is contained in the prior likelihood ratio. But if that were the case, the argument from silence would demonstrate the prior likelihood dominates the overall expression. Just think about how absurd the argument would be if it didn't take into consideration total evidence. Suppose you have N sources, all confirming H. A new source N+1 is identified as potential evidence for H. Maybe it is a document written by some author presumed to be in a position to know whether H was true. Suppose conditions 1-3, listed above by the author, lead to P(~E|H) to be very close to zero. This would affirm P(~E|~H). But since every other source N-1 strongly confirms H, reasonable historians would affirm H because the total evidence strongly confirms H. But no one makes arguments like this when using arguments from silence, so presenting it that way seems like a straw man argument against the argument from silence. When someone asserts the argument from silence is fallacious, they are almost always neglecting the broader context of discussion in which the argument is presented against a body of total evidence. A bit more about this concept; according to the Stanford Encyclopedia of Philosophy:

In order to be justified in believing some proposition then, it is not enough that that proposition be well-supported by some proper subset of one's total evidence; rather, what is relevant is how well-supported the proposition is by one's total evidence. In insisting that facts about what one is justified in believing supervene on facts about one's evidence, the Evidentialist should be understood as holding that it is one's total evidence that is relevant. Of course, this leaves open questions about what relation one must bear to a piece of evidence E in order for E to count as part of one's total evidence, as well as the related question of what sorts of things are eligible for inclusion among one's total evidence.[6]

As mentioned earlier, what constitutes evidence depends on admissibility criteria and the evidence someone has access to. The requirement of total evidence is a principle in epistemology and philosophy of science that states we should base our beliefs, probabilities, or inferences on all available and relevant evidence, rather than on a subset of evidence or incomplete information. This principle plays a crucial role in Bayesian reasoning and the interpretation of Bayes factors, as Bayesian methods inherently rely on incorporating and updating beliefs in light of all relevant evidence. Bayesianism views belief updating as a normative model of reasoning. Total evidence aligns with this framework by insisting that all available information be used to evaluate hypotheses, ensuring that Bayesian reasoning is not distorted by partial or biased evidence. How we define the E in P(E|H) matters because if E is a subset of the total evidence, P(E|H) could be underestimated. This is a very important consideration because Bayesian reasoning does not explicitly guarantee the problem definition to consider whether E is comprehensive; it just specifies an update rule for rational inference. Someone can properly apply Bayesian logic but still arrive at an incorrect probability because of how they defined P(E). $P(E|H)$ is conditional on the subset of data and may be biased if the sample is not representative of the entire evidence set. This can lead to misleading posterior probabilities unless the missing evidence is either irrelevant or explicitly accounted for later. The quantity P(E|H) considers whatever is defined as E in the analysis. To align with the requirement of total evidence, E should ideally encompass all relevant evidence. If only a sample is used, the analysis may still be valid for that subset, but the results must be interpreted with caution to account for the potential impact of omitted evidence. In the case of Arguments from Silence, P(~E|H) should refer to the entire set of evidence E that is absent ~E but expected. 

What we count as evidence obviously matters, and will influence what is considered to be total evidence. In historical contexts, I am not sure if there are clearly defined admissibility rules like there are in legal contexts. Presumably, historians have some sort of procedures for categorizing and grading evidence, as well as rules of thumb for deciding whether something should be considered evidence. It most certainly is a broader subset than what was presented earlier by the author of that article. Admissibility criteria help determine what qualifies as E in the computation of  P(E∣H)

• Relevant Evidence: Admissibility criteria prioritize evidence that directly pertains to the hypothesis $H$. Irrelevant evidence, even if available, is excluded to prevent noise from distorting $P(E|H)$.
• Reliable Evidence: Evidence must be sufficiently credible or reliable. Unreliable evidence could skew $P(E|H)$, leading to incorrect posterior probabilities.
• Complete Evidence: Ideally, admissibility criteria should align with the requirement of total evidence by ensuring all relevant evidence is considered. Ignoring admissible evidence could result in incomplete likelihood calculations.

Admissibility criteria function as a filter to ensure P(E|H)  reflects an accurate probability based on valid evidence:

• Total Evidence Requirement: The total evidence principle mandates considering all admissible evidence. Evidence that does not meet the admissibility criteria (e.g., irrelevant, unreliable, or misleading data) should not be part of $E$.
• Selective Evidence Inclusion: If evidence selection is biased or admissibility criteria are overly restrictive, $P(E|H)$ may only reflect a subset of the true evidence. This violates the total evidence principle and can lead to misleading Bayesian inferences.

This is not always a straight forward process:

• Practical Challenges: Defining admissibility criteria can be subjective or context-dependent. In some cases, determining whether evidence is reliable, relevant, or complete may be unclear.
• Balancing Inclusion and Exclusion: While admissibility criteria prevent irrelevant or misleading evidence from distorting $P(E|H)$, overly strict criteria could result in the omission of valid evidence, violating the total evidence principle.
• Uncertainty in Evidence: Admissibility decisions sometimes involve probabilistic judgments. Bayesian reasoning can handle uncertainty in evidence (e.g., using hierarchical models), but this assumes that all admissible evidence has been included.

What is considered "admissible" is intimately connected to what is considered "relevant", which is a very elusive concept. Courts decide what is admissible based on something called Exclusionary restrictions. Exclusionary restrictions are criteria or rules used to exclude certain types of evidence, variables, or data from consideration in a particular analysis, argument, or decision-making process. These restrictions are often applied to ensure relevance, reliability, or fairness, but they can also reflect pragmatic or theoretical concerns. In essence, they define what is not admissible or allowable in the evaluation of a hypothesis, model, or decision. Exclusionary restrictions play a crucial role in controlling the quality, relevance, and appropriateness of evidence or variables in various domains. While they ensure rigor, reliability, and adherence to ethical or legal norms, they must be carefully designed to avoid over-exclusion or undue subjectivity that could undermine the integrity of reasoning or decision-making processes. Here are a few contexts and applications of exclusionary restrictions:

1. Philosophy of Science and Evidence Evaluation:
   • In scientific reasoning, exclusionary restrictions are used to filter out evidence that is considered irrelevant, unreliable, or biased.
   • For example:
     • Evidence not derived from proper experimental conditions might be excluded.
     • Anecdotal evidence may be restricted in favor of systematic data.
   • These restrictions help ensure the validity and robustness of inferences.
2. Bayesian Reasoning:
   • Exclusionary restrictions in Bayesian reasoning may determine which evidence $E$ is included in $P(E|H)$.
   • For example:
     • Evidence obtained through unreliable means or conflicting with prior constraints may be excluded.
     • Irrelevant evidence—evidence that has no bearing on the likelihood of a hypothesis $H$—is excluded to avoid inflating or deflating posterior probabilities.
3. Statistical Modeling:
   • Exclusionary restrictions can apply to variables or datasets in statistical models, such as:
     • Removing outliers or noise from the dataset.
     • Excluding variables that do not significantly contribute to the model or violate assumptions (e.g., multicollinearity in regression).
   • These restrictions ensure the model is parsimonious and interpretable.
4. Legal Contexts:
   • In legal reasoning, exclusionary restrictions often take the form of rules that bar certain types of evidence from being presented in court.
     • For example:
       • Hearsay evidence is often excluded because it is considered unreliable.
       • Evidence obtained unlawfully (e.g., through illegal searches) may be excluded under the exclusionary rule to protect rights and encourage lawful conduct by law enforcement.
5. Ethical and Policy Decisions:
   • Exclusionary restrictions are applied to uphold ethical norms or policy standards. For example:
     • Data obtained through unethical means, such as coercion or exploitation, may be excluded from consideration in decision-making.
     • Certain demographic factors, such as race or gender, may be excluded in hiring or admissions decisions to prevent discrimination.

Below is a list of different types of exclusionary rules:

1. Relevance-Based:
   • Evidence or variables that are not relevant to the hypothesis or decision are excluded to avoid distraction or overfitting.
   • Example: In Bayesian reasoning, $P(E|H)$ should only include evidence that can differentiate between $H$ and competing hypotheses.
2. Reliability-Based:
   • Evidence that is deemed unreliable (e.g., due to measurement error, biased sources, or incomplete data) is excluded.
   • Example: Excluding self-reported data when objective measures are available.
3. Legal or Procedural:
   • Evidence that violates procedural rules or legal principles is excluded.
   • Example: Illegally obtained evidence is inadmissible in many judicial systems.
4. Ethical or Normative:
   • Data or evidence obtained through unethical means or in violation of normative standards is excluded.
   • Example: Excluding data from studies that violate human rights.
5. Practical or Feasibility-Based:
   • Evidence or variables that are too costly, complex, or impractical to include may be excluded.
   • Example: Excluding high-dimensional variables in a statistical model to avoid computational challenges.

There are advantages of using exclusionary restrictions

• Focus and Relevance: Exclusionary restrictions ensure that only pertinent evidence or variables are considered, simplifying analysis and interpretation.
• Reliability and Validity: By filtering out unreliable evidence, these restrictions help maintain the credibility of inferences or decisions.
• Normative Consistency: In ethical or legal contexts, exclusionary restrictions reinforce adherence to moral and procedural principles.
• Pragmatism: They help manage complexity by reducing the scope of evidence or variables to those that are most impactful.

There are also challenges to properly employing exclusionary restrictions. In the context of an argument from silence, perhaps someone assumes P(~E|H) is large because they have significantly narrowed what constitutes E in relation to H. Perhaps they have violated the requirement of total evidence by ignoring evidence deemed relevant to H. This might be caused by a subjectivity inherent to the process of Bayesian reasoning, leading to information loss. Nevertheless, Argument from Silence is a legitimate form of reasoning provided certain conditions are satisfied. It can be assessed probabilistically using the framework above, along with considerations about what counts as evidence. It also depends crucially on how we define the search space with respect to a set of possible pieces of evidence. Consider a parallel argument:

1. If my keys were in this room, I would be able to find them
2. I cannot find them
3. Therefore my keys are not in this room 

Perhaps there is evidence that corroborates the hypothesis “the keys are in this room”. You search for this evidence, and the keys directly, but find nothing. You conclude the hypothesis “keys in the room” is false. This can also be seen as an argument from negative evidence:

• Major Premise: If A were true, A would be known to be true. 
• Minor Premise: A is not known to be true. 
• Conclusion: A is false. S

Such pattern of reasoning has been analyzed in computing as a relativistic form of deductive reasoning called autoepistemic reasoning. On Moore’s view (1985: 273) inferences of the kind Tweety is a bird. Most birds can fly. Therefore Tweety can fly can be analyzed considering the premise “Most birds can fly” as a consistency clause, providing that “the only birds than cannot fly are the ones that are asserted not to fly” (see also McDermott and Doyle 1981). Since Tweety is not asserted to fall within the group of birds that cannot fly, Tweety can fly. Therefore, the conclusion that “Tweety can fly” is not drawn absolutely (that is, it is not an ontological fact that birds fly, and if something does not fly it is not a bird) but only relative to a theory, or shared knowledge. Such a pattern of reasoning can be formalized as follows (Moore 1985: 275):

If P1,…, Pn are in T, and P1,…, Pn ⊢ Q, then Q is in T (where “⊢” means ordinary tautological consequence). 

If P is in T, then LP is in T. 

If P is not in T, then ~LP is in T.

The second and third clauses provide that if a proposition is (is not) in the theory, or domain of knowledge, such a proposition is (is not) believed (indicated by the logical operator ‘L’) to be in such theory.

Based on the structure above, evidence of absence depends on the completeness of negative knowledge. This depends on "how wide the paradigm of instances to be negated is" according to Walton. This makes sense, under normal circumstances in a situation we can conceive of a number of factors that would relate to the conclusion, and then check for each of them to see if they're satisfied. This is related to what I've described earlier about conceptualizing the set of possible evidence related to some hypothesis, or in the mundane form of reasoning exemplified above with the keys:

In computing, such principle has been developed under the name of the Closed World Assumption, setting forth that “if a ground atom A is not a logical consequence of a program P, then it is possible to infer ~A” (see Reiter 1978). This rule has been developed by Clark into the principle called “Negation as Failure” (1978: 114), stating that “To show that P is false, we do an exhaustive search for a proof of P. If every possible proof fails, ~P is ‘inferred’”

Walton provides an example of denying the predicate of no negative effects in the context of a medical substance:

The argument depends on how exhaustive the implicit reasoning stage is. This is also a common form of reasoning about information in databases known to be relatively complete and efficient at tracking information. Suppose we search an enterprise information system for some fact, but fail to find the fact. We can reasonably infer the fact is not the case, given the track record of logging the information. Someone could reason that the information was removed, but this action would generate evidence. We could track a metadata log to identify any changes to the database. This would verify the inference. This is related to the burden of proof. Someone can assert something about the information in the database being absent, and therefore false. An interlocutor could then assert the information was deleted. This would shift the burden of proof to them. If this burden is not satisfied, we accept the conclusion that the initial assertion is false. It is a bit more difficult when it comes to historical reasoning about ancient events because the presupposed set of clearly defined alternatives is rather large. But if we rule out information based on inquiry from established disciplines or other legitimate forms of inquiry, the assertion ~H is very reasonable.

Consider the statement “Absence of evidence is not evidence of absence”; if we relate this to "correlation does not equal causation" all this is telling us is that the subset of causation does not equal the superset of correlation. But correlation seems to be a requirement of causation. There are additional assumptions needing to be satisfied prior to concluding causation, likewise there are additional assumptions needed to be satisfied before concluding that absent evidence is indeed evidence of absence. Evidence of absence is possible, provided certain conditions are reasonably satisfied. This is the entire point of belaboring on the points above about the felicity conditions of argument from silence. So we can see, after considering the nuance of the argument, that it is indeed a valid probabilistic argument. It depends crucially on how we define the search space with respect to evidence. Think about the parallel argument about the keys: perhaps there is evidence that corroborates the hypothesis “the keys are in this room”. You search for this evidence, and the keys directly, but find nothing. You conclude the hypothesis “keys in the room” is false. 

I thought of an analogous situation , an instance where we would normally accept an argument from silence, because we identify it as proper inference. Suppose I tell you that someone harbors some unconscious racist bias. You search for evidence that indicates whether they are a racist, given some definition of racism and the types of behavior we expect to manifest. You don’t find any. We do an exhaustive search and still find no indication of racism. Do you conclude the person is not a racist? The this has the same structure ad argument from silence. Some data is expected under some hypothesis, the evidence fails to actualize under multiple expected scenarios, therefore we conclude the hypothesis is false. If H is the hypothesis that this person is racist: P(-E|-H) > P(-E|H). We would conclude they are not racist. We would not be skeptical, and say they “might still be a racist” and come up with ad hoc unverifiable conditions that explain the lack of evidence, such as some scheme deliberately being employed to suppress the data. We would simply conclude they are not racist. We would rightly recognize that if a person still holds to the assertion that H , they would be doing so for ideologically motivated reasons (such as a new definition of racism that asserts someone is racist by definition). 

This is indeed a relevant and practical example. It highlights the fact that priors can fix an outcome independent of evidence. Suppose I redefine racism such that it’s a systematic effect that manifests in the aggregate. Then by definition, someone can be racist (by participation in a racist system) while not exhibiting any specific racist behaviors under conventional definitions. This is directly analogous to the situation theists find themselves in. If by definition, or assumption, some biblical event must have happened, absent evidence is irrelevant. 

This is also important because it highlights the argumentative role of evidence, i.e. what counts as established evidence. If a definition of racism assumes racism is subliminal, then absent overt evidence, predicted by conventional definitions of racism, is actually expected (we expect absent evidence). Under the new conceptualization, conventional evidence of racism is not relevant. It may manifest, but the evidence is decisive only in a single direction; it only strengthens the case. This new hypothesis would imply a set of data, such as micro aggressions, to be present conditional on H. This holds true for theistic reasoning as well. If we expected some evidence for some historical event H, and it is absent, the theist can simply rephrase H such that it’s compatible with the absent evidence. Suppose someone asserts that the hebrews were not enslaved, because we would expect to see evidence of slavery. Since this is crucial theologically for the theist, they can simply explain away the absent evidence by claiming “the enemies of god deliberately suppressed the data, this is expected since we are in a constant spiritual battle against the forces of evil. Satan is cunning, he knows that be suppressing the data, people would lose faith in god”. Or they can redefine slavery such that it wouldn’t leave traces we traditionally would identify as such. This type of ad hoc explanation is very common. It amounts to modifying H after the fact, to account for discrepancies between H and E. It expands the scope of H, inserting unverifiable conditionals into H, asserting they have been “plausibly” fulfilled.

Notice that we would immediately find this reasoning unacceptable in most cases. Suppose we assume a conventional definition of racism, expecting to find evidence of racism in some individual, but evidence fails to instantiate. Many people immediately recognize that redefining H to mean “subliminal”, implying that racism won’t emit any data (since measuring something latent is quite tricky) , seems off. I’m not saying that it’s absolutely unacceptable to reasonably adjust definitions, but sometimes concepts can be persuasively redefined (written at length in Walton). Analogously, if there is no evidence your wife is cheating, you conclude she is not cheating. You do not define ad hoc conditions that allegedly prove she is suppressing the information. That will land you in marriage counseling or divorce. The degree to which someone constructs ad hoc explanations to support H when there is no confirming evidence , reflects their implicit commitment to H , which can be motivated for ideological reasons, among others (such as paranoia in the case of the cheating spouse). Even in the case where someone ought to remain agnostic, ideological motivations can incentivize someone to mistake something as evidence by rationalization, bolster H with implausible ad hoc auxiliary assumptions, or in many cases shift the burden to some interlocutor to find evidence for -H.

Think about the variety of evidence, if we expect different varieties of evidence to positively confirm some hypothesis, and they fail to instantiate, this dramatically increases the probability of -H. This is because the converse is true. If we identify many different types of evidence that positive affirm some hypothesis, this radically increases our confidence in H. Think about it this way: suppose there is a variety of evidence that can (possibly) affirm H, we will place it in this set called E=(X,Y,Z,A,B,C….). The letters represent different kinds of evidence. If we only expect X, even if X is confirmed, we could argue that there is some systematic bias generating X, thus undermining the inference from E to H. But if the rest of the evidence is present, we would need to postulate multiple different systematic effects that bias all categories of evidence, to explain its absence. This would be radically improbable, so therefore we would conclude H on the basis of E. In the case of arguments from silence, we not only lack X, but also the rest of the categories. Someone affirming H or being agnostic about H, would have to identify and argue for systemic mechanisms that suppress every category of evidence, all acting independently. The burden would on them to explain why absolutely no evidence exists in the set E.  Of course they could argue for some meta condition , call it M, that explains the suppression of all categories of evidence. But this borders on conspiratorial reasoning and we would normally reject it.

Back to my main point of this; why do apologists seem to be the only ones who care about this? It should be obvious: many claims asserted in the Bible literally have no evidence substantiating them, and in many cases the evidence is mere hearsay. Apologists focus on this argument to dismiss its credibility. Earlier I covered a few conditions mentioned by the author to decide whether the probability of evidence is low. I agree that there are conditions that might prevent a historical author from transcribing some historical fact. However, we cannot merely assume that these negating conditions were present, simply because they did not transcribe the fact. These conditions need to be shown to have been instantiated; if they are not then that is an argument from ignorance. Nevertheless, apologists will shift the burden of proof, arguing from ignorance about the plausibility of these conditions. Take a condition listed on a famous catholic apologetics website: the subject under discussion is "embarrassing". I agree this is possible, but if we assume, in absence of evidence that this condition is satisfied, the author didn’t transcribe because of this embarrassment, that’s an argument from ignorance. It is tantamount to saying "I don't know what conditions prevented the author from transcribing the information, but I know something must have, therefore the hypothesis is true". It’s possible to construct an inference to best explanation that explains why an author wouldn’t transcribe something; but this is rife with issues. For starters, these arguments depend heavily on what’s considered plausible. This is inherently connected with the assumed world view of the person advancing the argument; I've discussed this at length in my other blog posts. For example, if you argue embarrassment is the explanation, based on some unestablished interpretation of a biblical passage, I can reject it because the biblical assumptions are not plausible to me. We do not share the same background assumptions, so what seems plausible to A might not seem plausible to B. We need data to understand whether these conditions are satisfied.

The inverse is usually assumed by apologists. What I mean by this, is suppose some argument from silence concludes P(~E|~H) is by far relatively higher than P(~E|H), and that P(~H) > P(H) is our base rate; many apologists will still hold to H regardless of where the total evidence points. Remember that if the argument is weak, then at best P(~E|~H) == P(~E|H), which would imply a position of absolute agnosticism about the proposition. This would mean that statements severely lacking historical evidence such as those asserted in Exodus for example, would require apologists to be agnostic about historical events relevant to their faith. This is obviously unacceptable, and contravenes the statements of faith they're required to sign prior to embarking on their little apologetics journey. They may get around this by asserting P(H) is a strong prior. However, these "priors" significantly lack any theoretical rigor, evidential adequacy, or consistency to be considered as strong. So what this really amounts to is prior manipulation. The prior probability (the initial belief about the likelihood of an event or proposition before considering evidence) is chosen in a way that unduly favors a particular outcome. This is often done arbitrarily or with a motivated bias, rather than based on objective or reasonable grounds. They do this by assigning an unjustifiably high probability to the prior, such that the evidence (likelihood) has little influence on the posterior, leading to a skewed conclusion. So in other words, P(H) strongly outweighs any form of evidence or absent evidence, such that P(~H) becomes implausible by definition. 

I am well aware of the many selection biases that plague non-experimental fields of study such as history. Something like the survivorship bias could be systematically skewing the historical evidence rendering P(H) extremely low. However, if multiple independent measures render P(H) low, and there is no evidence of some selection mechanism, it's reasonable to infer ~H. Not allowing this form of inference leads to some highly counterintuitive results such as the one I listed above about a cheating spouse. On the topic of selection bias, the reason we see apologetics websites listed in the google search for "argument from silence" definitely sheds light on the mechanisms that generated those search results. Why is it, that in the result set, our sample is systematically skewed towards these particular results, and not anything from professional academics? There is a strong motivational force to dismiss this argument as ipso-facto fallacious, or to quickly conclude that a particular instance of the argument is fallacious, due to the motivational reasoning inherent in apologetics. Earlier I called this the "Fallacy Fallacy" but that's not quite right. Instead, its a form of premature fallacy attribution, motived by defensive reasoning. 

The act of misidentifying fallacies prematurely is related to motivated reasoning, which is the tendency to process information in a way that aligns with one's preexisting beliefs, emotions, or desires. When someone engages in motivated reasoning, they might misidentify or over-interpret arguments as fallacious to dismiss them more easily, often without fully engaging with the substance of the argument. A person might be motivated to label an argument as fallacious because it contradicts their beliefs. For instance: If someone strongly opposes a viewpoint, they may quickly declare the argument supporting it as a "strawman" or "ad hominem," even if it isn't, to justify disregarding it. Misidentifying a fallacy prematurely can be a shortcut to avoid the cognitive effort of critically analyzing the argument. Motivated reasoning makes this shortcut appealing because it reinforces existing attitudes without the need for deeper reflection. Consider this scenario, The Causes of the Fall of the Roman Empire:

Argument:

• Historian A argues, "The Roman Empire fell because of the over-expansion of its borders, which stretched resources too thin and made the empire vulnerable to outside invasions."

Response (Potential Misidentification of a Fallacy):

• Historian B accuses Historian A of committing a post hoc fallacy (assuming that because over-expansion preceded the fall, it caused the fall) and dismisses the argument entirely.

How Motivated Reasoning May Play a Role:

• Motivation to Defend a Preexisting Belief:
  Historian B might be motivated by their belief in another explanation for Rome’s fall, such as internal political corruption or economic collapse. Instead of engaging with the argument about over-expansion, they dismiss it outright by prematurely accusing Historian A of a post hoc fallacy.

• Cognitive Shortcut:
  Declaring "post hoc fallacy!" allows Historian B to sidestep deeper engagement with the evidence (e.g., examining whether over-expansion indeed led to overtaxation, logistical issues, or weakened defense strategies).

• Potential Error:
  In reality, over-expansion might have been one of several contributing factors to Rome’s fall. While Historian A's argument may not explain the entire phenomenon, labeling it as a fallacy prematurely could result in the loss of valuable insights about the complex interplay of causes.

Motivated reasoning often emerges in historical debates because the stakes can be ideological. For instance:

• Defenders of Western civilization might be motivated to downplay "internal decay" explanations, as they could be seen as undermining the perceived greatness of Rome.
• Others might emphasize external invasions to draw parallels to modern political concerns, such as immigration or military defense.

In such cases, accusations of fallacies (like "post hoc" or "slippery slope") may be wielded as rhetorical tools to dismiss opposing views rather than engage with them critically. To avoid misidentifying fallacies prematurely, historical reasoning requires:

1. A nuanced understanding of fallacies and when they genuinely apply.
2. An openness to complex, multifaceted explanations that don't fit neatly into one narrative.
3. Self-awareness about motivated reasoning, especially in ideologically charged debates.

What I am suggesting, is the sample results from the google query, can be explained by this phenomenon, and not some issue overlooked by professional historians. Apologists recognize that the severe lack of data substantiating even mundane biblical claims is problematic if they are seeking to establish an evidential grounding for the bible. These search results indicate, not a problem with this form of reasoning, but a defensive strategy. Even the way apologists represent the arguments presented by historians is overtly fallacious and unfaithful. Consider this example by the catholic apologetics website: "The Exodus never happened. There’s no evidence that it did". Obviously, something like that would never get published. This is probably what a historian would tell you if you caught them on an elevator and had ten seconds to speak to them. It overlooks the depth and breadth of the reasoning behind their conclusion. But by prematurely attributing it as fallacious, reveals more about the structures motivating such an assertion, not the lack of rigor behind the argument itself.

Among apologists, it is common for someone to add ad hoc explanations to $H$  when the likelihood $P(-E|H)$ (the probability of observing the absence of evidence $-E$ if $H$ were true) is very low. This typically occurs because the low value of $P(-E|H)$ creates tension for an agent committed to $H$, as the lack of expected evidence undermines the hypothesis. Motivating Factors for adding ad hoc explanations include:

1. Cognitive Dissonance: When someone is deeply committed to a hypothesis, the absence of expected evidence creates cognitive dissonance—the psychological discomfort caused by holding contradictory beliefs or evidence. To reduce this dissonance, they may introduce ad hoc assumptions that reconcile the absence of evidence with their commitment to H. Example: A scientist might assume that an experiment failed not because their theory is flawed, but because the conditions were somehow atypical or the measurement tools were inadequate.
2. Emotional Investment: People may have emotional attachments to  H because it aligns with their personal beliefs, identity, or values. For example, a historical theory that supports one's cultural heritage might lead someone to explain away contradictory evidence (e.g., "The records were likely lost or destroyed").
3. Confirmation Bias: The tendency to seek, interpret, or create information in a way that confirms one’s pre-existing beliefs can motivate ad hoc reasoning. In this case, when −E appears, the agent may construct unverifiable explanations to preserve their belief in H.  Example: A believer in extraterrestrial visitation might argue that the absence of credible UFO evidence is due to a government conspiracy suppressing the truth.
4. Epistemic Inertia: People are often resistant to revising or discarding long-held beliefs because doing so requires significant effort and acknowledgment of past errors. Adding an ad hoc explanation is a way to "patch" a theory without the more disruptive process of abandoning or revising it.
5. Social Pressures: Commitment to  H may be reinforced by social or institutional pressures, especially when  H is central to a group’s identity, ideology, or goals. In such cases, adding ad hoc assumptions may be motivated by a desire to maintain credibility, avoid alienation, or protect group cohesion.
6. Theory Tenacity: In science and philosophy, some theories are considered too important or elegant to abandon lightly. Proponents might justify temporary ad hoc fixes by arguing that  H has a strong track record and will ultimately be vindicated. Thomas Kuhn called this "normal science" in his analysis of scientific paradigms—scientists work to reconcile anomalies within a dominant paradigm rather than discarding it prematurely. Example: In the Ptolemaic model of astronomy, epicycles were added to account for anomalous planetary motion because the geocentric paradigm was deeply entrenched. It is important to note, however, that since theism is not well defined, it does not operate as a scientific theory.

7. A Priori Commitment: Sometimes, an agent's commitment to  H is based on non-empirical factors, such as religious or metaphysical beliefs, that are insulated from empirical scrutiny. In such cases, ad hoc explanations are used to harmonize  H with contradictory evidence, as abandoning H might threaten a broader worldview. Example: A creationist might explain the absence of certain fossil evidence by invoking unverifiable claims like a "testing" or "deceptive" design by a deity.

I've alluded to this earlier in the post actually, in the case of racism. There might be strong motivations to insert ad hoc assumptions to save H. These manifest multiple ways, including but not limited to the following:

1. Conspiracy Theories: Claiming that evidence is intentionally suppressed or hidden (e.g., "The lack of documents is due to a cover-up").
2. Hypothetical Entities: Postulating unobservable factors to explain the absence of evidence (e.g., "An unknown mechanism prevents us from detecting X").
3. Unfalsifiable Assumptions: Introducing assumptions that cannot be tested independently (e.g., "We haven’t found evidence yet, but we will eventually").
4. Shifting Goalposts: Adjusting criteria for what counts as evidence so that the absence of evidence no longer appears problematic.

The tendency to add ad hoc assumptions is often a symptom of over-committing to a hypothesis. While it can sometimes be reasonable (e.g., temporarily preserving a theory with strong prior success), it risks undermining the hypothesis’s explanatory power, parsimony, and falsifiability. Philosophers and scientists emphasize the importance of letting evidence guide beliefs, rather than twisting explanations to fit preconceptions. As Karl Popper warned, too much reliance on ad hoc reasoning can render a hypothesis unscientific, as it becomes immune to empirical refutation.

When someone inserts an ad hoc assumption (e.g., $h_1, h_2, \dots, h_N$) into a hypothesis $H$, the burden of proof typically shifts to them to demonstrate that these auxiliary assumptions are independently plausible and supported by evidence. This is because the auxiliary assumptions are being introduced to explain away what would otherwise undermine $H$, and without justification, they risk reducing $H$'s credibility and falsifiability.  For example, if someone argues $P(-E\mathrm{\mid }H)$ is low because "evidence was suppressed", it is now their burden to show that evidence was probably suppressed, by appealing to evidence that positively affirms the auxiliary assumption. 

1. Shift in the Explanation: When someone asserts that  P(−E∣H)is low because of an auxiliary assumption (e.g., "evidence was suppressed"), they are effectively introducing a new component to the explanation that needs to be justified independently. Without supporting evidence for the auxiliary assumption, the explanation becomes speculative and untestable.
2. Avoiding Arbitrary Complexity: According to principles like Ockham’s Razor, we should avoid introducing unnecessary assumptions unless they are independently justified. If the auxiliary assumption cannot be substantiated, it is an arbitrary addition and risks making H overly complex and less credible.
3. Maintaining Epistemic Accountability: Scientific and philosophical discourse relies on participants being accountable for claims they introduce. If someone adds h1,…,h_n to H, they take on the burden to provide evidence or reasoning that demonstrates these assumptions are likely true or at least plausible.

 If someone argues P(-E|H) is low because "evidence was suppressed," they must:

1. Provide positive evidence for the claim that suppression occurred.
2. Show that this suppression is consistent with what is observed.
3. Establish that suppression is a plausible and sufficient explanation for the lack of evidence.

Without these steps, the explanation becomes circular or unfalsifiable. The lack of evidence −E is explained by suppression, and the suppression claim itself is justified by the lack of evidence. This undermines the explanatory value of the hypothesis. In practice, someone doing will need to provide evidence for auxiliary assumptions. For example, to argue that evidence was suppressed, one might point to leaked documents showing suppression, testimonies from credible sources or a pattern of behavior by relevant agents that supports suppression. If the auxiliary assumptions (e.g., suppression) cannot be tested independently, they weaken the hypothesis because they reduce its falsifiability. Even if direct evidence for suppression is unavailable, an agent must at least argue that suppression is more probable than alternative explanations for −E, based on background knowledge and reasoning. Introducing unverified auxiliary assumptions reduces falsifiability by shielding H from disconfirmation. To preserve H 's integrity, these assumptions must be justified. By adding auxiliary assumptions, the agent shifts the debate from H to h1,h2,…,h_n. Failure to substantiate these assumptions undermines both the auxiliary assumptions and the original hypothesis. Without requiring proof for auxiliary assumptions, one could introduce an endless chain of further assumptions to explain any anomaly, rendering H untestable. By not following these types of inferential rules, there are essentially no bounds on what someone can believe. When someone introduces auxiliary assumptions to rescue  H , they must bear the burden of proof to justify those assumptions. This involves providing positive evidence or strong reasoning for their plausibility. If they fail to meet this burden, the original hypothesis  H remains unpersuasive, as it relies on unverifiable components that weaken its explanatory power. 

In this case of arguing from silence, the modus operandum of an apologist is to flood the internet with content dismissing the credibility of the argument, introduce ridiculous auxiliary assumptions, and shift the burden of proof. This is motivated by strong a-priori commitments to biblical historicity and inerrancy, stemming from institutional and ideological forces. This explains the strongly biased search results. If the biblical stories lack evidential support, they begin to appear as mere myths, dethroning their cultural status, with nothing distinguishing them from the plethora of alternative (often contradictory) myths we have deemed as useless, mere entertainment, or literature. Anyway, that's all I really have to say for now.