Arguments from Silence

The Puzzle: Why “Argument from Silence” Leads to Apologetics

The question that motivates this essay is simple: if arguments from silence are a general form of historical and evidential reasoning, why does searching for the term so often lead to apologetics? Answering that requires separating the logic of the argument from the ways people deploy or dismiss it.

Different fields tend to rely on different argument forms. Statistical sciences emphasize questions of causality and correlation; legal reasoning frequently appeals to precedent; and analogical arguments allow us to evaluate a case using criteria developed for related cases. Search results can provide a rough indication of where an argument form is most prominent. Searches for arguments from precedent, for example, are dominated by legal material, while searches for correlational arguments return material on data analysis.

That makes one result pattern striking: searches for “Argument from Silence” return a surprising number of apologetics websites. At first glance, that is not what we would expect from a general form of historical reasoning. Search engines rank pages using signals such as relevance, authority, content quality, and freshness, so the pattern raises a substantive question about why apologetics content has become so prominent for this particular query.

Ranking factor	What it contributes
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.

The central question, then, is why apologetics appears so prominently instead of fields in which this argumentative structure is routinely used. Why are apologetics pages discussing the argument so often, and why are so many other pages linking to them? My argument is that this pattern is better explained by premature fallacy attribution and motivated reasoning than by a defect in the argument form itself. To make that case, however, we first need to establish what an argument from silence is and under what conditions it is a reasonable inference.

What an Argument from Silence Actually Is

Before addressing that search-result pattern, we need a clear account of the argument itself. The crucial distinction is between a mere absence of evidence and an absence that would be surprising if a hypothesis were true.

Orientation

Arguments from silence arise frequently in the historical analysis of classical texts, but the underlying structure is much more general. It is useful to begin with the informal argument and then add the conditions that make the inference stronger or weaker.

Informal Form

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.

Historical Conditions

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

An extant document D in which no reference to an event E appears.
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
Event E is assumed to be a type of event which the author of D would not have overlooked, had the event taken place.

Evidence of Absence

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.

General Form and Limitations

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".

When Silence Is Evidence

That distinction shifts the discussion from logical form to evidential conditions. Silence matters only when evidence was reasonably expected, and the force of the inference depends on how strongly that expectation can be defended.

Default and Defeasible Reasoning

This type of inference is related to the closed-world assumption: what is not currently known to be true may be treated as false by default. It therefore resembles default logic and is non-monotonic and defeasible. Viewed as a deductive argument in classical logic, the inference is invalid. As practical reasoning, however, it is familiar and often reasonable. If there is no evidence that my wife is cheating on me, for example, I normally infer from that absence that she is not cheating. Although arguments from silence are often discussed in historical contexts, the same default pattern appears in ordinary reasoning.

Burden of Proof and Presumption

Arguments from silence also interact with the burden of proof. If a hypothesis has not been supported by evidence, it is generally unreasonable to accept it unless background considerations shift the presumption in its favor. In Bayesian terms, those background considerations are reflected in prior probabilities, while observed or missing evidence affects the likelihoods. For present purposes, let H represent a historical hypothesis and E the evidence we would expect if H were true.

If H is false, the absence of E may be unsurprising. If H is true and E would normally be expected, however, failure to observe E counts against H. This is the default inference behind an argument from silence. The inference remains defeasible because H might still be true even if E is absent—for example, if evidence was suppressed, destroyed, or never recorded. But once those alternatives are ruled out or shown to be unlikely, the absence of expected evidence provides stronger support for the negation of H.

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.”

Historical reasoning resembles legal reasoning in one important respect: a claim that fails to meet a basic evidential threshold may never reach the stage of serious consideration. Wigmore’s distinction is useful here. In litigation, insufficient production can prevent an issue from reaching the jury; in historical inquiry, a hypothesis unsupported by evidence may likewise fail to merit sustained consideration. This matters for arguments from silence because the absence itself must satisfy a burden of persuasion: the reasoner has to show that the missing evidence is meaningful within the wider evidential context.

A defendant relying on silence, for example, must show that the absence is meaningful rather than accidental or the result of missing records. The inference could fail if it focuses only on eyewitness testimony while ignoring available physical evidence. The same principle applies to historical reasoning: asserting that evidence was not expected is inadequate if alternative evidential pathways have not been considered. A persuasive argument from silence therefore depends not merely on one missing source, but on a defensible account of the relevant evidence that should have existed.

Taken together, these points show why an argument from silence is presumptive rather than deductive. The inference begins with an expectation: if the event occurred, some relevant evidence should exist. An opponent can weaken the argument by identifying a credible alternative explanation for the silence, such as poor record-keeping, lost sources, censorship, or lack of access.

Those alternatives do not defeat the inference merely by being imaginable; they also require justification. Silence becomes more probative when the evidence was strongly expected, the historical record is comparatively rich, and the omission is anomalous or surprising—for example, when official records omit an event that should have been highly salient.

A Simple Parallel: The Keys Example

If my keys were in this room, I would be able to find them
I cannot find them
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

Negative Evidence and Autoepistemic Reasoning

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 $\neg H$ 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? This has the same structure as an 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: $P(\neg E \mid \neg H) > P(\neg E \mid H)$. 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).

The example also shows how a prior commitment or a change in definition can insulate a hypothesis from evidence. Suppose racism is redefined as a systemic effect that can exist even when an individual exhibits none of the behaviors predicted by a conventional definition. Under that revised hypothesis, the absence of those behaviors no longer counts strongly against H; the evidential expectations have changed. The same structural move is available in theological reasoning: if a biblical event is treated as true by definition or prior commitment, then missing evidence can be made irrelevant by revising the auxiliary assumptions around H.

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 positively confirm 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.

What Counts as Evidence?

The phrase “expected evidence” hides another problem: what qualifies as evidence in the first place? Before formalizing the inference, we need to clarify how relevance, reliability, and admissibility determine the evidential set.

Evidence as an Argumentative Achievement

Before moving to a formal analysis, we need to ask what “evidence” actually is. That question is often taken for granted, but it is crucial here. In The Evidential Foundations of Probabilistic Reasoning, David Schum argues that an observation becomes evidence through an argumentative process that establishes its relevance and reliability. This differs from a purely probabilistic account, on which evidence is simply whatever raises the probability of a hypothesis. Silent evidence makes the distinction especially important: before an absence can matter, we have to establish that the missing item would have counted as evidence in the first place.

Suppose someone says that E is evidence for H because E is a historical document. That claim can still be challenged: the document may be unreliable, irrelevant, or inadmissible for the purpose at hand. Hearsay illustrates the broader point. People are often persuaded by hearsay, yet legal systems exclude some hearsay because its evidential value has not been established in the required way. Schum’s account therefore highlights a distinction between merely possessing an observation and successfully establishing that observation as evidence.

This view emphasizes the social, interpretive, 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, regardless of how it is interpreted or contextualized. On that account, $$E$$ counts as evidence for $$H$$ when $P(H \mid E) > P(H)$ . The criterion is therefore probabilistic: if observing $$E$$ raises the probability of $$H$$ , then $$E$$ counts as evidence for $$H$$ . Unlike Schum’s approach, this account does not first require a separate argument that $$E$$ is reliable or relevant. The difference is important: the probability account asks whether an observation changes the probability of the hypothesis, whereas Schum’s approach also asks whether that observation should enter the evidential dataset at all.

That distinction matters for arguments from silence because $P(\neg E \mid H)$ depends on how we characterize the absence of evidence. An interlocutor might insist that evidence is not absent because some observation E exists. If E is rejected as irrelevant or unreliable, however, and no alternative evidence remains, the evidential situation reduces to $\neg E$ . This is a limitation of a bare probability account: the update rule does not itself explain how observations earn a place in the evidential dataset. Probability assignments therefore depend on a prior argumentative process that establishes what is credible and relevant.

Silent evidence makes this especially clear. For an absence to matter, someone must identify what is missing and explain why its absence is significant. In historical reasoning, the absence of records for a supposedly major event can count against the event only if such records would normally have existed. The same point applies to corroboration: a document offered as evidence for H may lose probative force when the expected corroborating material is absent. On Schum’s view, argumentation determines whether the positive evidence survives scrutiny and whether the silence itself deserves evidential weight.

This distinction also answers a common objection: we often lack direct records for ordinary people in history even though we have good reason to believe such people existed. But that conclusion is not evidence-free. It is supported by background and indirect evidence, including the persistent and recurring fact of human reproduction. The relevant question is therefore never simply whether one specific record is absent, but what evidence the hypothesis predicts across the available evidential space.

Admissibility, Relevance, and Exclusionary Restrictions

What counts as evidence determines what can count as total evidence. Historical inquiry may not use admissibility rules as formally specified as those in law, but historians still rely on practices for judging relevance, provenance, reliability, and evidential weight. The admissible evidential set is therefore broader than the single-source pathway considered in the model above. Those criteria matter directly to the computation of $P(E \mid H)$ , because changing what qualifies as E changes the probability being assessed.

Criterion	Function
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 $P(E \mid H)$.
Reliable Evidence	Evidence must be sufficiently credible or reliable. Unreliable evidence could skew $P(E \mid H)$, 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 $P(E \mid H)$ 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, $P(E \mid H)$ may only reflect a subset of the true evidence. This violates the total evidence principle and can lead to misleading Bayesian inferences.

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 $P(E \mid H)$, 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:

Domain	How exclusionary restrictions operate
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.
Bayesian Reasoning	: Exclusionary restrictions in Bayesian reasoning may determine which evidence $E$ is included in $P(E \mid H)$. 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.
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.
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.
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:

Restriction type	What it excludes
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, $P(E \mid H)$ should only include evidence that can differentiate between $H$ and competing hypotheses.
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.
Legal or Procedural	: Evidence that violates procedural rules or legal principles is excluded. Example: Illegally obtained evidence is inadmissible in many judicial systems.
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.
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 $P(\neg E \mid H)$ 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, the 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:

The Bayesian Formulation

With those epistemic conditions in view, the argument can be expressed in Bayesian terms. The formalism is useful because it makes explicit which probabilities determine whether silence favors a hypothesis or its negation.

With the evidential framework in place, we can now formalize the argument. A Bayesian analysis asks how observing the absence of expected evidence changes the odds of a hypothesis. The central comparison is between the probability of silence if H is true and the probability of silence if H is false.

Definitions

$H$: The hypothesis that the event or entity in question exists/occurred.
$\neg H$: 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).
$\neg E$: 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(\neg E \mid \neg H)}{P(\neg E \mid H)}.

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

Step 1: Assign Probabilities

$P(\neg E \mid H)$ 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), $P(\neg E \mid H)$ will be low.
$P(\neg E \mid \neg H)$: The probability of silence given that$H$ is false.
- This depends on the context. If no evidence would arise regardless of $H$’s truth, $P(\neg E \mid \neg H)$ will be high.
Prior Odds $P(\neg H) / P(H)$ Our initial belief in the likelihood of $H$ vs. $\neg H$.

Step 2: Likelihood Ratio

The argument from silence is strongest when:

P(\neg E \mid \neg H) > P(\neg E \mid H).

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

Step 3: Odds Update

Using Bayes' theorem, we update the odds:

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

Here are some general implications of the argument:

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

Here is a fictitious example:

Quantity	Assigned value / interpretation
$P(H)$	Prior probability of $H$ (e.g., the event occurred) = 0.5.
$P(\neg H)$	Prior probability of $\neg H$ = 0.5.
$P(\neg E \mid H)$	Probability of silence if $H$ is true = 0.2.
$P(\neg E \mid \neg H)$: Probability of silence if	$H$ is false = 0.8.

Likelihood Ratio:

\text{Likelihood Ratio} = \frac{P(\neg E \mid \neg H)}{P(\neg 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 $\neg E$ decreases significantly because the silence is more likely if $\neg H$ is true. By comparing $P(\neg E \mid H)$ and $P(\neg E \mid \neg H)$, this Bayesian framework provides a systematic way to evaluate arguments from silence.

The formalism clarifies the core claim: an argument from silence gains force when the likelihood ratio favoring the negation of H is substantially greater than 1. It also makes the assumptions visible, which is precisely why the framework is useful for criticism. The next question is not whether the algebra is valid, but whether the probabilities and evidential categories used in the model have been defined in a defensible way.

Where the Bayesian Treatment Can Go Wrong

Formalization, however, does not guarantee a sound analysis. The mathematics can be correct while the evidential categories, conditional probabilities, or search space are defined in ways that prejudge the result.

The Proposed Model

In "The Argument from Silence," 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, $P(\neg E \mid \neg H)/P(\neg E \mid H)$. 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 $\neg E$ 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 $P(E \mid H)$ is rather small, say less than 0.5, the denominator of the likelihood ratio, which is equal by definition to 1 – $P(E \mid H)$, 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.

The third consequence is the one I want to examine. The mathematics is straightforward, but establishing the magnitude of

P(E \mid H)

is not. That probability can be inflated or deflated by how the reasoner defines the relevant conditions and the evidential search space. McGrew analyzes it using three conditions:

If H (the event or fact in question) were true, how probable is it that the author in question would have noticed it (N)?
If H were true and the author had noticed it, how probable is it that he would record it (R)?
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:

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

$P(N \mid H) \times P(R \mid H \land N) \times P(S \mid H \land N \land R)$

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

Exhaustiveness, Independence, and Probability Dilution

The chain-rule decomposition on page 13 is mathematically correct if the listed conditions are the relevant conditions and are handled with the appropriate dependence structure. If any conditional probability is low, the product can become low; correspondingly, a low

P(E \mid H)

makes

P(\neg E \mid H)

high and pushes the likelihood ratio closer to 1, weakening the argument from silence. The mathematical step is not my objection. The problem is how the conditions are chosen. Why assume that these three conditions capture the relevant evidential pathways? Why center the analysis on a single author? And why treat the survival condition as though it exhausts the other ways evidence could reach us?

The author partly acknowledges the point by stating that $P(N \land R \land S \mid H) \le P(E \mid H)$ . If N, R, and S exhaust the relevant routes to E, then the two quantities would be equal. My objection is that there is no good reason to assume they exhaust the evidential space. More realistically, $P(E \mid H)$ includes those conditions plus other possible evidential pathways, and the listed conditions may themselves be dependent. Narrowing E to this conjunction can therefore depress $P(E \mid H)$ by construction.

I also think that by stating $P(N \land R \land S \mid H)$, the author is arbitrarily stipulating events that should be aggregated together, and is therefore arbitrarily deflating $P(E \mid H)$ 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 $P(E \mid H)$ and therefore inflating $P(\neg E \mid H)$.

There is 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:

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

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:

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

The danger is that added detail can quietly impose extra probabilistic constraints. Reasoners may treat dependent events as independent, overlook the base rate of the primary event, or define a condition so narrowly that its probability becomes artificially small. Suppose, for example, that

\(P(N)\)

is treated as the joint probability that an author was awake, had functioning eyes, possessed a pen and paper, and knew how to write. Packaging all of those background conditions into N can radically deflate

\(P(N)\)

, even though several of them may already be implicit in the context. This is one reason Bayesian analyses become fragile when the underlying probabilities are not anchored by defensible base rates or a clearly specified model.

Arbitrarily adding conditions, especially highly specific ones, can therefore produce a form of probability dilution. In the example above, the analysis makes

P(\neg E \mid H)

high because a joint condition such as

P(N,R,A \mid H)=P(E \mid H)

has first been made low through the selected conditions. My concern is that N may be redundant and that S itself depends on many further conditions. The calculation can be formally valid while the model construction does the substantive work. The following steps make that concern more explicit.

The Awareness and Survival Conditions

1. N (Awareness) Redundant

If the ultimate goal is to analyze $P(E \mid H)$ or $P(\neg E \mid H)$—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: $P(E \mid H) = P(R, A \mid H) = P(R \mid H) \cdot P(A \mid R, H)$ $N$ 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 $P(A \mid R, H)$ seem arbitrarily low if the dependencies aren’t accounted for properly.

Why This Matters:

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 $P(A \mid R, H)$ can feel arbitrarily small.
This small value for $P(A \mid R, H)$ might unfairly dominate the overall $P(E \mid H)$, even if $R$ (writing it down) was highly probable.
The Fallacy of Over-Specification: Treating $A$ as a single, unified condition hides the fact that $P(A \mid R, H)$ is actually a conjunction of many events, each of which reduces the total probability: $P(A \mid R, H) = P(A_1 \cap A_2 \cap A_3 \dots \mid R, H)$ where $A_1, A_2, A_3, \dots$ represent conditions like physical preservation, political continuity, and access to historical archives.
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 $P(A \mid R, H)$ 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:

$R$: The fact is written down.
S: The writing survives through history.

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

Probability of No Evidence:

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

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:

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

For example:

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

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

$P(\neg E \mid H)$ is high because $P(E \mid H) = P(R, A \mid H)$ is low:

I would argue that $A$ (survival) depends on overly complicated and arbitrary assumptions. Suggest using a broader, more reasonable $P(A \mid R, H)$.
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 $P(E \mid H)$.

The analysis is overly restrictive if it focuses exclusively on one individual's written account as the sole determinant of $P(E \mid H)$ or$P(\neg E \mid H)$. 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.

The Total-Evidence Problem

This leads to the central technical concern: the evidential variable E should not be restricted to one author’s testimony when a historical event could have left many independent traces. The relevant question is what the total evidential landscape should look like if H were true.

Evidence Is Multimodal

1. Evidence is Often Multimodal

Historical evidence rarely relies on a single source or pathway. The absence of evidence $\neg E$ cannot reasonably hinge on just one author’s potential recording of an event. Historical evidence often emerges from multiple pathways, so $P(\neg E \mid H)$ must account for all of these potential sources. $P(\neg E \mid H)$ is not determined solely by one author. Instead, it typically arises from a combination of:

Evidence pathway	Role in the analysis
Direct Written Records	The specific author’s mention (e.g., $R$ and S in the current model).
Other Independent Written Accounts	Records by other authors or cultures, often reinforcing or corroborating the event.
Physical or Archaeological Evidence	Artifacts, ruins, or environmental traces that indirectly suggest the event happened.
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 $P(E \mid H)$ and reduce $P(\neg E \mid H)$.

2. Expanding $P(E \mid H)$ to Include Multiple Pathways

To account for these additional sources, $P(E \mid H)$ 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:

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

General Formula:

Let:

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

The expanded $P(E \mid H)$ is:

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

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

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

3. Reassessing $P(\neg E \mid H)$

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

$P(\neg E \mid H) = 1 - P(E \mid H).$

If $P(E \mid H)$ becomes significantly larger because of multiple pathways, $P(\neg E \mid H)$ becomes much smaller.

4. Addressing the Redundancy of Focus on One Source

By restricting $P(E \mid H)$ 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 $P(E \mid H)$, 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:

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

These indirect forms of evidence are not captured by $P(\neg E \mid H)$ 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 ($\neg E_1$), the physical data $E_2$ and contextual evidence $E_3$ could still strongly support $H$. Ignoring these sources severely underestimates $P(E \mid H)$ and inflates $P(\neg E \mid H)$.

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 $P(E \mid H)$.

6. Accounting for Uncertainty in Alternative Pathways

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

Assign Probabilities to Additional Pathways: Estimate $P(\text{Other Written Records} \mid H)$, $P(\text{Physical Data} \mid H)$, etc., based on the likelihood of independent documentation or physical traces.
Example: If the hypothesis concerns a widely known historical event, $P(\text{Other Written Records} \mid H)$ might be high.
Incorporate Conditional Independence: Treat the different sources as conditionally independent given $H$, unless strong dependencies are known.
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:

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

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

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

This broader perspective ensures a more accurate and balanced assessment of $P(E \mid H)$ 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 $P(E \mid H)$. Focusing solely on one individual's record (and whether or not it survives) as the determinant of $\neg E$ (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 $P(\neg E \mid H)$ to an unreasonable degree. Essentially, we can challenge the restrictiveness of $P(\neg E \mid H)$ by addressing the complementarity of evidence and presenting a broader framework for evaluating $P(E \mid H)$. Even if one pathway fails to produce evidence, others can compensate, thereby reducing

$P(\neg E \mid H)$. This broader view ensures that the absence of one type of evidence (e.g., written records) doesn’t overly inflate $P(\neg E \mid H)$. Below is a summary of what we have discussed thus far:

Redundant Evidential Pathways

1. Written Records Are Not the Sole Source

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

$E$ depends solely on the existence, recording, and survival of a single written account.
$\neg E$ 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 $P(E \mid H)$.

3. Redundancy of Evidence Reduces $P(\neg E \mid H)$

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

$P(\neg E \mid H) = P(\neg E_1 \cap \neg E_2 \cap \neg E_3 \cap \dots \mid H)$

Where:

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

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

$P(\neg E \mid H) \approx P(\neg E_1 \mid H) \cdot P(\neg E_2 \mid H) \cdot P(\neg E_3 \mid H)$

For example:

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

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

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

Where:

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

Practical Simplifications:

If the pathways are roughly independent:

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

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

4. Likelihood Ratio and the Absence of Evidence

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

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

The Requirement of Total Evidence

Example:

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

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

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 $P(\neg E \mid H)$ to be very close to zero. This would affirm $P(\neg E \mid \neg H)$. 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]

The requirement of total evidence extends this point. Beliefs and probability assignments should be based on all available and relevant evidence rather than on a convenient subset. That principle fits naturally with Bayesian updating: the update rule is only as informative as the evidence supplied to it. If relevant evidence is omitted or the evidential variable is defined too narrowly, a mathematically correct update can still produce a misleading posterior.

How we define the E in $P(E \mid H)$ matters because if E is a subset of the total evidence, $P(E \mid H)$ 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 $P(E)$. $P(E \mid H)$ 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 $P(E \mid H)$ 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, $P(\neg E \mid H)$ should refer to the entire set of evidence E that is absent $\neg E$ but expected.

Priors, Auxiliary Assumptions, and Ad Hoc Rescue

Even a well-defined evidential set does not prevent a hypothesis from being insulated from disconfirmation. Strong priors and auxiliary assumptions can absorb contrary evidence unless those assumptions are independently justified.

Why Auxiliary Assumptions Are Introduced

A further complication appears when an agent is already strongly committed to H. If $P(\neg E \mid H)$ is very low, the absence of expected evidence creates pressure to revise the hypothesis. One way to resist that pressure is to add ad hoc auxiliary assumptions that make the silence less surprising. The important issue is not that auxiliary assumptions are always illegitimate, but that they require independent justification rather than being introduced solely to protect H from disconfirmation.

Motivating factor	How it can preserve H
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.
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").
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 $\neg E$ 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.
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.
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.
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.
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.

Common Rescue Strategies

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:

Rescue strategy	How it works
Conspiracy Theories	: Claiming that evidence is intentionally suppressed or hidden (e.g., "The lack of documents is due to a cover-up").
Hypothetical Entities	: Postulating unobservable factors to explain the absence of evidence (e.g., "An unknown mechanism prevents us from detecting X").
Unfalsifiable Assumptions	: Introducing assumptions that cannot be tested independently (e.g., "We haven’t found evidence yet, but we will eventually").
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.

The Burden of Proof for Auxiliary Assumptions

When someone inserts an ad hoc assumption—say $h_1, h_2, \dots, h_N$ —into a hypothesis $$H$$ , the burden shifts to that person to show that the auxiliary assumptions are independently plausible and evidentially supported. The reason is straightforward: the new assumptions are being introduced to explain away what would otherwise count against H. Without independent support, they reduce the hypothesis’s credibility and falsifiability. If someone claims, for example, that $P(\neg E \mid H)$ is low because “the evidence was suppressed,” they now bear a burden to show that suppression probably occurred.

Shift in the Explanation: : When someone asserts that $P(\neg E \mid H)$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.
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.
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 $P(\neg E \mid H)$ is low because "evidence was suppressed," they must:

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

Without that support, the explanation risks becoming circular or unfalsifiable: the absence of evidence is explained by suppression, while suppression is inferred from the very same absence. A credible suppression hypothesis would need independent support—for example, leaked documents, reliable testimony, or a demonstrated pattern of behavior by relevant agents. Even when direct evidence of suppression is unavailable, the reasoner must at least show that suppression is more probable than competing explanations for the missing evidence. Otherwise the auxiliary assumption simply shields H from disconfirmation.

More generally, unverified auxiliary assumptions reduce falsifiability by shielding H from disconfirmation. Once such assumptions are added, the debate shifts from H itself to the auxiliary claims $h_1, h_2, \dots, h_n$ . Those claims inherit their own burden of proof. If no such burden were imposed, one could add an endless chain of further assumptions to explain any anomaly, making the original hypothesis effectively untestable. Auxiliary assumptions can therefore preserve a hypothesis only when they are supported by positive evidence or strong independent reasoning; otherwise they weaken, rather than rescue, the explanatory case for H.

Why Apologetics Is Especially Interested in the Argument

These points bring the analysis back to the opening search-result puzzle. The issue is not simply whether an argument from silence can fail; it is how standards for evidence, priors, and auxiliary explanations are applied when a hypothesis is tied to prior ideological commitments.

Returning to the Search-Result Puzzle

This brings us back to the question that opened the essay: why are apologists so unusually interested in arguments from silence? My answer is that many biblical historical claims face sparse or disputed evidential support, so the legitimacy of reasoning from missing evidence matters directly to apologetic projects. I agree that many conditions can prevent a historical author from recording a fact. The problem arises when those conditions are merely assumed because the expected record is absent.

Those negating conditions must themselves be established. Otherwise the response risks becoming an argument from ignorance. Consider a condition proposed by a Catholic apologetics source: perhaps the subject was too “embarrassing” for an author to record. That is certainly possible. But if embarrassment is inferred solely from the absence of the record, the reasoning becomes circular: “I do not know what prevented the author from recording the information, but something must have prevented it; therefore the hypothesis remains true.”

An inference to the best explanation could, in principle, support embarrassment or some other omission mechanism. But then the explanation has to be defended on independent grounds. Plausibility judgments are shaped by background assumptions, and two interlocutors may not share the same worldview or interpretive commitments. If one person treats a theological interpretation as background knowledge and another does not, merely calling the resulting explanation “plausible” will not resolve the disagreement. We need evidence that the proposed condition was actually operative.

Strong Priors and Prior Manipulation

The same problem appears in the opposite direction when a strong prior commitment to H dominates the evidential update. Suppose an argument from silence yields $P(\neg E \mid \neg H) > P(\neg E \mid H)$ , while the base rate also favors $\neg H$ . An agent may nevertheless continue to affirm H. If the likelihoods were equal, the silence itself would be uninformative and the rational position could be agnostic; but when both the prior and the evidential pattern count against H, preserving H requires some additional justification.

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. In other words, $P(H)$ strongly outweighs any form of evidence or absent evidence, such that $P(\neg H)$ becomes implausible by definition.

Selection Bias and Premature Fallacy Attribution

Selection effects complicate historical inference, and non-experimental fields such as history are especially vulnerable to them. Selection bias and survivorship bias, for example, can distort the evidential record. But if multiple independent lines of inquiry make H unlikely and there is no evidence for a selection mechanism that would explain the missing data, inferring the negation of H can still be reasonable. Rejecting that form of inference altogether produces counterintuitive results, including ordinary cases such as the cheating-spouse example discussed earlier. The search-result pattern itself can also be treated as a selection question: why is the visible sample so heavily weighted toward apologetics rather than professional historical methodology?

My suggestion is that there is a strong motivational incentive to dismiss the argument as ipso facto fallacious, or to label particular instances fallacious before engaging with the underlying evidential expectations. Earlier I called this the “Fallacy Fallacy,” but that label is not quite precise. A better description is premature fallacy attribution driven by defensive reasoning.

Motivated Reasoning

Premature fallacy attribution is closely related to motivated reasoning: the tendency to process information in ways that protect existing beliefs, emotions, or commitments. A person who strongly opposes a conclusion may be tempted to identify a straw man, an ad hominem, or another named fallacy before asking whether the argument actually has that structure. The label then becomes a cognitive shortcut: it licenses dismissal without the harder work of assessing the evidence. The following Roman Empire example illustrates the mechanism.

Illustrative Example: Premature Fallacy Attribution

Stage	Content
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:

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

What the Search Results Suggest

I am suggesting that the Google search results can be explained by this pattern rather than by a basic error that professional historians have somehow overlooked. For apologetic projects, sparse evidence for biblical claims creates pressure to weaken the legitimacy of arguments from silence. That pressure can also encourage caricatures of historical reasoning—for example, reducing a complex evidential case to the slogan “The Exodus never happened because there is no evidence.” A historian might use that sentence as an elevator summary, but the actual argument depends on the expected varieties of evidence, the quality of the available record, alternative explanations for silence, and the total evidential context. Calling the compressed slogan fallacious does not answer the fuller argument.

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.

Conclusion

The central conclusion is not that every argument from silence succeeds. It is that the form is a legitimate kind of defeasible, probabilistic reasoning when the missing evidence was genuinely expected under H. Its force depends on how the evidential search space is defined, whether the analysis incorporates total relevant evidence, and whether alternative explanations for the silence are independently supported.

The Bayesian framework helps expose those dependencies, but it does not remove judgment from the analysis. A formally correct calculation can still mislead if $$E$$ is defined too narrowly, if conditional probabilities are diluted by artificial conjunctions, if priors are fixed to protect a favored conclusion, or if auxiliary assumptions are introduced only after the expected evidence fails to appear. Those moves shift the real dispute from the arithmetic to the construction of the model.

That is why the apologetics search pattern matters. My claim is that the prominence of these discussions reflects strong prior commitments to biblical historicity and inerrancy, together with incentives to challenge the credibility of evidential silence, introduce auxiliary explanations, or shift the burden of proof when expected evidence is missing. If those moves are not independently justified, they do not defeat the argument from silence; they instead reveal where the evidential burden has moved.

Quantity	Assigned value / interpretation
\(P(H)\)	Prior probability of \(H\) (e.g., the event occurred) = 0.5.
\(P(\neg H)\)	Prior probability of \(\neg H\) = 0.5.
\(P(\neg E \mid H)\)	Probability of silence if \(H\) is true = 0.2.
\(P(\neg E \mid \neg H)\): Probability of silence if	\(H\) is false = 0.8.

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