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Fix $r>1$ ravens and $M>1$ nonblack objects in a finite population. Under $H_0$, all ravens are black; under $H_1$, exactly one is nonblack. The total numbers $r,M$ are held fixed under both hypotheses, so the classification of one nonraven also changes between hypotheses. This is a deliberately specified model for Bayesian confirmation, not a universal rule about how bird observations supply evidence.

A uniform draw among nonblack objects returns a nonraven with probabilities $1$ under $H_0$ and $1-1/M$ under $H_1$. A uniform draw among ravens returns a black object with probabilities $1$ and $1-1/r$. The respective likelihood ratios for $H_0$ over $H_1$ are therefore $(1-1/M{)}^{-1}$ and $(1-1/r{)}^{-1}$. When $M>r$, the former is closer to one, so supplies less support under these assumptions.

With equal prior probabilities, Bayes' theorem gives posterior probabilities $1/(2-1/M)$ and $1/(2-1/r)$. More generally, with prior probability $p$ for $H_0$, replace the two expressions by $p/[p+(1-p)(1-1/M)]$ and $p/[p+(1-p)(1-1/r)]$. Previously known information or a different sampling protocol requires different likelihoods; selecting an object already known to be a nonraven need not supply new evidence.

Let $R(x)$ mean that $x$ is a raven and $B(x)$ that it is black. The hypothesisis logically equivalent to $\forall x[\neg B(x)\Rightarrow \neg R(x)]$. If the observed paper is both nonblack and a nonraven, it is a positive instance of the second formulation. Combining an unrestricted positive-instance rule with preservation of scientific confirmation under logical equivalence produces the raven paradox.^[ref]

Whether the observation actually supplies evidence depends on what was already known and how the object was selected. In Bayesian confirmation, with background $K$, support means $P(H\mid E,K)>P(H\mid K)$. If the observer already knows the selected item is a nonraven and its colour is conditionally independent of $H$, then $P(E\mid H,K)=P(E\mid \neg H,K)$: the likelihood ratio is one and there is no confirmation. If all of $E$ was already known, learning it again also gives no update.

Under a different sampling scheme, the same classification can provide slight support. In the finite-population model of raven confirmation, fix $r$ ravens and $M$ nonblack objects. Compare no nonblack raven with exactly one. Sampling uniformly among the nonblack objects gives a nonraven with probabilities $1$ and $1-1/M$, hence likelihood ratio $(1-1/M{)}^{-1}>1$. If $M$ is large this is tiny evidence, not a proof. This example establishes possibility under explicit assumptions, not that every white-paper observation has that evidential role.

The appropriate answer is therefore conditional: mere logical instantiation does not settle the support supplied by observing this paper. State the background and sampling procedure. A familiar paper deliberately selected as a known nonraven normally supplies no new evidence about raven colour; a specified alternative sampling model may yield a small positive update.

A Bayesian confirmation account can dissolve the apparent conflict between logical equivalence and ordinary judgements about relevant evidence. It does so by rejecting an unrestricted positive-instance rule and specifying how the evidence was sampled. Bayes' theorem alone does not guarantee that every white shoe is irrelevant, or that every black raven provides strong support.

Let $H$ assert that every raven is black. Its contrapositive is logically equivalent to $H$, so the two descriptions must receive the same conditional probability. The raven paradox arises if every positive instance of a universal statement automatically confirms it: a white nonraven then confirms the contrapositive and hence $H$. The disputed step is that unrestricted confirmation rule, not the logical equivalence.^[ref]

For background $K$, set $p=P(H\mid K)$, $a=P(E\mid H,K)$ and $b=P(E\mid \neg H,K)$. For $0<p<1$ and positive evidence probability,Thus it is the comparison of likelihoods, including the sampling protocol and alternatives, that determines support. Merely rewriting $H$ cannot change it.

The finite-population model of raven confirmation makes the intuitive difference quantitative. Fix $r$ ravens and $M$ nonblack objects. Compare an all-black-ravens hypothesis with the alternative that exactly one raven is nonblack. A uniform draw among nonblack objects that finds a nonraven has likelihood ratio $1/(1-1/M)$. A uniform draw among ravens that finds a black one has likelihood ratio $1/(1-1/r)$. If $M\gg r>1$, the former is much nearer one. With equal prior probabilities, $M=10^6$ and $r=100$ give posterior probabilities approximately $0.50000025$ and $0.50251256$. A tiny amount of confirmation need have no practical significance.

This example is not a theorem about every possible background. Inspecting an object already known to be a white shoe adds no relevant information about its being a nonraven. Other alternatives or selection mechanisms change the likelihoods and can remove or reverse the support. Conditioning only on the observed colour while silently changing how the object was selected would compare different experiments.^[ref]

The resolution is therefore qualified. A coherent probability model preserves equivalence and explains why superficially parallel observations need carry radically different evidence. It does not select a unique prior or sampling model without scientific judgement. Those remaining modelling questions are limitations of an application, rather than a contradiction in Bayesian inference.

Old evidence challenges a simple account of confirmation as actual conditional updating, but does not refute every probabilistic treatment of the raven paradox. The strongest response distinguishes support from newly acquired information and makes the sampling procedure explicit.

Let $H$ say that every raven is black. Its contrapositive says that every nonblack thing is a nonraven. Since these statements are logically equivalent, evidence should support them equally. Combining this equivalence requirement with the unrestricted rule that every positive instance confirms a universal generalization appears to make discovering a white shoe support $H$. The tension is between that result and the judgement that such a discovery is irrelevant.^[ref]

Bayesian confirmation replaces an automatic instance rule with a comparison of likelihoods. For background $K$, positive prior probability for both $H$ and its alternative, and evidence $E$ of positive probability,The equivalence follows directly from Bayes' theorem. A statement's grammatical appearance as a positive instance does not determine either likelihood.

Consider an intentionally simple finite model. There are $r$ ravens and $M$ nonblack objects, with these counts fixed under both hypotheses. Under $H$ none of the nonblack objects is a raven; under the sole alternative $H_1$, exactly one is. Selecting uniformly among the nonblack objects and discovering a nonraven has likelihoods $1$ and $1-1/M$. Selecting uniformly among the ravens and discovering a black one has likelihoods $1$ and $1-1/r$. With equal prior probabilities, the respective posterior probabilities of $H$ areIf $M\gg r$, the first discovery supplies far less support. For $M=10^6$ and $r=100$, the increases over $1/2$ are approximately $2.5\times 10^{-7}$ and $0.00251$. This illustrates a possible resolution of the intuitive disparity, not a theorem that every nonblack nonraven always confirms $H$. More complex alternatives can reverse the comparison, and examining something already known to be a shoe need supply no relevant new information.^[ref]

Now suppose $E$ is already in $K$. Then $P(E\mid K)=1$, and ordinary conditioning givesThere is no further increase from learning the same fact again. Yet an already established observation can become an important reason to accept a subsequently developed theory. Clark Glymour's problem of old evidence exposes this mismatch between actual updating and scientific assessments of evidential support.^[ref]

One response evaluates the observation relative to an earlier or counterfactual background that does not contain it. That can represent how strongly a theory accounts for the observation compared with alternatives. It must specify that background rather than pretend to remove knowledge uniquely from a person's present beliefs. Another response concerns learning that a theory successfully accounts for the observation. That discovery can be new even when the observation is old. Modelling it requires attention to limited reasoning and changing hypotheses; an ideal agent already knowing every logical consequence cannot simply learn an entailment it previously overlooked.

Neither manoeuvre licenses counting the same evidence twice, or ignoring that parameters were selected to fit it. Neither establishes a unique complete theory of confirmation. Nevertheless, the old-evidence objection concerns the interpretation and completeness of a probabilistic framework; it does not invalidate the likelihood calculation for genuinely new sampling information.

The preferable solution combines a background-sensitive account of support with rejection of unrestricted instance confirmation. It preserves logical equivalence while allowing a white shoe to provide tiny support, no support or contrary support under different justified models. This is more explanatory than assigning confirmation solely from logical form, although assessing old evidence requires additional work beyond routine conditioning.

Non-human animals show evidence for several components of social-state tracking, but the evidence does not establish a uniform, human-like theory of mind. Theory of mind should be decomposed into sensitivity to goals, visual access, knowledge and belief, rather than treated as a single ability that an animal either possesses or lacks. The central evidential question is whether behaviour requires representing another's mental state, or can be explained by learned relationships among observable actions, contexts and outcomes.

Goal attribution has support from experiments distinguishing similar outcomes produced by different actions. Chimpanzees responded differently to a person who was unwilling to transfer food and one who was willing but unable: they persisted differently and left earlier in the unwilling condition. ^[ref] Holding the absence of food transfer broadly constant prevents the explanation that any unrewarded trial produces the same reaction. The contrast supports sensitivity to the organization or apparent purpose of actions. However, differences in movement and affect remain observable cues, so the finding need not imply a fully explicit concept of intention. Generalization to new obstacles and action sequences is a stronger test than repeating familiar signals.

Visual perspective taking concerns another individual having a different view. When dominant and subordinate chimpanzees competed for food, subordinates preferentially obtained food to which they had better visual access. Giving the subordinate a headstart forced its choice before the dominant's approach could reveal the answer, reducing a straightforward movement-following explanation. ^[ref] Such controls support tracking of another's visual access. They do not show that the subject represents every aspect of how the scene looks to the other animal; access to an object is a narrower problem than its appearance from a different viewpoint.

The distinction extends from seeing to knowing. An animal can potentially remember whether another witnessed hiding even after neither currently sees the food. Controlled competitive tasks found chimpanzees sensitive to knowledge versus ignorance, yet not successful in closely related false-belief conditions. ^[ref] Knowledge attribution therefore has stronger support in those tasks than an unrestricted ability to predict action from mistaken beliefs. Negative performance is not proof of incapacity: memory, inhibitory demands, motivation and task comprehension may limit the response. But a positive result for knowledge should not be relabelled a positive result for belief simply because both are social.

Bird studies provide evidence beyond primates. Ravens that had experienced a peephole altered cache-protection behaviour when a potential observer could be heard but not seen. The use of recorded sounds removed an actual competitor's gaze and movements as immediate guides, and the manipulation of available visual access made the test more informative than comparison of merely occupied and empty rooms. ^[ref] This supports generalization from access conditions to a possible observer. It remains possible to model performance through learned relations between openings, sounds and risk; novel transfer tests are needed to distinguish how abstractly the bird represents “being seen”.

Experience with one's own actions can also change responses to others. Scrub jays with previous pilfering experience showed stronger re-caching responses after having been observed caching. ^[ref] One interpretation is projection of the bird's own pilfering experience onto a potential competitor. Another is acquisition of a more general rule about cache vulnerability. The finding makes prior experience a causal candidate and suggests a relation between self and other, but it does not establish that the bird consciously imagines another individual's intentions. Observed and unobserved caching, prior experience and the opportunity to relocate food all need to be distinguished.

False-belief understanding sets a particularly demanding criterion: the subject must predict an agent using information that the subject knows is wrong. In great apes, an anticipatory looking paradigm found looks towards the location where an agent falsely believed an object remained, even when the subject had seen the object's removal. ^[ref] This bypasses some demands of an active food-choice response. Yet gaze towards a previously attended location can also arise from action prediction, attentional cues or associations with where an actor last saw an object. Anticipation is therefore evidence about prediction, whose representational basis needs further controls.

A subsequent experiment manipulated the ape's own experience of a barrier that looked the same from a distance but was either opaque or translucent. All subjects then watched the same test actions. Those with the opaque experience were more likely to anticipate the actor searching according to the outdated location. ^[ref] Because the visible test behaviour was held constant, an explanation relying only on a difference in the actor's immediate behaviour was weakened. The result supports transfer from self-experience to another's informational situation. It still leaves questions about the breadth of the represented belief: a limited model of visual registration and action expectation may succeed without the flexible propositional reasoning characteristic of human language.

Taken together, positive anticipatory-looking results and negative active-choice results need not be forced into a single yes/no answer. The tasks measure different outputs and impose different demands. Stronger evidence would show the same individuals generalizing across novel agents, objects and routes of access, distinguishing true belief, false belief and ignorance while physical events, attention and rewards are controlled. Eye-tracking analyses should specify regions and timing in advance, report the subjects who produce no usable anticipatory look, and avoid inferring population-wide competence solely from a selected subset of trials. Replication and uncertainty around effects matter especially with small captive samples.

The relation to self-representation is similarly componential. Using one's experience of a transparent barrier to predict another's access suggests some correspondence between self and other. It does not demonstrate an autobiographical self or reflection on one's own beliefs. Metacognition, such as selective information seeking when knowledge is missing, concerns monitoring one's own processing and is not equivalent to attributing knowledge to somebody else. Mirror self-recognition concerns one's body and is neither a necessary consequence nor a sufficient demonstration of belief attribution. These distinctions prevent a visually based test from becoming a universal scale of mentality.

The defensible conclusion is a mosaic: animals can track goals and access, some experiments support knowledge attribution and belief-sensitive predictions, and the most elaborate interpretation remains contested. The experiments constrain possible representations of self and others; their value lies in progressively excluding alternatives, rather than naming a mental state after any apparently intelligent social response.

A researcher who coauthored experiments on cache protection when ravens could hear but not see competitors. Prior peephole experience helped test the role of inferred visual access.^[ref]