Having an intelligent conversation with a computer that has superhuman knowledge is still something I can’t stop marveling at and thinking about… Here’s a small excerpt from one of my conversations. I realize that unlike a Wikipedia search, or using a search engine there is the danger of it being much easier to get swindled into believing made up facts, and that there are other dangers related to AI but all that doesn’t diminish from my utter amazement of how far we have come in technological advancements, and my curiosity on where does the exponent lead to…

**Has Hempel’s raven paradox been settled in a way that is agreed upon?**

No, there is no universally agreed upon resolution to Hempel’s raven paradox among philosophers of science and logicians. The paradox arises from an apparent inconsistency in how we confirm or refute hypotheses through observations.

The paradox is based on the following argument:

1) The hypothesis: All ravens are black.

2) Observing a black raven confirms this hypothesis (a positive instance).

3) But logically, observing a non-black non-raven (e.g. a green apple) also confirms the same hypothesis, because it is logically equivalent to “All non-black things are non-ravens.”

This leads to the paradoxical conclusion that confirming instances of “All ravens are black” include not just black ravens, but any object that is not a raven. This seems to go against scientific practice.

There have been various proposed resolutions or dissolutions, but none is universally accepted:

1) Some argue the paradox arises from an impoverished account of confirmation and that a more refined theory of inductive logic would resolve it.

2) Others see the paradox as pointing out flaws in our intuitive notions of confirmation rather than logic itself.

3) Some propose distinguishing between different kinds of evidence as a way to defuse the paradox.

4) A Bayesian perspective views the paradox as arising from improper application of Bayesian inference.

So while insightful, the paradox remains an active topic of debate in philosophy of science, without a consensus resolution as of yet. It continues to spur discussions about inductive reasoning and scientific methodology.

**I assume this has been also linked to the problem of induction. Have there been any advancements in resolving that problem?**

You’re correct that Hempel’s raven paradox is closely linked to the broader problem of induction, which questions the justification for making inductive inferences based on empirical observations to arrive at general conclusions or theories.

While there is still no universally accepted solution to the problem of induction, there have been some notable perspectives and advancements in grappling with it:

- Karl Popper’s Falsificationism: Popper argued that we can never ultimately verify or prove a theory through induction, but we can falsify or refute theories through counterexamples. This shifts the focus to rigorously attempting to falsify theories.
- Bayesian Confirmation Theory: This probabilistic approach, pioneered by Thomas Bayes, allows evidence to quantitatively confirm or disconfirm hypotheses to certain degrees based on prior probabilities. It offers a way to make inductive inferences without claiming certainty.
- Inference to the Best Explanation: This view claims that we can justifiably infer to the theory that best explains the available evidence, even if not proven with certainty. Pioneered by philosophers like Peter Lipton.
- Explanationist Defense of Induction: Some have argued that inductive inferences are inductively supportable in the meta-inductive sense – our successful reliance on induction in the past provides an inductive basis for its future reliability.
- Computational Views: Approaches drawing from information theory, algorithmic complexity, and minimum description lengths have provided new perspectives on justified inductions.

However, none of these approaches fully resolves the philosophical problem of induction as posed by Hume and others. The problem continues to be an active area of research and debate in philosophy of science, epistemology, and other fields. But the new frameworks offer insights into constraining and justifying inductive practices provisionally, even if an airtight deductive solution remains elusive.

**Have there been any efforts to view and perhaps make progress on the induction problem in light of the efficacy of supervised machine learning?**

Yes, there have been interesting discussions and proposals from philosophers and computer scientists on how the success of machine learning, especially supervised learning, might inform or provide new perspectives on the problem of induction.

Here are some of the key ideas that have been explored:

- The No Free Lunch Theorems: These theorems from computational learning theory show that no single inductive learning algorithm can be optimal across all possible learning tasks. This highlights the need for inductive biases or assumptions in machine learning, which has parallels to philosophical debates around the justification of inductive premises.
- Pragmatic Vindication of Induction: Some argue that the remarkable empirical success of machine learning vindicates inductive inference in a pragmatic sense, even if not providing a deductive philosophical solution. ML’s ability to find patterns and make accurate predictions from data can be viewed as a practical resolution.
- Bayesian Interpretations: Bayesian models of inductive inference have gained influence in philosophy and machine learning. The ability of Bayesian methods to successfully incorporate prior knowledge and uncertainty has been proposed as a framework for inductive reasoning.
- The Role of Simplicity: Principles like Occam’s razor about preferring simpler hypotheses that fit the data are operationalized in techniques like regularization in ML. This has sparked discussions around simplicity principles potentially justifying induction.
- Bootstrapping Induction: The idea that pattern recognition and induction can become more reliable by being grounded in higher levels of abstraction derived from previous inductions. Deep learning can be viewed as bootstrapping simple inductions into more powerful representations.

While machine learning doesn’t address the deductive philosophical issues, its effectiveness has led some philosophers and scientists to argue for a more pragmatic or model-based approach to induction. However, the induction problem remains an active area of conceptual exploration across multiple fields.**How does the Bayesian perspective (which claims that observing a non-black object which is not a raven adds some infinitesimal evidence that all ravens are black) defend the fact that by the same logic, observing a non-red object which is not a raven should add infinitesimal evidence that all ravens are red?**

The Bayesian perspective on the raven paradox does face some challenges in fully resolving the issue you raise about the apparent symmetry between confirming “all ravens are black” and “all ravens are red” through non-raven observations.

However, Bayesian proponents argue that the degree of confirmation provided by those non-raven observations is not actually symmetric between the two hypotheses, because the hypotheses have different prior probabilities to begin with.

Specifically:

1) The hypothesis “all ravens are black” has a relatively high prior probability based on our background knowledge that ravens are a type of bird and birds are typically black/dark in color.

2) The hypothesis “all ravens are red” has a very low prior probability, as red is a highly atypical color for ravens or any bird species.

So while observing a green apple does boost the posterior probability of both hypotheses in a Bayesian analysis, the degree to which it boosts “all ravens are black” is much higher because that hypothesis already had a considerably higher prior probability to begin with.

Additionally, Bayesians argue that in practice, scientific hypothesis testing does not occur in a vacuum – we use additional constraints like background knowledge, assumptions of uniformity, temporal and spatial proximity of observations etc. Within such a constrained model, non-raven observations provide stronger confirmation to the more plausible “all ravens are black” hypothesis.

That said, the symmetry issue does expose limitations to the pure Bayesian account. Some Bayesian philosophers have argued for adopting a hierarchical model with different hypothesis spaces or language constraints to fully resolve paradoxes like this.

So in summary, while not a complete resolution, the Bayesian framework attempts to defuse the paradox by asymmetric priors and realistically constrained hypothesis spaces. But it remains an interesting challenge case.**So what this means is that the Bayesian approach claims that the non-black non-raven sighting contributing to evidence for all ravens are black is only relevant after at least one positive black raven sighting.**

Yes, that’s an accurate way to characterize the Bayesian response to the raven paradox regarding non-black non-raven observations providing confirming evidence for “all ravens are black.”

The key points are:

1) Before any observations, the prior probability assigned to the hypothesis “all ravens are black” is relatively high based on background knowledge about ravens being a type of black bird.

2) Observing an actual black raven provides a significant boost to the posterior probability of this hypothesis in the Bayesian calculation.

3) Only after establishing this initial evidential support from positive raven instances, can non-black non-raven observations then provide any confirming evidence at all (however infinitesimal) for the “all ravens are black” hypothesis.

This is because in the Bayesian framework, observing a green apple is consistent with but does not directly confirm or disconfirm the hypothesis until we have established some initial plausibility. Only then can the non-raven observation complement the positive evidence.

So the Bayesian view is that non-black non-raven observations are entirely irrelevant for confirming “all ravens are black” in a context with zero prior positive instances. They only become relevant as supplementary confirmations after black raven sightings have first raised the posterior probability substantially.

This asymmetry between requiring positive instances first, before allowing non-black non-ravens to complement, is key to the Bayesian resolution of appearing to confirm hypotheses too promiscuously based just on non-instances.**How would a Bayesian respond to a hypothetical scenario where half of the raven population are black and half are red. All the red ravens are put in one place which cannot be accessed. Now suppose that it is confirmed that all the seen ravens are black, and then confirmed that all accessible non-black objects in the universe are non-ravens. According to the Bayesian perspective, this should give immense evidence that all ravens are black. But we know that half of the raven population that wasn’t accessed is red. How is this accounted for using the Bayesian explanation?**

You raise an excellent challenge to the Bayesian account with this hypothetical scenario involving a hidden population of red ravens. Let me walk through how a Bayesian perspective might try to address this:

1) Initially, without knowledge of the hidden red raven population, accumulating evidence that all observed ravens are black and all observed non-black things are non-ravens would indeed raise the posterior probability extremely high for the hypothesis “all ravens are black” in a Bayesian calculation.

2) However, the key assumption that gets violated in this scenario is that the observations are representative samples from the total raven population. Bayesian updating explicitly or implicitly assumes exchangeability – that the observations aren’t biased samples.

3) If we incorporate the knowledge that half the total raven population was completely unobserved and potentially different, this breaks the exchangeability assumption required for straight Bayesian updating from the observed data.

4) A Bayesian could account for this by introducing uncertainty about the representative sampling process itself into their prior probabilities and likelihood calculations. This may involve hierarchical modeling.

5) Essentially, they would have to factor in a non-trivial prior probability that the observed sample is misleading about the total population, given the knowledge that an entire other raven sub-population was cordoned off.

So while the Bayesian machinery can potentially account for this via complex representation of sampling uncertainties, it hinges on accurately specifying those prior probabilities about sampling bias in the first place, which requires incorporating the knowledge about the hidden sub-populations upfront.

If this prior knowledge is not accounted for appropriately, then yes, the Bayesian can fall victim to making overconfident inductive inferences from an unwittingly biased sample, exactly as your hypothetical example demonstrates.

This acts as a caution that simply updating from data in a Bayesian way is not sufficient – one also needs robust priors about potential sampling and observation biases. Getting this part right is critical for Bayesian inference to remain coherent in practice.