Andre Ye

Hume's Problem Solved, Schurz (*)

Last updated on 29 Jul 2022.

Review, notes, and reflections on Hume’s Problem Solved: The Optimality of Meta-Induction by Gerhard Schurz

Table of Contents

1: The Problem of Induction

1.1: The Notion of Induction

  • Concept of an ‘inductive inference’ - can be used in a wide and narrow context.
    • Wide interpretation - deduction preserves guarantees through logical inference and derivation from axioms; induction is associated with necessary uncertainty and assume regularity and uniformity.
      • Content expanding - inductive inference, in a wide interpretation.
      • Content preserving - deductive inference, in a wide interpretation.
    • Narrow interpretation - a regularity or pattern present in current observations is used to derive a prediction for a future instance. ‘Humean’ induction.
      • The book will adopt the narrow interpretation of induction henceforth.
  • Inductive prediction and inductive generalization - formulated probabilistically.
  • Focus of the book is justifying induction in the narrow sense - Hume’s problem. This book’s focus is on optimally justifications.
  • Separation of two problems: justifying induction and justifying abduction. Abductive problems are more complex and goes beyond Hume’s problem.

1.2: David Hume and the Problem of Justifying Induction

  • The Englightenment intended to liberate rational thought from dogmatic prejudice.
  • Hume introduced skepticism of the justifiability of induction in terms of causality (we know that event \(F\) is regularly followed by event \(G\) because \(F\) causes \(G\)).
  • Hume’s argument:
    1. Induction cannot directly be justified by observation.
    2. Induction cannot be justified by deductive logic.
    3. Induction cannot be justified by induction, for such is circular reasoning.
    4. Reformulation in terms of probability does not aid the case of justifying induction. To believe that “most observed \(F\)s have been \(G\)s, therefore the next \(F\) will be \(G\)” will be true in most future cases, we must infer that future probabilities can be inferred from past frequencies.
    5. There is no possible rational epistemic justification of induction, and it is rather psychological habit.

2: On Failed Attempts to Solve the Problem of Induction

2.1: Can Induction Be Avoided?

  • Karl Popper: rational thought does not need inductive arguments. To test a theoretical hypothesis, one must derive empirically testable consequences through deductive means. However, this argument fails because it does not address the inductivity of determining corroboration, not to mention that the testable consequences may not be derivable through deductive means in statistical contexts.
  • Epistemic induction is a subset of meta-induction. Concerns the induction of meta-hypotheses about the confirmation success of object-level hypotheses.

2.2: Is Induction Rational by Definition?

  • Analytic account - suggests induction is part of the meaning of the word ‘rational’.
  • We must derive an independent justification of induction rather than pointing towards ‘common sense’ to wave away problems.
  • Intuitionism and cognitive relativism. We cannot ground our epistemic account on intuitionist rationality.
  • We understand truth in a correspondence-theoretic but metaphysically neutral sense - the truth of a statement is understood in terms of its relationship with reality, but is not necessarily wholly outside of teh subject.
  • We need a system of arguments which favor the truth of predictions in comparison to noninductive methods. Arguments must be noncircular.

2.3: Can Induction Be Justified by Assumptions of Uniformity?

  • John Stuart Mill, Bertrand Russell - reliability of induction should be justified by the uniformity of nature.
  • How do we define uniformity?
  • The circularity of induction remains.
    1. Nature appears uniform because it has acted in consistent ways.
    2. Because it has appeared uniform in the past, it will continue to be uniform.
    3. Because nature has and will be uniform, it is uniform.
    4. Because nature is uniform, induction is valid.
  • Norton - localized uniformity, inductive reasoning is not governed by quantifier rules but local induction. Yet domain-specific locality assumptions are still subject to circularity.

2.4: Can Circular Justifications of Induction Have Epistemic Value?

  • Some suggest that circular justification can have epistemic value.
  • Premise circular - premise is identical with conclusion.
  • Rule circular - truth of the conclusion is presupposed by the underlying inference rule (e.g. rule of induction).
  • Inductive justification of induction is rule circular.
  • Rule-circular arguments can be used to simultaneously justify induction and anti-induction.
Component Rule circular justification of induction. Rule-circular justification of anti-induction.
Premise Past inductions have been successful. Past anti-inductions have not been successful.
Therefore, by the rule of induction: by the rule of anti-induction:
Conclusion Inductions will be successful in the future. Anti-inductions will be successful in the future.
  • Rule-circular justifications are epistemically worthless.
  • Blind trust in authority rule can be justified using rule-circular reasoning:
    1. Inference rule: if my accepted authority tells me \(P\), I infer \(P\) is true.
    2. Premise: My accepted authority tells me that blind trust in authority rule is reliable.
    3. Conclusion: I infer by the inference rule that blind trust in authority is reliable.

2.5: Can Induction Be Justified by Abduction or Inference to the Best Explanation?

  • Some argue inductive inferences are justified because they are instances of abductive inferences/inferences to the best explanation (IBEs).
  • IBE: infers the premises of the best available explanation are presumably true from an observed phenomenon.
  • An inductive generalization is an IBE because the best available explanation of an observed regularity is the generalization.
  • This only shows that inductive assumptions semantically are equivalent to implicit law-like assumptions.
  • Lawlike generalizations which fall under IBEs already presume a uniformity of nature (i.e. the possibility of generalization as the ‘best’ available explanation is itself succumbing to circularity).
  • The justification for IBE is itself rule-circular.

2.6: The Role of Induction and Abduction for Instrumentalism and Realism.

  • Interaction of epistemic induction and abduction:
    1. Evidence: a theory is the most empirically successful so far.
    2. Instrumentalist conclusion: this theory is the most empirically adequate, and therefore also the most empriically successful in the future. (Meta-inductive inference to empirical adequacy.)
    3. Realistic conclusion: this theory is the closest to the truth. (Abductive inference to approximate truth.)
  • Meta-inductive inferences allow us to transfer past empriical successes into the future.
  • Instrumentalism - accept the meta-inductive inference, but not the abductive inference. Theories can only be said to be more or less empirically fitting, not true or false with respect to reality.
  • Realism - the abductive inference is essential to science.
  • This book offers a convincing justification of meta-induction.

3. The Significance of Hume’s Problem for Contemporary Epistemology

3.1: The Aims of Epistemology

  • Justification played a key role in epistemological inquiry from the Enlightenment.
  • Justification distinguishes knowledge from accidental true belief.
  • The rationalists and the empiricists of Enlightenment epistemology both subscribed to an axiom-derivation methodology of knowledge derivation.
  • In postmodernity, foundation-oriented epistemology was questioned: it is pretentious in that its assumptions are ‘too good to be true’.
  • Challenge of foundation-oriented epistemology: justificational regress - the need to base each justification on premises which are themselves in need of justification.
  • This book attempted to account the modern account of foundation-oriented epistemology.

3.2: Foundation-Oriented Epistemology and its Main Problems

  • Foundation oriented justification requires:
    1. All beliefs are justified via arguments whose ultimate premises consist of directly evidence elementary beliefs. This bars infinite justification chains.
    2. Justification circles are avoided, as opposed to coherentist accounts which accept circularity.
    3. Justifications provide justifications for the reliability of the arguments employed in 1), or at least their optimality w.r.t. reliability. An argument pattern is reliable iff the objective probability of the conclusions is sufficiently high.
  • Three major types of argument patterns in analytic epistemology - deduction, induction, abduction (IBE).
  • For a justification to be complete, the legitimacy of the basic beliefs must be noncircularly established.
    • Deductive arguments preserve truth strictly in all cases.
  • Deriving a higher-order justification for induction or abduction which is not circular and does not fall into infinite regress is very difficult.
  • Classical solution to the problem of basic beliefs (PBB):
    1. Introspective beliefs, which derive from one’s experience, may be in error from an externalist perspective. They can be re-formulated in subjective language such that what is known is that the subject is experiencing something (Decartes - cogito, ergo sum).
    2. Analytic beliefs are true because of the laws of classical logic, or derive from semantic definitions.
    3. Basic beliefs consist of introspective and analytic beliefs, both of which are strictly correct in their formulations.
  • Objections to the classical solution to PBB: even introspective beliefs can be prone to error.
    • We restrict introspective beliefs to the present and the private (semantically). Any strictly ‘incorrect’ beliefs are correct in that they reflect the current state of thought.
  • The main problem of the minimalist solution to the PBB is the difficulty in inferring anything nontrivial about the world which lies outside of our consciousness from the class of BBs.
    • Deduction can only prove statements represented in the predicate set.
  • One needs induction to overcome this problem. The burden of solving BPP shifts to justifying inference through induction and abduction.

3.3: Coherentism and its Shortcomings

  • Coherentistic accounts - the beliefs of an epistemic agent are confirmed when they mutually support each other. This accepts circular justifications.
  • Circular justifications, however, are without epistemic value because one can justify mutually contradicting propositions with it.
  • Inductive and anti-inductive accounts can both be justified with circular justifications.
  • There mere existence of circular justification relations does not increase the probability of being true.
  • This likewise applies to probabilistic coherentist arguments. A statement and its opposite can have the same distributions.
  • For every belief system contining \(n\) elementary propositions, there exist \(2^n\) cohrent but mutually contradictory belief systems.
  • This applies to complete justifcatory circles but not partial circles.

3.4: Externalism and its Shortcomings

  • Internal - a subject has cognitive accessibility to the relevant state.
  • External - a subject does not have cognitive accessibility to the relevant state.
  • Goal externalism - the goal of knowledge is truth. Not problematic given that truth is metaphysically neutral.
  • Justification externalism: replace the internalist concpet of justification with an external one.
  • Reliability externalism - a belief is externally justified iff this belief was formed by a cognitive process reliable in our world.
  • Externalist justifications deprive the subject of cognitive accessibility, which hinders their meliorative function.
  • Justification externalism can be used to argue in favor of both induction and anti-induction.

3.5: The Necessity of Reliability Indicators for the Social Spread of Knowledge

  • Reliably produced beliefs must be recognizable to members of the population by means of reliability indicators.
  • The ability to discriminate between reliable information and unreliable information requires reliability indicators, which can be based on the conditions of the social spread of knowledge in cultural evolution.

4. Are Probabilistic Justifications of Induction Possible?

4.1: Why Genuine Confirmation Needs Induction Axioms

  • Many probabilistic accounts of Hume’s problem attempt to reconcile it probabilistically: inductive inferences lead from true premises to true conclusions with high probability.
  • Inductions can only be successful if the observed frequencies can be inductively projected into the future, which is itself an inductive assumption.
  • Beware of Bayesian pseudo-confirmation.
    • Justification by conjunction. Let \(G =\) grass is green and $X =\(God exists. Then,\)G\(confirms\)G \wedge X\(, but this is trivial because\)P(X G) = P(X)$$.
    • Justification by content cutting. We consider latent variables which are not considered.
\[H_1 = E \wedge H_1^*, H_1^* = \forall x (\neg O x \wedge Ex \to G x)\] \[H_2 = E \wedge H_2^*, H_2^* = \forall x ( \neg O x \wedge E x \to \neg G x)\]

\(H_1\) is the iductive generalization, \(H_2\) is the anti-inductive. \(Ox\) expresses that the individual 44x\(has been observed;\)E := \forall x (O x \wedge Ex \to Gx)\((evidence). Both hypotheses are Bayes-confirmed by\)E$$.

4.2: Digression - Goodman’s Paradox and the Problem of Language Relativity

  • Without inductive assumptoins, content-transcending confirmation is impossible.

Goodman’s grue predicate. An object \(x\) is called ‘grue’ iff \(x\) has been observed previous to some future time \(t_k\) and is green (\(Gx\)), or it has not been observed previous to time \(t_k\) and is blue \(B(x)\).

  • If we apply the inductive generalization to all existing emeralds, we have both that all emeralds are green and all emerlads are grue; yet these are contradictory predictions.
  • One cannot apply principles of induction to both ordinary and Goodman predicates without producing a contradictory.
  • Why should the grue predicate not be projected?
    • Being grue entails information about observation; induction is by definition an inference rom the observed to the unobserved.
    • Being grue refers to a point in time and refers to a change in prpoerties over time.
  • Positional predicates - predicates which contain logically essential information about particular indiviudals or space-time points. We should not project these. Induction works by transferring the same properties from the observed to the unobserved; we cannot have things changing all over the place. Rather, we can only project qualitative predicates.
  • The problem of language dependence is independent of Hume’s problem.

4.3: Statistical Principal Principle and Narrowest Reference Classes

  • Statistical probabilies are always already weak inductive principles.
  • Principal principle (PP) - David Lewis, bridges epistemic and objective probabilities.
  • Narrowest reference class: we always want to associate the subject probability of an event with the most constraints on the information which is used to inductively determine such the probability.
  • Too complicated for me to understand :’)

4.4: Statistical Principal Principle and Exchangeability as Weak Induction Axioms

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