Posts Tagged ‘problem of induction’


In critical rationalism on 19/11/2011 at 7:05 am

As I was sitting in my chair,
I knew the bottom wasn’t there,
Nor legs nor back, but I just sat,
Ignoring little things like that.

How apt at describing most justificationist programs and their belief that all will be resolved. Acting as if some hypotheses are to be preferred is indispensable to the conduct of ordinary life, but they are a matter of convention, and to believe that one may reliably choose the right properties is a belief without warrant. Therefore, they do not follow their own proposed procedure.


It’s Not Easy Being Grue

In empiricism, experiments, induction, quine, skepticism, underdetermination on 10/10/2011 at 1:52 am

One solution to Goodman’s new riddle of induction, as proposed by David Lewis and WVO Quine, is that certain languages describe natural properties, which have a special metaphysical status. All things being equal, the evidence will favor the hypothesis that uses languages that have natural properties over any other language in all cases. The problem of choosing between hypotheses that will be favored by the evidence and hypotheses that will not is solved by choosing a hypothesis expressed in a language that uses natural properties. There is, however, a problem with this solution: how can a scientist decide whether a language is using a natural property?

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Induction Machines

In critical rationalism, induction on 23/08/2011 at 12:26 pm

Imagine that a computer is built to make empirical generalizations with inductive logic (whatever that may be) and that this computer is in a simple universe with a limited number of individuals,number of properties, and relationships between these properties the individuals can have. Furthermore, the universe operates with a limited number of ‘natural laws’. In this universe a computer can be created such that in some reasonable period of time it will discover the ‘natural laws’. If the laws were modified, then the computer would find a new set of laws. If this universe were further complicated, then this computer could be enhanced to be able to formulate hypotheses, to test these hypotheses, and to eliminate those that do not survive testing.

This induction machine is limited insofar as it is limited by its programmer’s intellectual horizon: the programmer decides what is or is not a property or relation; the programmer decides what the induction machine can recognize as repetitions; it is the programmer that decides what kinds of questions the machine should address. All the most important and difficult problems are already solved by the programmer, and this induction machine is little more than a speeding-up process of a room full of bean-counters or punch-card holders.

Here we have today’s work in artificial intelligence, which is precisely limited by this constraint. The theories that these computer programs develop are conditional on the initial conditions that are needed for in an induction machine. Inductive inferences does not then occur within the context of discovery; the programmer provides these. Inductive inferences occur within the context of justification, and even then it still does not satisfactorily solve the problem of induction, for the problem cannot logically be solved. These computers have become problem-solving machines that operate on conjecturing the most parsimonious theory and attempted refutation of that theory.



In experiments, induction, justificationism on 20/08/2011 at 7:29 am

A philosophical problem has the form: I don’t know my way about. (Ludwig Wittgenstein)

Up until the late 19th century every observation was compatible with Newton’s theory of gravity. All these observations are also compatible with Einstein’s General Theory of Relativity. Two quite different theories were compatible with the same set of observations; therefore, one cannot know they have derived true theories from observations.

Assume we have a long series of numbers. They go on: 2, 4, 8 … What is the next number in the series?

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Strevens on Induction

In induction on 04/08/2011 at 4:38 pm

When two theories are empirically equivalent, their likelihoods relative to any given body of evidence are equal. Thus the difference in anyone’s subjective probabilities for the theories must be due entirely to the difference in the prior probabilities that were assigned to the theories before any evidence came in. Bayesian confirmation theory preserves a prior bias towards simplicity, but it implements no additional bias of its own. …

Bayesian confirmation theory does impose an objective constraint on inductive inference, in the form of the likelihood lover’s principle, but this is not sufficient to commit the Bayesian to assuming the uniformity of nature, or the superiority of “non-grueish” vocabulary or simple theories. The first of these failures, in particular, implies that [Bayesian confirmation theory] does not solve the problem of induction in its old-fashioned sense.

If the old-fashioned problem of induction cannot be solved, what can we nevertheless say about [Bayesian confirmation theory]’s contribution to the justification of induction? There are two kinds of comments that can be made. First, we can identify unconditional, though relatively weak, constraints that [Bayesian confirmation theory] puts on induction, most notably the likelihood lover’s principle. Second, we can identify conditional constraints on induction, that is, constraints that hold given other, reasonable, or at least psychologically compelling, assumptions. We can say, for example, that if we assign low priors to grueish hypotheses, [Bayesian confirmation theory] directs us to expect a future that resembles the past. This is, remember, considerably more than we had before we began. (Michael Strevens, Notes on Bayesian Confirmation Theory [.pdf], 66)

Strevens is admirable, for he is upfront about the inadequacies of Bayesianism. That said, Strevens at times overstates his case. For instance, tet me simplify the bolded passages: “The evidence for any particular theory is underdetermined … however, if we reject theories that are incompatible with our assumption that the future will resemble the past, we will expect a future that resembles the past.” At least two problems for Strevens:

(1) The future does not resemble the past in all domains: black swans, white ravens — in fact, all falsified scientific theories — should give us pause before assuming something that is demonstrably false when applied to all domains. Therefore, if the future resembles the past in only some domains, why we should assume that the future will resemble the past in any particular domain?

(2) Even if the future should resemble the past in this particular domain, it does not follow that any theory that assumes a future that resembles the past is true, for while ‘grueish hypotheses’ are ruled out, there may still be an alternative theory that follows from that same assumption that the future will resemble the past that is in fact true. After all, it’s happened in the past (Einstein replacing Newton).


Practical Prediction

In experiments, induction, justificationism, popper, salmon on 01/08/2011 at 5:24 am

Wesley Salmon objects to Popper’s theory of knowledge on the grounds that, contrary to its stated rejection of a principle of induction, in order to explain how one can rationally decide between competing unfalsified theories, it requires the adoption of a principle of induction. The advice to an applied scientist or engineer to act as if the best-tested theories are probably true and the untested theories are probably false, though no doubt excellent advice, does not have any claim to be dubbed ‘rational’ unless a pragmatic principle of induction is adopted.

If the applied scientist’s choice is guided by the best-tested scientific theories available to him, then it appears that he is assuming that what was successful in the past will remain successful in the future. This would be an  assumption rejected by Popper, for it employs the principle of induction. However, if a scientist, following Popper’s theory of knowledge, renounces a principle of induction, then he is not allowed to say that ‘future unobserved events will resemble past observed events.’

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In critical rationalism, popper on 29/07/2011 at 1:10 am

Peter Singer’s 1974 article in the New York Review of Books, Discovering Karl Popper is extremely favorable of Popper’s philosophy of science–except for three paragraphs in the middle, which are highly informed criticism. I’ve reproduced them below along with some limited comments in light of that criticism.

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Stating the Obvious

In fallibilism on 20/07/2011 at 1:46 am

We may certainly define truth by reference to the criteria of efficacy; such a definition is not self-contradictory and does not lead into an infinite regress; nevertheless, it is arbitrary; to accept it requires an act of faith and therefore the principle credo ut intelligam operates over the entire field of knowledge; this is hardly more than to say that we are incapable of producing an epistemological absolute or that our intelligence is finite: not exactly a world-shaking discovery. (Leszek Kolakowski, Religion: If There Is No God– : On God, the Devil, Sin, and Other Worries of the So-Called Philosophy of Religion, p. 79)


What is This Thing Called Knowledge?

In critical rationalism, duhem, experiments, induction on 13/07/2011 at 12:21 pm

Duncan Pritchard, who holds a chair in epistemology at the University of Edinburgh, published What is This Thing Called Knowledge? some years ago. He has three textbooks, two published books on epistemology, and approximately fifty journal articles to his name. Let me make this clear: Pritchard is no first-year undergrad at a community college. And yet, his What is This Thing Called Knowledge? has a short section on Popper’s response to the problem of induction that is … shameful. Just shameful.

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Wherefore art thou Induction?

In critical rationalism, induction, quine, underdetermination on 11/07/2011 at 10:12 am

The word ‘induction’ takes on many meanings, always when most convenient. Like a slippery eel, just when a critic of induction has their hands around its neck, it wiggles out once more.

Does induction refer to the ‘context of discovery’ or the ‘context of justification’?

If ‘induction’ refers to the context of discovery, the critic of induction need only point to the greatest historical developments in science. Without blinders on, the critic points out that these theories are birthed in the heat of dealing with significant scientific problems. The framework comes before observation (read: Einstein). How then could enumerative induction work? Theories are then imaginative creations–possible solutions to problems. Even if enumerative induction is permitted during the context of discovery, it does not help the scientist any more than dreaming next to a raging fire (read: Kekulé’s oroboros), drug use (read: Feynman, Kary Mullis), &c., which is to say that is has no privileged position over even the most arbitrary ‘methods.’

If ‘induction’ refers to the context of justification, is this a process of objective inductive verification à la Carnap? If so, then this program is defunct, for no number of verifications can increase the probability assigned to a strictly universal statement. Is this the process of subjective certitude after repeated verifications? Then it contradicts the probability calculus and fails to solve the problem of underdetermination.

If ‘induction’ refers to the metaphysical assumption of regularity of systems, which we may approximate if enough inductions of the system are collected, then the inductivist retreats to asserting only that there exists regularities, calling this assumption ‘induction.’ If a proposed regularity should turn out to be false, then this was either a mistaken induction or not induction at all. If it is not an induction, then this is little more than wordplay: we cannot tell this type of induction apart from a conjecture. If it is a mistaken induction, this type of induction should only be known to be mistaken in hindsight: it tells us nothing until we learn that we are wrong.

And what is that but a falsification?


A Woven Web

In fallibilism, xenophanes on 10/07/2011 at 6:29 am

The gods did not reveal, from the beginning,/ All things to us, but in the course of time/ Through seeking we may learn and know things better./ But as for certain truth, no man has known it,/ Nor shall he know it, neither of the gods/ Nor yet of all the things of which I speak./ For even if by chance he were to utter/ The final truth, he would himself not know it: For all is but a woven web of guesses (Xenophanes, Fragments, DK, B 18;35; & 34)

Assume 1, 2:

  1. All observation is theory-impregnated. (Hanson’s Patterns of Discovery is a good introduction.)
  2. If all observation is theory-impregnated, observations cannot definitely refute scientific theories: they are conjectural.
  3. Therefore, if observation (conjecture A) conflicts with scientific theory (conjecture B), there is no way to justify conjecture A or B as wrong.

Surely, some times A will be wrong, other times B will be wrong. Either A or B (or both) could agree with our background knowledge (conjecture C), but that does not make A or B any more justified, since that’s pushing the regress back one more step.

If observation is repeatedly corroborated, that is no sign that it is correct (the problem of induction, natch). Even if observation is known to be correct, that does not entail the falsity of conjecture B: there could be a problem with the experimental setup (conjecture D) or an auxiliary hypothesis (conjecture E).

Maybe Democritus was on to something when he said, “We know nothing in reality; for truth lies in an abyss.” Which conjecture is refuting which? Or is everything we know (or think we know) tentative and provisional, subject to future change? A “woven web of guesses” sits fine with me.


The Problem of Induction

In critical rationalism, popper on 01/07/2011 at 1:45 pm

One of the aims of science, perhaps its most fundamental aim, is knowledge — not of past events — but of future events. Scientists want to ‘read the book of nature’, to borrow a phrase from Bacon. Think of the laws of nature as being general truths, or as they’re known in predicate logic, universal statements (“for all x, y”). So the question, to rephrase David Byrne is, how do I get there? How can scientists grasp hold of the laws of nature?

The popular answer is by ‘inductive inference’, called by Aristotle “the passage from individuals to universals.” [1] Inductive inference usually takes the following form: “This bacon is crispy. That bacon is crispy. … Therefore, all bacon is crispy.”

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Gardner’s Misstep

In carnap, critical rationalism, empiricism, experiments, fallibilism, gardner, holism, justificationism, popper, the ancient greeks, wittgenstein on 28/06/2011 at 10:33 am

Martin Gardner wrote A Skeptical Look at Karl Popper back in 2001. I decided to give it a read then put down some provisional comments …

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Philosophy of Science v. Epistemology

In duhem, induction, quine on 23/06/2011 at 10:41 am

In light of Einstein, Rutherford, and Maxwell, if we assume the knowledge-acquiring process S employs in everyday affairs is distilled or refined in scientific practice, then the problem of induction and the Duhem-Quine thesis should have long ago put to rest any theory of knowledge that claims S can know theory 1 has a greater objective verisimilitude than theory 2.


Rand’s Problem with the Problem of Induction

In induction, rand on 21/06/2011 at 3:16 am


By careful observation – free from preconception – we are able to discover the identities of the entities we observe. Thereafter, we are fully entitled to assume that like entities will cause like events, the form of inference we call induction. And, because it rests on the axiom of the Law of Identity, correct induction – free from contradiction – is a valid route to knowledge. (¶ 11)

I must address this paragraph, line by line: “By careful observation – free from preconception – we are able to discover the identities of the entities we observe.” (¶ 11) The assumption that we may have unmediated observation, ‘free from preconception’, is just that: an assumption that such an observation may take place. From what we know in neuroscience and basic biology, it appears that all sensory qualities we have are not in any way immediate. It is dubious, to say the least, that it is possible to observe ‘free from preconception’, for it would require a mind wiped clean even of its structure, and perhaps eliminating all its previous content. Simply put, the mind is not in any way a blank slate. To counter the fact that it is impossible to know if one is observing ‘free from preconception’ by declaring that we have observation ‘free from preconception’ is absurd.

Thus, Dykes must first argue that observation is ‘free from preconception,’ and that we may come to know which observations are ‘free from preconception’ and which observations are not.

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Rand’s Law of Identitiy

In induction, rand on 21/06/2011 at 2:59 am

I’ve learned very few truly valuable things in life. I won’t list them all, and they may be repugnant or less than valuable to some, but I will list one: argument is not about winning. If you win an argument, you lose. Arguments are about getting closer, no matter how hard they are, to the truth. Of course, I choose not to go into a lengthy argument about why this is the case, simply because I’m not out to convert anyone.

That said, there are times that I see arguments that are just wrong. In these cases, I do not mean to say that their conclusions are therefore false, only that the argument is fallacious — not manifestly so, as is often the case. Some times the wrongness is hidden deep within, and only by prying carefully at the edges can we get a glimpse at where the argument runs afoul.

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Probability, pt. 2

In induction, skepticism on 15/06/2011 at 4:11 pm

Take the statement A, “It will snow on Friday” and the statement B, “It will not snow on Saturday”. The content of the conjunction AB (“It will snow on Friday and it will not snow on Saturday”) will be greater than or equal to any of its components. The more a statements says about the state of affairs, the greater its content. However, note that the probabilities assigned to either A or B require that the probability of the conjunction AB will be smaller than either of the conjuncts A or B.

If we define 0 as ‘false’ (such as a contradiction: for instance “The ball is both all blue and all red at once.”) and 1 as ‘true’ (such as a tautology: for instance, “A prince is a prince.”) and the possible values between 0 and 1 are all the possible assigned probabilities, then AB will always be more improbable or as probable than either the conjuncts A or B. For instance, if we assign the number .5 to A and .5 to B, then AB equals .25. The more a statement says, the less probable it is.

That is to say, with progressive content of our theories (or when a succession of theories increase in their predictive power) they become more and more improbable.

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The Unexpected

In critical rationalism, induction on 15/06/2011 at 3:01 pm

But in all my experience, I have never been in any accident … of any sort worth speaking about. I have seen but one vessel in distress in all my years at sea. I never saw a wreck and never have been wrecked nor was I ever in any predicament that threatened to end in disaster of any sort. (E. J. Smith, 1907, Captain, RMS Titanic)

Yesterday when I was out riding my bike, my bicycle’s front tube popped with a loud BANG! After examining it later, I realized that it was no fault of the road — no sharp rock or shard of glass — but of the tube: it was worn out so much that it split at the seam. This lead me to write once again on the topic of induction. This time, I will focus on everyday examples in order to illustrate the point that inductive inferences are as unjustified as any other conjecture about the unobserved.

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In induction on 15/06/2011 at 7:37 am
The inductivist chooses to solve the problem of induction by assigning probabilities to theories. They are in search of theories that are immensely probable, not improbable: their goal is to adopt theories only if they are probably true. However, they are caught in a reductio due to an unforeseen consequence: if they adopt probable theories, they ought to adopt theories with as little content as possible; one ought to adopt tautologies and historical claims of the form “I have observed …” — and nothing more.This goes against the stated aims of science and any sort of process of induction, since it shies away from making any sort of predictions. Assigning probabilities to theories is then at odds with the process of induction.
Consider the theory “all ravens are black”. This theory is equivalent to the infinite conjunction of statements which the structure “The raven R is black.” However, since the probability of each unobserved raven cannot be assigned 1 without assuming a priori the statement “the next observed raven R will be black,” it follows that every unobserved raven must be assigned a probability of X<1 of having blackness.

Thus, the probability of the infinite conjunction with each conjunct being <1 approaches zero.



In irrationalism, justificationism, kuhn, underdetermination on 15/06/2011 at 6:41 am

If argument is to provide sufficient reasons for accepting or rejecting a claim, then why is disagreement possible?

Agreement is valued everywhere: it sounds friendlier and builds communities of like-minded individuals. Valuing unanimity, we may accept tradition without asking too many questions. Agreement is easy, and invites acceptance of ideas without much thought, whereas disagreement is criticism, involving a great deal of creativity and willingness to go against the grain, and more often than not produces conflict. Kuhn advocated agreement in his ‘normal science’, knowing that science and a measure of dogmatism occur naturally: there are far worse traditions than science. There is an alternative to this admittedly irrationalist solution, namely a rational justification: as Bacon said, by far the best proof is experience. The problem then becomes how does experience justify theory?

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