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Posts Tagged ‘induction’

Chairs

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.

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It’s Worse Being Green

In empiricism, induction, justificationism, underdetermination on 12/10/2011 at 5:19 am

In It’s Not Easy Being Grue, I argued for skepticism — or at least incredulity — towards any inductive inference made solely by appealing to a posteriori evidence. Two hypotheses, as long as they have a logical content greater than the evidence and are not yet refuted are, as a matter of following the rules of logic, necessarily equally favored by the evidence. Even if one should appeal to one of the two hypotheses having a natural property, this problem still stands, since it cannot be uncovered through a posteriori investigation. Of course, more than two hypotheses fit this criteria — any number of empirically adequate hypotheses with greater logical content than the evidence may be constructed. In sum, favoring one hypothesis over another, even with an a prior warrant, cannot be determined from a posteriori evidence at all.

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Rules

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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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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Dogma

In fideism, induction on 23/07/2011 at 6:24 am

Nothing is more characteristic of a dogmatist epistemology than its theory of error. For if some truths are manifest, one must explain how anyone can be mistaken about them, in other words, why the truths are not manifest to everybody. According to its particular theory of error, each dogmatist epistemology offers its particular therapeutics to purge minds from error. (Imre Lakatos)

I’ve heard it said from followers of Rand that a theory (usually one of Rand’s own, or a variation thereof) is unassailable, for any criticism of the theory must necessarily assume the theory in order to criticize it. This, somehow, invalidates all criticism.

Is the supposition “Any criticism must assume the validity of the theory being criticized” self-evident?

One problem: how does one know that all possible criticisms employ that theory? Is anyone familiar with all potential arguments against the theory? Of course not: novel ideas are created every day. Therefore, this assertion, that all criticism must assume the theory is true, is based on an inductive inference, which cannot, as a matter of logic, be as demonstrably self-evident or unconditionally immune to criticism as it first appears.

It might be the case that it is true, but it is hardly evident to me, especially once this doubt is raised. Furthermore, whatever theory is used to demonstrate how the initial theory is self-evident must, of course, be scrutinized to determine if it suffers from the same problem: is this new theory self-evident as well? A regress of ‘unassailable’ theories begins in earnest.

The world is far more interesting than we can imagine: asserting that no criticism could possibly exist speaks only to, I think, their limited intellectual horizon. I conjecture that it is better for an idea to stick its neck out as far as it can, therefore inviting many criticisms, and taking them serious. One criticism, if accepted, is enough. As the followers of Rand would have it, the world can only be a constant construction of sandcastles following the blueprints of the Master, and yet no helpful criticism of the blueprints or their faithful execution is permitted. I might go so far as to say that this meta-theory is self-evident, but of course, I don’t.

Assume that everything I have just said is not the case: assume that the Randian (for they are such an easy punching bag, no?) now says that by any criticism that does not assume the same things as Objectivism is then starting from different — incompatible — assumptions, and is not a viable criticism. This might be a possible defensive maneuver for the Randian, for it disallows criticism of its assumptions and criticism of its coherence. Here we have the gestation of the most uninteresting post-modernists within the Randian (or the religious presuppositionalists like Van Til), for the Randian must not be aware of a reductio ad absurdum.

And this, I should note, is a point that deserves no further clarification on my part, for pointing out incoherence is one of the most powerful criticisms available.

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Evidence

In empiricism, induction, justificationism on 19/07/2011 at 6:49 am

There is endless conjecture, and certainty is not to be counted upon (Kant, Critique of Pure Reason)

Some people treat evidence as something that accumulates over time, like sap from a tree. Once enough evidence is collected, you need only synthesize it into syrup, and then you’ve proved your point. “I have X amount of evidence for Y, therefore you ought to believe Y, otherwise you are behaving irrationally.” So the story goes.

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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?

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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.

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Reduced to Twelve Lines of Dialogue

In critical rationalism, empiricism, induction, popper on 05/07/2011 at 12:32 pm

Logical Positivist: Popper, we know we didn’t let you in our club, but what you do you think of our plan on eliminating metaphysics by reducing all meaningful statements to elementary statements of experience or analytic truths? Isn’t it swell?

Popper: Are you blind?

Logical Positivist: What?

Popper: You define ‘meaningful’ as ‘possible to empirically investigate’ while you define ‘meaningless’ as ‘impossible to empirically investigate,’ but metaphysics has usually been defined as non-empirical. Your use of the word ‘meaningless’ is derogatory, rather than descriptive. I call your very plan into question as merely restricting definitions.

Logical Positivist: No, it’s not!

Popper: Fine, if that will not turn you, put that criticism aside. Does this criticism work? Your very plan is not analytic, nor is it reducible to an elementary statement of experience. Therefore, there exists at least one meaningful metaphysical statement: your plan.

Logical Positivist: … could you try something more … palatable?

Popper: Sure, try this on for size. If we assume that you are successful in eliminating all metaphysics–by that very criterion of meaning, scientific laws cannot be reduced to elementary statements of experience, and ought to be rejected as meaningless.

Logical Positivist: … um … Let me get back to you …

Popper: Take as much time as you want. Put all my previous objections aside and assume for the moment that you have solved them all. How about this? You accept an inductive logic, right?

Logical Positivist: Sure!

Popper: Your proposed inductive logics are not reducible to elementary statements of experience or analytic truths. Your plan is clearly incoherent.

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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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Positive and Negative

In bartley, critical rationalism, duhem, experiments, fallibilism, fideism, irrationalism, skepticism on 29/06/2011 at 11:53 am

There is a significant difference between what I will call ‘negative’ and ‘positive’ thinking. Positive thinking rests on the assumption that a solution’s past success (the ‘is’) guarantees or increases the probability of the solution’s future success (the ‘ought’): past success ought to show future success. Negative thinking, however, does not run into the is/ought problem: if a universal statement contradicts an existential statement, and the existential statement corresponds with the facts, then the existential statement is false.

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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.

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Salmon and Corroboration

In bartley, critical rationalism, induction, popper, salmon on 22/06/2011 at 2:14 am

If inductive evidence is ampliative evidence, then it is clear what would count as a successful outcome of the inductivist project. Given hypothesis h, and evidence e, one must show that evidence e makes p(h if e, e), greater than p(h if e). Evidence e can be anything one cares to name, including repeated sightings of white swans, black raven, or blue hats.

Popper and Miller proved in 1983 that, following from the rules of probability, no e can satisfy this requirement. Until this proof is answered, inductivists are tilting at windmills.

” … if the hypothesis h logically implies the evidence e in the presence b [background knowledge] (so that he is equivalent to h) then p(h, eb) is proportional to p(h, b) … suppose that e is some such evidence statement as ‘All swans in Vienna in 1986 are white’, h  the supposedly inductive generalization ‘All swans are white’ and k the counterinductive generalization ‘All swans are black, except those in Vienna in 1986, which are white’. Then p(h, eb) = p(h, b)/p(k, b). No matter how h and k generalize on the evidence e, this evidence is unable to disturb the ration of their probabilities …. Supporting evidence points in all directions at once, and therefore points usefully in no direction. (Popper & Miller, Why Probabilistic Support is not Inductive, Phil. Trans. of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 321, No. 1562 (Apr. 30, 1987))

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Rand’s Problem with the Problem of Induction

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

Dykes:

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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Priors

In induction, skepticism on 16/06/2011 at 5:52 pm

Assign a prior probability to a hypothesis h that takes into account our ignorance of the truth or falsity of the hypothesis. Even though a good deal of the predictions (p), in conjunction with some basic statements and initial conditions, of h may be true, either the hypothesis is true or is false. Outcomes of testing are transmitted back to the hypothesis: corroborating evidence e implies that h is true (for true hypotheses will always have their predictions corroborated), while evidence e that conflicts with prediction p, a logical consequence of h, implies that h is false.

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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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Newton & Einstein

In induction, popper on 15/06/2011 at 4:05 pm

“… (1) Newton’s theory is exceedingly well corroborated. (2) Einstein’s theory is at least equally well corroborated. (3) Newton’s and Einstein’s theories largely agree with each other; nevertheless, they are logically inconsistent with each other because, as for instance in the case of strongly eccentric planetary orbits, they lead to conflicting predictions. (4) Therefore, corroboration cannot be a probability (in the sense of the calculus of probabilities).

“… The proof is simple. If corroboration were a probability, then the corroboration of ‘Either Newton or Einstein’ would be equal to the sum of the two corroborations, for the two logically exclude each other. But as both are exceedingly well corroborated, they would both have had a greater probability than ½ (½ would mean: no corroboration). Thus, their sum would be greater than 1, which is impossible. It follows that corroboration cannot be a probability.

“… It would be interesting to hear what the theoreticians of induction … who identify the degree of corroboration (or the ‘degree of rational belief’) with a degree of probability — would have to say about this simple refutation of their theory.” (Popper, Karl. 2009. The Two Fundamental Problems of the Theory of Knowledge, xxivn. New York: Routledge.)

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Duhem

In duhem, holism, induction, popper on 15/06/2011 at 3:21 pm

The Duhem problem can be expressed as follows:

A physicist disputes a certain law; he calls into doubt a certain theoretical point. How will be justify these doubts? From the proposition under indictment he will derive the prediction of an experimental fact; he will bring into existence the conditions under which this fact should be produced; if the predicted fact is not produced, the proposition which served as the basis of the prediction will be irremediably condemned. (Duhem, The Aim and Structure of Physical Theory, Princeton University Press. Translated from the French by Philip P. Wiener.1954, p. 184)

By means of this mode of inference we falsify the whole system (the theory as well as the initial conditions) which was required for the deduction of the statement p, i.e. of the falsified statement. Thus it cannot be asserted of any one statement of the system that it is, or is not, specifically upset by the falsification. Only if p is independent of some part of the system can we say that this part is not involved in the falsification. (Popper, The Logic of Scientific Discovery, 76)

A naive theory of science might say that when testing a theory T, if an observation-statement O is found to agree or disagree with the logical consequences of T, O either supports or refutes T. This can be expressed as follows:

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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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