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Excursion 2 Tour I: Induction and Confirmation (Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars)
Tour Blurb. The roots of rival statistical accounts go back to the logical Problem of Induction. (2.1) The logical problem of induction is a matter of finding an argument to justify a type of argument (enumerative induction), so it is important to be clear on arguments, their soundness versus their validity. These are key concepts of fundamental importance to our journey. Given that any attempt to solve the logical problem of induction leads to circularity, philosophers turned instead to building logics that seemed to capture our intuitions about induction. This led to confirmation theory and some projects in today’s formal epistemology. There’s an analogy between contrasting views in philosophy and statistics: Carnapian confirmation is to Bayesian statistics, as Popperian falsification is to frequentist error statistics. Logics of confirmation take the form of probabilisms, either in the form of raising the probability of a hypothesis, or arriving at a posterior probability. (2.2) The contrast between these types of probabilisms, and the problems each is found to have in confirmation theory are directly relevant to the types of probabilisms in statistics. Notably, Harold Jeffreys’ non-subjective Bayesianism, and current spin-offs, share features with Carnapian inductive logics. We examine the problem of irrelevant conjunctions: that if x confirms H, it confirms (H & J) for any J. This also leads to what’s called the tacking paradox.
Quiz on 2.1 Soundness vs Validity in Deductive Logic. Let ~C be the denial of claim C. For each of the following argument, indicate whether it is valid and sound, valid but unsound, invalid.
Remember, validity is a matter of form. Any argument with the same form as a valid argument is itself valid. If an argument is not deductively valid, then it is invalid. An invalid argument is one where it’s possible to have an argument with its same form where all the premises are true and the conclusion false. A deductively sound argument must be both valid and have all true premises.
(1) All U.S. senators are male.
Dianne Feinstein is a U.S. senator who is female.
Therefore, it’s not true that all U.S. senators are male. __________________
(2) All U.S. senators are male.
Dianne Feinstein is a U.S. senator.
Therefore, Feinstein is male. __________________
(3) All numbers are either even or odd.
3 is a number but is neither even nor odd.
Therefore, it’s not true that all numbers are even or odd.________________________
So as not to duplicate too closely the form of (1), I had actually wanted the following form for (3). I’ll call it (3)’:
(3)’ If all numbers are either even or odd, then 3 is either even or odd.
3 is neither even nor odd.
Therefore, it’s not true that all numbers are even or odd.________________________
(4) All U.S. senators are female.
Dianne Feinstein is female.
Therefore, Dianne Feinstein is a U.S. senator. ______________________
(5) If all senators are female, then Senators Feinstein and Warren are female.
Senators Feinstein and Warren are female.
Therefore, all senators are female.___________________
(6) If a Normal model M gave good predictions in the 3 cases I applied it, then M will always give good predictions.
Normal model M gave good predictions in the 3 cases I applied it.
Therefore, model M will always give good predictions. __________________________________
(1) – (6) follow patterns in 2.1. Here’s one that’s a bit different for extra credit:
If Normal model M gave good predictions in all the 5 cases I applied it, then M will always give good predictions.
Therefore, if Normal model M ever fails to give good predictions, then M would have failed in at least 1 of the 5 cases I applied it. (Is it valid or invalid?) ____________________
(Answers will be posted in the comments next week, I invite you to post yours in the mean time)
Excursion Tour 1 concepts: the asymmetry of induction and falsification; argument, sound and valid; enumerative induction (straight rule); problem of induction; confirmation theory (and formal epistemology); statistical affirming the consequent; guide to life; paradox of irrelevant conjunction, tacking paradox; Likelihood ratio [LR] between H and ~ H; the concept “entails severely”, Bayes-Boost (B-boost), absolute vs incremental confirmation; Fisher and Peirce on the faulty analogy between deduction and induction; Likelihood Ratio [LR]
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Where you are in the Journey:
Excursion 1 Tour I: I posted all of Excursion 1 Tour I, here, here, and here.
Excursion 1 Tour II. Except for the Souvenir C: A severe tester’s translation Guide, I did not post Excursion 1 Tour II. For the material on Royall and the Law of Likelihood in 1.4 (including a link to an article by Royall), see this post; for stopping rules and the likelihood principle, see this post. That post also offers Museum links to the Savage Forum!
Excursion 2 Tour I: I posted the first stop of Tour I (2.1) here. Material from 2.2 (irrelevant conjunction/tacking paradox) may be found in these blogposts here and here. For the full Itinerary: SIST Itinerary.
Error Statistics Philosophy 
1. valid and unsound
2. valid and unsound
3. valid and unsound
4. invalid
5. invalid
6. valid and unsound
7. invalid
Good. The EC is an example of begging the question or circular argument.
1 is valid but unsound. The first premise is false.
2 is valid but unsound. The first premise is false.
3 is valid but unsound. Both premises are false. The conclusion is true. Who says two wrongs don’t make a right?
4 is invalid. Even if the premises were true, the conclusion does not follow from the premises.
5 is invalid. Even supposing both premises to be true, the conclusion does not follow from the premises.
6 is valid, but the soundness of the premises is unclear. In general the first statement is false, so in general 6 is valid but unsound. If the first statement is true in the presence of some specific set of circumstances, then 6 is valid and sound.
The extra credit argument is invalid. The conclusion does not follow from the premises.
The extra credit is a valid argument. The conclusion is logically equivalent to its premise. In informal logic it would be considered a circular argument.
I had better keep my day job! I am most grateful to have adept philosophers awake at the helm.
Do you see that the conclusion is a rewrite of the (single) premise? It’s called contraposition. (If A then B) = (if not-B, then not-A)
Somewhat late but before reading any solutions:
2 – valid & unsound
4-6 Invalid
7 – valid and unsound
1 and 3, probably both valid and unsound, although I’m slightly confused about this form in which the conclusion can actually be that a premise is false, so the status of these premises seems to differ from the other cases, in which premises themselves are not questioned.
PS: Is there any particular reason why comments are closed for the 17 November entry?
No, it certainly wasn’t intentional. Try again. Corrected.
(6) is valid. Modes Ponens. I added (3)’. The answer is the same though, but (3) was too much like (1). Contradictory premises yield a valid argument because it’s impossible to have all premises true. Thus you can never have all premises true and the conclusion false.