This document provides an overview of Bayesian confirmation theory and discusses some key issues, including: 1. It defines Bayesian confirmation theory as assessing the degree to which data confirms or supports a hypothesis based on Bayes' theorem and likelihood ratios. 2. It notes Fitelson's account of confirmation as an increase in the likelihood of a hypothesis given the evidence. 3. It discusses the problem of irrelevant conjunction, also known as the "tacking paradox," where confirming irrelevant conjunctions of hypotheses seems to undermine confirmation. 4. It explains how Fitelson and Hawthorne avoid the conclusion that any evidence confirming some hypothesis H also confirms any other hypothesis J by requiring the evidence to be independently probable on H

1. 24
PART II OF MAYO

2. 25
Little Bit of Logic
(Double purpose: both for arguing philosophically and for
understanding inductive/deductive methods)
Argument:
A group of statements, one of which (the conclusion) is claimed to
follow from one or more others (the premises), which are
regarded as supplying evidence for the truth of that one.
This is written:
P1, P2,…Pn/ ∴ C.
In a 2-value logic, any statement A is regarded as true or false.

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A deductively valid argument: if the premises are all true then,
necessarily, the conclusion is true.
To use the “⊨” (double turnstile) symbol:
P1, P2,…Pn ⊨ C.
Note: Deductive validity is a matter of form—any argument with
the same form or pattern as a valid argument is also a valid
argument.
(Simple truth tables provide an algorithm or computable method
to determine validity; no knowledge of context needed)

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EXAMPLES (listing premises followed by the conclusion)
Modus Ponens Modus Tollens
If H then E If H then E
H_______ Not-E____
 E Not-H
If (H) GTR, then (E) deflection effect.
(not-E) No light deflection observed.
 (not-H) GTR is false
(falsification)
These results depend on the English meaning of “if then” and of
“not”. In context, sentence meanings aren’t always so clear. But if
interpreted to have these forms, they are valid.

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Disjunctive syllogism:
(1) Either the (A) experiment is flawed or (B)
GTR is false
(2) GTR is true (i.e., not-B).
Conclusion: Therefore, (A) experiment is flawed.
If it’s either A or B, and it’s not A, then it must be B.
Since if it’s not B, you can’t also hold the two
premises—without contradiction. (soup or salad)
Either A or B. (disjunction)
Not-B
Therefore, A

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So we have 3 valid forms:
Modus Ponens
Modus Tollens
Disjunctive Syllogism

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Deductively Valid Argument (argument form):
Three equivalent definitions:
• An argument where if the premises are all true, then
necessarily, the conclusion is true. (i.e., if the conclusion is
false, then (necessarily) one of the premises is false.)
• An argument where it’s (logically) impossible for the premises
to be all true and the conclusion false. (i.e., to have the
conclusion false with the premises true leads to a logical
contradiction: A & not-A.)
• An argument that maps true premises into a true conclusion
with 100% reliability. (i.e., if the premises are all true, then
100% of the time the conclusion is true).

8. 31
The categorical syllogism version of modus ponens
In the above, A and B are statements, here it’s a property
(but arguments in the real world blend them)
All A’s are B’s.
x is an A.
Therefore x is a B.
All swans are white
x is a swan.
Therefore, x is white.

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True or False?
If an argument is deductively valid, then its conclusion must be
true.
Here’s an instance of the valid form:
All philosophers can fly.
Mayo is a philosopher.
Therefore Mayo can fly.
To detach the conclusion of a deductively valid argument as true,
the premises must be true.
(Deductively) Sound argument: deductively valid + premises are
true/approximately true.

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Invalid Argument: Consider this argument:
If H then E
E________
 H
If (H) everyone is a Bayesian, then (E) Lindley is a Bayesian.
(E) Lindley is a Bayesian.
So, (H) everyone is a Bayesian.
Affirming the consequent.

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Invalid argument: An argument where it’s possible to have
all true premises and a false conclusion without contradiction.
We just illustrated that affirming the consequent is invalid.
If all numbers are even then 4 is even.
4 is even.
Therefore all numbers are even.
Invalid arguments let us go from all true premises to false
conclusions.

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Exercise: showing deductively valid arguments can have false
conclusions.
(1) (I did this above) Give an example of an argument that
follows the valid pattern of modus tollens that has a false
conclusion.
All philosophers can fly.
Mayo is a philosopher.
Therefore Mayo can fly.
Valid but unsound.

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(2) Exercise (YOU do this one) Show that an argument following
the form of a disjunctive syllogism in I (A) can have a false
conclusion.
Either A or B. (disjunction)
Not-B
Therefore, A
(3) If the premises are contradictory is the argument valid or
invalid? (pertains to next argument)

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If (H) all swans are white, then the next swan I
see will be white.
The next swan I see x is black.
Therefore (not-H) Not all swans are white
(falsification)
(prefer the above to ex (1) on p. 60)
(H) all swans are white.
The next swan I see x is black.
Therefore (not-H) Not all swans are white

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Logic of Simple Significance Tests: Statistical Modus
Tollens
(Statistical analogy to the deductively valid pattern modus
tollens)
If the hypothesis H0 is correct then, with high probability,
1-p, the data would not be statistically significant at
level p.
x0 is statistically significant at level p.
____________________________
Thus, x0 is evidence against H0, or x0 indicates the falsity of
H0.

16. 39
January 30, 2019 (Due next class, Feb 6)
Ex 1 a.
Suppose there are 2 successes in n = 5 Bernoulli trials, i.e., the observed proportion of successes is .4. Order doesn’t matter,
but if you like to envision the outcome, say the first and last are successes.
x0 = <X1=1, X2 =0, X3=0, X4=0, X5=1>
To determine which is better “supported” (in the likelihood sense) by outcome x0 , 𝜃 = .3 or 𝜃 = .5, compute
Lik(𝜃 = .3; x0)_____ and
Lik(𝜃 = .5; x0)_____
(The sample size is small enough to easily compute each, just for practice). Then form the LR. Which is more likely?
What’s the maximally likely hypothesis? 𝜃 = _____. How likely is it? _____
(See us in our offices if you have questions. If you ask someone for help, just have them do the ex. At the top of p. 36, or the
one in Dr. Spanos’ notes)
Questions to Direct Your Reading (Q-0) (not to be collected)
• What is Bayesian confirmation theory (Bayesian epistemology)? What is Bayes’ theorem?
• What is “confirmation” on Fitelson’s account?
• What is the problem of irrelevant conjunction (tacking paradox)?
• What is the argument (Glymour) that if x confirms some hypothesis H, then x confirms any J?
• How does Fitelson (Fitelson and Hawthorne) avoid this conclusion? Discuss.
• Why does Popper’s example show (or seem to show) that confirmation cannot be a probability?
October 19, and 25 (2013) blog: errorstatistics.com
http://errorstatistics.com/2013/10/19/bayesian-confirmation-philosophy-and-the-tacking-paradox-in-i/
http://errorstatistics.com/2013/10/25/bayesian-confirmation-theory-example-from-last-post/