Lecture 6 Confirmation

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Lecture 6Confirmation
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Inductive logic?

• Logic is about evidence or good reasons. In
  deductive logic, it is all clear and simple a
  good reason is something that guarantees the
  truth of the conclusion.
• But in inductive logic we deal with weaker
  reasons, weaker type of evidence. If e is
  inductive evidence for h, it is not guaranteed
  that if e is true, h will also be true. It only
  means that with e we will have more reason to
  believe that h is true than before.
• Someone might think that if h deductively implies
  e, then we can take e to be inductive evidence
  that h is true.
• For example, if hypothesis h (all metals expand
  when heated) deductively implies e (this piece of
  metal will expand when heated), an observation
  that e is true will confirm h.
• But this is a logical fallacy, known as affirming
  the consequent.
• Wesley Salmon once joked that in deductive logic
  we learn how to avoid fallacies, and in inductive
  logic we learn how to commit these fallacies!

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What confirms what?

• In fact, no one thinks that for any hypothesis h,
  every deductive consequence of h confirms h. If
  this were true, every hypothesis would be
  confirmed by every observation.
• Proof
• Take a hypothesis h. Now, h deductively implies
  (h v p), where p is any statement. (This is an
  application of the rule of addition in natural
  deduction.) Or, symbolically h ? (h v p).
• Take any observational statement o. If h is true,
  then (h v o) must also be true. In other words, h
  ? (h v o).
• But if o is true, then (h v o) is true as well.
  But then o makes true one of the deductive
  consequences of h, and this means that o confirms
  h.
• How about limiting confirmation to instances of
  generalizations? That is, maybe should take
  statement All Fs are G to be confirmed only by
  observations of things that are both F and G.

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The ravens paradox

• Carl Hempel showed that there are problems with
  that idea too. He showed that two intuitively
  very plausible assumptions lead to the conclusion
  that sounds unacceptable. His problem is called
  the ravens paradox.
• Nicods Condition (NC)The proposition that a
  given object a has both characteristics F and G
  confirms the proposition that every F has G.
• Equivalence Condition (EC)If proposition P
  confirms Q, and Q is logically equivalent to Q,
  then P confirms Q.
• Paradoxical Conclusion (PC) The proposition (E)
  that a is both non-black and a non-raven confirms
  the proposition (H) that every raven is black
  (because All R are B All B are R).
• A white shoe confirms that all ravens are black!

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All ravens are black All non-black things
are non-ravens
Ravens
Black things
R B Empty!!!
R B
R B
R B
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Solutions to the ravens paradox

• The argument is valid. PC deductively follows
  from NC and EC (assuming the equivalence of All
  R are B and All B are R).
• There are three possible strategies to solve this
  paradox
• One can reject NC deny that every generalization
  is confirmed by any of its instances.
• One can reject EC claim that an observation may
  confirm h, but not a logically equivalent
  hypothesis h.
• One can accept the conclusion claim that
  observing a brown shoe confirms that all ravens
  are black (indoor ornithology!).
• Hempel opted for strategy 3. He thinks that it
  just seems that a white shoe does not confirm
  All ravens are black. The reason is that we
  tacitly introduce some information, and we are
  therefore no longer looking at the relation only
  between the evidence and the hypothesis.

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Hempels idea

• Hempel thought that since we assume that there
  are so many more non-black things than ravens,
  then if we take a non-black thing we expect it to
  be a non-raven.
• This explains why we do not take a non-black
  (white) thing that is a non-raven (shoe) as
  significantly confirming h. He thinks that
  finding a white shoe actually does confirm h, but
  to a very small degree, which we mistake for no
  confirmation at all.
• Non-black thing is a falsification opportunity
  for h, so if this thing turns out not to be a
  raven, this slightly confirms h.
• Hempels idea is that if we remove all tacitly
  introduced information that is influencing our
  judgment of the degree of confirmation of h, and
  if we focus only on the relation of the evidence
  and hypothesis, we will see that a black raven
  and a white shoe both confirm All ravens are
  black.

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Against Hempel

• Hempel was searching for the pure relation
  between e and h. The goal was to discover a
  degree of confirmation that e give to h,
  abstracting from all other information.
• But many contemporary philosophers think that
  there is no pure degree confirmation. They
  believe that confirmation is not a binary
  relation between e and h, but a relation between
  three things e, h, and the background knowledge.
• They think that without knowing anything else we
  simply cannot know whether e confirms h or not.
• There is a possibility that e confirms h, or that
  e is irrelevant for h, or that e even disconfirms
  h.
• Without more information the degree of
  confirmation is simply undefined.

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NC is false!

• An instance of a generalization does not
  necessarily support it. Object a that is F and G
  may not be evidence for All F are G.
• Example All humans are shorter than 2.5
  meters. But finding an instance of that
  generalization can actually decrease our
  confidence in that generalization.
• Let H be human, and S be shorter than 2.5
  meters. Now imagine that we find a person who is
  2.48 meters tall. We found a person who is H and
  S. But now we would be less ready to believe that
  all H are S!
• Another example There are three people (a, b and
  c) who just left (L), and there is a hypothesis
  that each left wearing someone elses hat (H)
  All L are H.
• Suppose that a left with bs hat, and b left with
  as hat. So, these two instances of All L are
  H refute the generalization.

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I. J. Goods counter-example

• I. J. Good invented the following possible
  situation, in which observing a black raves seems
  to disconfirm that all ravens are black.
• We observe a black raven. Our background
  knowledge K says that exactly one of the
  following hypotheses is true (H) there are 100
  black ravens, no non-black ravens, and 1 million
  other birds, or else (H) there are 1,000 black
  ravens, 1 white raven, and 1 million other birds.
  And K also states that an object a is selected at
  random from all the birds.
• Observing a black raven in a random selection is
  much more likely if H is true (than if H is
  true), because there are more ravens under H
  than under H.
• Under the conditions, observing a black raven
  gives you a reason to believe that not all ravens
  are black.

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The Wason Selection Task
These four cards have a letter and number on two
sides.
D
F
7
3
If a card has a D on one side, the other side
must have a 3.
Which card or cards must you turn over in order
to test this rule?
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Solution

• Card D (a) if it has 3 on the other side, no
  problem. (b) if it doesnt have 3
  on the other side, the rule falsified.
• Card F (a) if it has 3 on the other side, no
  problem. (b) if it doesnt have 3
  on the other side, no problem.
• Card 3 (a) if it has D on the other side, no
  problem. (b) if it doesnt have D
  on the other side, no problem.
• Card 7 (a) if it has D on the other side, the
  rule is falsified. (b) if it
  doesnt have D on the other side, no problem.
• Cards 1 and 4 give different outcomes, depending
  on what is on the other side, so they have to be
  turned over.
• Cards 2 and 3 always have the same outcome (no
  problem), independently on what is on the other
  side, so they are useless.

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Application to the raven paradox

• Whether a black raven confirms All R are B,
  depends not only on the background information
  but also on which characteristic is first
  discovered.
• If I have a raven behind my back and ask you
  whether you would like to see it (and check
  whether it is a black), you should say yes (if
  you want to get more evidence about whether all R
  are B).
• But if I have a black thing behind my back and
  ask you whether you would like to see (and check
  whether it is a raven) you should say that you
  are not interested, because whatever it is, it
  cannot disprove that all R are B.
• Notice, however, that the two situations may be
  the same, in the sense that in both cases what I
  actually have behind my back is a black raven.

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Four ways of observing R and B

• If we put an asterisk for an information that is
  obtained first (out of the two pieces of
  information), here are four possibilities
  (similar to 4 cards in the Wason selection task)
• RB - Confirming All R are B (For example you
  see a raven in a dim light and cannot recognize
  the color. But then you realize its black. There
  was a possibility of falsification, so the result
  is confirmation.)
• BR - Useless
• RB - Useless
• BR - Confirming All R are B (You see
  something white in a tree, and you are not sure
  what it is, a bird or something else. Then you
  realize it is a shoe. Again, there was a
  possibility of falsification, so the result is
  confirmation. White shoe confirms that all ravens
  are black.)

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Four ways of sampling

• We can apply the same way of thinking to the
  problem of sampling. There are four categories
  that can be sampled ravens, non-ravens, black
  things, non-black things.
• Ravens Finding that all sampled ravens are black
  supports the generalization All R are B.
• Non-ravens Finding that all sampled non-ravens
  are non-black does not support the generalization
  All R are B.
• Black things Finding that all sampled black
  things are ravens does not support the
  generalization All R are B.
• Non-black things Finding that all sampled
  non-black things are non-ravens support the
  generalization All R are B.

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Sampling options graphical representation
(1) R B Empty!!!
(2) R B
(3) R B
(4) R B
Ravens (1,2) Useful because it includes area
1.Black things (2,3) Useless because it does
not include area 1.Non-ravens (3,4) Useless
because it does not include area 1.Non-black
things (1,4) Useful because it includes area 1.