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Phl246 Exercise 11-15
Exercise 11: Show that HD-confirmation satisfies the converse consequence condition CCC,
where you may suppress the background assumption in the definition of HD-confirmation so that
sentence E HD-confirms sentence H just in case H l
Please hand in ONLY ONE homework assignment: 3a or else 3b, but NOT BOTH!
Homework assignment 3a
The swan-hypothesis H is a universal if-then sentence and says that all swans are white. H is logically
equivalent to the universal if-then sentence H that ev
Homework assignment 5
A probability space is a triple (W, A, Pr) consisting of an arbitrary, non-empty set of worlds or possibilities
W, an algebra A of propositions over W, and a probability measure Pr(): A . (The dot in Pr()
indicates that the function
A Logical Introduction to Probability and
Induction
Franz Huber
University of Toronto
PHL 246 (Probability and Inductive Logic)
Fall 2016
This is a draft of a textbook for students at the University of Toronto that still includes numerous flaws and still
UNIVERSITY OF TORONTO, Faculty of Arts and Science
Mid-Term Exam, PHL246, Duration 50 minutes, No Aids Allowed
Please keep your answers to the following 8 questions (each worth 2.5% points) as short as possible.
Question 1: What does the Principle of Indu
REVIEW OF PROBABILITY
In Probability, three items (W, A, Pr)
Step 0 W
Step 1-3 Find A
Algebra is a subset of the powerset of W
1. W is an element of the Algebra (W is a proposition)
2. If B is an element of the Algebra, then so its complement: (W\B) is an
PHL246 FINAL Review Notes
Proving Theorems
1. Conditional Derivation: To prove that the consequence follows from the antecedent.
a. Example: If A confirms B, then A confirms B C, for any C.
2. Direct Derivation: Make both sides look the same to show they
Homework assignment 1
Intuitively, a set is a collection of objects (things, entities). For instance, the set C of Canadian cities with
a population of more than 1 million is the set containing Toronto, Montreal, and Calgary. We use curly
brackets cfw_ an
Homework assignment 4
A partition P of an arbitrary, non-empty set W is a set of subsets of W, P (W), such that any two
members B and C of P are mutually exclusive (have no members in common), BC = , and the
members of P are jointly exhaustive (every memb
Kexin Wang
1001742153
phl246 Exercise 6-10
Exercise 6: Show that, in set theory, the following is true of all sets P and Q (1
point):
PQ=QP
1. P Q = cfw_ x: x (P Q)
from Extensionality
2. P Q = cfw_ x: (x P) ^ (x Q)
from 1. and the
definition of
3. P Q =
A Logical Introduction to Probability and
Induction
Franz Huber
University of Toronto
PHL 246 (Probability and Inductive Logic)
Fall 2016
This is a draft of a textbook for students at the University of Toronto that still
includes numerous flaws and still
Outline
1. Induction
2. Probability
Goodmans view and.
According to Goodman the same is true of inductive logic:
An inductive inference, too, is justied by conformity to
general rules, and a general rule by conformity to accepted
inductive inferences. (65
Instructors Manual
The instructors manual contains the solutions to the 50 exercises as well as 34
exam questions.
1
Exercises chapter 1
Exercise 1: Writedown the truth table for the following formula p q q ,
or simply p q q.
Solution:
p
T
T
F
F
q
T
F
T
F
Homework assignment 8
Suppose your background assumptions B are such that your degree of belief that all swans are white
is independent of a being a swan:
Pr(a is a swan|all swans are white, and B) = Pr(a is a swan|B)
Suppose further that Pr(a is white|a
Exercise 6: Show that, in set theory, the following is true of all sets P and Q: PQ=QP
Exercise 7: Show that, in set theory, the following is true of all sets P, Q, and R:
P (Q R) = (P Q) R
Exercise 8: Show that, in set theory, the following is true of
Homework assignment 2
The truth table for the formula (pq)(q) is obtained by first listing all the possible assignments of
truth-values to the propositional variables p and q that occur in the formula.
p
q
T
T
T
F
F
T
F
F
Then these truth values are writt
I
1. Sample Spaces
Experiment. An experiment is any change that actualizes one of many possible
outcomes. A first example of an experiment is throwing a cubical die. An
outcome is the number on the top side of the die. There are six possible
outcomes, and
Homework assignment 4
A partition P of an arbitrary, non-empty set W is a set of subsets of W, P (W), such that any two
members B and C of P are mutually exclusive (have no members in common), BC = , and the
members of P are jointly exhaustive (every memb
453 vrr. COMPARATIVE INDUCTIVE LOGIC
87. Hempeis Analysis of the Concept of Confirming Evidence
Some interesting investigations by Hempel concerning the concept of con-
tinuing evidence are here discussed. Hempel shows correctly that two wide.
spread con
Homework assignment 7
Consider two objects, viz. today and tomorrow, and one property they can have, viz. whether or not the
sun rises on them. There are four possibilities:
s1 = the sun rises today and the sun rises tomorrow
s2 = the sun rises today, but
Homework assignment 1
Intuitively, a set is a collection of objects (things, entities). For instance, the set C of Canadian cities with
a population of more than 1 million is the set containing Toronto, Montreal, and Calgary. We use curly
brackets cfw_ an
Outline
1. Induction
2. Probability
The probability calculus
A function Pr : A R on a(n) (-) algebra A of
propositions over W into the real numbers R is a
(-additive) probability measure on A iff for all
A, B, Ai A, i N:
1. Pr (A) 0 (non-negativity)
2. Pr
Outline
1. Induction
2. Probability
Elementary consequences
Pr () = 0 and Pr (W \ A) = Pr A = 1 Pr (A)
Pr (A B) = Pr (B) Pr (A | B)
Pr (A B) = Pr (A) + Pr (B) Pr (A B) (picture!)
Franz Huber
PHL 246: Probability and Inductive Logic
Outline
1. Induction
2.
Exercise 16: We are considering an algebra A over a non-empty set of possible worlds W.
Show that the intersection of A and B is a proposition if both A and B are propositions, i.e.
(A B) A if A A and B A. (1 point)
Show (A B) A if A A, B A
1. W is a prop
Exercise 31: Consider two objects or individuals, viz. today a and tomorrow b, and one
property S they can have, viz. whether or not the sun rises on them. List the four state
descriptions and three structure descriptions that the two individual constants
Outline
1. Induction
2. Probability
SKIP Milnes theorem
M4a: c (H, E F , B) c (H, E G, B) is determined by
c (H, E, B) and c (H, F , E B) c (H, G, E B).
M4b: If c(H, E F , B) = 0, then
c (H, E, B) + c (H, F , E B) = 0.
Franz Huber
PHL 246: Probability and
Outline
1. Induction
2. Probability
SKIP Catch-all counterparts
The catch-all counterpart of (M3a) is L3a: If
Pr (E | H B) < Pr (F | H B) and
Pr E | H B = Pr F | H B , then
c (H, E, B) c (H, F , B). Here we hold xed the catch-alls
Pr E, F | H B rather tha
Outline
1. Induction
2. Probability
Measures of incremental conrmation
Earman (Bayes or Bust?,1992): difference measure
d (H, E, B) = Pr (H | E B) Pr (H | B)
Franz Huber
PHL 246: Probability and Inductive Logic
Outline
1. Induction
2. Probability
Measures
Outline
1. Induction
2. Probability
Hjeks illustration of Bronfmans objection
Sophia lives in Pasadena. All indices agree that she
should move to Canada.
The big-city index tells her to move to Toronto (but to stay
in Pasadena before moving to Whistler or
Exercise 1
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(p
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Exercise 2
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Exercise 3
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Logic Exercises #11-15
Student # 1003201395
Relevant Definitions from Textbook
Sentence e HD-confirms sentence h given sentence b just in case the conjunction of h and b,
hb, logically implies e, but b does not.
However, it is permitted for this question
2
More Precisely
We start with the things that go into sets. After all, we can't have collections of
things if we don't have any things to collect. We start with things that aren't
sets. An individual is any thing that isn't a set. Sometimes individuals a
Homework assignment 1
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a population of more than 1 million is the set containing Toronto, Montreal, and Calgary. We use curly
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di
a
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