LPS/PHILOS 105A/205A
MIDTERM STUDY GUIDE
The midterm exam is an in-class exam that we will do in-class on Thursday October 27.
The exam is closed-book, and so please come with just pencils we will provide blue books.
The exam consists of two parts.
In par

Lecture 05
Kazimierz Kuratowski, defined ordered pair function we use
1
Recall: Functions & their Properties
Recall from Lecture 2:
a function f:XY is an operation
which, when given an input
xX, returns an output yY.
Here, X is called domain and Y
called

Lecture 6
Picture of Equivalence Classes
1
Other Binary Relations:
Equivalence Relations
An important kind of binary
relation is an equivalence relation.
A binary relation E on XX is a
is an equivalence relation on X if
it satisfies three properties, for

Lecture 7
Modal Logic
1
Showing Invalidity
. . . . The Method, Generalized
2
Extending the method to other
conditions
A formula is C-invalid if there
is a C-frame <W,R>, a model
M=<W,R,I>, and a world w in
W such that VM(, w)=F.
Intuitively, a formula is

Lecture 6
Modal Logic
1
Validities Concerning Interaction of
Negation, Box, and Diamond
2
The Short Story
So the short story about the
relationship between necessity and
possibility is that the following are
valid:
i)
ii)
~ ~
~ ~
So formally, these shoul

Lecture 4
Modal Logic
1
Models: Definition and Examples
2
Definition of a Model
A model M is given by three
pieces of information:
3. An interpretation function I,
that species which basic
letters are true at which
worlds.
Last time we already saw ve
exam

Rule for computing ' in models
p is true at a world w if p is true at all the
worlds accessible from w.
Rule for computing ' in models
p is true at a world w if p is true at some
world accessible from w.
Reflexive
A frame W, R is reflexive if wRw holds fo

Lecture 5
Modal Logic
1
Validity: Recalling the Definition
2
Validity: Definition
A formula is valid if for all
models and all worlds in that
model, is true at the world.
In terms of our notation:
a formula is valid if for all
models M=<W,R,I> and all
wor

Lecture 3
Modal Logic
1
Outline of Todays Lecture
I. Kripke Semantics for Propositional Modal Logic
II. Motivating Examples
2
Outline of Todays Lecture
I. Kripke Semantics for Propositional Modal Logic
II. Motivating Examples
3
Kripke Semantics =
Truth T

Modal Logic
Lecture 1
1
Outline
I. Brief Introduction to the basic idea of modal logic
II. Propositional Logic Review: the Propositional Connectives
III. Propositional Logic Review: More Complex Truth Tables
IV. Propositional Logic Review: The Language of

Lecture 3
Bertrand Russell, of the Russell Paradox
1
Goals Today
Review what we know about sets so far: extensionality, abstraction,
lists, and set operations.
Describe the Russell paradox and the related result that there is no set
of all sets.
Introduce

Lecture 07
Marshall Stone, of Stones Theorem
1
Plan
Define isomorphic linear orders, define well-orders,
look at some examples
Define Boolean algebras, look at examples, define
isomorphism of Boolean algebras, state Stones
theorem.
2
Isomorphisms of Linea

Lecture 6
Georg Cantor, Founder of Set Theory
1
Goals Today
Define cardinal addition, multiplication, and exponentiation
2
Cardinal Addition, Multiplication,
and Exponentiation
So the way that were going to proceed is by defining, for arbitrary
sets A and

MODAL LOGIC FINAL STUDY GUIDE
The nal exam is Friday December 11 from 10:30-12:30pm. The exam is closed-book, and
so please come with just pencils we will provide blue books. The exam consists of four
parts.
In part I, worth 50%, you will be given 10 of t

Lewis' Kangaroo Example
If kangaroos did not have tails, they
would fall over.
Example signifies hot the counter
factual got its name, because where
the antecedent is false it is contrary
to the fact.
true counterfactual with true
antecedent
Example
true

MODAL LOGIC MIDTERM STUDY GUIDE
The midterm exam is an in-class exam that we will do in-class on Thursday November 5. The exam is
closed-book, and so please come with just pencils we will provide blue books. The exam consists of three
parts.
In part I, wo

Lecture 12-13
Cantor, one of the founders of set theory
1
Goals Today
Introduce the concept of ordinals by:
defining ordinals (= transitive set well-ordered by membership)
stating and proving structure theorem on ordinals
define union and successor operat

Lecture 8
Ernst Zermelo, first to describe cumulative hierarchy
1
Goals Today
Describe Transfinite Induction and Recursion
The Cumulative Hierarchy of Sets
Describe the Stage Axioms
2
Recall: Ordinal Structure
Theorem
For all , , $
1. Set-Transitivity: $.

Lecture 16
von Neumann, who had a role introducing
foundation and replacement
1
Outline
Last Three Axioms of Set Theory:
Foundation
Replacement
Choice
2
Outline, where we are at
Last Three Axioms of Set Theory:
Foundation
Replacement
Choice
3
Recall: The

Lecture 10
Cantor, one of the founders of set theory
1
Plan
Define isomorphic linear orders, define well-orders,
look at some examples
Define ordinals, look at simple examples.
2
Plan, where we are at
Define isomorphic linear orders, define well-orders, l

Lecture 08
Georg Cantor, Founder of Set Theory
1
Goals
Define the notion of cardinality, look at examples
Develop methods for calculating cardinalities
Prove Cantors Theorem on Cardinality of Powerset
2
Definition of Cardinality
Recall that f:XY is a bije

Lecture 4
Richard Dedekind, Axiomatized Arithmetic
1
Goals Today
Review ZFC axioms thus far: extensionality, comprehension, pairing,
powerset.
Introduce two more ZFC axioms: union and infinity.
Show how we can use this to recover our earlier knowledge of

Modal Logic
Lecture 2
1
Outline
I. Review: Translating with Ordinary Propositional Logic
II. Translating Knowledge/Belief into Modal Propositional Logic
III. Translating Possibility/Necessity using Modal Propositional Logic
IV. The well-formed formulas of

Modal Logic
Lecture 16
Date
1
Recall: Translating Knowledge
&Belief
Consider:
(1) it rained today but it didnt
have to rain today.
(2) it rained today but Anne
doesnt know that it rained
today.
We formalize these as follows,
with key p = it rained today.

LPS/PHILOS 105A/205A HOMEWORK 2
DUE FRIDAY OCTOBER 14 BY 5PM.
Name:
Student ID number:
Note 1. It is permissible and you are encouraged to work in groups of 2-3, but everyone
must write up their own solutions. Please indicate here the names of the other 1

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LPS/PHILOS 106 HOMEWORK 2
DUE FRIDAY OCTOBER 23 BY 5PM
Name‘ J eﬂ’rey V0220
Student ID number:
Note 1. It is permissible and you are encouraged to work in groups of 2—3, but everyone
must write up their own solutions. Please indicate here the names o

LPS/PHILOS 106 HOMEWORK 2
DUE FRIDAY OCTOBER 23 BY 5PM
Name:
Student ID number: go (“l—"0;
Note 1. It is permissible and you are encouraged to work in groups of 2-3, but everyone
must write up their own solutions. Please indicate here the names of the o

LPS/PHILOS 106 HOMEWORK 3
DUE WEDNESDAY NOVEMBER 25 BY 5PM
Name:
go i U‘L‘GQS
Student ID number:
Note 1. It is permissible and you are encouraged to work in groups of 2-3, but everyone
must write up their own solutions. Please indicate here the names of

LPS/PHILOS 106 HOMEWORK 3
DUE WEDNESDAY NOVEMBER 25 BY 5PM
Name:
J 8151‘ng V0210
Student ID number:
with whom you worked on this assignment. It is obligatory to indicate the names of the
other students with Whom you worked, if you worked wi

LPS/PHILOS 105A/205A-H01VIEWORK 2
DUE FRIDAY OCTOBER 14 BY 5PM.
Name: flLf/gy V0220
Student ID number: 3 / q 445 3,-
Note 1. It is permissible and you are encouraged to work in groups of 23, but everyone
must write up their own solutions. Please indic

LPS/PHILOS 105A/205A HOMEWORK 4
DUE FRIDAY DECEMBER 2 BY 5PM.
Name:
Student ID number: 5 (A");
Note 1. It is permissible and you are encouraged to work in groups of 23, but everyone
must write up their own solutions. Please indicate here the names of th

LPS/PHILOS 105A/205A HOMEWORK 3
DUE FRIDAY NOVEMBER 18 BY 5PM.
Name:
SLIM
Student ID number:
Note 1. It is permissible and you are encouraged to work in groups of 23, but everyone
must write up their own solutions. Please indicate here the names of the

LPS/PHILOS 105A/205A HOMEWORK 2
DUE FRIDAY OCTOBER 14 BY 5PM.
Name:
Student ID number: g; Lulcb 05
Note 1. It is permissible and you are encouraged to work in groups of 2-3, but everyone
must write up their own solutions. Please indicate here the names

LPS/PHILOS 105A/205A HOMEWORK 3
DUE FRIDAY NOVEMBER 18 BY 5PM.
lName:
Jag/67 M0210
Student ID numberi 3/ (/ng
Note 1. It is permissible and you are encouraged to work in groups of 2-3, but evegyone
must write up their own solutions. Please indicat