Formal Proofs
and Boolean Logic
Formal Proofs
We are now (finally!) going to add rules for doing formal Fitch
proofs involving the Boolean connectives (i.e., , , ).
Like =, each symbol with have two associated rules, an Intro
rule and an Elim rule.
Carefu
Day 5: More Proofs
Introducing ->s and RAA
Review from last time
P: Bowie plays a great concert
Q: Jim goes to the concert.
R: Mary goes to the concert.
S: The concert is sold out.
T: The light show is fantastic.
1. The concerts being sold out is sufficie
Practice lst Midterm
Philosophy 155 Spring 2014
Simmons, Dorst, Driggers, Losonsky, Svirsky
SYMBOLIZATIONS For questions 1 and 2, construct your own standard scheme of
abbreviation. ASSIGN SENTENCE LETTERS P, Q, TO THE BASIC ENGLISH
SEN TENCES IN THE ORDE
Practice Test
Are the following statements true or false? 3 points
1. If an argument has true premises and a true conclusion, then it is sound. F
2. If the conclusion of a valid argument is false, then all of its premises are false as
well. F
3. If the co
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ECONOMICS
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University, University of North Carolina
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Day 17: Expansions
and Countermodels
Last four days!
Questions about the Exam
Are there any?
The Final will be twice the length of the other two exams.
It will cover: wffs, proofs, truth tables and what well cover this
week.
I will let you know what t
Day 4: Introducing
Proofs
Review from last time
Translate the following three sentences using the following
scheme:
P: John knows Bill is cheating.
Q: Bill is cheating with Gary
R: Maureen is cheating with Gary
S: Maureen knows Gary is cheating
1. Bill a
Introduction to
Mathematical Logic
Summer Session II, 2015 with your host Kyle Driggers
What will the work look like in
this class?
Before we get started with the content of the class, I want to
show you what the coursework will look like.
The class is
Conditionals
If then
If you mow my lawn, then I will pay you $25.
If the Maple Leafs dont make some changes, then
they will not make it into the playoffs.
If the victim was killed with a chefs knife, then the
cook did it.
If there is lye in the coffee, t
Day 2: The Formal
Language
Review from last time
1. What do we call the connection between the premises and
the conclusion of a valid argument?
2. Whats causal consequence and why doesnt it interest us in
this class?
3. Whats soundness and why arent we st
Practice Final Phil 155
Translation Sentential
1.
2.
~( (PvQ)v(RvS) ) -> (~(TvU) -> W)
R -> (P & Q)
Translation Predicate
1. x( Dx->(FxvHx->Bx)
2. x(Px& y(My->Lxy)&Ix)
3. ~x(y(Sx->Lxy)&z(Pz &Oxz)
*close to the difficultly on final
Invalidating assignment
Day 19:
Countermodels with
Identity
Final day of new material!
Review from last time
Roundtable with 4.2, 4.3.1
4.2 (xiii-xvii)
4.3.1 (v-ix)
Break
Countermodels with Identity
When we construct an interpretation of a sequent without
identity, we provid
Day 18:
Countermodels
Review from last time
Perform expansions for the following with universes of two
objects. U:cfw_a,b
xFx<->x(Fx&~Hx)
x(Fx->xGx)
Countermodels
Today: Single- and (Introduction to) Multi-place without
Identity
xFx &xGx |-x(Fx&Gx)
Philosophy 155 I Spring 2012
Simmons, Dorst, Driggers, Losonsky, Svirsky
PRACTICE SECOND MIDTERM
Sym bolizations
Using standard schemes of abbreviation, provide structure-revealing symbolizations of
the following sentences of English.
(1) Jane wont win th
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Chapter 1 Section 1
Symbolic Notation
Simple sentences are capital letter, which can be thought of as abbreviation sentences of English.
The negation sign '~' is used as the word not
The conditional sign ' is used as the word if.then
Symbolic sentences of
Chapter 1 Section 11
Theorems
If the conclusion is true in every possible situation, it is a truth of logic
If a derivation is used to show an argument with no premises to be valid. That amounts to showing that
the conclusion is logically true. theorem, a
Chapter 1 section 6
Conditional Derivations
If Robert drives, Sam wont drive
If Sam doesnt drive, Teresa wont go
Willa will go only if Teresa does
Therefore, If Robert drives, Willa wont go
R~S
~S~T
WT
R~W
1. Show R~W
2. R
ass cd (assumption for condition
08/31/2015
Rules of inference- truth-preserving: they never take you from true input sentences to false output
sentences
Repetition: R
P
use the rule of repetition to get new sentence:
P
Modus Ponens: MP
(PQ)
P
Q
P
T
T
F
F
Q
T
F
T
F
(PQ)
T
yes
F no
T
no
T
09/21/15
(P(QR)S
PR
Therefore QS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Show QS
Q
(P(QR)S
PR
Show S
~S
~(P(QR)
Show (P(QR)
P
R
Show QR
Q
R
S
ASS CD
PR
PR
ASS ID
3 6 MT
ASS CD
4 9 MP
ASS CD
10 R
13 CD
11 CD
3 8 MP
6 16 ID
5 CD
P(PQ)
Program:
SymbolizationSelect: problem number
The Bruins fail to win
W: the bruins win
Click on box, click form of sentence
Type whats being negated
~The bruins win
Click, atomic
W
Answer at top, check, save, and submit
Ina will not fail to be chosen as ca
Phil 155
Chris Dorst
[email protected]
Office: Caldwell 206C
Office Hours: Tues 11-12 Friday 10-11 or by appointment
Text: An introduction to symbolic logic by Terence parsons
Software: Logic 2010 http:/logiclx.humnet.ucla.edu/
To register use student ID
1
PHIL 155 Test 2
Chapter 2
Conjunction- and ^
Disjunction- or v
Biconditional if and only if
P
T
T
F
F
Q
T
F
T
F
(P^Q)
T
F
F
F
(PvQ)
T
T
T
F
(PQ)
T
F
F
T
(PQ)^(QP)
T
T
T
F
F
T
T
F
F
T
T
T
Conventions for Informal Sentences
1. Drop outermost parentheses