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
All monkeys are mammals -P
All mammals are animals -Q
So, all monkeys are animals R
Valid argument, unprovable with old system
Some cats are friendly
Cats are household pets
So, its not the case that all household pets are unfriendly.
Sentential Calculus-
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a; 1A., 5 fgmd C i P " 90 Ug» "MA; $102K; 1). Keoka-Z.( ,2
bUXCck. Q"V-ll\; 5'Q1QS .~(. 2" 0 'Q' t. EZH y -
{N 1 ~r
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P
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
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
k on philosophy of social science? A better
'cern with human well-being is at the very
hrase 'sdence sodaie was coined by the
_ noel-Joseph Sieys in 1789 in his pam-
social science was to provide the justica-
based On majority rule, for this is the best
'
Oedipus
I am confronted with many references to eyes, seeing, blindness, light and darkness
while reading the play Oedipus Rex by Sophocles. The following three references,
however, leave me much impression.
The first reference is I see said the blind man
Title of experiment: Accelerated Motion(Lab 1)
Lab section: MoWe 6:50PM - 8:40PM
TA name Brandon Yost, Nolan Miller
Date the experiment was performed: Aug 29th
Your name: Junbo Guan
PID:
730132901
Your partners name: Ryan, Kyle
Honor pledge and your signa
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
Phil 155
Chris Dorst
dorst@live.unc.edu
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
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
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)
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
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
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 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
09/18/2015
R
(QS)~R
Therefore ~S
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Show ~S
S
R
(QS)~R
~R
~(QS)
Show QS
Q
S
ASS ID
PR
PR
3 DN
4 5 MT
ASS CD
2R
9 CD
6 7 ID
~(RQ)P
P(~QQ)
~Q
Therefore ~R
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Show ~R
R
~(RQ)P
P(~QQ)
Validity: the truth of the premises guarantees the truth of the conclusion.
The premises dont have to be true, its just about the relationship with the premises and
conclusion
Theres no gap (a way you could put in another premises to make the conclusion f