Some Useful Equivalences
xF x xF x
xF x xF x
x(F x P ) xF x P
x(F x P ) xF x P
x(P F x) P xF x
x(P F x) P xF x
x(F x P ) xF x P
x(F x P ) xF x P
x(F x Gx) xF x xGx
x(F x Gx) xF x xGx
xy(F x Gy) yx(F x Gy)
xy(F x Gy) yx(F x Gy)
Some Useful Theorems
xF x xF
Practice Proofs
1
Proofs that use only Stage 1 rules
1. Associativity of &
P&(Q&R)
(P&Q)&R
2. Commutativity of &
P&Q
Q&P
3. Idempotence of &
P
2
P&P
Proofs that use CP but not Elimination or
RAA
1. Weakening the Consequent
P (Q&R)
PQ
PQ
P (Q R)
2. Streng
Intro Logic
&I
d
e
d&e
d
e
e&d
I
SENTENTIAL LOGIC
&O
;&O
d&e d&e
;(d & e)
d
e
d ;e
O
d
d
de ed
I
de
;d
e
de
;e
d
O
de
ed
de
de
ed
ed
O
DN
DN
;(d e)
;d e
;O
df
d
f
d
;d
df
;f
;d
&D
: d&e
d

e

D
D
: df
d
: f

: ;d
d
:

d
d
DD
d
: de
d
RULES OF DEDUCTION
Rule P (Premise Introduction): On any line (n) of a deduction, we may put any schema,
with [n] as the premise number. There is no citation.
Rule D (Discharge): Suppose that on some line (m) we have a schema S, that one of the
premise nu
EMR17
DeductiveLogic
Lecture 25
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: I am a knight.
Betty says: One of us is a monk.
Clara sa
EMR17
DeductiveLogic
Lecture 24
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: I am either a knave or a monk.
Betty says: We are all di
EMR17
DeductiveLogic
Lecture 29
Rules for deduction
Our system of formal deduction for quantificational logic now
has five rules:
Premise Introduction (PI)
Conditional Introduction (CI)
Truthfunctional Implication (TF)
Universal Instantiation (UI)
Univer
EMR17
DeductiveLogic
Lecture 27
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: I am not a monk.
Betty says: At least one of us is a kna
EMR17
DeductiveLogic
Lecture 28
More rules
Our system of formal deduction for quantificational logic
currently has three rules:
I GOOFED last
Premise Introduction (PI)
time, in suggesting
that we could drop
this!
Conditional Introduction (CI)
Truthfuncti
EMR17
DeductiveLogic
Lecture 26
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: We are all the same.
Betty says: No, were not.
Clara say
EMR17
DeductiveLogic
Lecture 19
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Betty and Clara are different.
Betty says: Alfred tells
EMR17
DeductiveLogic
Lecture 23
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Betty tells the truth more often than Clara.
Betty says:
EMR17
DeductiveLogic
Lecture 21
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter five islanders: Alfred, Betty, Clara, David, and
Emma.
Alfred says: Someone tells the truth more often than me.
EMR17
DeductiveLogic
Lecture 20
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter some islanders: Alfred, Betty, and some large
number of other islanders.
Alfred says: Everyone tells the truth m
EMR17
DeductiveLogic
Lecture 22
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Betty tells the truth more often than Clara.
Betty says:
EMR17
DeductiveLogic
Lecture 18
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Either I am a knave, or Betty is a monk.
Clara says: Eit
EMR17
DeductiveLogic
Lecture 17
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: I am the same as either Betty or Clara.
Betty says: We a
EMR17
DeductiveLogic
Lecture 14
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Clara tells the truth less than any of us.
Betty says: I
EMR17
DeductiveLogic
Lecture 16
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: I am different from Betty.
Betty says: I am different fr
EMR17
DeductiveLogic
Lecture 15
Warmup puzzle
You are visiting the island of knights, knaves, and monks. The
usual rules apply.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: We are all the same.
Betty says: Alfred and Clara are di
EMR17
DeductiveLogic
Lecture 13
Warmup puzzle
You are visiting a different (but nearby) island. Like the first
island, some inhabitants are knights, and some are knaves.
But there is a third kind of inhabitant: monks. They say
whatever they feel like, wi
EMR17
DeductiveLogic
Lecture 9
Warmup puzzle
You are back on the Island; as before, each inhabitant is either
a knight or a knave.
You encounter four islanders: Alfred, Betty, Clara, and David.
Alfred says: The total number of knights among us is odd.
Be
EMR17
DeductiveLogic
Lecture 11
Warmup puzzle
You are visiting a different (but nearby) island. Like the first
island, some inhabitants are knights, and some are knaves.
But there is a third kind of inhabitant: monks. They say
whatever they feel like, wi
EMR17
DeductiveLogic
Lecture 12
Warmup puzzle
You are visiting a different (but nearby) island. Like the first
island, some inhabitants are knights, and some are knaves.
But there is a third kind of inhabitant: monks. They say
whatever they feel like, wi
EMR17
DeductiveLogic
Lecture 7
Warmup puzzle
The story: At the end of class one Friday afternoon, the professor announces to
her students that she will give them an exam during one of next weeks classes.
(Class meets every day during the week.) She adds
EMR17
DeductiveLogic
Lecture 8
Warmup puzzle
You are back on the Island; as before, each inhabitant is either
a knight or a knave.
You encounter four islanders: Alfred, Betty, Clara, and David.
Alfred says: Either Betty is the same as Clara, or I am a
kn
EMR17
DeductiveLogic
Lecture 10
Warmup puzzle
Consider the following statement:
Either the boxed statement on this slide is false,
or there will be cookies during todays quiz.
What can you deduce about that statement?
Is it true?
More importantly, will t
EMR17
DeductiveLogic
Lecture 4
Warmup puzzle
You are back on the Island; as before, each inhabitant is either
a knight or a knave.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Exactly one of us is a knight.
Betty says: We are all
EMR17
DeductiveLogic
Lecture 2
TAKE SOME M&Ms (in back)
A warmup puzzle
On the Island of Knights and Knaves, there are two kinds of
people: knights, who always tell the truth, and knaves, who
always tell falsehoods.
This is the only feature that distingu
EMR17
DeductiveLogic
Lecture 6
Warmup puzzle
You are back on the Island; as before, each inhabitant is either
a knight or a knave.
You encounter three islanders: Alfred, Betty, and Clara.
Alfred says: Betty is the same as Clara.
Betty says: We are not al