PL2003Symbolic Logic
Assignment #1key.
1a)
P
P
P'
(P P' )
1b) (P (Q R' ) S)  not a wff.
1c) (P' )  not a wff.
1d)
Q
P'
Q
(P' Q)
R'
(P' Q) R' )
1e) (P' )  not a wff.
2a)
P' T
QF
RT
(P' Q) F
R F
(P' Q) R) F
(b)
P' T
RT
P' F
R F
(P' R) F
(c)
QF
P' T
Q T
PLUY 2004 Symbolic Logic
Assignment #1. Due Thurs Feb 4.
1. Indicate which of the following strings of symbols qualify as wffs in PL. For those that
qualify, draw their construction trees.
(a)
(b)
(c)
(P P' )
(P (Q R' ) S)
(P' )
(d)
(e)
(P' Q) R' )
(P' )
Chapter 16: Introducing PL Trees
Ex1. (P Q), (P' Q'), (P' Q') R
Truth table would contain 27 = 128 rows!
Tautologically invalid!
R nowhere occurs in premises.
Premises are not taut. inconsistent.
But all we really need to construct is a counterexample
Chapter 14: The Language of PLC
Motivation:
Ex1. (1) If Jack bet on the Cardinals, then Jack lost his money.
(2) Jack did bet on the Cardinals.
VALID
(3) So Jack lost his money.
(1) If Jack bet on the Cardinals, then Jack lost his money.
(2) Jack lost his
Gdel's 1st Incompleteness Theorem
Gdel's 1st Incompleteness Theorem.
Let N be a firstorder formal theory of arithmetic that is recursively
axiomatizable. If N is consistent, then it is negation incomplete.
Questions:
1. What is a "firstorder formal theo
PL2004 Symbolic Logic
Assignment #2. Due Thus Feb 11.
1. For each of the pairs of wffs below, construct a truth table that includes them and indicate
whether or not they are truthfunctionally equivalent. Explain your answer by referring to
your truth tab
Chapter 36: Functions
Motivation: Ultimately, to formulate simple arithmetic as a formal theory.
Def. 1. A function f : D R is a map from one set of objects D to another
R such that one or more objects in D get mapped to a unique object in R.
 Input (on
Chapter 34: Definite Descriptions
A definite description is a phrase that doesn't contain
a name and that aims to designate a particular thing.
Ex.
"The Queen of England"
"The smallest prime number"
"The trombone in my basement"
(vs. "Elizabeth")
(vs. "1"
Chapter 32: Identity
Let R be a twoplace relation and suppose a, b, c are objects that stand in it.
Def 1. R is transitive just when, if a has
R to b, and b has R to c, then a has R to c.
Exs.
is
heavier than
is
an ancestor of
Def 2. R is symmetric just
Chapter 30: Soundness and Completeness for QL Trees
Soundness of QL Trees
If a QL argument is not qvalid, then the corresponding QL tree will never close.
Let:
"QL" mean "QL treeentails"
Let:
"QL" mean "qentails"
Then:
Soundness of QL Trees means: If A
Chapter 29: QL Trees
(a)
A

A
Add A to each open path containing A.
Check it off
(b)
(A B)

A
B
(c)
(A B)

A
B
Add A, B to each open path containing (A B).
Check it off
(d)
(A B)

A
B
Add A, B to each open path containing (A B).
Check it off
Add A
Chapter 27: QValuations
The vocabulary V of a set of QL wffs is the set of
constants and predicates that appear in those wffs.
A "qvaluation" for a set of wffs with vocabulary V does three things:
1.
Fixes a domain of objects D. Convention: Must be non
Chapter 28: QValidity
In PL:
The PL wffs A1, ., An tautologically entail the PL wff C just when
there is no PL valuation that makes A1, ., An true and C false.
An argument in PL is tautologically valid just
when its premises tautologically entail its con
Chapter 17: Rules for PL Trees
Unpacking Rules:
(a)
A

A
Add A to each open path containing A.
(b)
(A B)

A
B
(c)
(A B)

A
B
(d)
(A B)
Add A, B to each open path containing (A B).
A
(e)
Add A, B to each open path containing (A B).
B
(A B)
A
B
Add a for
Chapter 25: Introducting QL Trees (Informally)
Recall the tree method:
To show that A1, ., An C is tautologically valid, show that A1, ., An, C is
tautologically inconsistent.
How will this work for QL:
Ex1:
F
G
n
Fn, x(Fx Gx) Gn
Fn
x(Fx Gx)
means
means
m
Chapters 23, 24: QL Translations
Let:
"F"
means
"is wise"
Domain = people
xFx
"Everyone is unwise."
xFx
"Not everyone is unwise."
or
"Someone is wise."
xFx
vC(.v.v.) is true iff vC(.v.v.) is true.
xFx
"Someone is unwise."
xFx
"No one is unwise."
or
"Every
Chapter 18: PLC Trees
Further PLC Rules:
(f)
(A B)
A
B
(g)
(A B)

A
B
(h)
(A B)
A
B
(e)
A
B
(A B)
A
B
A
B
Add a fork with A, B as separate branches to each open
path containing (A B).
Add A, B to each open path containing (A B).
Add a fork with A, B and
Chapters 21, 22: The Language of QL ("Quantifier Logic")
Motivation:
(1)
Fido is a cat.
(2)
All cats are scary.
(3)
Fido is scary.
In PL:
Valid argument!
Let P = Fido is a cat.
Q = All cats are scary.
P, Q R
Not tautologically valid!
R = Fido is scary.
In
PLUY 2004 Symbolic Logic
Prof: Jonathan Bain
Ofce: LC124
Spring 2016
T/Th 8:3010:20
Hours: Weds 12
RH425
faculty.poly.edu/~jbain/logic
I. Description
This is an introduction to the methods and applications of 1storder symbolic logic, including both pr
Chapter 8: The Syntax of PL
I. Syntactic Rules for PL
A. Alphabet (13 symbols)
(1) P Q R S '
simple propositions
(2)
connectives
(3) ( )
punctuation
(4) ,
additional punctuation
B. Grammar: What counts as a "wellformed formula" (wff ).
Definition of an
Chapters 12, 13: Tautologies and Tautological Entailment
A wff of PL is a tautology if it takes the value true on every valuation of its atoms.
A wff of PL is a contradiction if it takes the
value false on every valuation of its atoms.
Examples of tautolo
Chapter 11: Truth Functions
A way of forming a complex sentence out of one or more constituent sentences
is truthfunctional if fixing the truthvalues of the constituent sentences is
always enough to determine the truthvalue of the complex sentence.
Cla