Class notes
Logic Lecture Notes
- Course
- Institution
Lecture notes of 72 pages for the course Logic and Sets at VU (Logic Lecture Notes)
[Show more]
Seller
Follow
mchelleh
Content preview
logic lecturesweek / propositional logic
VALID ARGUMENT
"
if the train arrives late and there are no taxis at the station ,
then jane is late for her
,
meeting " .
·
jane is not l a te for the
meeting .
·
the train does arrive l a te .
·
therefore ,
there a re taxis at the station .
DECLARATIVE SENTENCES
a declarative sentence (or proposition) is a statement that is either true or false.
I
57 3 true
513 false
grass is true
green
is and roses a re blue false
grass green
grass is green or roses are blue true
if X31 ,
then x*X true
ARGUMENT ABSTRACTION
"
if the train arrives late and there are no taxis if p and not a then r
.
-
, ,
station jane is late for her ."
at the ,
then
meeting .
·
jane is not l a te for the
meeting .
not r
·
the train does arrive l a te .
P
·
therefore ,
there a re taxis at the station . therefore , 9.
key of translation
·
P the train arrives l a te
a there are taxis at the station
r
Jane is l a te fo r her meeting
logical formalization :
(((p1 -9) -
r) - (ur -p)) -
d
Validity of the arguments is due to their logical form .
it does not depend on the ac tual con te n t of propositions p, 9 and r.
SYMBOLS
we wa n t to study logic without being distracted by the concrete contents of pro-
positions .
, ropositional variables
P p , a , ,
r . . .
connectives
1 and conjunction
v op disjunction
eithe r. . . exclusive
⑰
or . . . or
not negation
>
-
if
... then . . . implication
- if and only if bi-implication
SENTENCES AND FORMULAS
I
57 3 P
grass is
green P
is blue 1
grass green and roses a re p a
grass is green or roses are blue pra
if X31 ,
then x*X p-
-
a
↳ where p: X- 1 and 9 : x2 =
X
FORMULAS OF PROPOSITIONAL LOGIC
,
P 9 , ,
r . . .
a re propositional variables .
- is a unary connective ↑
1 V 0 -
connectives .
, , , , are binary
inductive definition :
.
1 base Step
every propositional variable is a formula .
.
2 construction steps
-
a
if is a for mula ,
then so is -
(4)
-
if $ and 4 a re formulas ,
then so a re (414) ,
(414) .
(404) ,
(4e4) ,
and
(P =t .
PARSING A FORMULA
( -
(p) -q) +
(p +
(av -
(r))
porcauer
·
↓
⑮ ↓
⑰
,OMITTING PARENTHESES
to omit parentheses from formulas . Without causing ambiguity , we use the priority
schema :
1v
>
- Ex
I
spra)
va -
P va
r +
pra
-
rv(p va)
-
( -
(p)) - -
p
(p1 -
(a)) =
r) (-
(r)1p))2q
↳ (p- -
q
+ r) -( r -
p) +
q
SYNTAX AND SEMANTICS
the syntax is how a word(or formula) is written .
↳ "cow"
the semantics is its meaning
↳ "a white mammal with black spots ,
that eats grass .
"
TRUTH VALUES
formulas of propositional logic are used to express declarative statements ,
which
are either true or false.
the truth value of a formula such as Put is determined by the truth
values of its components ,
& and ↑
.
NEGATION
[
$ true if I is false
·
negation -
"not $") is
false ifI is true
& -
O
1 0
S T
CONJUNCTION
[
/"D true I
if is true and I is true
a conjunction &1 ↑ and tr") is
false all other cases
a ↑ Out
11 I
1 O O
G 1 O
g G O
, DISJUNCTION -> inclusive "or"
[
/"D true I
if is true ↑ is true /or both
a disjunction & V to or tr") is
or
false all other cases
a ↑ Out
1 1 1
1 O 1
G 1 1
000
EXCLUSIVE OR
[
$ /"either true I
if is true ↑ is true (not
an exclusive or or r") is
or
both)
false all other cases
P ↓ out
1 1 O
1 O 1
G 1 1
0 O O
IMPLICATION
# - ↑ means : if I is true ,
then t is true .
↳ I
if true and I false then - false .
,
E
- false if I is true and ↑ is false
an implication ("if & then") is
true otherwise
P ↓ # - ↓
1 1 1
1 O O
G 1 1
0 O 1
B1- IMPLICATION
[
a bi-implication & 4 ("O if and iI
f ") is true if 4 and It have the
only
to
same truth va l u e
① ↑ est
false otherwise
1 1 1
10 O
G 1 S
g G 1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller mchelleh. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.98. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
62890 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now