Conditionals
Introduction to Logic
True or False?
If Amy lives in Los Angeles, than Amy
lives in Florida
A. True
B. False
C. Depends
True or False?
If bananas are yellow, then the sky is blue
A. True
B. False
C. Depends
Valid or Invalid?
It is false th
Proof Strategies
Introduction to Logic
Formal Proofs vs Truth Tables
Formal Proofs:
Correspond to step-bystep reasoning
Dont get so big fast
Require creativity
Can be used for
propositional as well as
predicate logic
Can demonstrate
consequence only
Truth Trees
Introduction to Logic
Drawback of the Short Truth Table
Method
A drawback of the short truth table method is that you
are not always forced to assign any further truth
values:
R (Q P)
T1 T2
(Q R) P
T2
T1
RP
T1
Q
T1
At this point, you can cho
Venn Diagrams
Introduction to Logic
Venn Diagrams
Venn diagrams consist of circles representing sets (or classes) of
objects. By overlapping the circles the right way, you can
get all possible objects. That is, there is a region for objects
that belong to
Proof by Picture
Introduction to Logic
Thabit ibn Qurra
Bhaskara
Bhaskara Animated
Perigals Dissection Proof
Annairizis Tiling Proof
Perigal Dissection from Tiling
Dissection from Annairizi Tiling
Thabit ibn Qurra from Annairizi
Tiling
Bhaskara from Tilin
Boolean Algebra
Introduction to Logic
From Last Time: Some Useful
Equivalences
Double Negation:
PP
Commutation:
PQQP
PQQP
DeMorgan:
(P Q) P Q
(P Q) P Q
Association:
P (Q R) (P Q) R
P (Q R) (P Q) R
Distribution:
P (Q R) (P Q) (P R)
P (Q R) (P
Logic
Introduction to Logic
Book and Software
Please make sure to get the book and software (it
comes as one package)!
You need the software as you will be doing much
of your HW using the software:
Using the software you will create files that you
elec
Logical Properties
Introduction to Logic
Another Ladies and Tigers
Same as before:
There are two rooms with signs on them
Either room contains either a lady or a tiger
You want to choose the room with the lady if there
is one!
Either both signs are t
Validity and Invalidity
Introduction to Logic
Surprise Quiz
During one of the classes this semester
there will be a surprise quiz.
The quiz wont count for your grade, but it
is a surprise quiz in that you dont know
which day I give that quiz.
Recap: Arg
Boolean Connectives
Introduction to Logic
Connectives: From Atomic
Claims to Complex Claims
So far, we have only seen atomic claims:
claims consisting of a single predicate. E.g.
a is to the right of b.
We can combine atomic claims using
connectives to
Boolean Logic and
Computer Circuitry
Introduction to Logic
The Digital (Binary) Computer
1s and 0s
Digital computers represent information using 1s
and 0s
A bit is something that is either 1 or 0. It
represents a single bit of information (what it
repre
Formal Proofs for
Propositional Logic
Introduction to Logic
Recap: The Method of Formal Logic
Step 1: Use certain symbols to express the
abstract form of premises and conclusion.
Step 2: Use a certain procedure to figure out
whether the conclusion follo
Short Truth Tables
Introduction to Logic
Truth Table Method
The truth table method systematically exhausts all
possible truth value combinations of the statements
involved.
In the truth-table we look for a row that reflects a
certain possibility, and th
Proof by Contradiction
Hall of Fame
Introduction to Logic
Minesweeper
Is there a mine here?
Minesweeper
No! Suppose there is
a mine there. Then
because of the 1s
to the upper left and
right of it, there will be no
mine to the left and right
of it. But the
Existential Graphs
Introduction to Logic
Existential Graphs
A graphical logic system developed by C.S. Peirce
almost 100 years ago.
Peirce studied semiotics: the relationship between
symbols, meanings, and users.
Peirce stressed the power of iconic rep
Eye Computation
Introduction to Logic
Eye computation, Changizi (2008)
Wires transmit signals in visual circuit.
Perceptually bistable, can perceive as
tilted away, or tilted toward you.
A spot on circuit has value 1 if tilted
toward you, and 0 if tilt
Proofs for Quantifiers
Introduction to Logic
Methods of Proof
Transylvania
Transylvania is inhabited by humans and
vampires. Humans are honest: whatever they tell
you is something they believe to be true. Vampires
are dishonest: whatever they will tell y
Quantificational Logic
Introduction to Logic
Quantificational Logic
Quantificational Logic is part II of the book
and the second half of the semester
Quantificational Logic is sometimes called
Predicate Logic or First-Order Logic
Quantificational Logic
Quantificational Logic:
Rewriting Statements
Introduction to Logic
Satisfying and Falsifying
Well-Formed Formulas
Well-formed formula: a grammatical expression
that may contain free variables
Examples:
Cube(x)
Tet(x) y(Cube(y) Smaller(x,y)
Object a s
Grid Logic
Introduction to Logic
Logic Education
Taught Introduction to Logic for many years
My question: Are students learning logic?
How do we measure this? What is logic?
Intuitively, logic is about logical reasoning.
So, do students get better at
Quantifier Proof Strategies
Introduction to Logic
Dont get lost in the symbols!
Keep in mind what the statements are claiming!
Dont just look at the symbols syntactically (how the
symbols are put together), but also semantically (what
the symbols actual
Symbolization with Quantifiers
Introduction to Logic
Multiple Uses of a Single
Quantifier
Quantificational statements get tricky when
there are multiple quantifiers involved.
Well start easy first, by considering
quantificational statements with multipl
Metalogic
Introduction to Logic
Logic and Metalogic
Metalogic is the study of logic. That is, where logic
has no specific subject matter (logic can be applied in
any field where reasoning takes place), the subject
matter of metalogic is logic itself.
Me
Atomic Sentences
Introduction to Logic
MC Question
A) There are no correct answers
B) All answers are correct
C) There is exactly one correct answer
D) None of the above
Recap: The Method of Formal Logic
Step 1: Use certain symbols to express the
abstrac