PHL 313K
Logic, Sets and Functions
Fall 2002
Prof. Robert Koons
Text and Software
Logic, Sets and Functions by Daniel
Bonevac, Nicholas Asher & Robert Koons
(Kendall/ Hunt, 2001).
Plato (for Windows and Macintosh).
Available with text and on the Plato web
site:
http://www.utexas.edu/courses/plato/
Grading
The course grade will be based on the
following components: homework (5
assignments), four tests, and an optional
final exam. The final exam can be used
to replace one's lowest test grade.
These components will be weighted in
the following manner:
Homework: 20%
Four Tests: 80%
All four tests are mandatory
Course Web Site
http://www.utexas.edu/courses/phl313k
Contains syllabus, homework
assignments, sample problems,
practice tests.
Instructor
Prof. Robert C. Koons
rkoons@phil.utexas.edu
Phone: 471-5530
Office: Waggener 405
Office hours:
Wed. 3-4
Thurs. 1-2, and by appointment.
Syllabus
Introduction to Logic
Ch. 1 & 2
Sentential Deduction
Ch. 3
Sentential Deduction
Ch. 3
HW #1, due Sept. 10
Quantifiers
Ch. 4
First Test: Sept. 19 (sentential logic)
Top 10 reasons for taking
PHL 313K
10. Riveting lectures.
9. Endlessly fascinating homework
assignments.
8. You get to say "modus ponens" over
and over again.
7. The kind of comraderie you missed by
not enlisting and going to boot camp.
6. It's your last chance to turn that lump
of gray goo you call a "brain" into
something capable of rubbing two
thoughts together.
5. Millions of trees will be saved because
of all the mindless crap you won't be
able to write anymore.
4. It's guaranteed to make you rich and
famous, or your money back. [Offer void in states
containing a, e, i, o or u.]
3. You'll be a big hit at parties, pointing
out all the fallacies in everyone else's
reasoning.
2. When you're bouncing your grandkids
on your knee, you'll be able to tell them,
"I studied logic under Robert Koons".
1. You won’t spend the rest of your life in
a cubicle writing code.
Nature of Logic
Logic: both very old and very new
Old: Aristotle, 2500 years ago.
New: most work completed in last 100
years. Knowledge explosion.
Three parts of logic
Sentential logic. Logical “chemistry”: the
logic of building molecular sentences with
connective bonds: and, or, if, not.
Predicate logic. Logical “atomic physics”:
the logic of simple sentences and clauses
including quantifiers: all, some, none.
Set theory. The universal language of
mathematics.
Basic concepts and
terminology
DEFINITION. An argument is a finite
sequence of sentences, called premises,
together with another sentence, called the
conclusion, which the premises are
purported to support.
Conclusion indicator words
therefore, thus, hence,
so, consequently, it follows that,
in conclusion,
as a result,
then,
it must be that
Premise indicator words
because
for
since
Defining deductive validity
DEFINITION. An argument is deductively
valid if and only if it's impossible for its
premises all to be true while its conclusion
is false.
Examples
Valid:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Invalid:
Grass is green.
Therefore, snow is white.
Three possibilities for valid
arguments
True premises, true conclusion.
All men are mortal, Socrates is a man,
therefore Socrates is mortal.
False premises, false conclusion.
No Democrat hunts, Bush is a Democrat,
therefore Bush doesn’t hunt.
False premises, true conclusion.
All Democrats hunt, Bush is a Democrat,
therefore Bush hunts.
What is impossible:
A valid argument with true premises and
a false conclusion.
This is excluded by the definition of
validity.
Some imperfections in this
definition.
According to our definition, the following
arguments are “valid”:
2+2=5. Therefore, snow is green.
Bush is a Democrat. Therefore, 2+2=4.
In the first case, it is mathematically
impossible for the premise to be true.
In the second case, it is mathematically
impossible for the conclusion to be false.
What’s wrong with our
definition
The logical validity of an argument should
depend only on the logical form of the
argument, not on whether it is “possible” in
some non-logical sense for the premises to
be true or the conclusion false.
We could say that an argument is valid if
it’s logically impossible for the premises
to be true and the conclusion false at once.
Validity applies to logical
forms
An argument is valid if it has a logical form
that is valid.
A logical form is valid if no argument
having that form has true premises and a
false conclusion.
Different kinds of logical
form
Ch. 2 & 3 -- sentential logical form. Focus on
words used to create compound sentences
from simple ones: “and”, “or”, “if ... then”,
“not”
Ch. 4 & 5 -- quantificational form. “Some”,
“all”, “none”. Subject/predicate structure of
sentences.
Ch. 6 -- the logic of identity. “is”, “=“
Soundness
DEFINITION. An argument is sound if
and only if (a) it is valid and (b) all its
premises are true.
We’re interested in validity,
not soundness
We're not concerned with truth, falsity of
sentences in logic. Logic pertains to all
subject matters, logicians can't be experts
on everything.
Some valid arguments are useful though
not sound.
Two uses of valid arguments
1. Start with known premises, deduce new
truths.
2. Start with doubtful premises, uncover
hidden implications.
A special case of the second: reductio ad
absurdum.
Reductio ad absurdum
A deductively valid argument with an
obviously false conclusion.
The reductio reveals to us that at least
one of its premises is false.
Usually, there is one premise (the
“hypothesis”) whose truth is doubtful.
The reduction “reduces” the hypothesis to
absurdity.
A classic example (Plato)
1. If you give back what you owe, you
always act rightly. (Hypothesis)
2. If you return a deadly weapon to a
temporarily insane friend, you give back
what you owe.
Therefore, if you return a deadly weapon
to a temporarily insane friend, you always
act rightly. (Obviously false)
What is “obviously false”?
As logicians, there is only one thing that
we can count as “obviously false”.
This is a contradiction: a statement of the
form ‘A and not A’.
For example: snow is white, and snow is
not white. Today is Wednesday, and today
is not Wednesday.
Definition of “implication”
DEFINITION. A set of sentences, S ,
implies a sentence, A , if and only if it's
impossible for every member of S to be
true while A is false.
DEFINITION. A sentence, A , implies a
sentence, B , if and only if it's impossible
for A to be true while B is false.
Connection: validity &
implication
If the argument “S1...Sn ˇ A” is valid, then
{S1… Sn} implies A.
And vice versa: if {S1… Sn} implies A,
then the argument “S1...Sn ˇ A” is valid.
The symbol ˇ separates the premises from
the conclusion of an argument.
Equivalence
DEFINITION. A sentence, A , is equivalent
to a sentence, B , if and only if it's
impossible for A and B to disagree in
truth value.
Equivalent: Nothing cheap is good =
nothing good is cheap
Not equivalent: Everything expensive is
good ± everything good is expensive.
Demonstrating lack of
equivalence
Imagine 2 stores:
(1) everything is good, some cheap, some
expensive.
(2) everything is expensive, some good,
some bad.
In (1), everything is expensive is good, but
not everything good is expensive.
In (2), conversely.
Logical properties of
sentences
DEFINITION. A sentence is contingent if
and only if it's possible for it to be true and
possible for it to be false.
DEFINITION. A sentence is logically true if
and only if it's impossible for it to be false.
DEFINITION. A sentence is contradictory
if and only if it's impossible for it to be true.
DEFINITION. A sentence is satisfiable if
and only if it's not contradictory.
Properties of sets
DEFINITION. A set of sentences is
contradictory if and only if it's impossible
for all of its members to be true. A set is
satisfiable otherwise.
Some examples
Valid:
A rose is a rose.
You are what you are.
A man's gotta do what a man's gotta do.
Contradictory:
Yogi Berra: nobody goes there anymore: it's
always too crowded.
Hegel: The moving body is both here and not
here at the same time.
Inclusion relationships
satisfiable
valid
contingent
contradictory
Two very old logical
principles
No satisfiable set of sentences can imply
a contradiction.
Only a contradictory set of sentences can
imply a contradiction.