Philosophy 120
Symbolic Logic I
H. Hamner Hill
CSTL-CLA.SEMO.EDU/HILL/PL120
Logic is the science of
arguments
• Separate good arguments from bad
ones
• Identify the characteristics of good
arguments (validity and soundness)
• Produce good arguments of our own
Student Objectives
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learn the vocabulary of logic
master methods and principles
explain important concepts in logic
improve communication skills
symbolize arguments using logical notation
test arguments for validity
evaluate reasoning using the tools of logic
Requirements
• 3 in class examinations
• 10 routine graded homework assignments
• a comprehensive final examination
Cell Phones
• Turn it off. We are in class, your call can
wait. Do not text message during class.
• Cell phones and logic do not mix.
• Hang up and derive!
• Read this column from the New York
Times.
Textbook and Associated
Computer Program
• The Power of Logic, 5th edition, available at
the Textbook Services.
• SUBJECT TO CHANGE: ALL homework
will be completed using the program The
Logic Tutor. The Logic Tutor is provided
by the textbook publisher and is available
online.
Logic is the science of arguments
• All rational inquiry turns on the ideal of a
logical consequence, the idea that some
claim must necessarily follow from others.
• Arguments are designed to show that one
claim logically follows from others.
• Logic allows to determine whether the
arguments succeed.
What is an argument?
• An argument is not a disagreement or a form
of verbal battle.
• An argument is a set of statements, one of
which (the conclusion) is supposed to follow
from the others (the premises).
Statement
• A sentence that has a truth value, i.e., a
sentence that is either true or false (but
never both).
• Statements are true when what they
assert about the world is the case.
• Can you think of a sentence that is not
a a statement?
Can you think of a sentence
that is not a a statement?
• OK, this is the sort of question logicians
love to ask, because the question itself is a
legitimate answer! The sentence “Can you
think of a sentence that is not a statement?”
is itself a sentence that is not a statement.
Questions are neither true nor false.
Commands, exclamations, and exhortations
(Let’s . . .) are other sentences that do not
express statements.
Types of statements
• Simple--A simple statement asserts
exactly one fact about the world
• Compound--A compound statement is
one or more simple statements plus
logical connectives.
• 5 logical connectives: not, and, or, ifthen, if and only if
NOTE: TRUTH is a property of
statements. VALIDITY is a
property of arguments
Conclusion
• A statement one is urged to accept on
the basis of reasons given.
Premise
• A statement given as a reason for
believing some other statement.
Identifying premises and the
conclusion
• Correctly identifying the premises and
conclusion of an argument are essential if
we are to evaluate it.
• English uses many discrete premise and
conclusion indicators (review your handout)
that serve as guideposts in arguments.
Deductive Validity
• A characteristic of arguments in which the
truth of the premises guarantees the truth of
the conclusion. It is impossible for both the
premises of a valid argument to be true and
the conclusion to be false.
• Any argument that is not valid is invalid or
non-valid
Validity does NOT guarantee the
truth of the conclusion
• It is possible for the conclusion of a valid
argument to be false. If this is the case,
then at least one premise must be false.
The following argument is VALID:
• All trout are mammals
All mammals have wings
SO, all trout have wings
• This argument is valid because IF the
premises are true THEN the conclusion
MUST be true. This holds even though the
premises are in fact false.
Soundness
• A characteristic of valid arguments
whose premises are in fact true. It is
impossible for the conclusion of a
sound argument to be false.
• It is irrational to reject the conclusion
of an argument one admits to be sound.
Logical Form and Grammatical
Form
• Logic is not a matter of grammar.
“Following logically’ is not a matter of
grammatical placement.
Logic is a matter of form
• Logic is a formal discipline. It is
concerned with the formal or structural
properties (patterns) and relations in
statements and arguments.
Argument Forms
• An argument form is a pattern of
argument, the logical structure of an
argument. Argument forms are either
valid or non-valid. Valid arguments
have valid argument forms.
Consistency
• Consistency is a property of sets of
statements
• A set of statements is consistent if, but only
if, it is possible for all of the statements in
the set to be true.
• A set of statements is inconsistent if, but
only if, it is impossible for all of the
statements in the set to be true.
Consistency and Validity
• We can use the concept of consistency to
test an argument for validity.
• How? Suppose I gave you a consistency
checking machine (a machine that tests a set
of statements for consistency). How could
you use that machine to determine whether
an argument is valid?
Hamner’s Helpful Home
Consistency Checker
• Input
• (set of statements)
•
Output
(verdict)
Consistent
Not Consistent
Using the Consistency Checker
• Negate the conclusion of the argument and
then ask whether the set of statements
consisting of the premises and the negation
of the conclusion is consistent. If yes, then
the argument is NON-VALID. If no, if that
set is inconsistent, then the argument is
VALID.
Historical Significance
• Indirect Proof (Reductio ad Absurdum)
• Euclidean and Non-Euclidean Geometry
•
Lobachevsky
Reimann
Indirect Proof
• Both Lobachevsky and Reimann tried to establish
the truth of all 5 of the core postulates of Euclidian
geometry using indirect proof. They succeeded in
proving 4 out of 5, but efforts to prove the parallel
postulate by indirect proof never led to a
contradiction.
• In fact, the failure to prove the parallel postulate
led to the development of Non-Euclidian
geometry.
Logic and Psychology
• Contexts of DISCOVERY and contexts of
JUSTIFICATION are different.
• LOGIC is concerned with the context of
justification, the business of defending
beliefs.
• The "logic" of discovery is a matter for the
discipline of psychology.
Justification and Discovery
• Ramanujan and the
difference between
justification and
discovery.
Justification and Discovery
• Ramanujan was one of the greatest
mathematicians of the 20th Century. Today’s
mathematicians are still trying to prove some of
his theorems.
• He insisted that his ideas came to him in dreams,
presented by the Goddess Namakaal. Even if this
is true, it doesn’t concern the logician.
• Logicians are interested in the justification of the
theorems (How they are proved), not how the are
discovered.
Arguments are often confused
with explanations
• Sometimes the language of arguments is
used when one is not arguing for a
conclusion but rather trying to explain a
phenomenon.
Arguments:
• Answer the question "Why should I believe
this?“
• Give reasons for believing that something is
the case.
Explanations:
• Answer the question "Why is this the case?“
• Give an account of something already
believed to be the case (the facts are not in
dispute).
Key Ideas
• Definition of “argument”
• Validity is a matter of form
• Validity does not guarantee the truth of
the conclusion
• Consistency as a test for validity
• Contexts of discovery and justification
• Arguments and explanations