Using Conditional and Indirect
Proof
Conditional and Indirect Proof
So now we have two kinds of proof where we
use an assumption.
Conditional proof.
Indirect proof.
In a conditional proof, we start by assuming the
a_ of the conditional we want to prov
Categorical Propositions
Categorical Propositions
A Categorical Proposition is a proposition that
relates two kinds, classes, or categories of
things.
For example:
All cats are animals.
This says that the class of cats is a part of the class of
animal
Logic
Logic
Logic is the study of correct reasoning.
Logic is the study of inferences.
Logic is the study of arguments.
Here is a correct inference:
Either Bob is going to the fairground or Susan will
meet him at the cinema.
Susan will not meet him a
Philosophy 173: Logic and Critical Thinking
Syllabus, Fall 2010
Tuesdays and Thursdays 10:30 11:45am
Science and Technology I 224
Course Description and Aims:
This course involves the development of analytical reasoning skills through the study of
fallaci
Exam 3 Practice
Recommended studying for Exam 3 includes the following items. Relevant practice problems
from the textbook are in brackets (Obviously, the more you do, the better you will be at that
particular area. If you are able to do the relevant exer
Exam 2 Practice
Recommended studying for Exam 2 includes the following items. Relevant practice problems
from the textbook are in brackets (Obviously, the more you do, the better you will be at that
particular area. If you are able to do the relevant exer
Exam 1 Practice
Exam 1 will be of multiple-choice format.
Recommended studying for Exam 1 includes the following items. Relevant practice problems
from the textbook are in brackets (Obviously, the more you do, the better you will be at that
particular are
Proofs Practice
ANSWERS
1.
1. A (premise)
Prove ~A
without using DN.
1. A
(Pr)
2.
~A
(AIP)
3.
A ~A (1,2, Conj)
4. ~A
(2-3, IP)
2.
1 A (B C)
Prove
(A B) C
without using exportation, IMPL, DeM or assoc.
1 A (B C)
2.
AB
3.
A
4.
BC
5.
B
6.
C
7. (A B) C
(Pr)
(
Proofs Practice
1.
1. A (Pr)
Prove ~A
without using DN.
2.
1 A (B C) (Pr)
Prove
(A B) C
without using Exp, Impl, DeM or Assc.
3.
1. ~(Q P) (Pr)
Prove
Q ~P
without using DeM
4.
1. (A B) C (Pr)
Prove
A (B C)
without using Assc.
5. Similar but different.
1.
Worksheet Dec. 7
Due Dec. 9
Name_
Construct proofs for the following logical truths using CP.
1.Prove:
AA
2. Prove:
(A v ~B) (~A ~B)
3. Prove:
A (B A)
4. Prove:
A (~A ~A)
Construct proofs for the following using IP.
5.
1. A (premise)
Prove ~A
without usin
Worksheet 4 Answers
Exercise 4.7
Part I
1. All banks that make too many risky loans are banks that will fail.
2. No women military officers are people eligible for combat duty.
3. All times security measures are lax are times terrorist attacks succeed.
4.
Chapter 7 18 Rules of Natural Deduction
For the first 8 rules of implication you must get the term you want to change/use on the
left hand side.
P v Q is True when one is false, exclusively, or when both are true, inclusively
In an indirect proof, we sta
Informal Fallacies
What is a Fallacy?
Fallacies: Making Bad Arguments Appear
Good
Formal Fallacies vs. Informal Fallacies
Formal Fallacies are about structure.
Only purportedly deductive arguments can
suffer from a formal fallacy.
Informal Fallacies a
More Categorical Propositions
The Creation of Logical Equivalencies.
For example, if it is true that No giraffe can fly,
then it is true that No flying thing is a giraffe.
Giraffes
Flying
Things
Flying
Things
Giraffes
The converse of an E statement is L
Indirect Proof
Indirect Proof (reductio ad absurdum)
Sometimes we prove things not by showing how
they follow from premises, but by showing how
their opposite leads to an absurdity.
We do this a lot in daily life. For example,
If we cut taxes, the econ
Conditional Proof
Conditional Proof
Many times in argument, we proceed by
supposing something to be the case.
We then show what follows from it.
By doing so, we prove a conditional.
Example
Either it will be sunny out tomorrow or it will
not be hot.
Proof Strategies
Practice and Review
Warm-up
1. A (B C)(Pr)
prove
(C v ~A) v ~B
1. A (B C) (Pr)
2. (A B) C (1, Exp)
?. (C v ~A) v ~B
1. A (B C) (Pr)
2. (A B) C (1, Exp)
3. ~(A B) v C (2, Impl)
?. (C v ~A) v ~B
1. A (B C)
2. (A B) C
3. ~(A B) v C
4. (~A v
Rules of Implication thus far
Modus Ponens(MP)
pq
p
_
q
Modus Tollens (MT)
pq
~q
_
~p
Disjunctive
Syllogism(DS)
pvq
~p
_
q
or
pvq
~q
_
p
Simplification(Simp)
pq
_
p
or
Hypothetical
Syllogism(HS)
pq
qr
_
pr
pq
_
q
Conjunction (Conj)
The next rule is conju
The Partial Truth Table Method
Suppose that you are looking for a
counterexample and you complete a truth table
thus far:
A
B
Premise
T
T
T
T
F
T
F
T
F
F
F
Premise
Conclusion
F
You dont have to complete the whole thing.
Which rows under column 2 are yo
Truth Tables and Validity
We can use a truth table to check the validity of
an argument.
Recall that we check the validity of an argument by
looking for a counterexample.
Review: What is a counterexample to an argument?
Its a case where the premises o
Truth Functionality
The operators of sentential logic are truth
functional.
This means that the truth of the compounds is a
function of the truth of the parts.
Thats another way of saying that if you know the
truth values of the parts, you know the tru
Conditionals
The main thing to get with conditionals is the use of
only if .
Remember that the antecedent comes before only if but
after lonely if.
Keep in mind that English sentences are not always written in the
same order as we symbolize (i.e. antec
Propositional Logic
Why is it called propositional
logic?
Because the basic units of it are
propositions or statements.
In categorical logic the basic units are terms
within a statement.
And in fact the compound units of it are
propositions too.
Propos
Symbolizing English Sentences
Symbolizing Simple Sentences
We use single letters for simple sentences.
Usually we use a letter that will remind us of
the sentence. For example, to symbolize:
Rob plays a mean harpsichord.
We might use the letter R.
Negat
Categorical Syllogisms
Syllogisms
A syllogism is an argument with 2 premises and a
conclusion.
As a first pass, a categorical syllogism is made up
of 3 categorical propositions, with 3 different
terms, each used twice in different propositions.
All men
Final Exam Study Guide
Disclaimer: This is a guide only and is not guaranteed to be complete.
To prepare for IT 103 final exam, go over the reading assignments, lab case studies, and
lecture notes.
Lecture 6: Systems Analysis
1. System development life cy