GEC104: Mathematics in the
Modern World
LOGIC AND REASONING
Source: DMS GEC104 LECTURE NOTES 2018
Ma. Cristina Duyaguit
Department of Mathematics and Statistics, CSM
Logic and Reasoning
MATHEMATICAL LOGIC
➢
study of valid reasoning
➢
deals with the methods of reasoning; provides rules and
techniques to determine whether an argument is valid
Math and Stat Department, CSM
1
Logic and Reasoning
Proposition (Statement)
➢
Ex.
a (complete) declarative sentence that has a truth value which
is either TRUE (denoted by T) or FALSE (denoted by F) (but
not both)
1. Today is Wednesday.
2. 12 < 7
3. Please lend me your ears.
(a statement)
(a statement)
(not a statement)
Math and Stat Department, CSM
2
Logic and Reasoning
● Negation of a Proposition
Ex.
➢
a proposition obtained by taking the negation of a given
proposition
➢
the “negation connective” uses the symbol ~ or  (to mean
not).
p : I like Pepsi.
𝑝:
I do not like Pepsi.
q : I like Coke.
𝑞:
I don’t like Coke.
Math and Stat Department, CSM
3
Logic and Reasoning
Argument (Mathematical Reasoning)
➢
a series of statements made in support of an assertion together
with the assertion drawn from these supporting statements
● Parts of an Argument
 premises – supporting statements
 conclusion – assertion drawn from these premises
Math and Stat Department, CSM
4
Logic and Reasoning
● The premises of an argument are offered in support of an argument.
➢
supposed to be reasons to believe the conclusion
● Argument can be classified according to how strong the relation of
support between the premises and the conclusion.
Math and Stat Department, CSM
5
Logic and Reasoning
● Types of Argument (Mathematical Reasoning)
➢
Deductive Reasoning
➢
Inductive Reasoning
● Main Differences
➢
Structure
➢
Strength
Math and Stat Department, CSM
6
Logic and Reasoning
● Structure
❖ Deductive Reasoning
➢ moving from general premises to specific conclusion
Ex. 1. All dogs are animals,
so my dog is an animal.
2.
All men are mortal.
I am a man.
Therefore, I am mortal.
Math and Stat Department, CSM
7
Logic and Reasoning
● Structure
❖
Inductive Reasoning
➢ making a general conclusion based on specific
premises
Ex.
Every object that I release from my hand falls to the ground.
Therefore, the next object I release from my hand will fall to
the ground.
Math and Stat Department, CSM
8
Logic and Reasoning
● Strength
❖ Inductive Reasoning
➢ If the premise is true, conclusion is MORE LIKELY to be
true (but could be false).
Ex.
Suppose you wanted to prove that all swans are white. How would
you do it?
Math and Stat Department, CSM
9
Logic and Reasoning
● Using inductive argument:
❖ I have seen a lot of swans and they were all white.
( OR may say)
❖ Every swan I have seen so far has been white, so the next
swan I see will be white.

Question: Are all swans really white?
➢ Answer: No! (Black swans exist!)
Math and Stat Department, CSM
10
Logic and Reasoning
•
Validity of an Argument
❖ A deductive argument is valid if the truth of its premises
guarantees the truth of the conclusion, i.e., if the premises are
true, then the conclusion MUST BE true.
➢ This is a strong argument.
❖ Invalid argument – an argument which is not valid
Math and Stat Department, CSM
11
Logic and Reasoning
Remarks:
1. The premises of a valid argument need not have to be true. In fact,
a valid argument may have a false conclusion.
Ex.
Romy likes either coffee or beer.
Romy does not like coffee.
Therefore, Romy likes beer.
2. Inductive argument is never valid!
Math and Stat Department, CSM
12
Logic and Reasoning
● Soundness of an Argument
❖ An argument is sound if it is valid and has premises which are
true.

A MUST-HAVE GOAL:
❖ When we construct an argument, we must aim to construct one
that is not only valid, but sound.
Math and Stat Department, CSM
13
Logic and Reasoning
Remarks:
1. Only valid arguments can be sound or unsound.
2. An invalid argument is never sound!
Math and Stat Department, CSM
14
Logic and Reasoning
● Test of Validity
❖ To determine if an argument is valid, we need to know
1.
the form of the argument and
2.
whether it is possible for that form to have true
premises and a false conclusion.
Math and Stat Department, CSM
15
Logic and Reasoning
● Counter-example to an Argument
❖ Form
➢
an argument of the same form that has true premises
and a false conclusion
 True Premise
 True Premise
 False Conclusion
 If the argument form is valid, it is impossible to find a counter-example.
Math and Stat Department, CSM
16
Logic and Reasoning
Consider the ff. argument:
 All men are mortal.
 Socrates is not a man.
 Therefore, Socrates is not mortal.
● Question: Is this argument valid?
● Hint:
One may use the counter-example to an argument to show
that the given argument is invalid.
Math and Stat Department, CSM
17
Logic and Reasoning
● Euler Diagram
➢ may be used to visually evaluate the validity of a deductive
argument
➢ essentially the same thing as a Venn diagram
Math and Stat Department, CSM
18
Logic and Reasoning
● Analyzing an Argument with the Euler Diagram
 Draw an Euler diagram based on the premises of the
argument.
 The argument is invalid if there is a way to draw the diagram
that makes the conclusion false.
Math and Stat Department, CSM
19
Logic and Reasoning
 The argument is valid if the diagram cannot be drawn to make
the conclusion false.
 If the premises are insufficient to determine the location of an
element or a set mentioned in the conclusion, then the
argument is invalid.
Math and Stat Department, CSM
20
Logic and Reasoning
● Consider the deductive argument
“All cats are mammals and a tiger is a cat, so a tiger is a
mammal.” Is this argument valid?
Math and Stat Department, CSM
21
Logic and Reasoning
● Premise:
All firefighters know CPR.
● Premise:
Jill knows CPR.
● Conclusion: Jill is a firefighter.
Math and Stat Department, CSM
22
Logic and Reasoning
● Evaluating Deductive Arguments with Truth Tables
❖
Arguments can also be analyzed using truth tables, although
this can be a lot of work.
➢ Steps:
1.
Represent each of the premises symbolically.
Math and Stat Department, CSM
23
Logic and Reasoning
2. Create a conditional statement, joining all the premises to
form the antecedent, and using the conclusion as the
consequent.
3. Create a truth table for the statement. If it is always true, then
the argument is valid.
Math and Stat Department, CSM
24
Logic and Reasoning
Basic truth tables: Let 𝑝 and 𝑞 be propositions (statements)
Conjunction
Disjunction
Conditional
Biconditional
𝑝
𝑞
𝑝∧𝒒
𝑝
𝑞
𝑝∨𝒒
𝑝
𝑞
𝑝→𝒒
𝑝
𝑞
𝑝↔𝒒
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
T
F
T
T
F
F
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
F
F
F
F
F
F
F
T
F
F
T
Negation
𝑝
¬𝑝
T
F
F
T
Math and Stat Department, CSM
25
Logic and Reasoning
● Consider the following argument:
Premise: If you bought bread, then you went to the store.
Premise: You bought bread.
Conclusion: You went to the store.

Let b represent “you bought bread”
and s represent “you went to the store”.
Math and Stat Department, CSM
26
Logic and Reasoning
Then the argument becomes:
Premise:
b→s
Premise:
b
Conclusion: s
 To test the validity, we determine if it is true that [(b → s) ⋀ b] → s.
Math and Stat Department, CSM
27
Logic and Reasoning
𝒃
𝒔
𝒃→𝒔
𝒃
𝒔
𝒃→𝒔
(𝒃 → 𝒔) ∧ 𝒃
T
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
T
F
T
T
F
F
F
T
F
F
T
F
𝒃
𝒔
𝒃→𝒔
(𝒃 → 𝒔) ∧ 𝒃
[(𝒃 → 𝒔) ∧ 𝒃] → 𝒔
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
Math and Stat Department, CSM
28
Logic and Reasoning
● Common Forms of Valid Arguments
➢ Determining the validity of an argument may be made easier if
the argument falls in one of the common forms of arguments
that are valid (or invalid).
Math and Stat Department, CSM
29
Logic and Reasoning
❖ The Law of Detachment (Modus Ponens- “mode that affirms”)
➢ The law of detachment applies when a conditional and its
antecedent are given as premises, and the consequent is the
conclusion.
➢ The general form is:
Premise:
p→q
Premise:
p
Conclusion: q
Math and Stat Department, CSM
30
Logic and Reasoning
❖ The Law of Contraposition (Modus Tollens- “mode that denies”)
➢ The law of contraposition applies when a conditional and the
negation of its consequent are given as premises, and the
negation of its antecedent is the conclusion.
➢ The general form is:
Premise:
p→q
Premise:
~q
Conclusion: ~p
Math and Stat Department, CSM
31
Logic and Reasoning
● Some Invalid Arguments (Fallacies)
❖ The Fallacy of the Converse
➢ arises when a conditional and its consequent are given as
premises, and the antecedent is the conclusion.
➢ The general form is:
Premise:
p→q
Premise:
q
Conclusion: p
Math and Stat Department, CSM
32
Logic and Reasoning
Example:
Premise: If a teacher is a PhD holder, then he/she is a trainer.
Premise: Teacher Maria is a trainer.
Conclusion: Maria is a PhD holder.
Math and Stat Department, CSM
33
Logic and Reasoning
❖ The Fallacy of the Inverse
➢ occurs when a conditional and the negation of its antecedent
are given as premises, and the negation of the consequent is
the conclusion.
➢ The general form is:
Premise:
p→q
Premise:
~p
Conclusion: ~q
Math and Stat Department, CSM
34
Logic and Reasoning
Example:
Premise: If a teacher is a PhD holder, then he or she is a trainer.
Premise: Teacher Maria is not a PhD holder.
Conclusion: Maria is not a trainer.
Math and Stat Department, CSM
35
Logic and Reasoning
● What type of argument does a mathematician use?
➢ mathematician uses inductive reasoning to make some
guesses or conjectures
➢ use deductive reasoning to prove these conjectures
 A statement that can be shown or proved to be true is called a
theorem.
Math and Stat Department, CSM
36
End Page
Source: DMS GEC104 LECTURE NOTES 2018
Click to edit subtitle style