A. Logic: What is Logic? Why must
learn it? How important is it?
When we talk about logic, it’s about
the statements either valid or invalid, proving, etc. The word “logic” is derived
from Greek “logikos/logike” means “intellectual, dialectical, argumentative,
and possessed of reason.” One of the implementation of logic is computing
machinery which is fundamental to computer science. Learning about logic is
very important, for example for learning Geometry; it requires us to have
knowledge about logic for proving the theorems, by learning about logic helps
us to do reasoning in solving problem in our daily life.
B. Logic in Indonesia Secondary
Mathematics Curriculum
In Indonesia, one of the goals for learning Math is that the
students have the ability to deduce generalization and to prove and explain
ideas and statement of Mathematics, for achieving that goal we need the
material of logic to be learned. The Standard Competence in learning Logic in
Indonesia consists of four;
-
Understanding
of Mathematics statements and its negations
-
Determining
the truth value of compound or quantified statements
-
Formulating
statements equivalent to the statements given
-
Using
principles of logic related to compound and quantified statements to draw
conclusion or to solve problem.
C. Characteristics and Principles of
Logic
There are two branches of logic:
1)
Deductive logic is about reasoning processes
in which the claim being reasoned is supposed to follow with certainty from
evidence presented.
2)
Inductive logic is about reasoning processes
in which the claim being reasoned is only supposed to follow with likelihood
from evidence presented.
Also, we have two different ways of
doing logic:
1) Formal logic is translating in reasoning
process from English into a symbolic language.
2) Informal logic is no translating, we simply
examine and evaluate in ordinary speech.
After
that, we can conclude from the branches and the ways of doing logic into four
different types:
1) Formal Deductive Logic
2) Formal Inductive Logic
3) Informal Deductive Logic
4) Informal Inductive Logic
D. Logic in Secondary School
1)
Standard
Competence
As I mentioned in part B about the standard competence of
learning logic, to achieve all of them the students would be given materials
about statements and its derivation, compound statements, convers, invers,
contraposition, existential and universal quantification, syllogism, modus
ponens, modus tollens, and proving mathematical statements.
2)
Sentence
and Statement
A statement is a sentence which either true or false and not
both, so a statement must be a sentence, but not vice versa. For example, “Susi
Susanti is a female athlete of badminton,” is a true statement. “The root of 4
is 4,” is the false statement. “The sentence 2x+2 = 1” is called an open
sentence which we can’t decide its truth or falsity.
3)
Create
New Statement
We can create (a) new statement(s) from its negation,
disjunction, conjunction, converse, inverse, and contraposition. For example,
the negation of “3 + 4x > 5” is “3 + 4x < 5”. We can compound two
statements by using “or” and “and”. The statements which are compounded using
“or” is called “disjunction,” it uses symbol “v” in example “p or q” is
symbolized by “p v q”.
Meanwhile the statements which are compounded using “and” is
called “conjunction,” it uses symbol “^” in example “p and q” is symbolized by
“p ^ q”. Other types of statement are “implication” and “bi-implication”. We
use “->” for “implication” and it’s read “if…then,” for example “if p=2, then
2p=4” can be written as “p=2 -> 2p = 4”. For instance, we use “<->”
for “bi-implication” and it’s read “if and only if,” for example “y=4 if and
only if 3y=12” can be written as “y=4 <-> 3y=12”.
4) Argument and Deduction
An argument
consists of some promises and a conclusion. For example:
Example 1
Nisa
loves blue.
The
book is blue.
So,
Nisa loves the book.
The
argument looks valid, but it doesn’t mean Nisa loves every single blue thing,
so it’s inductive inference.
Example 2
All
animals will extinct.
Cat
is an animal.
So,
cat will extinct.
The
argument must be true for there does not exist any possibility other than “cat
will extinct,” so it’s kind of deductive inference.
E. Problems and Difficulties in
Learning Logic and Its Solution: Research Based
The problems
that usually happen are:
1)
Difficulty
Varies with Principle,
Content, and
Complexity
Generally,
students better on logic task about valid arguments (implication and
bi-implication) than invalid arguments (converse and inverse)
2)
Generalizing
from a small sample
Students
often draw a general conclusion from a single example or a few examples.
3)
Dismissing
a constraint
Many
students are baffled by problems involving multiple constraints and often
simply disregard one or more of the constraint.
4)
Adding
Unwarranted Constraint
Students
commonly make assumptions about problems that are misleading, and so reduce
their flexibility in solving problem.
5)
Confusing
if with only if
Most of
them misinterpret and wrong in distinguishing “if-then” and “if and only if”
6)
Logic
Instruction:
-
Children
need ample and regular opportunities to practice using reasoning skills and
making conjectures.
-
Encourage
educated guessing
-
Help
children understand the value of negative feedback in deducing and answer
-
Underscore
the importance of intuitive thought and inductive reasoning and why and how
they should be checked
7) Informally Evaluating Reasoning:
-
Encourage
children to use counterexamples to identify invalid conjectures or deductive
arguments
-
Using
Euler diagrams to evaluate deductive arguments.
