MGF 1107
Mathematics of Social Choice
Part 1a – Introduction, Deductive
and Inductive Reasoning
Topics Covered
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Voting
Weighted voting systems
Apportionment
Game Theory
Fair division
• Textbook: For All Practical Purposes, by
COMAP (the Consortium for Mathematics and
its Applications), 7th edition
Voting Topics
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Deductive and inductive reasoning
Voting criteria
Voting methods
Which methods satisfy and fail which
criteria?
• Fundamental theorems of voting theory
Deductive and Inductive Reasoning
• Deductive reasoning is the process of
making specific conclusions based on
general principles.
• Inductive reasoning is the process of
making general conclusions based on
specific examples.
Examples of Deductive Reasoning
The following are examples of deductive reasoning…
1) If
6x = 18
then x = 3
2)
Here given 6x = 18 we can make the deductive
conclusion that x = 3 because we can use the
general principle that dividing both sides of an
equation by the same number (namely 6) will
yield an equivalent equation from which we
can identify the solution.
The measure of angle x is 30 degrees.
x
60◦
Here we can make the deductive conclusion
that the unknown angle will measure 30
degrees because we can use the general
principle that the sum of the angles of any
triangle is always 180 degrees.
Examples of Deductive Reasoning
More examples of deductive reasoning…
3) All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Socrates
men
mortal things
This argument is an example
of valid deductive reasoning.
The conclusion is valid
because it is based on a
fundamental principle
regarding a basic form of
deductive logic.
Examples of Deductive Reasoning
More examples of deductive reasoning…
4) If a fair division method is envy-free then it is proportional.
The Selfridge-Conway fair division method is envy-free.
Therefore the Selfridge-Conway fair division method is
proportional.
Selfridge-Conway
method
methods that
are envy-free
methods that are proportional
Again, this is a
valid
deduction.
The validity of
the deduction
is based on
the same
general
principle of
logical
deduction as
the previous
example.
Math = Deductive Reasoning
• Mathematics is essentially deductive reasoning applied to relations
among patterns, structures, forms, shapes, and change.
• Most people think of mathematics as only about numbers. But we are
doing mathematics in this course when we apply general principles to
make deductive conclusions. This course is entitled Social Choice
Mathematics because we use deductive reasoning in the study of
topics that involve decisions among groups of people – from voting to
strategies in games.
• Deductive reasoning is difficult! For most people it does not come
naturally. To make correct deductions requires a complete
understanding of the relevant abstract principles.
• Deductive reasoning is always valid. However, errors can occur either
by starting with incorrect assumptions or by incorrectly applying the
relevant principles.
Math = Deductive Reasoning
• We use deductive reasoning basically
because we learn some rules, or some
formula, then, when asked a question, we
apply the rules or formula and get a specific
answer to the question.
• That’s why this is mathematics. You learn a
rule, a formula, learn to apply the principles
correctly, and get the right answer.
Examples of Inductive Reasoning
The following are examples of inductive reasoning…
1)
2)
3)
4)
5)
Every swan I have ever seen is white. Therefore all swans are
white.
The teacher used powerpoint in the first few classes. Therefore,
the teacher will use powerpoint tomorrow.
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.
Every fall there have been hurricanes in the tropics. Therefore,
there will be hurricanes in the tropics next fall.
Based on the most accurate modern observations, the Earth has
rotated around the Sun in an elliptical curve for millions of years.
Therefore, the Earth will continue to rotate around the sun in an
elliptical curve next year.
Science = Inductive Reasoning
•
Science is essentially the application of inductive reasoning to form knowledge
based on evidence observed in the natural world.
•
Every scientific theory must explain all observable evidence and must make
falsifiable predictions. A falsifiable prediction is a prediction that can be proven
false by future evidence.
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Every statement in science is theory. For example, even the statement that the
Earth orbits around the sun is a theory based on evidence and the statement
that germs cause disease is a theory based on evidence. Every scientific
theory must be supported by evidence, must be falsifiable and could be proven
wrong but future evidence.
•
Inductive reasoning is not logically valid in the same way that deductive
reasoning is valid. In other words, even if a statement of inductive reasoning
(or science) is ultimately true the only way to “prove” that it is true is to collect
more evidence to support the statement. Nevertheless, regardless of the
strength of the evidence for a particular inductive conclusion, it will always
remain possible that future evidence could prove the conclusion false.
Side-Tracked on Scientific Theory
• The science taught in college is based on very well established
evidence and is not likely to be altered by future evidence. The
evidence is so well established that it is considered “fact”.
• The theory of gravitation and the theory of germs, the theory of light,
atomic theory, for example, the fundamentals of these theories are no
longer questioned.
• Of course evolution is also a scientific theory. Some people refer to it
as “only a theory” not understanding that, of course, everything in
science is theory, and, to be good science that theory must be
supported by lots of evidence and make falsifiable predictions.
Elements of a Deductive System
• Undefined terms
• Defined terms – formally defined terms are
made using undefined terms.
• Axioms – fundamental assumptions that
are self-evident and need not be proven.
• Theorems – statements of fact proven to
be true based on deductive logic.
An Example from Geometry
We define vertical angles to be the nonadjacent angles formed by
intersecting lines.
<A and <B are vertical angles.
A
B
C
In this example, the term “vertical angles” is a formally defined term.
However, “nonadjacent angles” and “intersecting lines” are
undefined terms.
The statement that the sum of the measures of angles A and C is 180 is
based on the axiom that a line forms a straight angle measuring 180
degrees. The same axiom implies that the sum of angles B and C is
180 degrees.
An Example from Geometry
Based on the relevant definitions and axioms – that is, based on the relevant
abstract principles we can make the following deductions…
1) m<A + m<C = 180 and
m<C + m<B = 180
( here m<A means measure of angle A )
2) Therefore m<A + m<C = m<C + m<B
A
B
C
3) Therefore m<A = m<B
4) Therefore vertical angles have the same measure.
Now we have an deductively proven a new fact – an example of a theorem:
Vertical angles have the same measure.
Social Choice Math
• When we study the mathematics of social choice, we will
learn …
– Undefined terms
– Defined terms
– Axioms
– Theorems
• And we will make deductive conclusions based on
relevant general principles.