Deductive Logic
Mathematics has traditionally been viewed as a study of logical systems―a set of axioms and
rules for creating concepts or theorems from the axioms― in which truth is transmitted from
premises to conclusions. For any particular system, such as geometry, certain fundamental but
unproven assumptions (called axioms or postulates) are accepted as absolutely true and rules of
logic are used to deduce new concepts or theorems. These new concepts add to the accepted
knowledge and can be used to further expand the store of mathematical knowledge in a particular
field.
axioms
(unproven
assumptions)
and/or
concepts/theorems
(previously proven
from axioms)

new concepts/theorems
(new proof)
For example, more than two thousand years ago, Euclid used known properties (axioms and
proven postulates) of lines, angles and shapes to prove the Pythagorean theorem―in a rightangle triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
This kind of thinking from accepted truths to specific new concepts is one example of reasoning
by deduction, called deductive logic. We can translate this deductive logic into a general form,
with a simple example, as follows:
General form
If p then q*
Example
If a number if divisible by 2, then the number is an even
number.
p
6 is divisible by 2
therefore q
therefore 6 is an even number
* Where p represents the hypothesis and q represents the conclusion.
Notice that the conclusion or product of this reasoning must be no less certain than the starting
hypothesis; that is, if the hypothesis is true then the conclusion must be true. In mathematical
proofs, the hypotheses are the axioms (assumed to be true) or previously proven concepts or
theorems. This means that mathematics can include absolute proofs and claim to discover new
truths based on previous or accepted truths.
Proof by Contradiction
The method described above is sometimes referred to as a direct proof. However,
mathematicians very often use deductive reasoning in a proof by contradiction (also known as
“reductio ad absurdum”―Latin, meaning reduction to the absurd). In this method, we assume
the opposite of what we are trying to prove and then derive a contradiction or error. This then
shows that the assumption is false and therefore the original hypothesis is true.
A well-known example of proof by contradiction is the proof that 2 is irrational―cannot be
expressed as a fraction a/b, where a and b are non-zero integers with no common factor; that is,
as a simple fraction. The proof by contradiction is outlined below.
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(1) If
2 is rational (assuming the opposite of the premise), then 2 
a
.
b
(2) This conclusion can be shown to be incorrect.
a
2  , b 2  a
b
2
2b  a 2 ,  a 2 is even and a is an even integer
let a  2c, where c is also an integer
 
2
 2b2  2c and simplifying to b2  2c 2
 b2 is even and b is an even integer
If both a and b are even integers, then they share a common factor of 2.
This is a contradiction for a rational number.
(3) Therefore,
2 is irrational.
In general symbols, proof by contradiction follows the outline:
General form
Notes
If –p then q*
assume the opposite of p, denoted by –p
–q
the conclusion q is shown to be false, denoted by –q
therefore p
the assumed opposite hypothesis, –p, is false because the
conclusion is shown to be false; therefore p is true
* Where p represents the hypothesis and q represents the conclusion.
A proof by contradiction is considered to be as rigorous (logically valid) as a direct proof.
Some Pitfalls of Deductive Logic


If one or more of the starting premises is not true, then the conclusion cannot be true even
if a valid argument is used.
If there is an error in logic (an invalid argument), then no reliable conclusion can be made
regardless of whether the premises are true.
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