Rules of Inference

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Rules of Inference
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Mathematical Reasoning

• We need mathematical reasoning to
• determine whether a mathematical argument is
  correct or incorrect and
• construct mathematical arguments.
• Mathematical reasoning is not only important for
  conducting proofs and program verification, but
  also for normal behavior, rational thinking, and
  being able to demonstrate common sense.
• Most importantly its important for detecting
  JUNK SCIENCE.

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Some Terminology
Some terminology to be used in the next two
classes Theorem, Proposition, Claim, Fact,
Result statement that can be proved
you can show that its true. Axioms,
postulates the basic assumptions on which the
proof us based. Lemma Intermediate result to be
proved on your way to proove a theorem.
A simple theorem used in the proof of another
theorem. Corollary Result that is directly
follows from a theorem you just proved. A
proposition that can be established from a
theorem that has been proved Conjecture A
Result you think is true, but cannot prove. A
statement whose truth value is
unknown. When a proof of a conjecture is found,
the conjecture becomes a
theorem Proof - A correct mathematical argument.
We use rules of inference to prove
theorems. By using them wrong, we create
fallacious proofs.
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Whats Wrong With This Logic?
People make mistakesMistakes are wrongJim is a
personJim is wrong
Humans are mammals.Dogs are mammals. Humans are
dogs.Then how am I typing this?
1. Theists define God as all-powerful. 2.
Therefore, God can lift any size rock. 3.
Therefore, there can be no rock too big for God
to lift.4. Therefore, God cannot create a rock
too big for him to lift.5. Therefore, there is
something God cannot do.6. Therefore, God is not
all powerful.7. Therefore God does not exist
(per the definition given by its believers).
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Some Fallacious Proofs
Whats wrong with this?
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Anything Wrong?
Premise 1 If Portland is the capital of Maine,
then it is in Maine. Premise 2 Portland is in
Maine. Conclusion Portland is the capital of
Maine.
Application of the death penalty is killing a
human being. Killing a human being is wrong.
Therefore, application of the death penalty is
wrong.
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Rules of Inference

• Rules of inference provide the justification of
  the steps used in a proof.
• One important rule is called modus ponens or the
  law of detachment. It is based on the tautology
  (p?(p?q)) ? q. We write it in the following way
• p
• p ? q
• ____
• ? q

The two hypotheses p and p ? q are written in a
column, and the conclusionbelow a bar, where ?
means therefore.
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Rules of Inference

• The general form of a rule of inference is
• p1
• p2
• .
• .
• .
• pn
• ____
• ? q

The rule states that if p1 and p2 and and pn
are all true, then q is true as well. These
rules of inference can be used in any
mathematical argument and do not require any
proof.
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Arguments

• Just like a rule of inference, an argument
  consists of one or more hypotheses and a
  conclusion.
• We say that an argument is valid, if whenever all
  its hypotheses are true, its conclusion is also
  true.
• However, if any hypothesis is false, even a valid
  argument can lead to an incorrect conclusion.

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Rules of Inference(modus ponens law of
detachment)
always true its a tautology
Conclusion if the premises p and p?q are both
true, then q can only be true. However, if the
premises do not hold, q can still be true or
false.
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Rules of Inference
simplification
modus tollens
conjunction
addition
disjunctive syllogism
resolution
hypothetical syllogism
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Rules of Inference
Examples It snows today If it snows today we go
skiing Therefore we go skiing
If it rains we do not have a barbeque today If we
dont have a barbeque today, well have one
tomorrow Therefore If it rains today, well have
a BBQ tomorrow.
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Valid arguments.
All inference rules were of the form premise 1
is true, premise 2 is true, therefore conclusion
is true. In general this looks like
For an argument to be true all the premises must
be true.
Example if n gt 1 then n2 gt 1 (True) We cannot
conclude (½)2 gt 1 because the premise is not
true.
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Fallacies (Part Deux)
If you do every problem in this book then youll
learn discrete math. Joe did not do every problem
in the book, therefore he did not learn discrete
math.
p you do all problems in the book. q
you learned discrete math.
correct
wrong
fallacy of denying hypothesis
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More Fallacies
If you do every problem in this book then youll
learn discrete math. Joe learned discrete
math, therefore he did every problem in this
book....
p you do all problems in the book. q
do learned discrete math.
correct
wrong
fallacy of affirming conclusion
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Inference for Quantified Statements
universal instantiation
universal generalization
existential generalization
existential instantiation
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Quantified Statements
Example Everyone in this math class has taken a
CS course Marla is in this class Therefore Marla
has taken a course in CS D(x) x has taken a
math class C(x) x has taken a CS
class. premises
conclusion
Reasoning
universal instantiation
modus ponens
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Rules of Inference

• Universal Instantiation - ?xP(x) ? P(c) if c ? U
• Universal Generalization - P(c) for an arbitrary
  c ? U ? ?xP(x)
• Existential Instantiation - ?xP(x) ? P(c) for
  some element c ? U
• Existential Generalization - P(c) for some
  element c ? U ? ?xP(x)

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Homework

• P. 72
• 1, 3, 5, 9abc, 15, 19