F22H1 Logic and Proof Week 6

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F22H1 Logic and Proof Week 6

Reasoning
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How can we show that this is a tautology (section
11.2)

The hard way logical calculation The easy
way reasoning Thats what chapter 11 is all
about Logical Calculation use well defined
rules one by one and you will slowly get towards
the thing that you are trying to
prove. Reasoning (logical calculation, but with
a few hints, tips and additional rules) is more
like common sense.
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Of course, we can do a truth table to show that
this is always T however you cant do that
with, e.g., 100 logical variables. So we need to
learn reasoning.

We start with this.
Why?? Because Lemma 7.3.4. Says this
So, if our calculation eventually gets to P gt
R, using steps that are either or
then we will have proved what we want to prove
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• This is the next step

Understand? Next
Tricky! But straightforward
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• Next this

Understand it? Also known as False-elimination.
Next, we can go a long way with rules from chap 7
Actually this only uses AND-weakening
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• The remainder is fairly quick and easy

So from Lemma 7.3.4. We have now proved the
tautology we wanted to prove.
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Koyaanasqatsi
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A brand new dawn
Lets just assume, for the time being, that the
whole first bit is true
Assumption (1)
Now lets make another assumption

• Assumption
• P
• from (1) and (2)
• Q
• from (1) and (3)
• R
• we assumed P (2) and derived R (4), this means
• 5) P gt R

We assumed (1), and got (5), this means 6)
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Inference
This is the general rule for inference
You can be creative about when to make an
inference and what to base it on, but of course
it has to be valid.
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Some valid inference rules

• Tautology
• Implication and double negation
• (2)
• Follows from (1) and (2) called
  
• (3)

• Assume
• follows from (1) --
  
• (m)
• follows from (1) and (m) if there are no
  assumptions between (1)
• and (m) that the derivation of (m) relies on --
• (m1)

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Reasoning by Contradiction
This is an extremely useful inference rule, very
often used, often called reductio ad absurdum
The idea is assume the negation of what you want
to prove If reasoning then leads to a
contradiction (i.e. leads to False), then you can
infer (i.e. you have proved) the negation of your
assumption.

• Assume
• via valid inferences but no other assumptions
  we get
• (m) False
• it follows that the original assumption must be
  False, so
• we can infer the following
• (m1)

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• E.g. Lets prove that 10 is an even number.
• To do this by Contradiction, we first assume that
  
• 10 is even is False
• Assumption
• 10 is an odd number
• this follows from (1)
• (2) 10 2k 1 for some k in N
• this follows from (2), by simple algebra
• (3) 9 2k for some k in N
• simple algebra, follows from (3)
• (4) k 4.5 and k in N
• This follows from (4), since it is a
  contradiction
• (5) False
• We have now proved this, via proof by
  contradiction
• (6) 10 is an even number

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Prove that this
Implies
It stands to reason, when you look at it. Lets
prove it by contradiction

• Assume
• Implication
• De Morgan, double negation, and associativity
• follows from (3) by AND-elimination, twice
• (4)
• we can see that (4) is a contradiction, so we
  have proved this
• (5)

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