F22H1 Logic and Proof Week 5

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F22H1 Logic and Proof Week 5

Overview of weeks 59 Predicates and Quantifiers

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Overview of rest of module

• Contacting me dwcorne_at_macs.hw.ac.uk
• Monday lecture lecture
• Thursday/Friday practice
• We will cover much but not all of
• Nederpelt Kamareddine Chapters 8, 9, 11, 16,
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• Two lectures (not from book) on the topics of
  clausal form and resolution.

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What we will do this week

• Chapters 8 and 9, and the exercises therein.
• What you will learn about
• Predicates and Quantifiers predicate logic
• Translating english into predicate logic
• Equivalences in predicate logic
• Examinable material
• Sections 8.2, 8.3, 8.4, 9.19.9

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Propositions vs Predicates

• Let A stand for Bob is a student
• A is a proposition it is either True or False
  definitely one or the other.
• Now let S Anna, Bill, Callum, Dave, the
  set of students, and let P be the set of all
  people, and let x be a variable.
• x is a student is not a proposition. What is
  it?

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Propositions vs Predicates

• Instead of x is a student, we typically invent
  suitable function names and write predicates as
  functions. E.g. here are a few examples
• Which of these are predicates and which are
  propositions?
• student(x)
• cat(Tom)
• taller_than(Madonna, x)
• prime_minister_of(x, y)
• hates(Garfield, Mondays)

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Quantifiers

• Let S be the set of all students x is a
  student is a predicate we can decide to write
  it as x in S instead, or maybe student(x)
• Suppose P is the set of all people.
• With quantifiers, we can express other possible
  statements
• This is universal quantification
• ?x P(x) student(x) -- every person is a
  student
• ?x S(x) has_a_brain(x) -- every student
  has a brain

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Quantifiers

• ?x P(x) student(x) there exists a member
  of P who is a student --- i.e. at least one
  person is a student i.e. there is a student
• ?x P(x) student(x) AND ?attends_lectures(x)
• ?x S(x) ?attends_lectures(x)
• ??x S(x) attends_lectures(x)

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Exercises
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Many-place predicates
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This is what chapter 9 is all about
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This means, for example
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All this means is Every man or woman has an
ID card is equivalent to Every man has an
ID card AND Every woman has an ID
card There is a man or a woman at the top of
the mountain is equivalent to There is a man
at the top of the mountain OR there is a woman
at the top of the mountain
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This just means Every person who is Tom lives
in Edinburgh is equivalent to Tom lives in
Edinburgh And There is someone who is Tom who
lives in Edinburgh is equivalent to Tom lives
in Edinburgh Obviously, the quantification
isnt needed in such cases all the rule does is
remove it.
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These seem rather daft. But the expressions on
the LHS might arise in the middle of trying to
prove something, and so the rule helps
to automatically simplify it. E.g. we may
have The domain is equivalent to False (why?),
so the whole thing True Dont worry about it
logic provides a language that enables us to
express anything at all in a formal way that
computers can manipulate But logic is powerful
enough to also express crazy things like this
that we would never say, however they may arise
during calculation.
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Proving the first of the Empty domain rules
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Domain Weakening
We can move the domain statements to the right,
and make them part of the proposition.
This is obvious when you think of it.
Means For all x in D, A(x)
Means For all, if x is in D then A(x)
Logically, there is no difference
There is an x in D for which A(x) is
equivalent to There is an x in D and A(x)
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These observations lead to the following rules
E.g. Every student who attends lectures and
works hard will pass is equivalent to For
every student who attends lectures, if they work
hard theyll pass There is a closed passenger
door with a red light flashing is equivalent
to There is a closed door, which is a
passenger door and has a red light
flashing but not equivalent to There is a
closed door, and if it is a passenger door then
it has a red light flashing
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It is NOT true that every student eats cheese
is equivalent to There exists a student who
does not eat cheese There does NOT exist a
camel that can dance the samba is equivalent
to Every camel cannot dance the samba
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Some practice with logical calculation
Showing
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Term Splitting
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Some exercises
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