Formal Methods in Computer Science CS1502 Aristotelian Forms

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Formal Methods in Computer ScienceCS1502Aristot
elian Forms

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To understand how to relate two claims with a
  quantifier using Aristotelian forms.
• To understand the Aristotelian forms through sets.

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Why Aristotelian Forms?

• Natural ways to connect 2 claims using a
  quantifier
• A way to limit the scope of the domain of
  discourse.

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Convention

• When I say standard forms, I mean Aristotelian
  forms.

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Aristotelian forms

• All Ps are Qs
• Some Ps are Qs
• No Ps are Qs
• Some Ps are not Qs

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The Square is Even

• Domain set of all even numbers
• How to claim that the square of any number in the
  set is even?
• In English
• The square of any number (in the domain) is even.
• For any object x (in the domain), x2 is even.
• In FOL
• ?x Even(x2)
• Domain set of all integers
• How to make the same claim?
• In English
• In FOL
• Domain set of everything

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1st Aristotelian Form

• Everything is a Q.
• ?x Q(x)

• All Ps are Qs.
• ?x (P(x) ? Q(x))
• (1) this effectively restrict the domain to P.
• (2) the set of Ps ? the set of Qs

Q
Q
P
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1st Aristotelian Form

• All Ps are Qs.
• ?x (P(x) ? Q(x))
• There could be other things that are not Ps. We
  dont care about them.
• There could be other things that are Qs, but are
  not Ps. This is fine. We are only interested
  in those that are Ps.
• Why not ?x (P(x) /\ Q(x)) ?
• Everything is both a P and a Q.
• Why not ?x (P(x) \/ Q(x)) ?
• Everything is either a P or a Q (or both).
• Why not ?x P(x) /\ ?x Q(x) ?
• Everything is a P and everything is a Q.
• Why not ?x P(x) ? ?x Q(x) ?
• If everything is a P, then everything is a Q.

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Everything

• Everything is a Q.
• ?x Q(x)

• Everything is small
• ?x Small(x)

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1st Aristotelian Form

• All Ps are Qs.
• ?x (P(x) ? Q(x))

• All cubes are small
• ?x (Cube(x) ? Small(x))

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All cubes are small ???

• All cubes are small.
• ?x (Cube(x) ? Small(x)) True or False?

Vacuously True
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Vacuously True

• ?x (Cube(x) ? Small(x))
• For every object x, if x is a cube, it is
  small.
• For every object we pick,it satisfies Cube(x)
  ? Small(x).
• Thus, ?x (Cube(x) ? Small(x)) is true.
• When there is no object that satisfies the
  antecedent, this is called vacuously true.

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Domain is empty

• ?x Small(x) True or False?

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Vacuously True

• ?x (Cube(x) ? Small(x))
• For every object x, if x is a cube, it is
  small.
• ?x Small(x)
• For every object x, x is small.
• Put it in Aristotelian form.
• For every object x, if x is in the domain, x is
  small.

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Summary All Ps are Qs

• ?x Small(x) in a non-empty world
• ?x (Cube(x) ? Small(x)) in a world with cubes.
• ?x (Cube(x) ? Small(x)) in a world with no cubes.
• ?x Small(x) in an empty world.

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2nd Aristotelian Form

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are not Qs
• ?x (P(x) /\ Q(x))

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Some numbers are primes.

• Domain set of all odd integers
• Translate Some numbers are primes.
• Predicate Prime(x) means x is a prime.
• For some object x, x is a prime.
• ?x Prime(x)
• Domain set of all integers
• Suppose we are still interested in only odd
  numbers.
• How to make the same claim?
• Some odd numbers are primes.
• For some object x, x is odd and x is prime.
• ?x (Odd(x) /\ Prime(x))
• Domain set of everything
• ?x (Odd(x) /\ Prime(x))

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2nd Aristotelian Form

• Something is a Q.
• ?x Q(x)

• Some Ps are Qs.
• ?x (P(x) /\ Q(x))
• (1) A way to restrict the domain
• (2) set of Ps intersects set of Qs

Q
P
Q
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2nd Aristotelian Form

• Why not ?x (P(x) ? Q(x)) ?
• For some object x, if x is a P then it is a Q.
• Some cubes are small. True or False?
• ?x (Cube(x) /\ Small(x)) True or False?
• ?x (Cube(x) ? Small(x)) True or False?

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• Why not ?x (P(x) \/ Q(x)) ?
• Why not ?x P(x) /\ ?x Q(x) ?
• Why not ?x P(x) ? ?x Q(x) ?
• Can a sentence in 2nd Aristotelian form be
  vacuously true?

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?x Q(x)
?x (P(x) ? Q(x)) (1st form)
Q
Q
P
?x (P(x) /\ Q(x)) (2nd form)
?x Q(x)
P
Q
Q
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3rd Aristotelian Form

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are not Qs
• ?x (P(x) /\ Q(x))

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None

• Nothing is a Q.
• (Everything is not a Q.)
• ?x Q(x)
• ( ?x Q(x) )
• (It is not the case that there is an object x
  that is a Q.)

• Nothing is small.
• (Everything is not small.)
• ?x Small(x)
• ( ?x Small(x) )
• (It is not the case that there is a small
  object.)

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3rd Aristotelian Form

• No Ps are Qs.
• (All Ps are not Qs)
• ?x (P(x) ? Q(x))
• ( ?x (P(x) /\ Q(x)) )
• (Its not the case that there is an object that
  is both a P and a Q.)

• No cubes are small
• (All cubes are not small)
• ?x (Cube(x) ? Small(x))
• ( ?x (Cube(x) /\ Small(x)) )
• (Its not the case that there is a small cube.)

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3rd Aristotelian Form

• Nothing is a Q.
• ?x Q(x)

• No Ps are Qs.
• ?x (P(x) ? Q(x))
• (1) A way to restrict the domain of discourse
• (2) set of Ps and set of Qs do not intersect

P
Q
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?x (P(x) /\ Q(x)) (2nd form)
?x Q(x)
P
Q
Q
negation
?x (P(x) /\ Q(x)) (3nd form) ?x (P(x) ? Q(x))
?x Q(x) ?x Q(x)
P
Q
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4th Aristotelian Form

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are not Qs
• ?x (P(x) /\ Q(x))

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Not all

• Something is not Q
• (Not all things are Qs).
• ?x Q(x)
• ( ?x Q(x) )
• (It is not the case that everything is a Q.)

• Something is not small
• (Not all things are smalls)
• ?x Small(x)
• ( ?x Small(x) )
• (It is not the case that everything is small.)

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4th Aristotelian Form

• Some Ps are not Qs.
• (Not all Ps are Qs)
• ?x (P(x) /\ Q(x))
• ( ?x (P(x) ? Q(x)) )
• (It is not the case that all Ps are Qs.)

• Some cubes are not small
• (Not all cubes are small)
• ?x (Cube(x) /\ Small(x))
• ( ?x (Cube(x) ? Small(x)) )
• (It is not the case that all Cubes are small.)

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Aristotelian forms

• Something is not Q
• ?x Q(x)

• Some Ps are not Qs.
• ?x (P(x) /\ Q(x))
• (1) A way to restrict the domain of discourse
• (2) set of Ps is not a subset of Qs

P
Q
Q
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?x Q(x)
?x (P(x) ? Q(x)) (1st form)
Q
Q
P
negation
?x Q(x) ?x Q(x)
?x (P(x) /\ Q(x)) (4th form) ?x (P(x) ? Q(x))
P
Q
Q
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4 Aristotelian Forms

• All Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are Qs
• ?x (P(x) /\ Q(x))

• No Ps are Qs
• ?x (P(x) ? Q(x))
• Some Ps are not Qs
• ?x (P(x) /\ Q(x))

Use ? with ? Use /\ with ?