FOL Practice

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FOL Practice
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Models

• A model for FOL requires 3 things
• A set of things in the world called the UD
• A list of constants
• A list of predicates, relations, or functions

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The Marble World

• UD Ashley, Clarence, Rhoda, Terry, and their
  marbles
• a Ashley
• c Clarence
• r Rhoda
• t Terry

• B(x) x is blue
• G(x) x is green
• R(x) x is red
• S(x) x is a shooter
• C(x) x is a cats-eye
• T(x) x is a steely
• M(x) x is a marble
• B(x,y) x belongs to y
• W(x,y) x wins y
• G(x, y, z) x gives y to z

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Some Examples

• All the cats-eyes belong to Rhoda.
• All the marbles but the shooters are cats eyes.
• Some, but not all, of the cats-eyes are green.
• All of the shooters that are steelies belong to
  Terry.

• (?x)(C(x) ? B(x,r))
• (?x)(M(x) S(x)) ? C(x)
• (?x)(C(x) G(x)) (?y)(C(y) ? G(y))
• (?x)(S(x) T(x)) ? B(x,t)

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WFFs and Truth in FOL

• Before we can decide if a sentence in FOL is
  true, we need to make sure it is well-formed
• Once this is determined, we can test if a
  sentence is true in the model.
• Lets look at a less complicated model to
  practice this

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The Positive Integer Model

• UD positive integers
• a 1
• b 2
• E(x) x is even
• O(x) x is odd
• L(x,y) x is larger than y

• Eb (Ob ? Lba)
• WFF?
• True?
• Test for truth by looking at the truth of each
  sub-sentence
• Eb 2 is even (TRUE)
• Ob 2 is odd (FALSE)
• Lba 2 is larger than 1 (TRUE)
• T (F ? T) gt T T gtT

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More Truth

• When quantifiers are involved, it is more
  complicated
• ?x means No matter what you choose for x
• UD Giant Bag o Stuff
• A universal statement has to be true for every
  item in the UD
• ?x means There is at least one x such that
• An existential statement is true as long as it is
  true for at least one thing in the UD

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Practice

• Every positive integer is either odd or even and
  no positive integer is both.
• Paraphrase No matter what x you choose, either x
  is even or x is odd and it is not the case that
  there is a y such that both y is even and y is
  odd.
• Symbolization ?x(Ex ? Ox) ?y(Ey Oy)

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?x(Ex ? Ox) ?y(Ey Oy)

• Is this true in the model?
• Remember that -statements are true if both
  conjuncts are true.
• Is ?x(Ex ? Ox) true?
• This statement is false if we can find just one
  item from the UD that makes it false, which we
  cant! Its TRUE.
• Is ?y(Ey Oy) true?
• This statement is true if ?y(Ey Oy) is false.
  Is there something in the UD that is both even
  and odd? There isnt, so ?y(Ey Oy) is false
  which makes ?y(Ey Oy) TRUE.
• Therefore, the entire statement is true in the
  model.

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Creating Dummy Models

• So far, we have looked at whether a sentence is
  true in a given model.
• Models dont have to be complicated things
  dealing with real properties of real world
  things.
• We can create a dummy model using a diagram to
  help predict if a FOL sentence or argument has
  particular properties.

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What We Can Do with Models

• True/False on a model
• We can give a model that makes a sentence
  true/false
• True/False on all models
• We can show that a sentence is true/false on all
  models or if it is indeterminate
• Consistency
• We can check if a set of sentences is consistent
  by giving a model where they are all true
• Validity
• We can check if an argument is invalid by giving
  a model where the premises are true but the
  conclusion is false

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Strategy for Model Creation

• Represent constants with a dot
• Represent properties with a circle
• Anything that falls in the circle has that
  property anything not in the circle doesnt
• Represent relations with an arrow
• Getting a model from the picture
• Put any dots in the UD
• List the dots as constants
• For each predicate, give a set that lists the
  dots in the predicates circle
• For each relation, give a set of ordered pairs

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Example 1

• Give a model that makes the sentence
• Nad -gt Nda true.

• Give a model that makes the sentence
• Iap -gt (Ipa -gt Iaa) false.

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Example 2

• Show that (Ga ?xBx) ? ?yBy is not true for
  every model
• Paraphrase If both a is in G and there is
  something in B, then everything is in B.
• UD a, b
• G(x) a
• B(x) b
• a a
• b b

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Example 3

• Show that ?xGx ?y?z(Fyz ? Gz) is not true for
  every model
• Paraphrase There is something in G and
  everything that is pointed to with F is in G.
• UD a, b
• a a
• b b
• G(x) a
• F(x,y) lta,bgt, ltb,agt

G
b
a
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What good are dummy models?

• Dummy models are quick way to discover a property
  of a sentence
• Once we know that property, it will still hold no
  matter what the model is
• So, if we can show that a sentence is true on at
  least one model, we know we can come up with a
  real world model that it is also true on

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Back to Example 2

• We know that (Ga ?xBx) ? ?yBy is not true for
  every model.
• We can use the dummy model as a template for a
  real world model
• UD a, b UD Homer, Otis
• Gx a Gx x is yellow
• Bx b Bx x is black
• a a a Homer
• b b b Otis

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Arguments

• An argument consists of a set of premises and a
  conclusion
• An argument is valid if and only if it is not
  possible to give a model that makes the premises
  all true and the conclusion false
• So, to show that an argument is invalid, give a
  model where the premises are true and the
  conclusion is false
• An invalid argument is always invalid! It may be
  possible to give a model that makes it seem like
  a good argument (politicians do this all the
  time!), but you can use the dummy model method to
  easily figure out if the argument is really any
  good.

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Example 4

• Show that the following argument is invalid
• ?x(Fx ? Gx) Everything in F is in G
• ?xFx Nothing is in F
• ??xGx Nothing is in G
• UD a, b UD Homer, Shai (the very furry
  bunny)
• Fx Fx x is a cat
• Gx b Gx x is furry
• a a a Homer
• b b b Shai

F
G
a
b
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Thats it!

• As you are working on the argument you give in
  your paper, you may want to think about how it
  might look in FOL
• Is your argument valid???
• Remember Paper outlines are due on Friday by the
  end of the day (e-mail before 1159 pm is OK)
• No reading for this week!