PART TWO

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PART TWO

• PREDICATE LOGIC

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Chapter Seven

• Predicate Logic Symbolization

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• Predicate Logic, or Quantifier Logic, is
  concerned with the interior structure of both
  atomic and compound sentences.

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1. Individuals and Properties

• In atomic sentences a property is ascribed to
  some individual.
• For example, Art is happy has the property of
  being happy ascribed to Art.
• We can symbolize this as Ha, with the uppercase
  letter denoting the property and the lowercase
  letter denoting the individual.

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Individuals and Properties, continued

• Properties that hold between two or more entities
  are called relational properties.
• Properties such as is happy which are ascribed
  to only one individual are monadic properties.

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Individuals and Properties, continued

• Capital letters used to denote properties are
  called property constants, and lowercase letters
  (up to an including the letter t) used to denote
  things, objects, and individual entities are
  called individual constants.
• The lowercase variables u through z are used as
  individual variables, replaceable by individual
  constants.

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2. Quantifiers and Free Variables

• In predicate logic two symbols, called
  quantifiers, are used to state how many.
• The universal quantifier is used to state that
  all entities have some property or properties.
• The existential quantifier is used to assert that
  some individual or individuals have one or more
  properties.

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Quantifiers and Free Variables, continued

• In predicate logic, parentheses indicate the
  scope of a quantifier.
• For example, the sentence Everything has mass
  and is extended is symbolized as (x)(Mx . Ex)
  (where Mx x has mass and Ex x is
  extended).
• The parentheses around the expression (Mx . Ex)
  indicate that the scope of the (x) quantifier is
  the remaining part of the sentence.

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Quantifiers and Free Variables, continued

• The expression (x) (Mx . Ex) is a sentence, but
  (Mx . Ex) is a sentence form, not a sentence.
• The expression (x)(Mx) . Ex is not a sentence as
  it contains an individual variable that is not
  quantified, the Ex.
• Unquantified variables are called free variables.
  Quantified variables are called bound variables.

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3. Universal Quantifiers

• When we use the universal quantifier, (x), we are
  saying something about all the individuals
  represented by the variable in the quantifier.
• Since few properties can be ascribed to
  everything, we might need to restrict our domain
  of discourse.

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4. Existential Quantifiers

• The existential quantifier (?x) is used to assert
  that some entities (at least one) have a given
  property.
• For example, Something is heavy can be
  symbolized as (?x)Hx

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5. Basic Predicate Logic Symbolizations

• Sentences that begin with words such as all
  every or any can be symbolized using the
  universal quantifier.
• (x) (Dx ? Fx)
• Sentences of the form Some As are Bs can be
  symbolized using the existential quantifier.
• (?x) (Dx . Fx)

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Basic Predicate Logic Symbolizations, continued

• Sentences of the form No As are Bs can be
  symbolized using either the existential
  quantifier
• (?x) (Dx . Fx)
• Or the universal quantifier
• (x) (Dx ? Fx)

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Basic Predicate Logic Symbolizations, continued

• Sentences of the form Not all As are Bs can
  be symbolized using the universal quantifier
• (x) (Dx ? Fx)
• Or the existential quantifier
• (?x) (Dx . Fx)

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6. The Square of Opposition

• Any sentence that can be symbolized with a
  universal quantifier can be symbolized with an
  existential quantifier, and vice versa.
• A traditional way of illustrating the
  relationship between the quantifiers is known as
  the square of opposition.

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7. Common Pitfalls in Symbolizing with Quantifiers

• Some English expressions that look like compound
  subjects or predicates should not be symbolized
  that way.
• Be careful with sentences that contain the word
  a or any. These sometimes operate logically
  like the particle all, but often they do not.
• Be careful when symbolizing conjunctions!

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8. Expansions

• A reliable guide to translation is to compare the
  truth conditions of the ordinary language
  sentence we are translating with those of its
  translation.
• In a universe with four individual entities (x)Fx
  would be true if (Fa . Fb) . (Fc . Fd), the
  expansion of (x) Fx with respect to that
  universe, was true.

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Expansions, continued

• A model universe is a domain of a small number of
  individuals about which we shall construe our
  quantified statements.

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9. Symbolizing Only, None but, and Unless

• Statements such as Only those who study will
  pass the test should be symbolized as (x)(Px ?
  Sx), with the universe of discourse restricted to
  the students in question.
• Sentences such as None but the good die young
  should be symbolized as (x)(Yx ? Gx)
• Sentences such as No one will pass the test
  unless he or she studies should be symbolized as
  (x)(Px ? Sx).

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Key Terms

• Bound variable
• Domain of discourse
• Existential quantifier
• Expansion
• Free variable
• Individual constant
• Individual variable
• Model universe
• Predicate logic

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Key Terms, continued

• Property constants
• Quantifier
• Relational property
• Scope of a quantifier
• Square of opposition
• Substitution
• Universal quantifier
• Universe of discourse