Predicate Logic and the language PL

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Predicate Logic and the language PL

• In SL, the smallest unit is the simple
  declarative sentence.
• But many arguments (and other relationships
  between sentences) are actually based on
  sub-sentential units. Including
• None of Marys friends supports Libertarians.
• Sarah supports Matlow and Matlow is a
  Libertarian.
• So Sarah is no friend of Marys.
• Although we could symbolize this argument in SL,
  its logic would be lost.

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Predicate Logic

• Sub-sentential units in predicate logic
• 1. Singular terms
• Names (The Washington Monument, Boston, Marie
  Curie, Harry Reid, Henry, Sherlock Holmes)
• Definite descriptions (the Senate majority
  leader, the discoverer of radium, Michaels only
  brother, the present king of France, the person
  Mary is now talking to)
• Issues
• Non-designating singular terms
• Singular terms and context
• Pronouns

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Predicate Logic

• Singular terms and pronouns
• If John voted for Hillary Clinton, then hes no
  Libertarian.
• If John voted for Hillary Clinton, then Johns
  no Libertarian.
• This test is so easy that if anyone fails it,
  its his or her own fault.
• We cant use This test is so easy that if
  anyone fails it its Johns or Cynthias own
  fault

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Predicate Logic

• Sub-sentential units continued
• 2. Predicates
• Sentences can have more than one singular term,
  for example
• New York is between Philadelphia and Boston
• Predicates of English are parts of English
  sentences that are obtained by removing one or
  more singular terms from an English sentence.

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Predicate Logic

• 2. Predicates
• (Or, a predicate is a string of words with one or
  more blanks in it such that when the blanks are
  filled in, a sentence results.)
• New York is between Philadelphia and Boston
• _______ is between Philadelphia and Boston
• New York is between _______ and Boston.
• _______ is between ________ and Boston.
• _______ is between Philadelphia and ______.
• _______ is between _________ and _______.

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Predicate Logic

• 2. Predicates
• So there are one-place predicates such as
• _______ is between Philadelphia and Boston
• New York is between _______ and Boston.
• And there are many-place predicates.
• This is a two-place predicate
• _______ is between ________ and Boston.
• This is a three-place predicate
• _______ is between _________ and _______.

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Predicate Logic

• 2. Predicates
• In general, where n is a positive integer, a
  predicate with n blanks is an n-place predicate.
• One way of generating a sentence from a predicate
  is filling the blanks with singular terms any
  singular term may be put in any blank, and the
  same singular term can be put in more than one
  blank.

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Predicate Logic

• Using variables
• Instead of blanks, we use the lower case letters
  w, x, y and z (with numerical subscripts
  when necessary) to mark the blanks in predicates.
• So one predicate early discussed can be displayed
  as x is between y and z.
• Another can be displayed as x is taller than y.

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Predicate Logic

• So from the two-place predicate x is taller than
  y, and the singular terms The Washington
  Monument, Mary, John, and the smallest
  prime number, we can generate
• Mary is taller than The Washington Monument.
• John is taller than Mary.
• Mary is taller than John.
• The Washington Monument is taller than John.
• The smallest prime number is taller than Mary.
• And so forth

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Predicate Logic

• We also retain the sentential connectives and,
  or, if then, if and only if, and not
• So given a stock of predicates, singular terms,
  and the sentential connectives, we can generate a
  wide variety of sentences of English.
• From the sentential connectives, the singular
  terms Michael, Sue and Rita, and the
  predicates x is easygoing, x likes y, and x
  is taller than y, we can generate

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Predicate Logic

• From the sentential connectives, the singular
  terms Michael, Sue and Rita, and the
  predicates x is easygoing, x likes y, and x
  is taller than y, we can generate
• Michael is easygoing.
• Michael is easygoing but Sue isnt easygoing.
• Sue likes Rita and Rita likes Michael.
• If Rita likes Michael, then Michael is taller
  than Sue and he is easygoing.
• Either Rita or Sue is taller than Michael, but
  not both.

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Predicate Logic

• Except when our domain is limited, what we cant
  yet generate (but eventually will) are claims
  such as
• Everyone is easygoing.
• No one is easygoing.
• Someone is easygoing.
• Someone is not easygoing.
• No one is taller than his or herself.
• Everyone likes him or herself.
• every, some, all, each, and none are
  quantity terms and quantity terms are not
  singular terms.

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The language PL

• Vocabulary
• The sentential connectives , v, ?, ? and
  .
• Individual constants (lowercase Roman letters a
  through v, with or without subscripts) to
  symbolize singular terms that denote (names and
  definite descriptions)
• Predicates of PL uppercase Roman letters A
  through Z with or without subscripts and
  followed by variables

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The language PL

• Vocabulary
• Predicates of PL uppercase Roman letters A
  through Z with or without subscripts and
  followed by one or more variables, n of the
  letters w, x, y and z after the predicate
  letter.
• Fx is a one place predicate
• Fxy is a two place predicate
• Fxyz is a three place predicate..

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The language PL

• As with sentence letters in SL, we can use a
  predicate (say, Lxy) to symbolize, on different
  occasions, a variety of 2 place predicates of
  English, including
• x loathes y
• x loves y
• x is larger than y
• x is less than y

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The language PL

• Vocabulary
• Constants of PL lower case Roman letters a
  through v are used to symbolize singular terms
• a is a constant
• b is a constant
• Sentential connectives and punctuation
  (parentheses and brackets)

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The language PL

• Symbolizing in PL
• 1. We begin with a symbolization key
• a. Specify the universe of discourse (abbreviated
  UD) for the occasion.
• Examples of UDs
• the positive integers
• the jellybeans in the jar on my desk
• all persons
• everything
• people in Michaels office

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The language PL

• Symbolizing in PL
• 1. We begin with a symbolization key
• b. Specify symbols for the predicates
• Ex x is easygoing
• Txy x is taller than y
• Lxy x likes why
• c. Specify symbols for constants (if there are
  any)
• a Anita
• b The Brooklyn Bridge

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The language PL

• A symbolization key
• UD People in Michaels office
• Lxy x likes y
• Ex x is easygoing
• Txy x is taller than y
• m Michael
• r Rita
• s Sue

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• Sue is easygoing
• Es
• Sue is taller than Michael, and Michael is
  taller than Rita
• Tsm Tmr
• If Rita likes Sue, then Rita is taller than Sue
• Lrs ? Trs
• If Michael is easygoing, Sue is not
• Em ? Es

• UD People in Michaels office
• Lxy x likes y
• Ex x is easygoing
• Txy x is taller than y
• m Michael
• r Rita
• s Sue

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The language PL

• We can symbolize some English sentences involving
  quantity terms with the resources we have so far
  if we have a UD that makes it possible. Given the
• UD People in Michaels office
• and the predicates and constants we have in the
  symbolization key which include a constant for
  each of the people)
• we can symbolize Michael likes everyone as
• (Lmm Lmr) Lms

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The language PL

• UD People in Michaels office
• We can symbolize Michael likes someone as
• (Lmm v Lmr) v Lms
• And Michael likes no one as
• (Lmm Lmr) Lms
• or
• (Lmm v Lmr) v Lms
• And Everyone is easygoing as
• (Em Er) Es

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• Given the symbolization key shown or handed out,
  symbolize
• Alice was born in Boston, so she wasnt born in
  Seattle.
• Bonnie was born in Cleveland but she lives in
  Philadelphia.
• Philadelphia is larger than Seattle, but Boston
  is larger than Philadelphia.
• If Bonnie is taller than Charles, and Charles is
  taller than Alice, then Bonnie is taller than
  Alice.
• No one lives in Boston.
• Everyone was born in Cleveland.

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• Create a symbolization key that has Alex, Bruce,
  Cathy and Danielle as its UD
• And the predicates
• x is attracted to y
• x is intimidated by y
• x is intelligent
• x is shorter than y
• x is sitting between y and z
• And the singular terms
• Alex
• Bruce
• Cathy
• Danielle

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• UD Alex, Bruce, Cathy and Danielle
• Axy x is attracted to y
• Ixy x is intimidated by y
• Cx x is curious
• Sxy x is shorter than y
• Bxyz x is sitting between y and z
• a Alex
• b Bruce
• c Cathy
• d Danielle

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• Symbolize
• Cathy is attracted to Bruce, but she is
  intimidated by him
• (Acb) (Icb)
• Bruce is sitting between Alex and Danielle
• Bbad
• If Cathy is shorter than Alex, she is attracted
  to him
• Sca ? Aca
• No one is curious
• (Ca Cb) (Cc Cd)
• or
• (Ca v Cb) v (Cc v Cd)

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• Homework
• 7.2E (all)
• 7.3E As much as you can of exercises 1-3.