PL%20continued:%20Quantifiers%20and%20the%20syntax%20of%20PL

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PL continuedQuantifiers and the syntax of PL

• Quantifiers of PL
• Quantifier symbols
• Variables used with quantifiers
• Truth functional compounds of PL
• The formal syntax of PL
• Vocabulary
• Quantifier of PL
• Atomic formula of PL
• Recursive definition of formula of PL

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PL continuedQuantifiers and the syntax of PL

• Kinds of formulas of PL
• Main logical operators
• Subformulas (immediate and otherwise)
• Variables bound and free
• Sentences of PL

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PL continuedQuantifiers and the syntax of PL

• Quantity terms all, each, everyone, everything,
  someone, something, no one, none, nothing
• Quantifier symbols ? and ?
• Quantifiers
• (?x) A universal quantifier
• Each x is such that
• (?x) An existential quantifier
• There is at least one x such that (or there
  is some x such that )

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PL continued Quantifiers and the syntax of PL

• UD people in Michaels office
• Lxy x likes y
• m Michael
• r Rita
• s Sue
• Michael likes everyone
• Each person is such that Michael likes them
• (?x) Lmx
• Note the variable x appears twice as part of
  the quantifier and as part of the expression that
  follows it.

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PL continued Quantifiers and the syntax of PL

• Michael likes someone
• (?x) Lmx
• Everyone likes Michael
• (?x) Lxm
• Michael doesnt like anyone.
• (?x) Lmx
• Michael doesnt like some people.
• (?x) Lmx

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PL continued Quantifiers and the syntax of PL

• Variables used in quantifiers and in sentences of
  PL
• w, x, y and z
• They serve as placeholders for individual
  variables in the specification of predicates,
  such as
• Lxy x likes y
• They can be replaced by constants, such as m in
• Lmm (Michael likes himself)
• And they serve as placeholders for terms such as
  thing in something,
• one in someone
• body in somebody

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PL continued Quantifiers and the syntax of PL

• We can use any of the 4 variables in specifying
  predicates (for example Ey y is easygoing)
• And any of the four variables in quantifiers and
  expressions that include quantifiers
• (?y), (?w), and (?z) are all quantifiers
• Finally, we can use
• (?z) Lmz
• to symbolize Michael likes everyone
• Even if our symbolization key includes
• Lxy x likes y

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• So far, we have considered sentences that include
  only one quantifier, one predicate, and, in some
  cases, a tilde.
• But we can easily form truth-functional compounds
  of such sentences
• UD people in Michaels office
• Ex x is easygoing
• Lxy x likes y
• m Michael
• r Rita
• Either everyone is easygoing or no one is
• (?x) Ex v (?x) Ex OR
• (?x) Ex v (?x) Ex BUT NOT AS
• (?x) Ex v (?x) Ex

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• And because of what weve said about variables,
  we can also symbolize
• Either everyone is easygoing or no one is
• as
• (?x) Ex v (?w) Ew
• (?y) Ey v (?z) Ez
• We can symbolize If Michael is easygoing,
  everyone is
• as
• Em ? (?x) Ex
• and Michael likes everyone but Rita doesnt
• as
• (?x) Lmx (?y) Lry

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• Again, using the symbolization key, we can
  symbolize
• If anyone is easygoing, Michael is
• as
• (?x) Ex ? Em
• and
• Rita is easygoing if and only if everyone is
• as
• Er ? (?x) Ex

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• Why we dont actually need both existential and
  universal quantifiers.
• Any sentence of the form
• Everything is this or that, say Everyone likes
  Michael, which can be symbolized as
• (?x) Lxm
• Can be rephrased as
• There is nothing that is not this or that
  (There isnt anyone who doesnt like Michael)
• (?x) Lxm

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• And any sentence of the form Something is this
  or that (such as Michael likes someone), which
  can be symoblized as
• (?x) Lmx
• can be paraphrased as
• It is not the case that Michael doesnt like
  everyone or It is not the case that Michael
  likes no one
• and symbolized as
• (?x) Lmx

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The syntax of PL

• Vocabulary
• Sentence letters of PL Capital Roman letters, A
  through Z, with or without subscripts (just the
  sentence letters of SL)
• Predicates of PL Capital letters, A through Z,
  with or without subscripts followed by one or
  more variables Ax, Axy, Axyz Bx, Bwz, Bwzx..
• Individual constants of PL lowercase Roman
  letters, a through v, without or without
  subscripts
• Individual variables of PL the lowercase Roman
  letters, w through z with or without
  subscripts.
• Truth-functional connectives v ? ?
• Quantifier symbols ? ?
• Punctuation ( )

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The syntax of PL

• PL contains expressions.
• PL contains formulas. Not all expressions of PL
  are formulas.
• PL contains sentences. Not all formulas of PL are
  sentences.
• An expression of PL a sequence of not
  necessarily distinct elements of the vocabulary
  of PL.
• Examples
• (((a ? bba)
• A ? Fx
• (?x) Txx
• (?x) (?y) Fxy

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The formal syntax of PL

• We use the bold letters P, Q, and R as meta
  variables ranging over expressions of PL.
• We use bold a as a meta variable to range over
  individual constants of PL.
• We use bold x as a meta variable to range over
  individual variables of PL.

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The formal syntax of PL

• Quantifier of PL An expression of PL of the form
  (?x) or (?x).
• A quantifier contains a variable (?y) and (?y)
  contain the variable y and are y quantifiers
  (?x) and (?x) contain the variable x and are
  x quantifiers and so forth for w and z
  variables and quantifiers.
• Atomic formulas of PL Every expression of PL
  that is either a sentence letter of PL or an
  n-place predicate of PL followed by n individual
  terms of PL. (E.g., B, Fab, and Gxx are
  atomic formulas)

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Recursive definition of a formula of PL

• Every atomic formula of PL is a formula of PL.
• If P is a formula of PL, so is P.
• If P is a formula of PL, so are (P Q), (P v Q),
  
• (P ? Q), and (P ? Q).
• If P is a formula of PL that contains at least
  one occurrence of x and no x-quantifier, then
• (?x) and (?x) are both formulas of PL.
• Nothing else is a formula of PL.
• Step 4 is to rule out expressions such as
• (?x) (?x) Lx

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Logical operators of PL

• Logical operator of PL An expression of PL that
  is either a quantifier or a truth functional
  connective.
• Every formula of PL is either atomic, quantified,
  or a truth-functional compound.
• It is atomic if it contains no logical
  operations.
• It is quantified if its main logical operator is
  a quantifier.
• It is truth-functional if its main logical
  operator is a truth-functional connective.
• Cabz No logical operator
• Cabz Gwx is the main logical operator
• (?y) (Gy ? Fya) (?y) is the main
  logical operator
• (?x) Gx v Cabz v is the main logical operator

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Logical operators of PL

• Main logical operators and subformulas
• Defined by cases (see p. 300 in text)
• 1. If P is an atomic formula of PL, then P
  contains no logical operator, and hence no main
  logical operator, and P is the only subformula of
  P.
• 2. If P is a formula of PL of the form Q, then
  the tilde that precedes Q is the main logical
  operator of P and Q is the immediate subformula
  of P.
• 3. If P is a formula of PL of the form Q R, Q v
  R, Q ? R, or Q? R, then the binary connective
  between Q and R is the main logical operator of
  P, and Q and R are the immediate subformulas of
  P.

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Logical operators of PL

• Main logical operators and subformulas
• Defined by cases (see p. 300 in text)
• 4. If P is a formula of PL of the form (?x) Q or
  (?x) Q, then the quantifier that occurs before Q
  is the main logical operator of P and Q is the
  immediate subformula of P.
• 5. If P is a formula of PL, then every
  subformula (immediate or not) of a subformula of
  P is a subformula of P and P is a subformula of
  itself.

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• Immediate
• Formula Subformula MLO Type
• Gadz Gadz none atomic
• Gadz Gadz truth-functional
• (?z) Gadz Gadz (?z) quantified
• (?z) Gadz (?z) Gadz truth-functional
• Scope of a quantifier The scope of a quantifier
  in a formula P of SL is the subformula Q of which
  that quantifier is the main connective.
• (?z) Gadz the scope of (?z) is all of (?z) Gadz
  .
• (?z) Gadz Em the scope of (?z) is (?z) Gadz.
• (?z) (Gadz Ez) the scope of (?z) is the entire
  sentence.

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• Bound variable An occurrence of a variable x in
  a formula of PL that is within the scope of an
  x-quantifier.
• Free variable An occurrence of a variable x in a
  formula of PL that is not bound.
• Sentence of PL A formula P of PL is a sentence
  of PL if and only if no occurrence of a variable
  in P is free.
• (Gx) ? (?x) Fx is not a sentence of PL.
• (?x) Gx ? (?x) Fx is a sentence of PL.
• (?x) (Gx ? Fx)
• A formula of PL that is not a sentence of PL is
  called an open sentence of PL.

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Sentences of PL

• Which of the following are sentences of PL?
• (?x) Bx ? (?x) Bx
• (?x) Bx ? Bx
• (?x) (Bx ? Bx)
• (?x) Fxa
• (?x) Fya
• G (?y) Cyy
• (Bxa Bax) v (?x) (Gx)
• Dbc ? (?x) Dbx
• 1, 3, 4, 6, and 8

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Sentences of PL

• Substitution instances
• We use
• P(a/x)
• to specify the formula of PL that is like P
  except that it contains the individual constant a
  wherever P contains the individual variable x.
• So if P is (?x) Fx, then P (a/x) is Fa
• Substitution instance of P If P is a sentence of
  PL of the form (?x) Q or (?x) Q, and a is an
  individual constant, then Q (a/x) is a
  substitution instance of P. The constant a is the
  instantiating constant.

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Sentences of PL

• Homework
• 7.5E Exercises 1-3
• 7.6E Exercises 1 and 2
• Read section 7.7