March 5: more on quantifiers

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March 5 more on quantifiers

• We have encountered formulas and sentences that
  include multiple quantifiers
• Take, for example
• UD People in Michaels office
• Lxy x likes y
• m Michael
• There is someone that Michael likes and
  everyone likes Michael
• (?x) Lmx (?y) Lym

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March 5 more on quantifiers

• But there are also sentences that include
  multiple quantifiers of overlapping scope
• UD People in Michaels office
• Lxy x likes y
• m Michael
• So weve symbolized
• Everyone likes Michael
• (?x) Lxm
• Now suppose we want to symbolize
• Everyone likes someone

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Multiple quantifiers with overlapping scope

• And suppose our UD is now all people
• UD all people
• Lxy x likes y
• Everyone likes someone
• (?x) there is someone such that
• (?x) (?y) such that
• (?x) (?y) Lxy

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Multiple quantifiers with overlapping scope

• Now suppose
• UD all people
• Lxy x likes y
• And we want to symbolize
• Everyone likes everyone
• (?x) is such that for everyone
• and we add (?y) for another universal quantifier
• so that we have
• (?x) is such that for all y (?y)
• (?x) (?y) x likes y
• (?x) (?y) Lxy

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Multiple quantifiers with overlapping scope

• Suppose we keep
• UD all people
• Lxy x likes y
• And now we want to symbolize
• Someone likes everyone
• (?x) such that for everyone y, x
• We add (?y) for that everyone, and then
• (?x) (?y) Lxy

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Multiple quantifiers with overlapping scope

• Suppose we keep
• UD all people
• Lxy x likes y
• And we want to symbolize
• Someone likes someone
• (?x) such that there is someone
• We add (?y) for there is some y and get
• (?x) (?y) Lxy

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Multiple quantifiers with overlapping scope

• Pairs of quantifiers can occur in 4
  combinations
• (?x) (?y) There is an x and there is a y such
  that or
• There is a pair x and y such that
• (?x) (?y) For each x and for each y or For
  each
• pair x and y
• (?x) (?y) For each x there is a y such that
• (?x) (?y) There is an x such that for each y
• Although we wont deal with them, there can be
  more than 2 (or more than a pair) of quantifiers
  of overlapping scope.

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(Informal) Semantics of PL

• The extension of a 3-place predicate (e.g., Bwxy
  w is between x and y), is a set of 3-ordered
  objects, and the extension of a 4-place predicate
  (e.g., Twxyz w between x, y, and z) is a set of
  4-ordered objects, and so forth each extension
  is a set of n-tuple sets (in the first case, a
  set of 3 ordered objects and, in the second, a
  set of 4 ordered objects).

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Multiple quantifiers with overlapping scope the
difference a UD makes for symbolization

• 1.
• UD persons
• Lxy x likes y
• Everyone likes everyone
• (?x) (?y) Lxy
• Someone likes someone
• (?x) (?y) Lxy

• 2.
• UD Everything
• Lxy x likes y
• Px x is a person
• Everyone likes everyone
• (?x) (?y) (Px Py) ? Lxy
• Someone likes someone
• (?x) (?y) (Px Py) Lxy

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Multiple quantifiers with overlapping scope

• One has to learn to read the sentences of PL
  into quasi-English to check out a
  symbolization. In doing so, it is crucial to
  identify the role of each logical operator. For
  example, in
• (?x) (?y) (Px Py) ? Lxy
• (?x) is the main logical operator and (?y) is
  the
• main logical operator of the subformula
• (Px Py) ? Lxy
• So we read (?x) first and (?y) second.
• Every x is such that every y is such that or
  Every pair x and y is such that

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Multiple quantifiers with overlapping scope

• We read (?x) first and (?y) second.
• Every x is such that every y is such that or
  Every pair x and y is such that
• As the main logical operator of the next
  subformula, (Px Py) ? Lxy, is the ? we move to
  that next
• Every x is such that every y is such that or
  Every pair x and y is such that IF

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Multiple quantifiers with overlapping scope

• (?x) (?y) (Px Py) ? Lxy
• Every x is such that every y is such that or
  Every pair x and y is such that IF
• x is a person and y is a person
• THEN
• Lxy (x likes y)
• All together
• Every x is such that every y is such that, if x
  is a person and y is a person, then x likes y
  OR
• For every pair x and y, if x is a person and y
  is a person, then x likes y

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(Informal) Semantics of PL

• In SL, the semantic notion we used to determine
  truth status was the truth value assignment.
• This worked because the sentences of SL are such
  that their truth status is a function of truth
  functional assignments to their atomic components
  (atomic sentences) and, in compound sentences,
  the TVAs of atomic sentences and
  truth-functional connectives.
• This doesnt work in PL, except for those
  sentences that are just the atomic sentences (A,
  B, C) of SL.
• The basic semantic concept of PL is an
  interpretation.

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(Informal) Semantics of PL

• In PL, an interpretation interprets
• each individual constant (if any)
• each predicate
• each sentence letter of PL
• All of these are relative to some universe of
  discourse
• We can take symbolization keys as we have
  encountered them so far as embodying
  interpretations with the following caveat
• In an interpretation, a UD is always some
  non-empty set, and so we classify them as sets
  when specifying an interpretation

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(Informal) Semantics of PL

• There are 2 kinds of atomic sentences of PL
• Sentence letters (A through Z with or without
  subscripts the semantics involve truth
  functional assignments)
• A n-place predicate of PL followed by
  n-constants
• What are its semantics?
• Start with a one-place predicate, Fx, and the
  atomic sentence
• Fa

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(Informal) Semantics of PL

• The truth or falsity of Fa depends on an
  interpretation that interprets
• the predicate Fx
• the constant a
• and a UD a set of objects over which the
  predicates and variables range, and from which
  constants pick out objects

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(Informal) Semantics of PL

• Atomic sentences of PL
• Consider Fa on interpretation 1
• 1. UD set of living things
• Fx x is a human
• a is Socrates (the historical figure)
• On this interpretation, Fa is true

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(Informal) Semantics of PL

• Change the UD and/or the predicate and/or the
  constant, and the truth status of sentence on the
  interpretation also changes. Consider Fa on
  interpretation 2
• 2. UD set of living things
• Fx x is a potato
• a Socrates
• Assuming, again, that a denotes the historical
  person, Socrates, the sentence Fa is false on
  this interpretation.

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(Informal) Semantics of PL

• Atomic sentences of PL involving 2-place
  predicates
• Lxy x is larger than y
• And say the UD is the set of positive integers
• A two place predicate is a relational predicate
  it picks out sets of pairs of members of the UD
  whose order often matters and in the case of
  this predicate, order does matter.

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(Informal) Semantics of PL

• 3. UD the set of positive integers
• Lxy x is larger than y
• a 1
• b 2
• Lba is true on interpretation 3.
• The extension of Lxy on interpretation 3 are
  those pairs (4, 3 5, 2 201, 200 4000, 3999
  and so forth) of which it is true that the first
  member of the pair is larger than the second
  member of the pair.
• So, the pair 2, 3 is not a pair that is an
  extension of Lxy on interpretation 3.

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• 4. The set of all buildings and all people
• Lxy x is larger than y
• a The Empire State Building
• b George Bush
• Lba is false on interpretation 4.
• An interpretation may assign the same member of
  the UD to more than one constant
• 5. The set of planets in our solar system
• Cxy x is closer to the sun than y.
• m Venus
• n Venus
• Cmn is false on interpretation 5.

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(Informal) Semantics of PL

• Some interpretations of 2-place predicates mean
  that the extension of a predicate includes pairs
  in which the 1st and 2nd members are the same
• 6. UD the set of positive integers
• Exy x is less than or equal to y
• The extension of the predicate Exy (as defined
  above) includes not only 2, 3 but also 2, 2 4,
  4 and so forth.

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(Informal) Semantics of PL

• Compound sentences of PL that do not include
  quantifiers
• Cab v Cba
• As the main logical operator is v, this
  sentence is true or false on an interpretation
  depending on whether at least either Cab or
  Cba is true on that interpretation. We use the
  characteristic truth tables for the connectives
  to determine whether a truth functional sentence
  of PL is true on some interpretation.
• 7. UD the set of persons
• Cxy x likes y
• a Andrea
• b Bruce

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(Informal) Semantics of PL

• Compound sentences of PL whose main logical
  operator is not a quantifier
• (?x) (Wx ? Mx) (?x) (Ex Ox)
• As the main logical operator is , this
  sentence is true or false on an interpretation if
  and only if both the right and left conjuncts are
  true on that interpretation.
• 8. UD the set of all things
• Wx x is a whale
• Mx x is a mammal
• Ex x is an even positive integer
• Ox x is an odd positive integer

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(Informal) Semantics of PL

• For quantified sentences of PL
• (?x) (Wx ? Mx)
• Here the main logical operator is a universal
  quantifier.
• So the sentence is true on some interpretation if
  and only if it is true that all xs that are Ws
  are Ms.
• 9. UD the set of living things
• Wx x is a whale
• Mx x is a mammal.
• (?x) (Wx ? Mx) is true on this interpretation.

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(Informal) Semantics of PL

• Quantified sentences of PL
• (?x) (Wx ? Mx)
• Here the main logical operator is a universal
  quantifier.
• 10. UD the set of all things
• Wx x has a brain.
• Mx x is a car
• The sentence is false on this interpretation.

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(Informal) Semantics of PL

• Quantified sentences of PL
• (?x) (Ex Ox)
• Here the main logical operator is an existential
  quantifier and the sentence is true on an
  interpretation if there is at least one thing
  that it is a E and an O.
• 11. UD the set of positive integers
• Ex x is even
• Ox x is odd
• (?x) (Ex Ox) is false on this interpretation.

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(Informal) Semantics of PL

• Quantified sentences of PL
• (?x) (?y) Lxy
• 12. UD the set of all persons
• Lxy x likes y
• (?x) (?y) Lxy is true on interpretation 12 if and
  only if for each person there is someone that
  person likes. (Everyone likes someone.)
• It could just be him or herself.
• 13. UD the sent of all persons
• Lxy x knows y
• It is likely that (?x) (?y) Lxy is true on
  interpretation 13.

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(Informal) Semantics of PL

• Quantified sentences of PL
• (?x) (?y) Lxy
• 13. UD the set of all persons
• Lxy x likes y
• (?x) (?y) Lxy is true on interpretation 13 if and
  only each person likes every other person.
• Unlikely!
• 14. UD the set of all persons
• Lxy x knows y
• It is also unlikely that (?x) (?y) Lxy is true on
  interpretation 14.

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(Informal) Semantics of PL

• Summary of informal semantics of PL
• The truth conditions for sentences of PL are
  determined by interpretations.
• An interpretation consists of the specification
  of a UD, and the interpretation of each sentence
  letter, predicate, and individual constant
  relative to the UD designated. (Individual
  variables are not interpreted.)
• Homework 7.8E Exercise 1 and more to be provided
  for semantics.