Review Test 5

1
Review Test 5

• You need to know
• How to symbolize sentences that include
  quantifiers of overlapping scope
• Definitions
• Quantificational truth, falsity and
  indeterminacy
• Quantificational equivalence
• Quantificational validity
• Quantificational consistency
• Quantificational entailment

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Review Test 5

• How to symbolize sentences that include
  quantifiers of overlapping scope
• We have limited the number of quantifiers with
  overlapping scope that you need to know how to
  symbolize 2 for any given sentence
• (?x) (?y) For each x and for each y (or for
  every pair x and y)
• (?x) (?y) For each x there is some y such that
• (?w) (?z) There is some w such that for every z
• (?z) (?x) There is some z such that for some x
  (or there is some pair z and x such that)

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Review Test 5

• How to symbolize sentences that include
  quantifiers of overlapping scope
• (?x) (?y) For each x and for each y (or for
  every pair x and y)
• 1. UD the set of positive integers
• Dxy x is equal to, or smaller than, or
  larger than y.
• Symbolize
• For every positive integer x, every positive
  integer y is such that x is equal to, or smaller
  than, or larger than y
• (?x) (?y) Dxy

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Review Test 5

• (?x) (?y) For each x there is some y such that
• For each positive integer, there is some positive
  integer that is larger than it.
• 2. UD the set of positive integers
• Lxy x is larger than y
• (?x) (?y) Lyx
• Change the UD and predicates to
• 3. UD everything
• Px x is a positive integer
• Lxy x is larger than y
• (?x) Px ? (?y) (Py Lyx) or
• (?x) (?y) (Px Py) ? Lyx

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• (?w) (?z) There is some w such that for every z
• Some positive integer w is such that for every
  positive integer z, w is equal to or smaller than
  z.
• UD the set of positive integers
• Txy x is equal to or smaller than y
• (?w) (?z) Twz
• Change the UD and predicates
• UD everything
• Px x is a positive integer
• Txy x is equal to or smaller than y
• (?w) (?z) (Pw Pz) ?Twz or
• (?w) Px (?z) (Pz ?Twz)

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Review Test 5

• How to symbolize sentences that include
  quantifiers of overlapping scope
• (?z) (?x) There is some z such that for some x
  (or there is some pair z and x such that)
• The sum of some positive integers x and y is 4.
• UD the set of positive integers
• Exy the sum of x and y is 4
• (?z) (?x) Ezy

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Review Test 5

• Pop quiz!
• Symbolize the following sentences in PL using the
  following interpretation
• UD the set of all things
• Px x is a professor
• Sxy x is a student of y
• Bxy x bores y
• Wx x is wasting his or her time
• Any student who is bored by all of his or her
  professors is wasting her or time.
• If a professor bores all of his or her students,
  then the professor is wasting his or her time.

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Review Test 5

• You need to know Definitions
• Quantificational truth, falsity and
  indeterminacy and so forth
• The basic semantic notion in predicate logic is
  an interpretation, and all of quantificational
  definitions are in terms of one or more
  interpretations.

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• What an interpretation is
• It includes a UD which is a nonempty set (it has
  at least one member)
• An interpretation of every predicate of PL
• An interpretation of every individual constant
  of PL
• As there are an infinite number of
  interpretations of any of the infinite number of
  predicates of PL and an infinite number of
  interpretations of the infinite number of
  individual constants of PL
• And an infinite number of UDs
• So, PL includes an infinite number of
  interpretations

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• To construct an interpretation so as to
  demonstrate that some quantificational notion
  holds or does not (and you cannot use this method
  to prove all claims but only some!), you need to
  specify
• A UD a nonempty set (the domain over which
  predicates and variables range, and members of
  which individual constants refer to or denote)
• An interpretation of each (relevant) predicate
  that helps you to demonstrate that a
  quantificational notion does or does not hold
  (except in terms of equivalence when you need 2)
• An interpretation of any (relevant) individual
  constants.

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Review Test 5

• You need to be able to
• Identify an interpretation that shows that a
  sentence is not quantificationally true
• Identify an interpretation that shows that a
  sentence is not quantificationally false
• Identify an interpretation that shows that a set
  of sentences is quantificationally consistent
• Identify an interpretation that shows that 2
  sentences are not quantificationally equivalent
• Identify 2 interpretations that show that a
  sentence is quantificationally indeterminate

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• Cases in which identifying one interpretation or
  two wont do the work you need
• If told to show that a sentence is
  quantificationally true, provide the reasoning
  that demonstrates this (no one or more
  interpretations can show this)
• If told to show that a sentence is
  quantificationally false, provide the reasoning
  that demonstrates this (same as above)
• If told an argument is quantificationally valid,
  provide the reasoning that demonstrates this
• If told a set quantificationally entails some
  sentence, provide the reasoning that demonstrates
  this

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• In general
• To disprove that some characteristic applies to
  all and any interpretations (when you are told it
  does not), identify an interpretation that shows
  this
• For example, that P is not quantificationally
  true or
• That some set is not quantificationally
  consistent or
• That some argument is not quantificationally
  valid

14

• Pop quiz 2!
• a. Can you show that a sentence is
  quantificationally true by identifying an
  interpretation on which it is true?
• b. Can you show that a set is quantificationally
  consistent by citing an interpretation?
• c. Do you need one or more interpretations, or
  must you use reasoning, to show that
• (?x) (?y) Syx is quantificationally
  indeterminate?

15

• 2 different kinds of question and, so, 2
  different kinds of proof
• a. Show that the sentence (?y) (?x) Gyx is not
  quantificationally false.
• Try an interpretation with a UD of the set of
  positive integers
• Interpret Gxy so that the sentence is true on
  that interpretation. And you will have shown that
  the sentence is not quantificationally false.
  Example
• Gxy x is greater than y
• Gxy x multiplied by y is even (or odd)

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• 2 different kinds of question and, so, 2
  different kinds of proof
• b. Show that the sentence (?y) (Ay Ay) is
  quantificationally false.
• As we cannot demonstrate that the sentence is
  false on every possible interpretation, we use
  reasoning to show that whatever the UD, and
  however A is interpreted, the sentence will
  always be false hence, that it is
  quantificationally false.
• This means showing that for any y, Ay Ay is
  always false. This formula is truth functional,
  and to be true it requires that both conjuncts
  are true. But there is no interpretation of A on
  which some y can be both A and A. If Ay is true,
  Ay is false, and vice versa. So the sentence
  (?y) (Ay Ay) is quantificationally false.

17

• 2 different kinds of question and, so, 2
  different kinds of proof
• a. Show that the following argument is not
  quantificationally valid
• (?x) (Ax ? Bx)
• (?x) Ax
• --------------------
• (?x) Bx
• This means we need to identify an interpretation
  (just one) on which each of the premises is true
  and the conclusion is false.

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• (?x) (Ax ? Bx)
• (?x) Ax
• --------------------
• (?x) Bx
• Identify a UD and an interpretation of Ax and Bx
  so that the premises are true but the conclusion
  is false.
• In this case, interpret A first (because if the
  2nd premise is true, the first one will be as
  well) using a predicate that has no extension
  then interpret B as a predicate that does.
• UD the set of all things
• Ax x is a unicorn
• Bx x is a mammal
• As there are no unicorns, the premises are true
  but as there are mammals the conclusion is false.

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• b. Show that the following argument is
  quantificationally valid
• (?x) (Px ? Ex)
• ----------------------
• (?x) (Px Ex)
• Reason this way Part 1 If the premise is true,
  then it is not the case that each thing in the
  domain is such that if it is P, it is E.
• So the conclusion follows there is something in
  the domain that is P and is not E.

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• b. Show that the following argument is
  quantificationally valid
• (?x) (Px ? Ex)
• ----------------------
• (?x) (Px Ex)
• Part 2 If the premise is false , then the
  argument is also valid.
• As the premise must be true or false, and if its
  true so is the conclusion, and if its false then
  it is not possible for the premise to be true and
  the conclusion false, the argument is valid and
  because we assumed no particular interpretation,
  it is quantificationally valid.

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• Show that the following sentences are not
  quantificationally equivalent.
• (?y) By
• (?y) By
• Here we can use interpretations, but we need two.
  We need to identify an interpretation on which
  one is true and an interpretation on which the
  other is false.
• So consider some predicate that doesnt apply to
  everything, but does apply to some things and
  choose a UD accordingly.

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• (?y) By
• (?y) By
• 1. UD the set of living things
• By y is a mammal
• On this interpretation, sentence one is true and
  sentence two is false.
• So the sentences are not quantificationally
  equivalent. Another interpretation to show this
• 2. UD the set of positive integers
• By y is even.
• On this interpretation, sentence one is true and
  sentence two is false.
• So the sentences are not quantificationally
  equivalent.

23

• Show that the following sentences are
  quantificationally equivalent.
• (?x) (Wx ? Mx)
• (?x) (Wx Mx)
• Again, as this is a claim that covers an infinite
  number of interpretations, we have to demonstrate
  it by reasoning.
• Whatever the UD, and whatever the interpretations
  of W and M, the first sentence says that anything
  that is a W is an M. The second sentence says
  there is nothing that is both a W and an M.

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• (?x) (Wx ? Mx)
• (?x) (Wx Mx)
• Suppose the first sentence is true on some
  interpretation. Then every member of the UD which
  is W is also M. So no member is both W and M, so
  the second sentence is true.
• Suppose that the first sentence is false on some
  interpretation. Then some member of the UD is W
  but not M. So the second sentence is also false
  (because (?x) Wx Mx) is true on that
  interpretation).

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• Finally, consider quantificational consistency.
• Here, unlike some earlier cases, to show that
  some set of sentences is quantificationally
  consistent, we need only identify one
  interpretation on which all the members of the
  set are true.
• But to show that some set is quantificationally
  inconsistent, we need to use reasoning to show
  that it is not possible for all the members of
  the set to be true on any interpretation.

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• Quantificational consistency.
• a. Show that the following set is
  quantificationally consistent
• (?y) (Ey ? Oy), (?x) (Ex Dx), (?w) (Ow
  Dw)
• So we need an interpretation on which each
  sentence is true. The existentially quantified
  sentences might be the best to begin with (an
  interpretation on which both are true and an
  appropriate UD.

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• (?y) (Ey ? Oy), (?x) (Ex Dx), (?w) (Ow
  Dw)
• Again, it is often useful to try a UD of positive
  integers.
• UD the set of positive integers
• Ex x is even
• Dx x is evenly divisible by 2
• Ox x is odd
• The two existentially quantified sentences are
  true.
• And so it turns out is the universally quantified
  sentence.
• So weve shown that the set is quantificationally
  consistent.

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• Suppose were asked to show that the following
  set is not quantificationally consistent.
• (?x) (?y) Gxy, (?w) (?z) Gzw
• We cannot check every interpretation to
  demonstrate that there is none on which all the
  members of the set are true. So we need to use
  reasoning.
• For whatever UD, and whatever interpretation of
  G, the first sentence says that for any pair x
  and y, it is not the case that x bears the
  relationship G to y.

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• (?x) (?y) Gxy, (?w) (?z) Gzw
• If this is true, then the second sentence is
  false for it says that there is some z that bears
  the relationship G to w.
• And if the first sentence is false, then the
  second sentence is true.
• So there is no interpretation on which both
  sentences can be true and the set is
  quantificationally inconsistent.