March 10: Quantificational Notions

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March 10 Quantificational Notions

• Interpretations Every interpretation interprets
  every individual constant, predicate, and
  sentence letter of PL. Our partial interpretation
  may not include some predicates or individual
  constants, but it does interpret them.
• Lx, Lxy, Lxyz, Lwxyz
• a, b, c, d, a1, b2
• Full interpretations are infinitely long as PL
  includes an infinite of predicates, an infinite
  of individual constants, an infinite of
  sentence letters and an infinite of UDs.

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March 10 Quantificational Notions

• We use the concept of an interpretation to
  specify the quantificational counterparts of
  truth-functional concepts.
• Individual sentences of PL fall into 1 of 3
  categories
• A sentence P of PL is quantificationally true IFF
  P is true on every interpretation.
• A sentence P of PL is quantificationally false
  IFF P is false on every interpretation.
• A sentence P is quantificationally indeterminate
  IFF P is neither quantificationally true nor
  quantificationally false.

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March 10 Quantificational Notions

• A sentence P of PL is quantificationally true IFF
  P is true on every interpretation.
• How could we demonstrate that some sentence P is
  quantificationally true given that we cannot
  check all of the infinite interpretations of that
  sentence? We cannot do it with all sentences, but
  we can do it for some using reasoning.
• Consider for example
• (?y) (Cy v Cy)

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• (?y) (Cy v Cy)
• We may reason as follows
• Because the sentence is existentially quantified,
  it is true on an interpretation just in case at
  least one member of the UD satisfies the
  conditions specified by Cy v Cy that is, if
  at least one member of the UD either is C or is
  not C.
• Without knowing what the interpretation of C is,
  we know that every member of a UD satisfies this
  condition for every interpretation interprets
  C, and every member of the UD is or is not in the
  extension of C.
• And because, by definition, every interpretation
  has a nonempty set as its UD, then the UD for any
  interpretation has at least one member and hence
  at least one member that satisfies the open
  sentence. So the sentence is quanticationally
  true.

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• In general, to show that some sentence P is
  quantificationally true, we may use reasoning to
  show that no matter what the UD is and no matter
  how the individual constants, predicates, and
  sentence letters are interpreted, the sentence
  always turns out to be true. So consider
• (?x) (?y) Bxy ? (?y) ( ?x) Bxy
• Whatever the UD and however B is interpreted, we
  know that the antecedent is either true or false.
  If it is true, so is the consequent. If the
  antecedent is false, the sentence is true. So the
  sentence is quantificationally true.

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• A sentence P is quantificationally false IFF P is
  false on every interpretation.
• (?x) Bx (?z) Bz
• is quantificationally false.
• If an interpretation makes the left conjunct
  true, then it makes the right conjunct false. If
  it makes the left conjunct false, then the
  sentence is false.
• As every interpretation includes B as a
  predicate, and however it interprets it, it
  cannot be the case that every member of the UD is
  in the extension of B but one member is not.

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• It is not always as easy to demonstrate that a
  sentence is quantificationally true or that it is
  quantificationally false.
• But we can often show that a sentence is not
  quantificationally true by showing that it is
  false on at least one interpretation.
• (?x) (Wx ? Mx) ? (?x) Mx
• 1. UD the set of all living things
• Wx x is a whale
• Mx x is a mammal
• The antecedent is true on this interpretation but
  the consequent is false.

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• We can often show that a sentence is not
  quantificationally false by showing that it is
  true on at least one interpretation.
• (?x) (?y) (Syx ? Sxy)
• 2. UD the set of positive integers
• Sxy x is smaller than y
• IF There is a positive integer x such that all
  positive integers are smaller than x
• The antecedent is false on this interpretation
  (as no positive integer is smaller than 1), so
  the sentence is true on this interpretation, and
  so the sentence is not quantificationally false.

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• We can often show that a sentence is not
  quantificationally false by showing that it is
  true on at least one interpretation.
• (?x) Ex (?x ) Ex
• 3. UD the set of positive integers
• Ex x is even
• Or consider
• (Ga (?y) Gy)
• To construct an interpretation on which this
  sentence is true (and thus not quantificationally
  false), we must construct an interpretation on
  which Ga (?y) Gy is false. And that means
  making one or the other of the conjuncts false.

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• (Ga (?y) Gy)
• We must construct an interpretation on which Ga
  (?y) Gy is false. And that means making one or
  the other of the conjuncts false.
• 4. UD Set of positive integers
• Gx x is even
• a 2
• Here the left conjunct is false and so is the
  conjunction

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• Again, we can show a sentence is not
  quantificationally true, or that it is not
  quantificationally false, by constructing a
  single interpretation that demonstrates this.
• But we cannot construct a single interpretation
  that will demonstrate that a sentence is
  quantificationally true or that it is
  quantificationally false. For some sentences, we
  can arrive at such a conclusion by reasoning but
  for many, we cannot.

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• A sentence P of PL is quantificationally
  indeterminate IFF P is neither quantificationally
  true nor quantificationally false.
• We show that a sentence is quantificationally
  indeterminate by constructing two
  interpretations one on which it is true (to show
  that it is not quantificationally false) and one
  on which it is false (to show that it is not
  quantificationally true).

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• A sentence P of PL is quantificationally
  indeterminate IFF P is neither quantificationally
  true not quantificationally false.
• We showed, using interpretation 4, that the
  following sentence is not quantificationally
  false by finding an interpretation on which it is
  true (Ga (?y) Gy). Now we need an
  interpretation on which it is false to show that
  it is also not quantificationally true.
• This means we need an interpretation on which Ga
  (?y) Gy is true.

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• This means we need an interpretation on which Ga
  (?y) Gy is true.
• 5. UD set of positive integers
• Gx x is odd
• a 2
• On this interpretation, Ga is true (for 2 is not
  odd) and (?y) Gy) is true (for there is at least
  one odd positive integer). So the conjunction is
  true, and its negation is false.
• Interpretations 4 and 5 demonstrate that the
  sentence (Ga (?y) Gy is quantificationally
  indeterminate.

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• Finding an interpretation on which a sentence is
  true or on which it is false takes some
  ingenuity.
• Determine if the sentence is one whose
  quantificational status can be settled by an
  interpretation or can be settled by reasoning (or
  by neither).
• Guidelines Examine the kind of sentence it is.
• If it is a truth-functional compound, use the
  truth conditions for that kind of compound.
• If the sentence is universally quantified, then
  the sentence is true IFF the condition specified
  after the quantifier is satisfied by all members
  of the UD you choose.

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• If the sentence is existentially quantified, then
  it will be true IFF the condition specified after
  the quantifier is satisfied by at least one
  member of the UD.
• Sometimes, the desired interpretation can not be
  found. For example
• If a sentence is quantificationally true, there
  is no interpretation on which it is false, and
  any attempt to construct an interpretation on
  which the sentence is false will fail.
• The same point holds for quantificationally false
  sentences.
• A set of positive integers is always a good
  choice for your UD as you construct
  interpretations.

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Quantificational notions

• An argument of PL is quantificationally valid IFF
  there is no interpretation on which every premise
  is true and the conclusion is false.
• An argument of PL is quantificationally invalid
  IFF the argument is not quantificationally valid.

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• (?x) (Fx v Gx)
• (?x) Fx
• ------------------
• (?x) Gx
• is quantificationally valid.
• Suppose that on some interpretation both premises
  are true. If the first premise is true, then some
  member x of the UD is either F or G.
• If the second premise is true, then no member of
  the UD is F. Therefore, because the member that
  is either F or G is not F, it must be G. Thus
  (?x) Gx will be true on any such interpretation.

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• Demonstrating that an argument is not
  guantificationally valid find an interpretation
  in which all of its premises are true and its
  conclusion is false. Consider
• (?x) (?y) Fy ? Fx
• (?y) Fy
• -----------------------
• (?x) Fx
• We can make the first premise true by
  interpreting F so that at least one member of the
  UD is in its extension for then that object
  will satisfy the condition specified by (?y) Fy
  ? Fx beause it will satisfy its consequent.

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• (?x) (?y) Fy ? Fx
• (?y) Fy
• -----------------------
• (?x) Fx
• The second premise will be true if at least one
  member of the UD is not in the extension of F. So
  F will have some, but not all, members of the UD
  in its extension. And because some members will
  be in the extension, the conclusion will be
  false.
• 6 UD set of positive integers
• Fx x is odd.

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• Again, there are asymmetries in what we can
  prove.
• We can prove an argument is valid only on
  specific interpretations, not all possible
  interpretations, although some we can prove are
  valid by reasoning.
• But we can prove an argument is
  quantificationally invalid by finding a single
  interpretation on which it is not valid (as to be
  quantificationally valid, an argument must be
  such that there is no interpretation on which all
  its premises are true and its conclusion is
  false).

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• Again, there are asymmetries in what we can
  prove.
• We can also prove an argument is not
  quantificationally invalid by finding an
  interpretation on which it is valid.
• Quantificational equivalency
• Sentences P and Q are quantificationally
  equivalent IFF there is no interpretation on
  which P and Q have different truth values.
• The following are quantificationally equivalent
• (?x) Fx ? Ga
• (?x) (Fx ? Ga)

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• (?x) Fx ? Ga
• (?x) (Fx ? Ga)
• We reason as follows
• Suppose that (?x) Fx ? Ga is true on some
  interpretation. Then (?x) Fx is either true or
  false on this interpretation.
• If (?x) Fx is true, then so is Ga. But then since
  Ga is true, every object x in the UD is such that
  if x is F, then a is G. So (?x) (Fx ? Ga) is true.

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• (?x) Fx ? Ga
• (?x) (Fx ? Ga)
• If (?x) Fx is false, then every object x in the
  UD is such that if x is F (which we assume here
  it is not), then a is G and the whole sentence is
  true. Again, (?x) (Fx ? Ga) is also true on
  that interpretation.

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• (?x) Fx ? Ga
• (?x) (Fx ? Ga)
• Now suppose that (?x) Fx ? Ga is false on some
  interpretation.
• Then (?x) Fx is true and Ga is false. But if (?x)
  Fx is true, then some object x in the UD is in
  the extension of F. This object does not satisfy
  the condition that if it is F (which it is), then
  a is G (which it is not on our present
  assumption). So (?x) (Fx ? Ga) is false if (?x)
  Fx ? Ga is false.
• Taken together with the result that if one is
  true, so is the other, we have demonstrated that
  the 2 sentences are quantificationally equivalent.

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Quantificational consistency

• A set of sentences of PL is quantificationally
  consistent IFF there is at least one
  interpretation on which all the members of the
  set are true.
• A set of sentences of PL is quantificationally
  inconsistent IFF the set is not
  quantificationally consistent.
• The set (?x) Bax, Bba v (?x) Bax is
  quantificationally consistent.

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• (?x) Bax, Bba v (?x) Bax
• 7. UD set of possible integers
• Bxy x is less than or equal to y
• a 1
• b 2
• On this interpretation (?x) Bax is true since 1
  is less than or equal to every positive integer.
• So, too, Bba is true since 2 is neither less
  than nor equal to 1, so Bba v (?x) Bax is true