Inference Rules for Quantified Propositions

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Inference Rules for Quantified Propositions
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Universal Specification

• If a statement of the form ?x, P(x) is true, then
  P(c) is true for arbitrary c in the universe of
  discourse.
• This can be written
• ?x, P(x)
• ________
• ? P(c) for all c.
• Example M(x) x is mortal. From ?x, M(x), we
  infer that Socrates is mortal.

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Universal Generalization

• If a statement P(c) is true for each element c of
  the universe, then ?x, P(x).
• This can be written
• P(c) for all c
• ________
• ? ?x, P(x).

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Example of Universal Generalization

• Prove if R is an antisymmetric relation, so is
  R-1.
• Let xRy be an arbitrary element of R where x ? y.
• yR-1x. (Defn. of inverse relation.)
• (y,x) ? R. (R is antisymmetric)
• (x,y) ? R-1. (Step 3 defn. of inverse
  relation.)
• ?(x,y), (xR-1y ? yR-1x) ? x y. (UG)
• R-1 is antisymmetric. (Step 5 defn. of
  antisymmetric)

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Existential Specification

• If ?x, P(x) is true then there is an element c
  such that P(c) is true.
• This can be written
• ?x, P(x)
• ________
• ? P(c), for some c.
• Element c is not arbitrary.
• We know only that some c satisfies P.
• We do not necessarily know which one (e.g., from
  a non-constructive proof).

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Existential Generalization

• If P(c) is true for some c, then ?x, P(x).
• This can be written
• P(c), for some c
• ________
• ? ?x, P(x).

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Protocol

• To infer from quantified premises
• Properly remove quantifiers.
• Use existential or universal specification
• Argue with the resulting propositions.
• Properly prefix the correct quantifiers.
• Use existential universal generalization

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Example arguments

• All humans are fallible.
• All government agents are human.
• Therefore, all government agents are fallible.
• ______________________
• H(x) x is a human.
• F(x) x is fallible.
• G(x) x is a government agent.

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• ?x, ( H(x) ? F(x) ).
• ?x, ( G(x) ? H(x) ).
• _________________
• ?x, ( G(x) ? F(x) ).

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• Proof
• 1. ?x, ( H(x) ? F(x) ) premise 1
• 2. H(c) ? F(c) step 1 U.S.
• 3. ?x, ( G(x) ? H(x) ). premise 2
• 4. G(c) ? H(c) step 3 U.S.
• 5. G(c) ? F(c) steps 2, 4, transitivity
• 6. ?x, ( G(x) ? F(x) ). step 5 U.G.

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• In English
• All CCS classes are easy.
• This is a CCS class.
• Therefore, this class is easy.
• A more compact representation
• ?x, C(x) ? E(x).
• C(CCS CS 2).
• Therefore, E(CCS CS 2).

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Proof

• 1. ?x, C(x) ? E(x) premise 1
• 2. C(CCS CS 2) ? E(CCS CS 2) step 1, U.S.
• 3. C(CCS CS 2) premise 2
• 4. E(CCS CS 2) step 2, 3, modus
  ponens

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Characters

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