Sentential Logic

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Sentential Logic

• Symbolization and Syntax

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Complex Symbolizations

• Key
• A Arsenal will win a trophy.
• B Bolton will win a trophy.
• C Chelsea will win a trophy.

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Complex Symbolizations

• We already now know how to symbolize sentences
  such as
• Either Chelsea or Bolton will win a trophy.
• C v B
• Neither Chelsea nor Arsenal will win a trophy.
• (C v A)
• Not both Arsenal and Bolton will win a trophy.
• (A B)

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Complex Symbolizations

• How about
• At least one of them will win a trophy.
• Either Arsenal or Bolton or Chelsea will win a
  trophy.
• A v (B v C)

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Complex Symbolizations

• How about
• None of them will win a trophy.
• Neither Arsenal nor Bolton nor Chelsea will win a
  trophy.
• (A v (B v C))

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Complex Symbolizations

• How about
• They all wont win a trophy.
• Its not the case that Arsenal and Bolton and
  Chelsea will win a trophy.
• (A (B C))

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Complex Symbolizations

• So far we have seen
• At least one
• None
• Not all

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Complex Symbolizations

• Lets try
• At most one
• Exactly one
• At least two
• At most two
• Exactly two

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Complex Symbolizations

• At most, one team will win a trophy.
• What does this mean? What are the conditions
  under which this claim is true? When is it
  false?
• (A B) ((A C) (B C))

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Complex Symbolizations

• Exactly one team will win a trophy.
• This could be read as saying that at least one
  team will win a trophy, and at most one team.
• (A v (B v C)) ((A B) ((A C) (B C)))

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Complex Symbolizations

• At least two teams will win a trophy.
• Just disjoin the possible pairs of winners
• (A B) v ((A C) v (B C))

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Complex Symbolizations

• At most, two teams will win a trophy.
• Isnt this equivalent to saying that at least one
  of the teams wont win a trophy? Or that they
  wont all win a trophy?
• A v (B v C)
• (A (B C))

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Complex Symbolizations

• Exactly two teams will win a trophy.
• As is exactly one, we combine at least two
  and at most two.
• ((A B) v ((A C) v (B C))) (A v (B v C))