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# Notation s t indicates that s and t are logically

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Unformatted text preview: denoting “x is mortal”. Specify the domain as all people. ༉  The two premises are: ༉  The conclusion is: ༉  Later we will show how to prove that the conclusion follows from the premises. Equivalences in Predicate Logic ༉  Statements involving predicates and quantiﬁers are logically equivalent if and only if they have the same truth value ༉  for every predicate substituted into these statements and ༉  for every domain of discourse used for the variables. ༉  Notation S ≡T indicates that S and T are logically equivalent. ༉  Example: ∀x ¬¬S(x) ≡ ∀x S(x) Quan\$ﬁers: Conjunc\$ons and Disjunc\$ons ༉  If the domain is ﬁnite: ༉  A universally quantiﬁed proposition is equivalent to a conjunction of propositions without quantiﬁers. ༉  An existentially quantiﬁed proposition is equivalent to a disjunction of propositions without quantiﬁers. ༉  If domain consists of the integers 1,2, and 3: ༉  Even if the domains are inﬁnite, you can still think of the quantiﬁers in this fashion, but the equivalent expressions without quantiﬁers will be inﬁnitely long. Nega\$ng Quan\$ﬁed Expressions ༉  Consider ∀x J(x) “Every student in your class has taken a course in Java.” Here J(x) is “x has taken a course in Java” and the domain is students in your class. ༉  Negating the original statement gives “It is not the case that every student in your class has taken Java.” ༉  This implies that “There is a student in your...
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