4.7: Summary
- Page ID
- 62290
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- First-Order Logic includes all of Propositional Logic, plus the quantifiers \(\forall\) and \(\exists\).
- Translating between English and First-Order Logic.
- The equals sign (=) is automatically included in every symbolization key.
- The order of the quantifiers is important, because it can change the meaning of an assertion.
- Uniqueness (\(\exists !\))
- Every variable in an assertion must be bound by a quantifier.
- Rules for negating quantifiers:
- the negation of a “\(\forall\)” assertion is a “\(\exists\)” assertion;
- the negation of a “\(\exists\)” assertion is a “\(\forall\)” assertion;
- Any assertion about all elements of \(\varnothing\) is “vacuously” true.
- Introduction and elimination rules for quantifiers.
- Just as in Propositional Logic:
- To show that a deduction is valid, provide a proof.
- To show that a deduction is not valid, provide a counterexample.
- Notation:
- \(\forall x\) (universal quantifier; means “For all \(x\)”)
- \(\forall x \in X\) (universal quantifier; means “For all \(x\) in \(X\)”)
- \(\exists x\) (existential quantifier; means “There exists some \(x\), such that. . . ”)
- \(\exists x \in X\) (existential quantifier; means “There exists some \(x\) in \(X\), such that. . . ”)
- \(\exists ! x\) (means “There is a unique \(x\), such that. . . ”)
- \(\exists ! x \in X\) (means “There is a unique \(x\) in \(X\), such that. . . ”)