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Unformatted text preview: Yuri Balashov, PHIL 2500 Lecture Notes Quantifiers UD: Andrea, Bentley, Charles, and Deirdre Bx: x is beautiful a: Andrea Ix: x is intelligent b: Bentley Rx: x is rich c: Charles Axy: x is attracted to y d: Deirdre Dxy: x despises y Lxy: x loves y Sxy: x is shorter than y Everybody is intelligent: (Ia & Ib) & (Ic & Id) ? ↓ Everything (in the Universe of Discourse) is such that it is intelligent. → Each x is such that x is intelligent. → → ( ∀ x) Ix Someone is rich: (Ia ∨ Ib) ∨ (Ic ∨ Id) ? ↓ At least one thing (in UD) is such that it is rich → At least one x is such that x is rich → ( ∃ x) Rx • The variable in ‘( ∀ x)’ and ‘Ix’ can be any variable (i.e., ‘x’, ‘y’, ‘z’, or ‘w’), but it must be the same variable in both cases. • Thus ‘( ∀ x) Ix’ and ‘( ∀ y) Iy’ say the same: that every single thing in UD is rich. • ‘( ∀ x) Iy’ is unacceptable (a syntactical error). 1 Yuri Balashov, PHIL 2500 Lecture Notes • Quantifiers ‘( ∀ x)’ and ‘( ∃ x)’ apply to expressions of PL with a free variable (e.g., ‘Ix’). • When so applied, a quantifier binds that variable. • If it was the only free variable in the expression, the quantifier, by binding that variable, turns the expression into a sentence of PL. ‘( ∃ x)’ + ‘Rx’ = ‘( ∃ x) Rx’ Quantifier Expression of PL Sentence of PL With a free variable Not a sentence of PL Everyone is either rich or intelligent: ( ∀ y) (Ry ∨ Iy) scope of quantifier ‘( ∀ y) (Ry ∨ Iy)’ is a quantified sentence...
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- Spring '08
- Logic, Yuri Balashov