7.4-7.5 - Yuri Balashov, PHIL 2500 Lecture Notes...

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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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7.4-7.5 - Yuri Balashov, PHIL 2500 Lecture Notes...

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