unit3 - Hardegree, Intermediate Logic; Unit 3: Translations...

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Hardegree, Intermediate Logic ; Unit 3: Translations in Identity Logic 1 of 12 UNIT 3: TRANSLATIONS IN IDENTITY LOGIC 1. Translation Forms In Identity Logic; Simplified Versions 1. a is the only F 2200 x{Fx x = a} equ: Fa & ~ 5 x(x a & Fx) a and b are the only F's 2200 x{Fx [x = a x = b]} equ: Fa & Fb & ~ 5 x(x b & Fx) a,b,c are the only F's 2200 x{Fx [x = a x = b x = c]} equ: Fa & Fb Fc & ~ 5 x(x c & Fx) 2. there are at least two F's 5 x 5 y{x y & Fx & Fy} there are at least three F's 5 x 5 y 5 z{x z & Fx & Fy & Fz} there are at least four F's 5 w 5 x 5 y 5 z{w z & Fw & Fx & Fy & Fz} 3. there is at most one F 5 x 2200 y{Fy ² y = x} equ: ~ 5 x 5 y{x y & Fx & Fy} equ: 2200 x 2200 y{[Fx & Fy] ² x = y} there are at most two F's 5 x 5 y 2200 z{Fz ² [z = x z = y]} equ: ~ 5 x 5 y 5 z{x z & Fx & Fy & Fz} equ: 2200 x 2200 y 2200 z{[Fx & Fy & Fz] ² [x = y x = z y = z]} there are at most three F's 5 x 5 y 5 z 2200 w{Fw ² [w = x w = y w = z]} equ: ~ 5 w 5 x 5 y 5 z{w z & Fw & Fx & Fy & Fz} equ: 2200 w 2200 x 2200 y 2200 z{[Fw & Fx & Fy & Fz] ² [w = x w = y w = z x = y x = z y = z]} 4. there is exactly one F 5 x 2200 y{Fy y = x} there are exactly two F's 5 x 5 y{x y & 2200 z{Fz [z = x z = y]}} there are exactly three F's 5 x 5 y 5 z{x z & 2200 w{Fw [w = x w = y w = z]} 5. there is exactly one F, which is G 5 x{ 2200 y{Fy y = x} & Gx} there is exactly one F that is G; there is exactly one thing that is both F and G 5 x 2200 y{[Fy & Gy] y = x} there is exactly one thing, which is both F and G 5 x{ 2200 y[y = x] & [Fx & Gx]
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Hardegree, Intermediate Logic ; Unit 3: Translations in Identity Logic 2 of 12 2. Translation Forms In Identity Logic; General Versions 1. is/are the only a is the only v such that [ v ] 2200 v { [ v ] v=a } a and b are the only v such that [ v ] 2200 v { [ v ] ( v=a v " b } a , b , c are the only v such that [ v ] 2200 v { [ v ] ( v=a v " b v " c )} e.g., a is the only x such that Fx 2200 x { Fx x " a } 2. at least 1, at least 2, etc. there is at least 1 v such that [ v ] 5 v [ v ] there are at least 2 v such that [ v ] 5 v 1 5 v 2 { v 1 2 & [ v 1 ] & [ v 2 ]} there are at least 3 v such that [ v ] 5 v 1 5 v 2 5 v 3 { v 1 2 & v 1 3 v 2 v 3 & [ v 1 ] & [ v 2 ] & [ v 3 ]} e.g., there are at least 2 x such that Fx 5 x 1 5 x 2 { x 1 x 2 & Fx 1 & Fx 2 } 3. at most 1, at most 2, etc. there is at most 1
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This note was uploaded on 04/09/2008 for the course PSYCH 201 taught by Professor Hardegree during the Spring '08 term at UMass (Amherst).

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unit3 - Hardegree, Intermediate Logic; Unit 3: Translations...

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