# unit2 - Hardegree Intermediate Logic Unit 2 Derivations in...

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Unformatted text preview: Hardegree, Intermediate Logic ; Unit 2: Derivations in Function Logic 1 of 14 UNIT 2: DERIVATIONS IN FUNCTION LOGIC 1. EXERCISE SET A (Review of Predicate Logic) For each of the following arguments, construct a formal derivation of the conclusion (indicated by "/") from the premises. 1. f x(Fx ² Gx) / f xFx ² f xGx 2. f x(Fx ² Gx) ; ~F x(Gx & Hx) / ~F x(Fx & Hx) 3. F xFx ² F x ~ Gx / f x[Fx ² ~f xGx] 4. f x[Fx ² ~f xGx] / F xFx ² F x ~ Gx 5. F x ~F y(Fy & Ryx) / f x(Fx ² F y ~ Rxy) 6. f x F yRxy ; f x f y[Rxy ² Rxx] ; f x[Rxx ² f yRyx] / f x f yRxy 7. f x F yRxy ; f x[ F yRxy ² F yRyx] ; f x f y[Ryx ² f yRyx] / f x f yRxy 2. EXERCISE SET B 1.A. For each of the following three formulas, list the formula that is obtained by a single application of the rule Universal Elimination ( f O), using the singular term ‘ f(a) ’. 1.B. For each of the following three formulas, list the formula that is obtained by successive applications of the rule Universal Elimination ( f O), substituting ‘ f(a) ’ for ‘ x ’ and ‘ s(a,b) ’ for ‘ y ’. (1) f x f y(Rxy ² Rxx) (2) f x f y{(Nx & Ny) ² N[p(x,y)]} (3) f x f y(R[x,m(y)] ² R[m(y),x]) 2.A. For each of the following three formulas, list every (non-trivial) formula that can be obtained by a single application of the rule Existential Introduction ( F I). (1) R[a , f(a)] (2) R[f(a) , f(a)] (3) R[s(a) , p(a,b)] 2.B. For each formula obtained in 2.A, list every (non-trivial) formula that can be obtained by one further application of F I. Hardegree, Intermediate Logic ; Unit 2: Derivations in Function Logic 2 of 14 3. EXERCISE SET C Directions For All Derivation Exercises . For each of the following, construct a formal derivation of the conclusion (indicated by "/") from the premises (if any). In cases in which two formulas are separated by '//', construct a derivation of each formula from the other. 1. P[f(b)] / f xP[f(x)] 2. P[f(b)] / f xPx 3. f xF[f(x)] / f xFx 4. F x(Fx ² Hx) ; F[p(a,b)] / H[p(a,b)] 5. F x(Fx ² Hx) ; F[p(a,b)] / f xHx 6. F x(Fx ² Hx) ; ~ H[f(a)] / ~ F[f(a)] 7. F x(Fx ² Hx) ; ~ H[f(a)] / ~F xFx 8. F x(Fx ² Gx) ; F x[(Fx & Gx) ² ~ Hx] ; H[p(a,b)] / f x(Hx & ~ Fx) 9. F x(Fx ² Gx) ; F x(Gx ² Hx) ; F[m(a)] / f x(Gx & Hx) 10. F x(Fx ² Gx) ; F x(Gx ² Hx) ; F[m(a)] / f xG[m(x)] & f xH[m(x)] 11. F x{R[x,s(a)] ² ~ R[x,p(a,b)]} ; R[s(a),s(a)] / f x ~ R[x,p(a,b)] 12. f xR[x,f(n)] ² F xR[x,f(n)] ; ~ R[g(n),f(n)] / ~ R[f(n),f(n)] 13. F x(Fx ² Rxx) ; F[p(c,d)] / f xR[x,p(c,d)] 14. F x(Fx ² Rxx) ; F x ~ R[m(a),x] / ~ F[m(a)] 15. F x{Fx ² F yRxy} ; F[g(a)] / R[g(a),g(a)] 16. f xR[f(a),x] ² F xR[x,f(a)] ; ~ R[g(b),f(a)] / ~ R[f(a),f(a)] Hardegree, Intermediate Logic ; Unit 2: Derivations in Function Logic 3 of 14 4. EXERCISE SET D 17. f x[ F yRxy ² f yRyx] ; R[f(a,b),f(a,b)] / R[g(a,b),f(a,b)] 18. f x(Rxx ² Fx) ; f x f y(Rxy ² Rxx) ; ~ F[g(a,b)] / ~ R[g(a,b),b] 19. f x(Fx ² Gx) ; f xM[m(x),x)] / f x{Fx ² F y{Gy & M[m(x),y]}} 20. f x f y{(Nx & Ny) ² N[p(x,y)]} ; F x{Nx &...
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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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unit2 - Hardegree Intermediate Logic Unit 2 Derivations in...

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