;(in-package "OSCAR")
(setf *problems*
(eval-when (:compile-toplevel :execute)
(make-problem-list
"
Problem #1
This is a case of collective rebutting defeat
Given premises:
P justification = 1.0
A justification = 1.0
Ultimate epistemic interests:
R interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> Q strength = 1.0
pf-reason_2: {Q} ||=> R strength = 1.0
pf-reason_3: {C} ||=> ~R strength = 1.0
pf-reason_4: {B} ||=> C strength = 1.0
pf-reason_5: {A} ||=> B strength = 1.0
Problem #2
This is the same as #1 except that some reasons are backwards.
Given premises:
P justification = 1.0
A justification = 1.0
Ultimate epistemic interests:
R interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> Q strength = 1.0
pf-reason_2: {Q} ||=> R strength = 1.0
pf-reason_3: {A} ||=> B strength = 1.0
BACKWARDS PRIMA FACIE REASONS
pf-reason_4: {} {C} ||=> ~R strength = 1.0
pf-reason_5: {} {B} ||=> C strength = 1.0
Problem #3
Figure 2
Given premises:
A justification = 1.0
B justification = 1.0
C justification = 1.0
Ultimate epistemic interests:
J interest = 1.0
K interest = 1.0
L interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A} ||=> D strength = 1.0
pf-reason_2: {D} ||=> G strength = 1.0
pf-reason_3: {B} ||=> E strength = 1.0
pf-reason_4: {C} ||=> F strength = 1.0
pf-reason_5: {I} ||=> L strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {G} ||=> J strength = 1.0
con-reason_2: {E} ||=> H strength = 1.0
con-reason_3: {H} ||=> K strength = 1.0
con-reason_4: {F} ||=> I strength = 1.0
con-reason_5: {F} ||=> (B @ E) strength = 1.0
con-reason_6: {H} ||=> (D @ G) strength = 1.0
Problem #4
Figure 3
Given premises:
A justification = 1.0
B justification = 1.0
Ultimate epistemic interests:
J interest = 1.0
K interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A} ||=> D strength = 1.0
pf-reason_2: {D} ||=> G strength = 1.0
pf-reason_3: {B} ||=> ~D strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {G} ||=> J strength = 1.0
con-reason_2: {~D} ||=> H strength = 1.0
con-reason_3: {H} ||=> K strength = 1.0
Problem #5
Figure 4
Given premises:
A justification = 1.0
B justification = 1.0
C justification = 1.0
Ultimate epistemic interests:
J interest = 1.0
K interest = 1.0
L interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A} ||=> D strength = 1.0
pf-reason_2: {D} ||=> G strength = 1.0
pf-reason_3: {B} ||=> ~D strength = 1.0
pf-reason_4: {C} ||=> D strength = 1.0
pf-reason_5: {I} ||=> L strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {G} ||=> J strength = 1.0
con-reason_2: {~D} ||=> H strength = 1.0
con-reason_3: {H} ||=> K strength = 1.0
con-reason_4: {D} ||=> I strength = 1.0
Problem #6
Figure 5
Given premises:
A justification = 1.0
B justification = 1.0
C justification = 1.0
Ultimate epistemic interests:
J interest = 1.0
K interest = 1.0
L interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A} ||=> D strength = 1.0
pf-reason_2: {D} ||=> G strength = 1.0
pf-reason_3: {B} ||=> ~D strength = 1.0
pf-reason_4: {C} ||=> F strength = 1.0
pf-reason_5: {I} ||=> L strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {G} ||=> J strength = 1.0
con-reason_2: {~D} ||=> H strength = 1.0
con-reason_3: {H} ||=> K strength = 1.0
con-reason_4: {F} ||=> I strength = 1.0
con-reason_5: {~D} ||=> M strength = 1.0
con-reason_6: {M} ||=> N strength = 1.0
con-reason_7: {N} ||=> (C @ F) strength = 1.0
con-reason_8: {F} ||=> (B @ ~D) strength = 1.0
Problem #7
Figure 7 -- self-defeat
Given premises:
P justification = 1.0
Q justification = 1.0
S justification = 1.0
Ultimate epistemic interests:
T interest = 1.0
(R v ~T) interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> R strength = 1.0
pf-reason_2: {Q} ||=> ~R strength = 1.0
pf-reason_3: {S} ||=> T strength = 1.0
Problem #8
Figure 8 -- the lottery paradox paradox
Given premises:
P justification = 1.0
Ultimate epistemic interests:
~T1 interest = 1.0
~T2 interest = 1.0
~T3 interest = 1.0
R interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {R} ||=> ~T1 strength = 1.0
pf-reason_2: {R} ||=> ~T2 strength = 1.0
pf-reason_3: {R} ||=> ~T3 strength = 1.0
pf-reason_4: {P} ||=> R strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {R , ~T1 , ~T2} ||=> T3 strength = 1.0
con-reason_2: {R , ~T2 , ~T3} ||=> T1 strength = 1.0
con-reason_3: {R , ~T1 , ~T3} ||=> T2 strength = 1.0
con-reason_4: {~T1 , ~T2 , ~T3} ||=> ~R strength = 1.0
Problem #9
Figure 8 -- the lottery paradox paradox using logic
Given premises:
P justification = 1.0
Ultimate epistemic interests:
~T1 interest = 1.0
~T2 interest = 1.0
~T3 interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {R} ||=> ~T1 strength = 1.0
pf-reason_2: {R} ||=> ~T2 strength = 1.0
pf-reason_3: {R} ||=> ~T3 strength = 1.0
pf-reason_4: {P} ||=> R strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {R} ||=> (T1 v (T2 v T3)) strength = 1.0
Problem #10
Figure 9 -- No nearest defeasible ancestor is defeated.
Given premises:
P justification = 1.0
Ultimate epistemic interests:
R interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> Q strength = 1.0
pf-reason_2: {Q} ||=> R strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {R} ||=> (P @ Q) strength = 1.0
Problem #11
figure 10 -- Robert and the pink elephant.
Given premises:
P justification = 1.0
Q justification = 1.0
R justification = 1.0
Ultimate epistemic interests:
U interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P , Q} ||=> S strength = 1.0
pf-reason_2: {R} ||=> T strength = 1.0
pf-reason_3: {S} ||=> U strength = 1.0
pf-reason_4: {V} ||=> ((P & Q) @ S) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {T , U} ||=> V strength = 1.0
Problem #12
figure 11 -- a simple case of ancestor defeat.
Given premises:
P justification = 1.0
Q justification = 1.0
R justification = 1.0
Ultimate epistemic interests:
W interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> S strength = 1.0
pf-reason_2: {S} ||=> U strength = 1.0
pf-reason_3: {Q} ||=> T strength = 1.0
pf-reason_4: {R} ||=> W strength = 1.0
pf-reason_5: {V} ||=> (S @ U) strength = 1.0
pf-reason_6: {U} ||=> (R @ W) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {S , T} ||=> V strength = 1.0
Problem #13
figure 12 -- a more complicated case of ancestor defeat.
Given premises:
P justification = 1.0
Q justification = 1.0
R justification = 1.0
X justification = 1.0
Ultimate epistemic interests:
W interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> S strength = 1.0
pf-reason_2: {S} ||=> U strength = 1.0
pf-reason_3: {Q} ||=> T strength = 1.0
pf-reason_4: {R} ||=> W strength = 1.0
pf-reason_5: {X} ||=> ~S strength = 1.0
pf-reason_6: {V} ||=> (S @ U) strength = 1.0
pf-reason_7: {U} ||=> (R @ W) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {S , T} ||=> V strength = 1.0
Problem #14
figure 13 -- a still more complicated case of ancestor defeat.
Given premises:
P justification = 1.0
Q justification = 1.0
R justification = 1.0
X justification = 1.0
Ultimate epistemic interests:
W interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P} ||=> S strength = 1.0
pf-reason_2: {S} ||=> U strength = 1.0
pf-reason_3: {Q} ||=> T strength = 1.0
pf-reason_4: {R} ||=> W strength = 1.0
pf-reason_5: {S} ||=> Y strength = 1.0
pf-reason_6: {X} ||=> ~S strength = 1.0
pf-reason_7: {V} ||=> (S @ U) strength = 1.0
pf-reason_8: {U} ||=> (R @ W) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {Y , T} ||=> V strength = 1.0
Problem #15
figure 14 -- a three-membered defeat cycle.
Given premises:
A justification = 1.0
P justification = 1.0
R justification = 1.0
T justification = 1.0
Ultimate epistemic interests:
B interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A} ||=> B strength = 1.0
pf-reason_2: {P} ||=> Q strength = 1.0
pf-reason_3: {R} ||=> S strength = 1.0
pf-reason_4: {T} ||=> U strength = 1.0
pf-reason_5: {Q} ||=> (R @ S) strength = 1.0
pf-reason_6: {S} ||=> (T @ U) strength = 1.0
pf-reason_7: {U} ||=> (P @ Q) strength = 1.0
pf-reason_8: {Q} ||=> (A @ B) strength = 1.0
Problem #16
figure 18 -- the paradox of the preface.
Given premises:
P1 justification = 1.0
P2 justification = 1.0
P3 justification = 1.0
S justification = 1.0
T justification = 1.0
Ultimate epistemic interests:
(Q1 & (Q2 & Q3)) interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P1} ||=> Q1 strength = 1.0
pf-reason_2: {P2} ||=> Q2 strength = 1.0
pf-reason_3: {P3} ||=> Q3 strength = 1.0
pf-reason_4: {S} ||=> R strength = 1.0
pf-reason_5: {T} ||=> ~(Q1 & (Q2 & Q3)) strength = 1.0
pf-reason_6: {S1} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
pf-reason_7: {S2} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
pf-reason_8: {S3} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_1: {Q1 , Q2} ||=> (Q1 & Q2) strength = 1.0
con-reason_2: {Q2 , Q3} ||=> (Q2 & Q3) strength = 1.0
con-reason_3: {Q1 , Q3} ||=> (Q1 & Q3) strength = 1.0
con-reason_4: {R , (Q1 & Q3)} ||=> S2 strength = 1.0
con-reason_5: {R , (Q2 & Q3)} ||=> S1 strength = 1.0
con-reason_6: {R , (Q1 & Q2)} ||=> S3 strength = 1.0
con-reason_7: {(Q1 & Q2) , ~(Q1 & (Q2 & Q3))} ||=> ~Q3 strength = 1.0
con-reason_8: {(Q2 & Q3) , ~(Q1 & (Q2 & Q3))} ||=> ~Q1 strength = 1.0
con-reason_9: {(Q1 & Q3) , ~(Q1 & (Q2 & Q3))} ||=> ~Q2 strength = 1.0
BACKWARDS CONCLUSIVE REASONS
con-reason_11: {} {Q1 , Q2 , Q3} ||=> (Q1 & (Q2 & Q3)) strength = 1.0
Problem #17
figure 18 -- the paradox of the preface, using logic.
Given premises:
P1 justification = 1.0
P2 justification = 1.0
P3 justification = 1.0
S justification = 1.0
T justification = 1.0
Ultimate epistemic interests:
(Q1 & (Q2 & Q3)) interest = 1.0
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {P1} ||=> Q1 strength = 1.0
pf-reason_2: {P2} ||=> Q2 strength = 1.0
pf-reason_3: {P3} ||=> Q3 strength = 1.0
pf-reason_4: {S} ||=> R strength = 1.0
pf-reason_5: {T} ||=> ~(Q1 & (Q2 & Q3)) strength = 1.0
pf-reason_6: {S1} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
pf-reason_7: {S2} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
pf-reason_8: {S3} ||=> (T @ ~(Q1 & (Q2 & Q3))) strength = 1.0
FORWARDS CONCLUSIVE REASONS
con-reason_4: {R , Q1 , Q3} ||=> S2 strength = 1.0
con-reason_5: {R , Q2 , Q3} ||=> S1 strength = 1.0
con-reason_6: {R , Q1 , Q2} ||=> S3 strength = 1.0
Problem #18
This uses contradiction-inversion.
Given premises:
B justification = 1
A justification = 1
C justification = 1
Ultimate epistemic interests:
Q interest = .7
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A , B} ||=> P strength = .7
pf-reason_2: {C} ||=> ~Q strength = .8
FORWARDS CONCLUSIVE REASONS
con-reason_1: {P} ||=> Q strength = 1.0
Problem #19
Four-way collective defeat.
Given premises:
A1 justification = 1
B1 justification = 1
C1 justification = 1
D1 justification = 1
Ultimate epistemic interests:
P1 interest = .7
Q1 interest = .7
R1 interest = .7
S1 interest = .7
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A1} ||=> P1 strength = .7
pf-reason_2: {B1} ||=> Q1 strength = .7
pf-reason_3: {C1} ||=> R1 strength = .7
pf-reason_4: {D1} ||=> S1 strength = .7
FORWARDS CONCLUSIVE REASONS
con-reason_1: {P1,Q1,R1} ||=> ~S1 strength = 1.0
con-reason_2: {P1,Q1,S1} ||=> ~R1 strength = 1.0
con-reason_3: {S1,Q1,R1} ||=> ~P1 strength = 1.0
con-reason_4: {P1,S1,R1} ||=> ~Q1 strength = 1.0
Problem #20
Two copies of four-way collective defeat.
Given premises:
A1 justification = 1
B1 justification = 1
C1 justification = 1
D1 justification = 1
A2 justification = 1
B2 justification = 1
C2 justification = 1
D2 justification = 1
Ultimate epistemic interests:
P1 interest = .7
Q1 interest = .7
R1 interest = .7
S1 interest = .7
P2 interest = .7
Q2 interest = .7
R2 interest = .7
S2 interest = .7
FORWARDS PRIMA FACIE REASONS
pf-reason_1: {A1} ||=> P1 strength = .7
pf-reason_2: {B1} ||=> Q1 strength = .7
pf-reason_3: {C1} ||=> R1 strength = .7
pf-reason_4: {D1} ||=> S1 strength = .7
pf-reason_5: {A2} ||=> P2 strength = .7
pf-reason_6: {B2} ||=> Q2 strength = .7
pf-reason_7: {C2} ||=> R2 strength = .7
pf-reason_8: {D2} ||=> S2 strength = .7
FORWARDS CONCLUSIVE REASONS
con-reason_1: {P1,Q1,R1} ||=> ~S1 strength = 1.0
con-reason_2: {P1,Q1,S1} ||=> ~R1 strength = 1.0
con-reason_3: {S1,Q1,R1} ||=> ~P1 strength = 1.0
con-reason_4: {P1,S1,R1} ||=> ~Q1 strength = 1.0
con-reason_5: {P2,Q2,R2} ||=> ~S2 strength = 1.0
con-reason_6: {P2,Q2,S2} ||=> ~R2 strength = 1.0
con-reason_7: {S2,Q2,R2} ||=> ~P2 strength = 1.0
con-reason_8: {P2,S2,R2} ||=> ~Q2 strength = 1.0
Problem #21
Given premises:
Ultimate epistemic interests:
((p -> q) <-> (~q -> ~p)) interest = 1.0
Problem #22
Given premises:
Ultimate epistemic interests:
(~~p <-> p) interest = 1.0
Problem #23
Given premises:
Ultimate epistemic interests:
(~(p -> q) -> (q -> p)) interest = 1.0
Problem #24
Given premises:
Ultimate epistemic interests:
((~p -> q) <-> (~q -> p)) interest = 1.0
Problem #25
Given premises:
Ultimate epistemic interests:
(((p v q) -> (p v r)) -> (p v (q -> r))) interest = 1.0
Problem #26
Given premises:
Ultimate epistemic interests:
(p v ~p) interest = 1.0
Problem #27
Given premises:
Ultimate epistemic interests:
(p v ~~~p) interest = 1.0
Problem #28
Given premises:
Ultimate epistemic interests:
(((p -> q) -> p) -> p) interest = 1.0
Problem #29
Given premises:
Ultimate epistemic interests:
(((p v q) & ((~p v q) & (p v ~q))) -> ~(~p v ~q)) interest = 1.0
Problem #30
Given premises:
(q -> r) justification = 1.0
(r -> (p & q)) justification = 1.0
(p -> (q v r)) justification = 1.0
Ultimate epistemic interests:
(p <-> q) interest = 1.0
Problem #31
Given premises:
Ultimate epistemic interests:
(p <-> p) interest = 1.0
Problem #32
Given premises:
Ultimate epistemic interests:
(((p <-> q) <-> r) <-> (p <-> (q <-> r))) interest = 1.0
Problem #33
Given premises:
Ultimate epistemic interests:
((p v (q & r)) <-> ((p v q) & (p v r))) interest = 1.0
Problem #34
Given premises:
Ultimate epistemic interests:
((p <-> q) <-> ((q v ~p) & (~q v p))) interest = 1.0
Problem #35
Given premises:
Ultimate epistemic interests:
((p <-> q) -> (~p v q)) interest = 1.0
Problem #36
Given premises:
Ultimate epistemic interests:
((p -> q) v (q -> p)) interest = 1.0
Problem #37
Given premises:
Ultimate epistemic interests:
(((p & (q -> r)) -> s) <-> ((~p v (q v s)) & (~p v (~r v s)))) interest = 1.0
Problem #41
Given premises:
(all x)(F x) justification = 1.0
Ultimate epistemic interests:
(some x)(F x) interest = 1.0
Problem #42
Given premises:
(some x)(all y)(F x y) justification = 1.0
Ultimate epistemic interests:
(all y)(some x)(F x y) interest = 1.0
Problem #43
Given premises:
(all x)((P x) -> ~(P x)) justification = 1.0
Ultimate epistemic interests:
~(P a) interest = 1.0
Problem #44
Given premises:
(all x)[(F x) -> ((H x) & ~(G x))] justification = 1.0
Ultimate epistemic interests:
((G a) -> ~(F a)) interest = 1.0
Problem #45
Given premises:
(all x)((H x) -> (G x)) justification = 1.0
Ultimate epistemic interests:
[((H a) -> (G a)) & ~(~(G b) & (H b))] interest = 1.0
Problem #46
Given premises:
(all x)[(P x) <-> ((H x) & ~(P x))] justification = 1.0
Ultimate epistemic interests:
(all x)~(H x) interest = 1.0
Problem #47
Given premises:
(all x)(F x) justification = 1.0
(all x)((F x) -> (G x)) justification = 1.0
Ultimate epistemic interests:
(all x)(G x) interest = 1.0
Problem #48
Given premises:
(F a) justification = 1.0
Ultimate epistemic interests:
(some x)((F x) v (G x)) interest = 1.0
Problem #49
Given premises:
(some x)(F x) justification = 1.0
Ultimate epistemic interests:
(some x)((F x) v (G x)) interest = 1.0
Problem #50
Given premises:
(some x)((F x) v (G x)) justification = 1.0
~(some x)(F x) justification = 1.0
Ultimate epistemic interests:
(some x)(G x) interest = 1.0
Problem #51
Given premises:
[(some x)(F x) -> (all y)(G y)] justification = 1.0
Ultimate epistemic interests:
(all x)(all y)[(F x) -> (G y)] interest = 1.0
Problem #52
Given premises:
(all x)[(F x) -> ((G x) -> (H x))] justification = 1.0
Ultimate epistemic interests:
[(all x)((F x) -> (G x)) -> (all x)((F x) -> (H x))] interest = 1.0
Problem #53
Given premises:
(all x)[(F x) -> (some y)((F y) & (G x y))] justification = 1.0
Ultimate epistemic interests:
(all x)[(F x) -> (some y)(some z)((G x y) & (G y z))] interest = 1.0
Problem #54
Given premises:
(all x)(some y)(R x y) justification = 1.0
(all x)(all y)((R x y) -> (R y x)) justification = 1.0
(all x)(all y)(all z)([(R x y) & (R y z)] -> (R x z)) justification = 1.0
Ultimate epistemic interests:
(all x)(R x x) interest = 1.0
Problem #55
Given premises:
Ultimate epistemic interests:
[(all x)(F x) -> (some x)(F x)] interest = 1.0
Problem #56
Given premises:
Ultimate epistemic interests:
(some x)[(F x) -> (all y)(F y)] interest = 1.0
Problem #57
Given premises:
Ultimate epistemic interests:
[(all x)(all y)((R x y) -> ~(R y x)) -> ~(some x)(R x x)] interest = 1.0
Problem #58
Given premises:
Ultimate epistemic interests:
~(some x)(all y)((R x y) <-> ~(R y y)) interest = 1.0
Problem #59
Given premises:
Ultimate epistemic interests:
~(all x)[((F x) v ~(F x)) -> ~((F x) v ~(F x))] interest = 1.0
Problem #60
Given premises:
Ultimate epistemic interests:
[(some x)((F x) v (G x)) <-> [(some x)(F x) v (some x)(G x)]] interest = 1.0
Problem #61
Given premises:
Ultimate epistemic interests:
[(all x)((F x) & (G x)) <-> [(all x)(F x) & (all x)(G x)]] interest = 1.0
Problem #62
Given premises:
Ultimate epistemic interests:
[(all x)((F x) -> (G x)) -> ((all x)(F x) -> (all x)(G x))] interest = 1.0
Problem #63
Given premises:
Ultimate epistemic interests:
[(P -> (all x)(F x)) <-> (all x)(P -> (F x))] interest = 1.0
Problem #64
Given premises:
Ultimate epistemic interests:
[(P -> (some x)(F x)) <-> (some x)(P -> (F x))] interest = 1.0
Problem #65
Given premises:
Ultimate epistemic interests:
[((all x)(F x) -> P) <-> (some x)((F x) -> P)] interest = 1.0
Problem #66
Given premises:
Ultimate epistemic interests:
(all x)((F x) v ~(F x)) interest = 1.0
Problem #67
Given premises:
Ultimate epistemic interests:
(some x)((F x) v ~(F x)) interest = 1.0
Problem #68
Given premises:
Ultimate epistemic interests:
(some y)((F a y) <-> (F y y)) interest = 1.0
Problem #69
Given premises:
Ultimate epistemic interests:
(all x)(some y)((F x y) <-> (F y y)) interest = 1.0
Problem #70
Pelletier's problem 18
Given premises:
Ultimate epistemic interests:
(some y)(all x)((F y) -> (F x)) interest = 1.0
Problem #71
Pelletier's problem 19
Given premises:
Ultimate epistemic interests:
(some x)(all y)(all z)(((P y) -> (Q z)) -> ((P x) -> (Q x))) interest = 1.0
Problem #72
Pelletier's problem 20
Given premises:
Ultimate epistemic interests:
[(all x)(all y)(some z)(all w)(((P x) & (Q y)) -> ((R z) & (S w)))
-> ((some v1)(some u)((P v1) & (Q u)) -> (some s)(R s))] interest = 1.0
Problem #73
Pelletier's problem 21
Given premises:
(some x)(p -> (F x)) justification = 1.0
(some x)((F x) -> p) justification = 1.0
Ultimate epistemic interests:
(some x)(p <-> (F x)) interest = 1.0
Problem #74
Pelletier's problem 22
Given premises:
Ultimate epistemic interests:
[(all x)(p <-> (F x)) -> (p <-> (all y)(F y))] interest = 1.0
Problem #75
Pelletier's problem 23
Given premises:
Ultimate epistemic interests:
[(all x)(p v (F x)) <-> (p v (all y)(F y))] interest = 1.0
Problem #76
Pelletier's problem 24
Given premises:
~(some x)((S x) & (Q x)) justification = 1.0
(all x)((P x) -> ((Q x) v (R x))) justification = 1.0
[~(some x)(P x) -> (some y)(Q y)] justification = 1.0
(all x)(((Q x) v (R x)) -> (S x)) justification = 1.0
Ultimate epistemic interests:
(some x)((P x) & (R x)) interest = 1.0
Problem #77
Pelletier's problem 25
Given premises:
(some x)(P x) justification = 1.0
(all x)((F x) -> (~(G x) & (R x))) justification = 1.0
(all x)((P x) -> ((G x) & (F x))) justification = 1.0
[(all x)((P x) -> (Q x)) v (some y)((P y) & (R y))] justification = 1.0
Ultimate epistemic interests:
(some x)((Q x) & (P x)) interest = 1.0
Problem #78
Pelletier's problem 26
Given premises:
[(some x)(P x) <-> (some y)(Q y)] justification = 1.0
(all x)(all y)(((P x) & (Q y)) -> ((R x) <-> (S y))) justification = 1.0
Ultimate epistemic interests:
[(all x)((P x) -> (R x)) <-> (all y)((Q y) -> (S y))] interest = 1.0
Problem #79
Pelletier's problem 27
Given premises:
(some x)((F x) & ~(G x)) justification = 1.0
(all x)((F x) -> (H x)) justification = 1.0
(all x)(((J x) & (I x)) -> (F x)) justification = 1.0
[(some x)((H x) & ~(G x)) -> (all y)((I y) -> ~(H y))] justification = 1.0
Ultimate epistemic interests:
(all x)((J x) -> ~(I x)) interest = 1.0
Problem #80
Pelletier's problem 28
Given premises:
(all x)[(P x) -> (all x)(Q x)] justification = 1.0
[(all x)((Q x) v (R x)) -> (some y)((Q y) & (S y))] justification = 1.0
[(some x)(S x) -> (all x)((F x) -> (G x))] justification = 1.0
Ultimate epistemic interests:
(all x)[((P x) & (F x)) -> (G x)] interest = 1.0
Problem #81
Pelletier's problem 29
Given premises:
[(some x)(F x) & (some y)(G y)] justification = 1.0
Ultimate epistemic interests:
([(all x)((F x) -> (H x)) & (all y)((G y) -> (J y))] <->
(all z)(all w)(((F z) & (G w)) -> ((H z) & (J w)))) interest = 1.0
Problem #82
Pelletier's problem 30
Given premises:
(all x)(((F x) v (G x)) -> ~(H x)) justification = 1.0
(all x)(((G x) -> ~(I x)) -> ((F x) & (H x))) justification = 1.0
Ultimate epistemic interests:
(all x)(I x) interest = 1.0
Problem #83
Pelletier's problem 31
Given premises:
~(some x)((F x) & ((G x) v (H x))) justification = 1.0
(some x)((I x) & (F x)) justification = 1.0
(all x)(~(H x) -> (J x)) justification = 1.0
Ultimate epistemic interests:
(some x)((I x) & (J x)) interest = 1.0
Problem #84
Pelletier's problem 32
Given premises:
(all x)(((F x) & ((G x) v (H x))) -> (I x)) justification = 1.0
(all x)(((I x) & (H x)) -> (J x)) justification = 1.0
(all x)((K x) -> (H x)) justification = 1.0
Ultimate epistemic interests:
(all x)(((F x) & (K x)) -> (J x)) interest = 1.0
Problem #85
Pelletier's problem 33
Given premises:
Ultimate epistemic interests:
[(all x)[((P a) & ((P x) -> (P b))) -> (P c)] <->
(all x)((~(P a) v ((P x) v (P c))) & (~(P a) v (~(P b) v (P c))))] interest = 1.0
Problem #86
Half of Pelletier's problem 34
Given premises:
Ultimate epistemic interests:
[[(some x)(all y)((P x) <-> (P y)) <-> ((some z)(Q z) <-> (all w)(Q w))] ->
[(some u)(all v1)((Q u) <-> (Q v1)) <-> ((some r)(P r) <-> (all s)(P s))]] interest = 1.0
Problem #87
Pelletier's problem 35
Given premises:
Ultimate epistemic interests:
(some u)(some v1)[(P u v1) -> (all x)(all y)(P x y)] interest = 1.0
Problem #88
Pelletier's problem 36
Given premises:
(all x)(some y)(F x y) justification = 1.0
(all x)(some z)(G x z) justification = 1.0
(all x)(all y)[((F x y) v (G x y)) -> (all z)(((F y z) v (G y z)) -> (H x z))] justification = 1.0
Ultimate epistemic interests:
(all x)(some y)(H x y) interest = 1.0
Problem #89
Pelletier's problem 37
Given premises:
(all z)(some w)(all x)(some y)[[((P x z) -> (P y w)) & (P y z)] &
[(P y w) -> (some u)(Q u w)]] justification = 1.0
(all x)(all z)[~(P x z) -> (some v1)(Q v1 z)] justification = 1.0
[(some y)(some s)(Q y s) -> (all x)(R x x)] justification = 1.0
Ultimate epistemic interests:
(all x)(some y)(R x y) interest = 1.0
Problem #90
Pelletier's problem 38
Given premises:
Ultimate epistemic interests:
[(all x)[[(P a) & ((P x) -> (some y)((P y) & (R x y)))] ->
(some z)(some w)[(P z) & ((R x w) & (R w z))]] <->
(all x)[[(~(P a) v (P x)) v (some z)(some w)((P z) & ((R x w) & (R w z)))] &
[~(P a) v (~(some y)((P y) & (R x y)) v
(some z)(some w)((P z) & ((R x w) & (R w z))))]]] interest = 1.0
Problem #91
Pelletier's problem 39
Given premises:
Ultimate epistemic interests:
~(some x)(all y)((F y x) <-> ~(F y y)) interest = 1.0
Problem #92
Pelletier's problem 40
Given premises:
Ultimate epistemic interests:
[(some y)(all x)((F x y) <-> (F x x)) -> ~(all z)(some w)(all v1)((F v1 w) <-> ~(F v1 z))] interest = 1.0
Problem #93
Pelletier's problem 41
Given premises:
(all z)(some y)(all x)[(F x y) <-> ((F x z) & ~(F x x))] justification = 1.0
Ultimate epistemic interests:
~(some z)(all x)(F x z) interest = 1.0
Problem #94
Pelletier's problem 42
Given premises:
Ultimate epistemic interests:
~(some y)(all x)[(F x y) <-> ~(some z)((F x z) & (F z x))] interest = 1.0
Problem #95
Pelletier's problem 43
Given premises:
(all x)(all y)[(Q x y) <-> (all z)((F z x) <-> (F z y))] justification = 1.0
Ultimate epistemic interests:
(all x)(all y)[(Q x y) <-> (Q y x)] interest = 1.0
Problem #96
Pelletier's problem 44
Given premises:
(all x)[[(F x) -> (some y)((G y) & (H x y))] & (some y)((G y) & ~(H x y))] justification = 1.0
(some x)[(J x) & (all y)[(G y) -> (H x y)]] justification = 1.0
Ultimate epistemic interests:
(some x)((J x) & ~(F x)) interest = 1.0
Problem #97
Pelletier's problem 45
Given premises:
(all x)[[(F x) & (all y)[((G y) & (H x y)) -> (J x y)]] ->
(all y)[((G y) & (H x y)) -> (K y)]] justification = 1.0
~(some y)((L y) & (K y)) justification = 1.0
(some x)[[(F x) & (all y)((H x y) -> (L y))] &
(all y)(((G y) & (H x y)) -> (J x y))] justification = 1.0
Ultimate epistemic interests:
(some x)((F x) & ~(some y)((G y) & (H x y))) interest = 1.0
Problem #98
Pelletier's problem 46
Given premises:
(all x)([(F x) & (all y)[((F y) & (H y x)) -> (G y)]] -> (G x)) justification = 1.0
[(some x)((F x) & ~(G x)) ->
(some x)(((F x) & ~(G x)) & (all y)(((F y) & ~(G y)) -> (J x y)))] justification = 1.0
(all x)(all y)[[((F x) & (F y)) & (H x y)] -> ~(J y x)] justification = 1.0
Ultimate epistemic interests:
(all x)((F x) -> (G x)) interest = 1.0
Problem #99
Pelletier's problem 47
Given premises:
(all x)((W x) -> (A x)) justification = 1.0
(all x)((F x) -> (A x)) justification = 1.0
(all x)((B x) -> (A x)) justification = 1.0
(all x)((C x) -> (A x)) justification = 1.0
(all x)((S x) -> (A x)) justification = 1.0
(some w0)(W w0) justification = 1.0
(some f0)(F f0) justification = 1.0
(some b0)(B b0) justification = 1.0
(some c0)(C c0) justification = 1.0
(some s0)(S s0) justification = 1.0
(some g0)(G g0) justification = 1.0
(all x)((G x) -> (P x)) justification = 1.0
(all x)[(A x) -> [(all w)((P w) -> (E x w)) v
(all y)(((A y) & ((M y x) & (some z)((P z) & (E y z)))) -> (E x y))]] justification = 1.0
(all x)(all y)[((C x) & (B y)) -> (M x y)] justification = 1.0
(all x)(all y)[((S x) & (B y)) -> (M x y)] justification = 1.0
(all x)(all y)[((B x) & (F y)) -> (M x y)] justification = 1.0
(all x)(all y)[((F x) & (W y)) -> (M x y)] justification = 1.0
(all x)(all y)[((W x) & (F y)) -> ~(E x y)] justification = 1.0
(all x)(all y)[((W x) & (G y)) -> ~(E x y)] justification = 1.0
(all x)(all y)[((B x) & (C y)) -> (E x y)] justification = 1.0
(all x)(all y)[((B x) & (S y)) -> ~(E x y)] justification = 1.0
(all x)[(C x) -> (some y)((P y) & (E x y))] justification = 1.0
(all x)[(S x) -> (some y)((P y) & (E x y))] justification = 1.0
Ultimate epistemic interests:
(some x)(some y)[[(A x) & (A y)] & (some z)[(E x y) & ((G z) & (E y z))]] interest = 1.0
Problem #100
Pelletier's problem 57
Given premises:
(F (g a b) (g b c)) justification = 1.0
(F (g b c) (g a c)) justification = 1.0
(all x)(all y)(all z)[[(F x y) & (F y z)] -> (F x z)] justification = 1.0
Ultimate epistemic interests:
(F (g a b) (g a c)) interest = 1.0
Problem #102
Given premises:
Ultimate epistemic interests:
[(all x)[((F a) & ((F x) -> (F (g x)))) -> (F (g (g x)))] ->
(all x)[[(~(F a) v (F x)) v (F (g (g x)))] &
[(~(F a) v ~(F (g x))) v (F (g (g x)))]]] interest = 1.0
Problem #103
The unintuitive problem
Given premises:
(all x)(all y)(all z)([(P x y) & (P y z)] -> (P x z)) justification = 1.0
(all x)(all y)(all z)([(Q x y) & (Q y z)] -> (Q x z)) justification = 1.0
(all x)(all y)((Q x y) -> (Q y x)) justification = 1.0
(all x)(all y)(~(P x y) -> (Q x y)) justification = 1.0
~(P a b) justification = 1.0
Ultimate epistemic interests:
(Q c d) interest = 1.0
Problem #104
Chang and Lee problem 3
Given premises:
(all x)(P x e x) justification = 1.0
(all x)(P e x x) justification = 1.0
(all x)(all y)(all z)(all u)(all v1)(all w)[((P x y u) & ((P y z v1) & (P u z w))) -> (P x v1 w)] justification = 1.0
(all x)(all y)(all z)(all u)(all v1)(all w)[((P x y u) & ((P y z v1) & (P x v1 w))) -> (P u z w)] justification = 1.0
(all x)(P x x e) justification = 1.0
(P a b c) justification = 1.0
Ultimate epistemic interests:
(P b a c) interest = 1.0
"
))
)
(defunction test* () (test :skip 86))