RESOLUTION OF EPIMENIDES'
PARADOX
*********************************
The Cretan poet Epimenidies, once wrote, "Cretans (are) always
liars."
Assuming this claim is true
leads to a contradiction (lines 6-16). Therefore, it cannot be true.
We make use the following
predicates:
Stmt(x) = x is a
statement
True(x) = x is true
Cretan(x) = x is a Cretan
Said(x,y) = x said statement y
Constants:
epi = Epimenides
x = the statement that
"Cretans always lie"
Dan Christensen
2023-04-17
AXIOMS
******
Define: x (Epimenides'
claim)
1 Stmt(x)
Axiom
2 ALL(a):[True(a)
=> Stmt(a)]
Axiom
3 True(x) <=> ALL(a):ALL(b):[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)]
Axiom
Define: epi (Epimenides)
4 Cretan(epi)
Axiom
5 Said(epi,x)
Axiom
Prove: ~True(x) (i.e. that x is not true)
Suppose to the contrary...
6 True(x)
Premise
Apply the definition of x to obtain a contradiction
7 [True(x) => ALL(a):ALL(b):[Cretan(a) & Stmt(b) & Said(a,b) =>
~True(b)]]
&
[ALL(a):ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)] => True(x)]
Iff-And, 3
8 True(x) => ALL(a):ALL(b):[Cretan(a) & Stmt(b) & Said(a,b) =>
~True(b)]
Split, 7
9 ALL(a):ALL(b):[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)]
Detach, 8, 6
10 ALL(b):[Cretan(epi) & Stmt(b) & Said(epi,b) =>
~True(b)]
U Spec, 9
11 Cretan(epi) & Stmt(x) & Said(epi,x) => ~True(x)
U Spec, 10
12 Cretan(epi) & Stmt(x)
Join, 4, 1
13 Cretan(epi) & Stmt(x) & Said(epi,x)
Join, 12, 5
14 ~True(x)
Detach, 11, 13
15 True(x) & ~True(x)
Join, 6, 14
By contradiction, Epimenides's claim (x) is not true.
As Required:
16 ~True(x)
4 Conclusion, 6
Apply definition of True(x)
17 [True(x) => ALL(a):ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)]]
&
[ALL(a):ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)] => True(x)]
Iff-And, 3
18 ALL(a):ALL(b):[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)] => True(x)
Split, 17
19 ~True(x) => ~ALL(a):ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)]
Contra, 18
20 ~ALL(a):ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)]
Detach, 19, 16
21 ~~EXIST(a):~ALL(b):[Cretan(a) & Stmt(b)
& Said(a,b) => ~True(b)]
Quant, 20
22 EXIST(a):~ALL(b):[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)]
Rem DNeg, 21
23 EXIST(a):~~EXIST(b):~[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)]
Quant, 22
24 EXIST(a):EXIST(b):~[Cretan(a)
& Stmt(b) & Said(a,b)
=> ~True(b)]
Rem DNeg, 23
25 EXIST(a):EXIST(b):~~[Cretan(a)
& Stmt(b) & Said(a,b)
& ~~True(b)]
Imply-And, 24
26 EXIST(a):EXIST(b):[Cretan(a)
& Stmt(b) & Said(a,b)
& ~~True(b)]
Rem DNeg, 25
Contrary to Epimenides'
claim, at least one Cretan must have once said the truth.
As Required:
27 EXIST(a):EXIST(b):[Cretan(a) & Stmt(b)
& Said(a,b) & True(b)]
Rem DNeg, 26