Sémantique du calcul des prédicats

f R
Ddef ={a, b, c}
f fI:D→D fI(a) = a, fI(b) = c fI(c) = c
R RI={(a, a),(a, b),(a, c),(b, c)}ι
ι(y) = a ι(v) = b v y
f(y)f(x)f(f(y))
f(f(f(y)))
F0=∀x, R(x, y)
F1=∃x, ∀y, R(x, y)
F2=∃x, R(f(x), f(y))
a b
a b b c a
c
a b b a
a b b a a b
R E
t=u E(t, u)t u
R
Ddef ={a, b, c}
R E
RN
R
R(t, u)u t

(a)a
(r)r
(g)g
(x)x
(x, g)x g
(x, r)x r
(g, a, r)g a r
(x, a, r)x a r
x=y x y
∀x r, (x, , r)⇒(x, , r)
∃x, ∀a r, (r)⇒(a)⇒(x, a, r)
I
Ddef ={, , , , , , , , }
I={,}
I={,}
I={,}
I={,,}
I={(,),(,),(,)}
I={(,),(,)}
I={(, , ),(, , ),(, , )}
I

(,)
I
II
(g, a, c)g a c
R
F1:∀x, ((∃y, ¬R(x, y)) ⇒ ∃y, (R(x, y)∧R(y, x)))
F2:∀x, ∃y, (R(x, y)∨R(y, x))
F3:∀x y z, ((R(x, y)∧R(y, z)) ⇒R(x, z))
F4:∃x, R(x, x)
R
x y
R(x, y)
(x, y)R(x, y)
F1F2F3F4
F2F1
F4F1F3
A F4F2F3