In[]:=
sigmoid[x_,alpha_,beta_,k_]:=k/(1+Exp[-alpha-x*beta])
In[]:=
FullSimplify[Reduce[{sigmoid[-1,alpha,beta,k]==a,sigmoid[0,alpha,beta,k]==b,sigmoid[1,alpha,beta,k]==c,(0<a<b<c)||(0<c<b<a)},{alpha,beta,k},Reals]]
Out[]=
alphaLog+&&betaLog&&-ack&&a>0&&c<&&((b>0&&c>0&&b<a)||(b>a&&b<c))
a
a-b
b
-b+c
(a-b)c
a(b-c)
b(a(b-2c)+bc)
2
b
2
b
a