Handbook of the History of Logic: - Fordham University Faculty

The Nominalist Semantics of Ockham and Buridan 431
Supposition (value-assignment in a model)
1. SUP(a)(t) =SGT(a), if SGT(a) ∈ E(t),
otherwise SUP(a)(t) =0
2. SUP(x)(t) ∈ D
3. SUP(‘x.A’)(t) =SUP(x)(t), if SUP(A)(t) =1,
otherwise SUP(‘x.A’)(t) =0
4. SUP(‘F m n (t1)...(tm)’)(t) = 1, if 〈SUP(t1)(t),...,SUP(tm)(t)〉 ∈SGT(F m n )∩
E(t) m ,
otherwise SUP(‘F m n (t1)...(tm)’)(t) =0
5. SUP(‘t1 = t2’)(t) = 1, if SUP(t1)(t) =SUP(t2)(t) ∈ E(t),
otherwise SUP(‘t1 = t2’)(t) =0
6. SUP(‘∼ A’)(t) = 1, if SUP(A)(t) = 0, otherwise SUP(‘∼ A’)(t) =0
7. SUP(‘A&B’)(t) = 1, if SUP(A)(t) =SUP(B)(t) =1,
otherwise SUP(‘A&B’)(t) =0
8. SUP(‘(∀v)(A)’)(t) = 1, if for every u ∈ RSUP (v)(t),SUP[v : u](A)(t) =1,
otherwise SUP(‘(∀v)(A)’)(t) = 0, where RSUP (v)(t) :={u ∈ E(t) :u =
SUP(v)(t)}, if{u ∈ E(t) :u = SUP(v)(t)} = ∅, otherwise RSUP (v)(t) :=
{0}, andSUP[v : u](w)(t) =u, ifw = v, otherwise SUP[v : u](w)(t) =
SUP(v)(t) —[RSUP (v)(t) is called “the range of v at t”]
A is true in M iff for some SUP,SUP(A)(t) =1;A is valid iff for every M,A
is true in M.