Handbook of the History of Logic: - Fordham University Faculty

430 Gyula Klima
in which the object language is capable of “cannibalizing” its own meta-language,
by introducing distinguished semantic predicates into the object-language matching
those originally defined in the meta-language.
The payoff of such a project would be the ability to see exactly what it takes, in
precise model-theoretical terms, to construct a nominalist semantics, facilitating
its comparison both with the competing medieval conception and with a number
of modern conceptions. And this, in the end, might just get us closer to a genuine
understanding of the fundamental semantic relations between language, thought
and reality.
Language
APPENDIX
L := 〈C, P, V, Trm, F〉 [language : constants, parameters, variables, terms, formulae]
C := { ∼, &, =, ∀,.,(, )} [constants: negation, conjunction, identity, universal
quantifier, punctuation marks]
P := Pr∪I,Pr := ∪{F m n },I := ∪{an} [parameters: predicates, individual names]
V := X ∪ Xr,X := ∪{xi}, [variables: simple variables plus restricted variables]
Xr := {tn : tn =‘x.A’ andA ∈ F } [restricted variable: formula prefixed with
simple variable and dot]
Trm:=I∪V [terms: individual names plus variables]
Formulae:
1. t1...tm ∈ Trm, F m n ∈ Pr → ‘F m n (t1)...(tm)’ ∈ F
2. t1,t2 ∈ Trm → ‘t1 = t2’ ∈ F
3. A, B ∈ F → ‘∼ (A)’, ‘A&B’ ∈ F
4. A ∈ F, v ∈ V → ‘(∀v)(A)’ ∈ F
Model
M := 〈D, T, E, 0, 1,SGT〉, where
D = ∅,T = ∅,t ∈ T,E(t) ⊂ D, 0 /∈ D, 1 /∈ D, SGT : P → D ∪ P (D m ) [non-empty
domain, set of times, times, set of existents at time t, 0 and 1: False and True,
signification function: from parameters to the domain plus the power-set of the
m-th Cartesian product of D with itself]
1. SGT(an) ∈ D
2. SGT(F m n ) ∈ P (D m )