# Forming a Bridge between Proof Theory and Model Theory

Formalizing the internal semantic meaning of propositional (sentential) variables using Rudolf Carnap (1952) Meaning Postulates bridges the gap between syntactic consequence (formal proof) and semantic consequence (logical entailment).

Syntactic versus Semantic Logical Consequence
Meaning postulates specify semantic logical entailment syntactically.
The best two examples from his paper: Bachelor(x) and Warmer(x,y):

Bachelor(x) → ~Married(x)

For example, let ‘W’ be a primitive predicate designating the relation Warmer. Then ‘W’ is transitive, irreflexive, and hence asymmetric in virtue of its meaning.

In the previous example of the predicate ‘W’, we could lay down the following postulates (a) for transitivity and (b) for irreflexivity; then the statement (c) of asymmetry:

(a)  ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z)
(b)  ∀(x)      ~Warmer(x,x)
(c)  ∀(x,y)    Warmer(x,y) → ~( Warmer(y,x) )

Meaning Postulates Rudolf Carnap (1952)