Expressing Truth directly within a formal system with no need for model theory

First we lay the foundation of expressing semantic truth directly within a formal system. All of semantic truth has its ultimate ground of being in expressions of language that have been defined to be true.
The construction of a theory begins by specifying a definite non-empty conceptual class E the elements of which are called statements. These initial statements are often called the primitive elements or elementary statements of the theory, to distinguish them from other statements which may be derived from them.

A theory T is a conceptual class consisting of certain of these elementary statements. The elementary statements which belong to T are called the elementary theorems of T and said to be true. In this way, a theory is a way of designating a subset of E which consists entirely of true statements. (Haskell Curry, Foundations of Mathematical Logic, 2010).  

This is the same page from his 1977 Book:   

Semantic meaning of English sentences specified in First Order Logic
Meaning Postulates Rudolf_Carnap (1952) 
(1) Bachelor(Jack) → ~Married(Jack)
(2) Black(Fido) ∨ ~Black(Fido)

Expressing the (a) transitivity (b) irreflexivity and (c) asymmetry relations of the English word Warmer: 
(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)