6

W. J. BLOK AND D. PIGOZZI

In addition h$ is finitary in the sense

T\-s f = r h

5

if for some finite V C I\ (4)

and it is structural in the sense

T\-s p^ cr(T)hs crp (5)

for every substitution a. Conversely, Los and Suszko [25] have shown that

every relation satisfying conditions (l)-(5) is the consequence relation for some

deductive system S. Consequently, in the sequel by a deductive system we will

mean a pair (C,\~s) where \~s is a function from the powerset of Fmc into

Fmc that satisfies conditions (2)-(6); defining axioms and rules of inference

are not assumed.

Associated with any deductive system are its various extensions, subsys-

tems, and fragments. By an extension of a deductive system S over the lan-

guage C we mean any system S' —

(C^s1)

over the same language such that

r H$ V ? = r h$/ (f for all T U {£ } C Fmc] S is called a subsystem of S' in

this case. S' is an axiomatic extension of S if it is obtained by adjoining new

axioms but leaving the rules of inference fixed.

Let C be a sublanguage of £, and let \-& be the restriction of (-$ to C in

the sense that T \~s ^ iff T U {^} C Fm& and r h$ y. It is easy to see that

conditions (l)-(5) continue to hold with h$/ in place of h$, and hence that

5

;

= (£',1-5') is a deductive system over C. S' is called the C-fragment of 5 .

1.1 The Lattice of Theories

Let S be an arbitrary deductive system. A set T of formulas is called a theory

of S (a S-theory for short) if T h$ y? implies / ? £ T for every p £ Fm, or,

equivalently, if T contains all substitution instances of the axioms, and is closed

under the inference rules. For any T C Fm we take

CnsT = {pe Fm:T\-s p}.

CnsT is the smallest 5-theory including T, and T is said to generate CnsT.

Cns , treated as a function on the power set of Fm, is called the consequence op-

eratorofS (the S-consequence operator). The characteristic properties (l)-(5)

of the consequence relation h$ correspond to the following closure properties

of Cns:

rccn

5

r

;

(6)

r c A ^ CnsTC CnsA] (7)

CnsCnsTC CnsT] (8)