SOME MODEL-THEORETIC RESULTS FOR THE RELEVANT LOGIC WITH QUANTIFICATION by MIROSEAW SZATKOWSEI in Bydgoszcz (Poland) 0. In [ 5 ] , R. ROUTLEY and R . K. MEYER describe a semantics for the relevant logic with quantification (RQ). A proof of the Compactness theorem for RQ was given by J. B. FREEMAN in [3] in a standard way by formulating and proving EoS's ultraproduct theorem in a version for relevant quantificational model structures. I n this paper we demonstrate that every relevant quantificational model structure can be viewed as a classical model structure, and by using this fact we prove that many results for RQ are consequences from the classical model theory : 1. Compactness theorem, 2 . Downward Lowenheim-Skolem theorem, 3. Completeness test theorem, 4. Amalgamation theorem for elementary embeddings, 5 . EoS's theorem.
This paper consists of 3 paragraphs. I n the first paragraph we establish symbols and terminology. I n the second paragraph on the basis of a theory T formulated in a language L, which is valid on some relevant quantificational model structure, we describe a theory To on a language Lo and we construct a classical model structure in which To is valid. Finally, in the third paragraph we give proofs for above mentioned theorems.