Embeddings between the elementary ontology with an atom and the monadic second-order predicate logic

✍ Scribed by Mitio Takano

• Publisher: Springer Netherlands
• Year: 1987
• Tongue: English
• Weight: 304 KB
• Volume: 46
• Category: Article
• ISSN: 0039-3215
• DOI: 10.1007/bf00372549

✦ Synopsis

Let .EOA be the elementary ontology augmented by an additional axiom 3S (S 9 S), and let ~8 be the monadic second-order predicate logic. We show that the mapping ~ which was introduced by V. A. Smirnov is an embedding of EOA into LS. We also give an embedding of ZS into EOA.

In Smirnov [3], he defined a mapping ~ which transforms each formula of the elementary ontology E0 into that of the monadic second-order predicate logic LS and a mapping ~ of the converse direction. But [3, Lemma 2] which states that if ~-LsA then [-EO ~(A), for every formula A of ZS is false, since 3P3xPx is provable in LS but ~p(3P3xPx), that is, 3P3x(x~P) is not provable in .EO; so that [3, Theorem] which claims that F~o A if and only if bLs 9(A), for every formula A of E0 also fails to hold, since 3S(S~S) is not provable in EO but 9(3S(S~S)), that is, 3S[3! xSx^Vx(Sx o Sx)] is provable in LS.

In this paper we shall introduce the theory EOA (the elementary ontology with an atom) and show that 9 is an embedding of EOA into LS (Theorem l). Also we shall give an embedding 0 of LS into .EOA (Theorem 2).