Additional Comments on Conjectures, Hypotheses, and Consequences in Orthocomplemented Lattices

Abstract

This paper is a brief continuation of earlier work by the same authors [4] and [5] that deals with the concepts of conjecture, hypothesis and consequence in orthocomplemented complete lattices. It considers only the following three points: 1. Classical logic theorems of both deduction and contradiction are reinterpreted and proved by means of one specific operator C_^ defined in [4]. 2. Having shown that there is reason to consider the set C_^(P) of consequences of a set of premises P as too large, it is proven that C_^(P) is the largest set of consequences that can be assigned to P by means of a Tarski's consequences operator, provided that T is a Boolean algebra. 3. On the other hand, it is proven that, also in a Boolean algebra, the set Φ_^(P) of strict conjectures is the smallest of any Φ(P) such that P ⊆ Φ(P) and th at if P ⊆Q then Φ(Q) ⊆ Φ(P).

Paper partially supported by SpanishM inistry of Education and Culture under projects PB98-1379-C02-C02 and CICYT-TIC99-1151

∧P = Inf(P) = p_^ and ∨P = Sup(P) = p_^.