General Logic-Systems that Determine Significant Collections of   Consequence Operators
  In this paper, general logic-systems and a necessary and sufficient algorithm are used to substantiate significant consequence operator properties. It is shown, among other results, that, in certain cases, (1) if the number of steps in a deduction is restricted, then such deduction does not yield a consequence operator. (2) In general, for any non-organized infinite language L, there is a special class of finite consequence operators that is not meet-complete. (3) For classical deduction, three different examples of modified propositional deduction yield collections of finite consequence operators that are not meet-complete. Other general logic-system examples are given. In a final section, the notion of potentially finite is investigated. 
Robert A. Herrmann