Universal Logic

This paper is, essentially, a progress report on work involving conceptualisations of metalogic and metamathematics, alternative to the standard proof-theoretic or model-theoretic conceptualisation.   Apparently, the best known abstract setting for the latter is A. Tarski's consequence theory later re-named as a closure operator theory.
In what follows we deal with closure operators (hereafter symbolized as Cn), closure systems (Th), consistency properties (Cons), extension or Lindenbaum operators (Ln), systems of maximal sets (Max) and omission or separation operators (Sep).   For ease of use let X, Y be members of the set {Cn, Th, Cons, Ln, Max, Sep}.
To start, we develop a bit of a general X-based metalogic where no reference to any specific logical constant is necessary.   At this stage the object-language is treated just as a non-empty set of completely unstructured sentences.
Logical constants are specified at the next stage where we need to formally identify which sentences are made up of which simpler sentences.   We begin by specifying a sentential connective #, a 0-order logical constant.   Using the method of slightly modified Galois connections we prove that (X, #)-based metalogic is equivalent to (Y, #)-based metalogic.
Next we show how (X,#)-based metalogic can be used to generate logic Lgc(X, #) and justify that Lgc(X, #) is precisely the ordinary or classical 0-order logic.   In this context we also discuss the problem of how to modify conditions imposed on X in order to make logic Lgc(X, #), it generates, to be, say, the intuitionistic logic, Johansson minimal logc or Lukasiewicz 3-valued logic.
Finally, we upgrade the object-language to the 1st order level.   A slight modification of the language is defined here as a language with the witness property.   This, it will be seen, facilitates discussion of quantifier-dependent aspects within the framework of (X, #,Q(W))-based metalogic where Q(W) refers to quantifiers and correlated with them their witnesses.   More specifically, we prove within this framework that (X, #,Q(W))-based and (Y,#,Q(W))-based metalogics are pairwise equivalent.   And we also upgrade the earlier generated logic Lgc(X, #) to the logic Lgc(X, #,Q(W)) of the 1st order level.


	From Standard to Nonstandard Metalogics Stan Surma Department of Philosophy University of Auckland - New Zealand
	This paper is, essentially, a progress report on work involving conceptualisations of metalogic and metamathematics, alternative to the standard proof-theoretic or model-theoretic conceptualisation. Apparently, the best known abstract setting for the latter is A. Tarski's consequence theory later re-named as a closure operator theory. In what follows we deal with closure operators (hereafter symbolized as Cn), closure systems (Th), consistency properties (Cons), extension or Lindenbaum operators (Ln), systems of maximal sets (Max) and omission or separation operators (Sep). For ease of use let X, Y be members of the set {Cn, Th, Cons, Ln, Max, Sep}. To start, we develop a bit of a general X-based metalogic where no reference to any specific logical constant is necessary. At this stage the object-language is treated just as a non-empty set of completely unstructured sentences. Logical constants are specified at the next stage where we need to formally identify which sentences are made up of which simpler sentences. We begin by specifying a sentential connective #, a 0-order logical constant. Using the method of slightly modified Galois connections we prove that (X, #)-based metalogic is equivalent to (Y, #)-based metalogic. Next we show how (X,#)-based metalogic can be used to generate logic Lgc(X, #) and justify that Lgc(X, #) is precisely the ordinary or classical 0-order logic. In this context we also discuss the problem of how to modify conditions imposed on X in order to make logic Lgc(X, #), it generates, to be, say, the intuitionistic logic, Johansson minimal logc or Lukasiewicz 3-valued logic. Finally, we upgrade the object-language to the 1st order level. A slight modification of the language is defined here as a language with the witness property. This, it will be seen, facilitates discussion of quantifier-dependent aspects within the framework of (X, #,Q(W))-based metalogic where Q(W) refers to quantifiers and correlated with them their witnesses. More specifically, we prove within this framework that (X, #,Q(W))-based and (Y,#,Q(W))-based metalogics are pairwise equivalent. And we also upgrade the earlier generated logic Lgc(X, #) to the logic Lgc(X, #,Q(W)) of the 1st order level.