We present in this paper an abstract categorical logic based on an abstraction of quantifier. More precisely, the proposed logic is abstract because no structural constraints are imposed on models (semantics free). By contrast, formulas are inductively defined from an abstraction both of atomic formulas and of quantifiers. In this sense, the proposed approach differs from other works interested in formalizing the notion of abstract logic and of which the closest to our approach are the institutions, which in addition to (...) be semantics free do not also impose any syntactic contingencies on the structure of formulas. To define the semantical framework in which formulas will be interpreted, we propose to follow the idea from categorical logic which defines the semantical interpretation of formulas from context and as subobjects of an object of a given category. In the spirit of Lawvere’s hyperdoctrines, we use a more abstract notion which generalizes the notion of subobject, standard in category theory: Pitt’s prop-categories. Always in the spirit of categorical logic, we propose a sequent calculus of which we show correctness and completeness for all semantical frameworks defined over any prop-categories. We then study some conditions which allow us to get this completeness result for particular classes of prop-categories. (shrink)

The Ehrenfeucht–Fraïsse game for a logic usually provides an intuitive characterizarion of its expressive power while in abstract model theory, logics are compared by their expressive powers. In this paper, I explore this connection in details by proving a general Lindström theorem for logics which have certain types of Ehrenfeucht–Fraïsse games. The results generalize and uniform some known results and may be applied to get new Lindström theorems for logics.

In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski- and the Lindenbaum-types. The characterization theorems for the Tarski- and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaum-types and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski- and a Lindenbaum-type, show their separations, (...) and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski–Lindenbaum-type logical structures. (shrink)

It is argued that the intuitionistic conception of negation as implication of absurdity is inadequate for the proof-theoretic semantic analysis of negative predication and distinctness. Instead, it is suggested to construe negative predication proof-theoretically as subatomic derivation failure, and to define distinctness—understood as a qualified notion—by appeal to negative predication. This proposal is elaborated in terms of intuitionistic bipredicational subatomic natural deduction systems. It is shown that derivations in these systems normalize and that normal derivations have the subexpression (incl. subformula) (...) property. A proof-theoretic semantics for negative predication and distinctness is defined, and an intuitionistic conception of truth based on canonical derivations is proposed. An application to the doctrine of the Trinity, to which negative predication and distinctness are central, illustrates the systems and their proof-theoretic semantics. (shrink)