An algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical (...) equations in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed. (shrink)

The Tarski T-Schema has a propositional version. If we use ϕ as a metavariable for formulas and use terms of the form that-ϕ to denote propositions, then the propositional version of the T-Schema is: that-ϕ is true if and only if ϕ. For example, that Cameron is Prime Minister is true if and only if Cameron is Prime Minister. If that-ϕ is represented formally as [λ ϕ], then the T-Schema can be represented as the 0-place case of λ-Conversion. If we (...) interpret [λ…] as a truth-functional context, then using traditional logical techniques, one can prove that the propositional version of the T-Schema is a tautology, literally. Given how well-accepted these logical techniques are, we conclude that the T-Schema, in at least one of its forms, is a not just a logical truth but a tautology at that. (shrink)

The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables, upon the basis of the conception of model introduced in :27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models that a rule is derivable (...) in the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant. Finally, we exemplify this application by studying sequent calculi for some of such logics. (shrink)

In H-N. Castañeda's ontology, a fundamental Fregean distinction is drawn between unsaturated and saturated entities, the former corresponding to predicative aspects of reality, the latter to individuals, that is, to items which can be referred to by means of singular terms1. Within saturated entities, Castañeda attempts to distinguish between abstract and concrete individuals. Sets and Platonic Forms of the F-ness-type are the typical examples of the former category2. As to the latter category instead, concrete individual guises represent both the bottom (...) layer in a hypothetical ontological hierarchy of individuals which has Platonic Forms at its top, and the first step in the psychological process of world cognition, which moves from the most concrete object to the most abstract3. Among concrete individual guises, moreover, Castañeda draws a further ontological distinction between possible and impossible guises, according to the different behavior guises display with respect to the possibility of existing4. Both kinds of guises, however, share that feature which enables to qualify them as guises, i.e. that of being bundles of properties. Every guise is indeed constituted out of (a certain set of) properties, or better predicative aspects5. The predication to a guise of one of its constitutive properties is taken by Castañeda as the primary sense of predication. He calls it Meinongian, or internal, predication (in symbols: "a(F)", where "a" stands for a guise and "F" for one of its constitutive properties)6. For Castañeda, moreover, the main linguistic tool by means of which a guise is denoted is a definite description, insofar as such a description clearly reveals the properties which a guise internally possesses7. Thus, we can say that "the F- er is F" is a typical expression of an internal predication of a property F to the guise denoted by the definite description "the F". If we take "F" as a metavariable for properties, we can say that the above formulation corresponds to what Routley has called the Characterization Postulate8. Now what we want to scrutinize in this paper is, first, whether the following four theses of Castañeda's Guise Theory are really compatible.. (shrink)