On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Given an intermediate prepositional logic L, denote by L ^–d its disjuctionless fragment. We introduce an infinite sequence {J _n}_n1 of propositional formulas, and prove:(1)For any L: L ^–d =I ^–d (I=intuitionistic logic) if and only if J _n L for every n 1.Since it turns out that L{J _{n} n}1 = Ø for any L having the disjunction property, we obtain as a corollary that L ^–d = I ^–d for every L with d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the if part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J _n =I+ +J _n (n 1).     

A free IPC is a natural logic: Strong completeness for some intuitionistic free logics

IPC, the intuitionistic predicate calculus, has the property

1. Vc(Γ?A ^c /x) ? Γ??xA.Furthermore, for certain important Γ, IPC has the converse property
2. Γ??xA ? Vc(Γ?A ^c /x).
3. may be given up in various ways, corresponding to different philosophic intuitions and yielding different systems of intuitionistic free logic. The present paper proves the strong completeness of several of these with respect to Kripke style semantics. It also shows that giving up (i) need not force us to abandon the analogue of (ii).

     

A new formulation of discussive logic

S. Jakowski introduced the discussive prepositional calculus D ₂as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D ₂has afterwards been extended to a first-order predicate calculus and to a higher-order logic (cf. the quoted paper). In this paper we present a natural version of D ₂, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented.     

Category theory,logic and formal linguistics: Some connections,old and new

We seize the opportunity of the publication of selected papers from the Logic, categories, semantics workshop to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.     

A new semantics for intuitionistic predicate logic

The main part of the proof of Kripke's completeness theorem for intuitionistic logic is Henkin's construction. We introduce a new Kripke-type semantics with semilattice structures for intuitionistic logic. The completeness theorem for this semantics can he proved without Henkin's construction.     

On commerce,institutions, and underdevelopment: A comparative perspective

There are ample sources on the exuberance of economic life and the organization of society. Eventually the Middle East and China lost ground to Western Europe and Japan. This has been subject to much work and scholarly activity. Considering the multiplicity of explanations, in this paper I focus on the interactions among power dimensions, activities of various social actors, cultural forces, and how these interactions influenced formation of certain institutions and inhibited others. In particular, I allude to the role of the political center in inhibiting land and commerce-based foci of power in the East, and how this inhibition prevented formation of institutions conducive to economic development. In addition to international economics, he is interested in medieval history and its impact on modern institutions of economic and political life. Dr. Dibooglu was the co-Guest Editor of the KT&P theme issue “Endo/Exogenous,” Volume 13, Number 4, Winter 2001. He may be reached at 〈dibo@siu.edu〉. He would like to thank Richard Grabowski for his helpful comments on an earlier draft but states that all errors or omissions are his own.     

Confirmation as partial entailment: A representation theorem in inductive logic

The most prominent research program in inductive logic – here just labeled The Program, for simplicity – relies on probability theory as its main building block and aims at a proper generalization of deductive-logical relations by a theory of partial entailment. We prove a representation theorem by which a class of ordinally equivalent measures of inductive support or confirmation is singled out as providing a uniquely coherent way to work out these two major sources of inspiration of The Program.     

Substructural logics,pluralism and collapse

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainferential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper.