A pragmatic interpretation of intuitionistic propositional logic

We construct an extension ^P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of ^P;rfs preceded by theassertion sign constituteelementary assertive formulas of ^P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of ^P is endowed with atruth value defined classically, and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in ^P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of ^P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic.This paper is an enlarged and entirely revised version of the paper by Dalla Pozza (1991) worked out in the framework of C.N.R. project n. 89.02281.08, and published in Italian. The basic ideas in it have been propounded since 1986 by Dalla Pozza in a series of seminars given at the University of Lecce and in other Italian Universities. C. Garola collected the scattered parts of the work, helped in solving some conceptual difficulties and refining the formalism, yielded the proofs of some propositions (in particular, in Section 3) and provided physical examples (see in particular Remark 2.3.1).     

The information in intuitionistic logic

Issues about information spring up wherever one scratches the surface of logic. Here is a case that raises delicate issues of ‘factual’ versus ‘procedural’ information, or ‘statics’ versus ‘dynamics’. What does intuitionistic logic, perhaps the earliest source of informational and procedural thinking in contemporary logic, really tell us about information? How does its view relate to its ‘cousin’ epistemic logic? We discuss connections between intuitionistic models and recent protocol models for dynamic-epistemic logic, as well as more general issues that emerge.     

Partial isomorphisms and intuitionistic logic

A game for testing the equivalence of Kripke models with respect to finitary and infinitary intuitionistic predicate logic is introduced and applied to discuss a concept of categoricity for intuitionistic theories.     

The method of axiomatic rejection for the intuitionistic propositional logic

We prove that the intuitionistic sentential calculus is -decidable (decidable in the sense of ukasiewicz), i.e. the sets of theses of Int and of rejected formulas are disjoint and their union is equal to all formulas. A formula is rejected iff it is a sentential variable or is obtained from other formulas by means of three rejection rules. One of the rules is original, the remaining two are ukasiewicz's rejection rules: by detachement and by substitution. We extensively use the method of Beth's semantic tableaux.To the memory of Jerzy SupeckiTranslated from the Polish by Jan Zygmunt. Preparation of this paper was supported in part by C.P.B.P. 08-15.     

On modal logic with an intuitionistic base

A definition of the concept of Intuitionist Modal Analogue is presented and motivated through the existence of a theorem preserving translation fromMIPC (see [2]) to a bimodalS ⁴–S⁵ calculus.Allatum est die 9 Septembris 1975     

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.     

A weak intuitionistic propositional logic with purely constructive implication

We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.This work was partially supported by NSF Grant DCR85-13417     

On a second order propositional operator in intuitionistic logic

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by. In full topological models * is not generally definable, but over Cantor-space and the reals it can be classically shown that; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate logic.Over [0, 1], the operator * is (constructively and classically) undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness.Parameters of the required kind can be obtained as so-called projections of lawless sequences.