Sequent-systems and groupoid models. I

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.     

Models of Deduction*

In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a general adjoint situation. To motivate the notion of adjointness, one of the central notions of category theory, and of mathematics in general, it is first considered how some features of it occur in set-theoretical axioms and in the axioms of the lambda calculus. Next, it is explained how this notion arises in the context of deduction, where it characterizes logical constants. It is shown also how the categorial point of view suggests an analysis of propositional identity. The problem of propositional identity, i.e., the problem of identity of meaning for propositions, is no doubt a philosophical problem, but the spirit of the analysis proposed here will be rather mathematical. Finally, it is considered whether models of deductions can pretend to be a semantics. This question, which as so many questions having to do with meaning brings us to that wall that blocked linguists and philosophers during the whole of the twentieth century, is merely posed. At the very end, there is the example of a geometrical model of adjunction. Without pretending that it is a semantics, it is hoped that this model may prove illuminating and useful. *Since the text of this talk was written in 1999, the author has published several papers about related matters (see ‘Identity of proofs based on normalization and generality’, The Bulletin of Symbolic Logic 9 (2003), 477–503, corrected version available at: http://arXiv.org/math.LO/0208094; other titles are available in the same archive).     

Models for stronger normal intuitionistic modal logics

This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated.This paper presents results of an investigation of intuitionistic modal logic conducted in collaboration with Dr Milan Boi.     

Gödel’s Natural Deduction

This is a companion to a paper by the authors entitled “Gödel on deduction”, which examined the links between some philosophical views ascribed to Gödel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Gödel presented with the help of Gentzen’s sequents, which amounts to Ja?kowski’s natural deduction system of 1934, and which may be found in Gödel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Gödel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Gödel’s Notre Dame course, and is followed by comments concerning Gödel’s natural deduction system.     

Validity of projective drawing indices of male homosexuality

We investigated the hypothesis that certain signs in the Draw-A-Person projective technique reflect male homosexuality. Human figure drawings from gay and 88 heterosexual men, with no clinical psychological symptoms, were submitted to trained raters who were blind to the purpose of this research. The raters independently judged whether 21 signs, previously referenced in the literature as suggestive of male homosexuality, were present in the figure drawings. Only two signs, hair and hips, differentiated between groups. Results were interpreted in the light of changing social attitudes towards homosexuals and methodological problems of prior studies.     

Investigating the Links Between Cultural Values and Belief in Conspiracy Theories: The Key Roles of Collectivism and Masculinity

Research suggests that belief in conspiracy theories (CT) stems from basic psychological mechanisms and is linked to other belief systems (e.g., religious beliefs). While previous research has extensively examined individual and contextual variables associated with CT beliefs, it has not yet investigated the role of culture. In the current research, we tested, based on a situated cultural cognition perspective, the extent to which culture predicts CT beliefs. Using Hofstede's model of cultural values, three nation-level analyses of data from 25, 19, and 18 countries using different measures of CT beliefs (Study 1, N = 5323; Study 2a, N = 12,255; Study 2b, N = 30,994) revealed positive associations between masculinity, collectivism, and CT beliefs. A cross-sectional study among U.S. citizens (Study 3, N = 350), using individual-level measures of Hofstede's values, replicated these findings. A meta-analysis of correlations across studies corroborated the presence of positive links between CT beliefs, collectivism, r = .31, 95% CI = [.15; .47], and masculinity, r = .39, 95% CI = [.18; .59]. Our results suggest that in addition to individual differences and contextual variables, cultural factors also play an important role in shaping CT beliefs.     

Sequent-systems and groupoid models. II

The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part.     

Hemispheric effects of canonical views of category members with known typicality levels

Is there a preferred hemispheric canonical view of a concept? We investigated this question in a natural superordinate category membership decision task using a hemifield paradigm. Participants had to decide whether or not an image of an object lateralized in the left (LVF) or right (RVF) visual half-field is a member of a predesignated superordinate category. The objects represented high, medium, or low typicality levels, and each object had six different perspective views (front, front-right, front-left, side, back-left, and back-right). The latency responses revealed a significant interaction of Hemi Field X View X Typicality (there wasno hemifield difference in accuracy). The findings confirm the presence of asymmetry in stored concepts in long-term memory and suggest, in addition, a hemispheric canonical view of these concepts, a view strongly related to typicality level.     

Bicartesian Coherence

Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality.