Logic of Intuitionistic Interactive Proofs (Formal Theory of Perfect Knowledge Transfer)

We produce a decidable super-intuitionistic normal modal logic of internalised intuitionistic (and thus disjunctive and monotonic) interactive proofs (LIiP) from an existing classical counterpart of classical monotonic non-disjunctive interactive proofs (LiP). Intuitionistic interactive proofs effect a durable epistemic impact in the possibly adversarial communication medium CM (which is imagined as a distinguished agent), and only in that, that consists in the permanent induction of the perfect and thus disjunctive knowledge of their proof goal by means of CM's knowledge of the proof: If CM knew my proof then CM would persistently and also disjunctively know that my proof goal is true. So intuitionistic interactive proofs effect a lasting transfer of disjunctive propositional knowledge (disjunctively knowable facts) in the communication medium of multi-agent distributed systems via the transmission of certain individual knowledge (knowable intuitionistic proofs). Our (necessarily) CM-centred notion of proof is also a disjunctive explicit refinement of KD45-belief, and yields also such a refinement of standard S5-knowledge. Monotonicity but not communality is a commonality of LiP, LIiP, and their internalised notions of proof. As a side-effect, we offer a short internalised proof of the Disjunction Property of Intuitionistic Logic (originally proved by Goedel).Comment: continuation of arXiv:1201.3667; extended start of Section 1 and 2.1; extended paragraph after Fact 1; dropped the N-rule as primitive and proved it derivable; other, non-intuitionistic family members: arXiv:1208.1842, arXiv:1208.591

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Constructive Provability Logic

We present constructive provability logic, an intuitionstic modal logic that validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl

Logics for modelling collective attitudes

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained

Semantics and Ontology:\ud On the Modal Structure of an Epistemic Theory of Meaning

In this paper I shall confront three basic questions.\ud First, the relevance of epistemic structures, as formalized\ud and dealt with by current epistemic logics, for a\ud general Theory of meaning. Here I acknowledge M. Dummett"s\ud idea that a systematic account of what is meaning of\ud an arbitrary language subsystem must especially take into\ud account the inferential components of meaning itself. That\ud is, an analysis of meaning comprehension processes,\ud given in terms of epistemic logics and semantics for epistemic\ud notions.\ud The second and third questions relate to the ontological\ud and epistemological framework for this approach.\ud Concerning the epistemological aspects of an epistemic\ud theory of meaning, the question is: how epistemic logics\ud can eventually account for the informative character of\ud meaning comprehension processes. "Information� seems\ud to be built in the very formal structure of epistemic processes,\ud and should be exhibited in modal and possibleworld\ud semantics for propositional knowledge and belief.\ud However, it is not yet clear what is e.g. a possible world.\ud That is: how it can be defined semantically, other than by\ud accessibility rules which merely define it by considering its\ud set-theoretic relations with other sets-possible worlds.\ud Therefore, it is not clear which is the epistemological status\ud of propositional information contained in the structural\ud aspects of possible world semantics. The problem here\ud seems to be what kind of meaning one attributes to the\ud modal notion of possibility, thus allowing semantical and\ud synctactical selectors for possibilities. This is a typically\ud Dummett-style problem.\ud The third question is linked with this epistemological\ud problem, since it is its ontological counterpart. It concerns\ud the limits of the logical space and of logical semantics for a\ud of meaning. That is, it is concerned with the kind of\ud structure described by inferential processes, thought, in a\ud fregean perspective, as pre-conditions of estentional\ud treatment of meaning itself. The second and third questions\ud relate to some observations in Wittgenstein"s Tractatus.\ud I shall also try to show how their behaviour limits the\ud explicative power of some semantics for epistemic logics\ud (Konolige"s and Levesque"s for knowledge and belief)