Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Inducing syntactic cut-elimination for indexed nested sequents

The key to the proof-theoretic study of a logic is a proof calculus with a subformula property. Many different proof formalisms have been introduced (e.g. sequent, nested sequent, labelled sequent formalisms) in order to provide such calculi for the many logics of interest. The nested sequent formalism was recently generalised to indexed nested sequents in order to yield proof calculi with the subformula property for extensions of the modal logic K by (Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination therein were semantic and intricate. Here we show that derivations in the labelled sequent formalism whose sequents are `almost treelike' correspond exactly to indexed nested sequents. This correspondence is exploited to induce syntactic proofs for indexed nested sequent calculi making use of the elegant proofs that exist for the labelled sequent calculi. A larger goal of this work is to demonstrate how specialising existing proof-theoretic transformations alleviate the need for independent proofs in each formalism. Such coercion can also be used to induce new cutfree calculi. We employ this to present the first indexed nested sequent calculi for intermediate logics.Comment: This is an extended version of the conference paper [20

Tool support for reasoning in display calculi

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, that allows us to adapt the tool for a wide variety of user defined calculi

On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary

Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment

Type-driven semantic interpretation and feature dependencies in R-LFG

Once one has enriched LFG's formal machinery with the linear logic mechanisms needed for semantic interpretation as proposed by Dalrymple et. al., it is natural to ask whether these make any existing components of LFG redundant. As Dalrymple and her colleagues note, LFG's f-structure completeness and coherence constraints fall out as a by-product of the linear logic machinery they propose for semantic interpretation, thus making those f-structure mechanisms redundant. Given that linear logic machinery or something like it is independently needed for semantic interpretation, it seems reasonable to explore the extent to which it is capable of handling feature structure constraints as well. R-LFG represents the extreme position that all linguistically required feature structure dependencies can be captured by the resource-accounting machinery of a linear or similiar logic independently needed for semantic interpretation, making LFG's unification machinery redundant. The goal is to show that LFG linguistic analyses can be expressed as clearly and perspicuously using the smaller set of mechanisms of R-LFG as they can using the much larger set of unification-based mechanisms in LFG: if this is the case then we will have shown that positing these extra f-structure mechanisms is not linguistically warranted.Comment: 30 pages, to appear in the the ``Glue Language'' volume edited by Dalrymple, uses tree-dvips, ipa, epic, eepic, fullnam