Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

Is there a reliability challenge for logic?

There are many domains about which we think we are reliable. When there is prima facie reason to believe that there is no satisfying explanation of our reliability about a domain given our background views about the world, this generates a challenge to our reliability about the domain or to our background views. This is what is often called the reliability challenge for the domain. In previous work, I discussed the reliability challenges for logic and for deductive inference. I argued for four main claims: First, there are reliability challenges for logic and for deduction. Second, these reliability challenges cannot be answered merely by providing an explanation of how it is that we have the logical beliefs and employ the deductive rules that we do. Third, we can explain our reliability about logic by appealing to our reliability about deduction. Fourth, there is a good prospect for providing an evolutionary explanation of the reliability of our deductive reasoning. In recent years, a number of arguments have appeared in the literature that can be applied against one or more of these four theses. In this paper, I respond to some of these arguments. In particular, I discuss arguments by Paul Horwich, Jack Woods, Dan Baras, Justin Clarke-Doane, and Hartry Field

Judgment aggregation in general logics

Within social choice theory, the new field of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Commonly, judgment aggregation is studied using standard propositional logic, with a limited expressive power and a problematic representation of conditional statements ('if P then Q') as material conditionals. In this methodological paper, I present a generalised model, in which most realistic decision problems can be represented. The model is not restricted to a particular logic but is open to several logics, including standard propositional logic, predicate calculi, modal logics and conditional logics. To illustrate the model, I prove an impossibility theorem, which generalises earlier results.judgement aggregation, discursive dilemma, modelling methodology, formal logics, impossibility theorem

Refinement by interpretation in {\pi}-institutions

The paper discusses the role of interpretations, understood as multifunctions that preserve and reflect logical consequence, as refinement witnesses in the general setting of pi-institutions. This leads to a smooth generalization of the refinement-by-interpretation approach, recently introduced by the authors in more specific contexts. As a second, yet related contribution a basis is provided to build up a refinement calculus of structured specifications in and across arbitrary pi-institutions.Comment: In Proceedings Refine 2011, arXiv:1106.348

Coherence in Modal Logic

A variety is said to be coherent if the finitely generated subalgebras of its finitely presented members are also finitely presented. In a recent paper by the authors it was shown that coherence forms a key ingredient of the uniform deductive interpolation property for equational consequence in a variety, and a general criterion was given for the failure of coherence (and hence uniform deductive interpolation) in varieties of algebras with a term-definable semilattice reduct. In this paper, a more general criterion is obtained and used to prove the failure of coherence and uniform deductive interpolation for a broad family of modal logics, including K, KT, K4, and S4

Argument-based Belief in Topological Structures

This paper combines two studies: a topological semantics for epistemic notions and abstract argumentation theory. In our combined setting, we use a topological semantics to represent the structure of an agent's collection of evidence, and we use argumentation theory to single out the relevant sets of evidence through which a notion of beliefs grounded on arguments is defined. We discuss the formal properties of this newly defined notion, providing also a formal language with a matching modality together with a sound and complete axiom system for it. Despite the fact that our agent can combine her evidence in a 'rational' way (captured via the topological structure), argument-based beliefs are not closed under conjunction. This illustrates the difference between an agent's reasoning abilities (i.e. the way she is able to combine her available evidence) and the closure properties of her beliefs. We use this point to argue for why the failure of closure under conjunction of belief should not bear the burden of the failure of rationality.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Propositional dynamic logic for searching games with errors

We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the (n+1)-valued Kripke models and a corresponding language based on a modal extension of {\L}ukasiewicz many-valued logic. We illustrate the definitions by providing a framework for an analysis of the R\'enyi - Ulam searching game with errors. Our main result is the axiomatization of the theory of the (n+1)-valued Kripke models. This result is obtained through filtration of the canonical model of the smallest (n+1)-valued propositional dynamic logic