1,266 research outputs found
Binary Sequent Calculi for Truth-invariance Entailment of Finite Many-valued Logics
In this paper we consider the class of truth-functional many-valued logics
with a finite set of truth-values. The main result of this paper is the
development of a new \emph{binary} sequent calculi (each sequent is a pair of
formulae) for many valued logic with a finite set of truth values, and of
Kripke-like semantics for it that is both sound and complete. We did not use
the logic entailment based on matrix with a strict subset of designated truth
values, but a different new kind of semantics based on the generalization of
the classic 2-valued truth-invariance entailment. In order to define this
non-matrix based sequent calculi, we transform many-valued logic into positive
2-valued multi-modal logic with classic conjunction, disjunction and finite set
of modal connectives. In this algebraic framework we define an uniquely
determined axiom system, by extending the classic 2-valued distributive lattice
logic (DLL) by a new set of sequent axioms for many-valued logic connectives.
Dually, in an autoreferential Kripke-style framework we obtain a uniquely
determined frame, where each possible world is an equivalence class of
Lindenbaum algebra for a many-valued logic as well, represented by a truth
value.Comment: 21 page
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their
axiomatizations by Solomon Feferman via Solovay-style completeness results.
Given a fixed-point model , or an axiomatization thereof, we
find a modal logic such that a modal sentence is a theorem of
if and only if the sentence obtained by translating the modal
operator with the truth predicate is true in or a theorem of
under all such translations. To this end, we introduce a novel version of
possible worlds semantics featuring both classical and nonclassical worlds and
establish the completeness of a family of non-congruent modal logics whose
internal logic is subclassical with respect to this semantics
The Modal Logics of Kripke-Feferman Truth
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results
Goldblatt-Thomason for LE-logics
We prove a uniform version of the Goldblatt-Thomason theorem for logics
algebraically captured by normal lattice expansions (normal LE-logics)
Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support
A framework and methodology---termed LogiKEy---for the design and engineering
of ethical reasoners, normative theories and deontic logics is presented. The
overall motivation is the development of suitable means for the control and
governance of intelligent autonomous systems. LogiKEy's unifying formal
framework is based on semantical embeddings of deontic logics, logic
combinations and ethico-legal domain theories in expressive classic
higher-order logic (HOL). This meta-logical approach enables the provision of
powerful tool support in LogiKEy: off-the-shelf theorem provers and model
finders for HOL are assisting the LogiKEy designer of ethical intelligent
agents to flexibly experiment with underlying logics and their combinations,
with ethico-legal domain theories, and with concrete examples---all at the same
time. Continuous improvements of these off-the-shelf provers, without further
ado, leverage the reasoning performance in LogiKEy. Case studies, in which the
LogiKEy framework and methodology has been applied and tested, give evidence
that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure
Negation in context
The present essay includes six thematically connected papers on negation in the areas of the philosophy of logic, philosophical logic and metaphysics. Each of the chapters besides the first, which puts each the chapters to follow into context, highlights a central problem negation poses to a certain area of philosophy. Chapter 2 discusses the problem of logical revisionism and whether there is any room for genuine disagreement, and hence shared meaning, between the classicist and deviant's respective uses of 'not'. If there is not, revision is impossible. I argue that revision is indeed possible and provide an account of negation as contradictoriness according to which a number of alleged negations are declared genuine. Among them are the negations of FDE (First-Degree Entailment) and a wide family of other relevant logics, LP (Priest's dialetheic "Logic of Paradox"), Kleene weak and strong 3-valued logics with either "exclusion" or "choice" negation, and intuitionistic logic. Chapter 3 discusses the problem of furnishing intuitionistic logic with an empirical negation for adequately expressing claims of the form 'A is undecided at present' or 'A may never be decided' the latter of which has been argued to be intuitionistically inconsistent. Chapter 4 highlights the importance of various notions of consequence-as-s-preservation where s may be falsity (versus untruth), indeterminacy or some other semantic (or "algebraic") value, in formulating rationality constraints on speech acts and propositional attitudes such as rejection, denial and dubitability. Chapter 5 provides an account of the nature of truth values regarded as objects. It is argued that only truth exists as the maximal truthmaker. The consequences this has for semantics representationally construed are considered and it is argued that every logic, from classical to non-classical, is gappy. Moreover, a truthmaker theory is developed whereby only positive truths, an account of which is also developed therein, have truthmakers. Chapter 6 investigates the definability of negation as "absolute" impossibility, i.e. where the notion of necessity or possibility in question corresponds to the global modality. The modality is not readily definable in the usual Kripkean languages and so neither is impossibility taken in the broadest sense. The languages considered here include one with counterfactual operators and propositional quantification and another bimodal language with a modality and its complementary. Among the definability results we give some preservation and translation results as well
The logic of vague categories
We introduce a complete many-valued semantics for basic normal lattice-based
modal logic. This relational semantics is grounded on many-valued formal
contexts from Formal Concept Analysis. We discuss an interpretation and
possible applications of this logical framework for categorization theory to
the formal analysis of multi-market competition.Comment: 24 page
An Approach to Fuzzy Modal Logic of Time Intervals
Temporal reasoning based on intervals is nowadays ubiquitous in artificial intelligence, and the most representative interval temporal logic, called HS, was introduced by Halpern and Shoham in the eighties. There has been a great effort in the past in studying the expressive power and computational properties of the satisfiability problem for HS and its fragments, but only recently HS has been proposed as a suitable formalism for artificial intelligence applications. Such applications highlighted some of the intrinsic limits of HS: Sometimes, when dealing with real-life data one is not able to express temporal relations and propositional labels in a definite, crisp way. In this paper, following the seminal ideas of Fitting and Zadeh, among others, we present a fuzzy generalization of HS that partially solves such problems of expressive power, and we prove that, as in the crisp case, its satisfiability problem is generally undecidable
Modelling socio-political competition
This paper continues the investigation of the logic of competing theories, be
they scientific, social, political etc. We introduce a many-valued, multi-type
modal language which we endow with relational semantics based on enriched
reflexive graphs, inspired by Plo\v{s}\v{c}ica's representation of general
lattices. We axiomatize the resulting many-valued, non-distributive modal logic
of these structures and prove a completeness theorem. We illustrate the
application of this logic through a case study in which we model competition
among interacting political promises and social demands within an arena of
political parties social groups.Comment: 24 page
New Directions in Justification Logic
Justification logics are constructive analogues of modal logics. As such, they provide perspicuous models of those modalities that have inherently constructive character, such as intuitionistic mathematical provability or the knowledge operator of evidentialist epistemology. In this dissertation, I examine a variety of positions in epistemology, along with their associated ontological commitments, and develop various classical and non-classical justification logics that are suitable for use as models of these positions
- …