Validity, dialetheism and self-reference

It has been argued recently (Beall in Spandrels of truth, Oxford University Press, Oxford, 2009; Beall and Murzi J Philos 110:143–165, 2013) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that (Formula presented.) can be non-trivially expanded with a naive validity predicate. The resulting theory, (Formula presented.) reaches this goal by adopting a weak self-referential procedure. We show that (Formula presented.) is sound and complete with respect to the three-sided sequent calculus (Formula presented.). Moreover, (Formula presented.) can be safely expanded with a transparent truth predicate. We will also present an alternative theory (Formula presented.), which includes a non-deterministic validity predicate.Fil: Pailos, Federico Matias. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

The Class of All Natural Implicative Expansions of Kleene’s Strong Logic Functionally Equivalent to Łukasiewicz’s 3-Valued Logic Ł3

25 p.We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix (with both only one and two designated values) and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova (Rep Math Log 47:173–182, 2012).S

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

Is game semantics necessary?

We discuss the extent to which game semantics is implicit in the formalism of linear logic and in the intuitions underlying linear logic

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Some remarks on semantics and expressiveness of the Sentential Calculus with Identity

Suszko's Sentential Calculus with Identity SCI results from classical propositional calculus CPC by adding a new connective $\equiv$ and axioms for identity $\varphi\equiv\psi$ (which we interpret here as `propositional identity'). We reformulate the original semantics of SCI in terms of Boolean prealgebras establishing a connection to `hyperintensional semantics'. Furthermore, we define a general framework of dualities between certain SCI-theories and Lewis-style modal systems in the vicinity of S3. Suszko's original approach to two SCI-theories corresponding to S4 and S5 can be formulated as a special case. All these dualities rely particularly on the fact that Lewis' `strict equivalence' is axiomatized by the SCI-principles of `propositional identity'.Comment: 31 page