Claudio Pizzi

MODEL-BASED REASONING IN SCIENCE AND TECHNOLOGY

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Necessity and Relative Contingency

The paper introduces a contingential language extended with a propositional constant τ axiomatized in a system named KΔτ , which receives a semantical analysis via relational models. A definition of the necessity operator in terms of Δ and τ allows proving (i) that KΔτ is equivalent to a modal system named K□τ (ii) that both KΔτ and K□τ are tableau-decidable and complete with respect to the defined relational semantics (iii) that the modal τ -free fragment of KΔτ is exactly the deontic system KD. In §4 it is proved that the modal τ -free fragment of a system KΔτw weaker than KΔτ is exactly the minimal normal system K.

Generalization and Composition of Modal Squares of Oppositions

The first part of the paper aims at showing that the notion of an Aristotelian square may be seen as a special case of a variety of different more general notions: (1) the one of a subAristotelian square, (2) the one of a semiAristotelian square, (3) the one of an Aristotelian cube, which is a construction made up of six semiAristotelian squares, two of which are Aristotelian. Furthermore, if the standard Aristotelian square is seen as a special ordered 4-tuple of formulas, there are 4-tuples describing rotations of the original square which are non-standard Aristotelian squares. The second part of the paper focuses on the notion of a composition of squares. After a discussion of possible alternative definitions, a privileged notion of composition of squares is identified, thus opening the road to introducing and discussing the wider notion of composition of cubes.

Aristotle’s Cubes and Consequential Implication

Abstract.It is shown that the properties of so-called consequential implication allow to construct more than one aristotelian square relating implicative sentences of the consequential kind. As a result, if an aristotelian cube is an object consisting of two distinct aristotelian squares and four distinct “semiaristotelian” squares sharing corner edges, it is shown that there is a plurality of such cubes, which may also result from the composition of cubes of lower complexity.

Aristotle's Thesis between paraconsistency and modalization

Abstract If the arrow → stands for classical relevant implication, Aristotles Thesis ¬ ( A → ¬ A ) is inconsistent with the Law of Simplification ( A ∧ B ) → B accepted by relevantists, but yields an inconsistent non-trivial extension of the system of entailment E. Such paraconsistent extensions of relevant logics have been studied by R. Routley, C. Mortensen and R. Brady. After examining the semantics associated to such systems, it is stressed that there are nonclassical treatments of relevance which do not support Simplification. The paper aims at showing that Aristotles Thesis may receive a sense if the arrow is defined as strict implication endowed with the proviso that the clauses of the conditional have the same modal status, i.e. the same position in the Aristotelian square. It is so grasped, in different form, the basic idea of relevant logic that the clauses of a true conditional should have something in common. It is proved that thanks to such definition of the arrow Aristotles Thesis subjoined to the minimal normal system K yields a system equivalent to the deontic system KD.

Relative Contingency and Bimodality

In the first part of the paper it is proved that there exists a one–one mapping between a minimal contingential logic extended with a suitable axiom for a propositional constant τ, named KΔτw, and a logic of necessity

Gestalt Effects in Counterfactual and Abductive Inference

{K\square \tau{w}}

Fictionalism and the Logic of “As If” Conditionals

whose language contains